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# WEATHER OBSERV ANALYSIS ATMO 251

Texas A&M

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Weather Observation and Analysis John NielsenGammon CourseNotes These awse notes are eapyngnted If you are presently regmeredfar AYMU 251 at Texas AampM Unwetsxty permission 1S hereby A utyau may not download arprmt tnem thnaut tnepermlsslan aftne author Redlstnbutlan aftnese eawse notes Whether danefteely a a Chapter 6 REMOTE ENSIN 61 Satellite Thermal Sounders The Eanhis aplanet covered by ahlanhet ofwaxm gas Eeeept for when there are elouds that blanket is nearly transparent to sunlight whieh are not hot enough to glow thanselves at least not that we ean see They in the range ofwavelengths we eall infrared one neat thing about the atmosphere is that in the infrared range different gases absorb and emit radiation at different wavelengths If a i WW I s l 39 no nu my quotminions int 5 mumm m man s gas ean absorb at a particular wavelength it ean emit theie too So a satellite sensor out in space that deteets inhaied radiation might see a glowing hall of gas ifit s tunedto awavelength that a gas in the from the Earth s surface ifit s tuned to a wavelength that no gas in the atmosphere touehes ATMO 251 chapteis page l on3 A waveleI 139 h strongly A wavelength moderu rely ab W bEd any h Weakly eleng l d scrigled and emm l by absorbed and emitted by 9 armosphZTTed by a the OT asp ef39e The atmosphere A Sqfellife sees s A satellite can only see safellife sees mostly rnd39flfion eml 39edo1in e adluhon emi ed f quot quot radiation emi ed from ow quot39 The armos hm r ear 6 top of fhc middle layers of The em u mos y radigize 5 here afmos here 39 ed or a tmo P P The Ear Scattered by ATMOSPHERE EARTH Think about those two wavelengths for a moment Suppose you could adjust the satellite to change its detection wavelength gradually from the m h p 1 onetothe t h tone As the wavelength changes most of the radiation detected by the satellite will be coming from farther down in the atmosphere until eventually it starts seeing some of the Earth s surface and still later all of it At this poing we re still in the who cares portion of this exposition as in Who cares what the radiation is coming from The answer to that question involves two very key facm about electromagnetic radiation The first fact is that the intensity of emitted radiation depends directly on the temperature of whatever object or gas emitted the radiation I bet you can see why that might be useful since the temperature of the Earth and its atmosphere is kinda relevant to the ATMO 251 Chapter 8 page 2 of 23 weather But there s a potential problem with using that information to convert satellite measurements to temperature measurements what if there s something between the emissions source and the satellite receiver that s blocking part or all of the radiation In that case wouldn t you underestimate the temperature Well you would except for the second key fact anything that absorbs at a particular wavelength also emits at the same wavelength So take a layer of gas that absorbs 30 of the radiation passing through it It will also emit radiation at 30 efficiency In effect it replaces the radiation it absorbs with radiation of its own So all is well As long as you can figure out based on the composition of the atmosphere how much radiation is coming from which levels you re in business Here s how it works At any particular wavelength the amount of radiation detected by the satellite is a strong function of the temperature of the gases or surface that emitted the radiation In our experiment from the previous paragraph the satellite should be receiving stronger and stronger radiation as the wavelength changes because the atmosphere lower down is generally much warmer than the atmosphere higher up The strength of the radiation tells us the temperature of the air that emitted the radaition Unfortunately the laws of physics aren t completely cooperative By virtue of the nature of emissions from a gas the satellite doesn t see radiation coming from a particular level Instead it originates within a very fuzzy layer So the atmospheric measurements are not very precise in that sense they re accurate but blurry In practice satellite instruments designed for temperature measurements have several fixed channels that detect radiation at different wavelengths Those channels are chosen to be in uenced to varying degrees by emissions from a gas that s fairly evenly distributed in the atmosphere but which absorbs and emits quite a bit of radiation at certain wavelengths The gas most commonly used for this purpose is carbon dioxide Knowing the concentration of carbon dioxide it is possible to determine which layer of the atmosphere each channel will see From a set of such measurements a rough temperature profile can be determined Examples of images from three such wavelength channels are shown The first is from a channel for which the atmosphere is entirely opaque so the image represents the horizontal distribution of temperatures in the upper troposphere The second is from a channel in which the atmosphere is nearly transparent so the image represents the horizontal distribution of temperatures on the Earth s surface The third is from a channel for which the atmosphere is partially opaque so the image looks like a fuzzy view of the temperature of the underlying Earth ATMO 251 Chapter 8 page 3 of 23 COOLEST WARMEST F x m u A safellife image a an opaque wavelength um um ATMO 251 Chapter 8 page 4 of23 At all three wavelengths clouds are opaque and thus emit well too An extensive commashaped cloud mass over the eastern United States shows up in all three images The highest cloud tops appear to be emitting at a temperature of about 50 C The clouds show up as slightly colder in the opaque image because the atmosphere above the high clouds is absorbing some of the emitted radiation Where the atmosphere is cloudfree the three images tell different stories The opaque image shows mostly northtosouth temperature variations in the upper troposphere but the pattern is kind of wavy Temperatures are much warmer over Northern California for example compared to over Colorado and Kansas This is because there is a jet stream ridge over the West Coast and a trough over the central United States The trough is a bit to the west of the cloud mass that marks the lowlevel cyclone this westward displacement is typical of active midlatitude systems The image in the transparent or atmospheric window channel shows surface temperatures beneath cloudfree air Temperatures are warmest in the desert Southwest although the ground is starting to heat up a bit too in eastern Montana and Wyoming There is a sharp color contrast along the West Coast in California and Baja California where the ground is warm and the ocean is cool The even cooler temperatures farther offshore and over the IowaMissouriKansas area are low clouds rather than ground temperatures Because the clouds are low in the troposphere they are not much cooler than the ground beneath or beside them so the contrast in temperatures is not all that great and if you don t look carefully or look at a visible satellite image you might be fooled into thinking that there are no clouds there at all The image in the semitransparent window only shows a blurry view of the ground The Sierra Nevada Mountains show up in California as a line of cooler temperatures but the contrast between low cloud and bare ground offshore and in the central Plains is gone The warmest temperatures in this image are off the coast of Baja California indicating that the middle and lower troposphere is quite warm there Suppose the satellite detects radiation from a gas with variable concentrations such as water vapor In that case even a single channel might see some widely varying radiation intensities Where there s lots of water vapor the radiation would be coming from near the top of the atmosphere where it s cool Where the top of the atmosphere is dry the radiation would be coming from lower down in the atmosphere where it s warmer If you have some way of estimating the vertical distribution of temperature perhaps from the same satellite you can use these measurements to estimate the concentration of water vapor in the upper atmosphere And with several channels sensitive to water vapor in ATMO 251 Chapter 8 page 5 of 23 varying degrees you can extract information on the vertical distribution of water vapor and total amount of water vapor in the atmosphere One such retrieval of total column water vapor is shown below The water vapor is expressed as a quantity called precipitable water which is the amount of rain that would fall if all the water vapor were instantly condensed out of the atmosphere and brought to the ground The retrieval is not possible where it is cloudy since the liquid water droplets and ice crystals that form clouds are strongly opaque and block the view of the lower troposphere Clouds or otherwise impossible retrievals are shown in gray or white in the image Remember water vapor is the gaseous form of water and is everywhere clouds form when the gas condenses into liquid or solid particles p mtx A satellite based Total atmospheric column water vapor39 retrieval The greatest amount of water vapor is over the Gulf of Mexico in this image If all the water vapor were condensed and fell as rain it would cover the GulfofMexico to a depth of over 50 mm or 2 in By contrast the atmosphere is so dry over Maine that all the rain that could possibly fall would barer wet the surface There is little water vapor in the atmospheric column over the Rocky Mountains too partly because the relative humidity is low and partly because there s less atmosphere there because the mountains are so high 62 Images and Gradients The laws of physics make it very dif th to extract detailed temperature and humidity information from satellite observations On the other hand there s no such limitation to horizontal resolution T e horizontal resolution is limited by the size of the satellite receiver and the strength of the radiation Even the coarsest satellite sounders have a ATMO 251 Chapter 8 page 6 of23 horizontal resolution of a few tens of kilometers Satellites designed for i aging are even better The coarsest meteorological satellites have a pixel size of 4 km and many are down to 1 km or better With a lot of satellite data especially at wavelengths designed to measure water vapor or other constituents the data is not coming from a single level Yet one can plot the image on a twodimensional map What vocabulary do we then use for variations within the image Can we talk about gradients even though we re not looking at a horizontal plane in the atmosphere he answer is yes you can While the physical quantity may not lie on a plane in the atmosphere the image lies on a plane on the computer screen or piece of paper And normally the variations gradients that show up in the images have their counterparts on horizontal surfaces in the atmosphere even if we don t know exactly which horizontal surfaces are involved You can compute a gradient in a satellite image in pretty much the usual way You generally don t have contours but you have the value of the eld at every point so you don t need to do any interpolation You can compare two values of temperature and divide by the distance between them The only thing you might have to be careful about is your coordinate system Many satellite images are not remapped to a normal projection So distances near the center of the image might be quite different from distances near the edge ofthe image ATMO 251 Chapter 8 page 7 of23 Noise is often present in satellite images By noise I mean random or arti cial variations from pixel to pixel in a satellite image While with many applications you get a more accurate gradient estimate by looking over a small interval this is not so with most satellite images So as to estimate the gradient of the physical quantity rather than the gradient of the noise it is important that your interval span several pixels unless you have good reason for trusting the pixeltopixel variations 63 Water Vapor Images Because of the importance of the distribution of water vapor in the atmosphere most meteorological satellites have one or more channels that detect water vapor emissions An image from such a channel is called a water vapor image The water vapor channel is chosen so that under typical conditions the radiation measured by the satellite was emitted at about 300 mb to 500 mb or within the upper troposphere If there s lots of water vapor in that layer the radiation will come from near the top of that layer where temperatures are particularly cold so the radiation will be weak If there s little water vapor in that layer the radiation will come from even lower in the atmosphere where temperatures are warmer so the radiation will be strong All that s left is to convert the radiation measurements into a graphical product If white is associated with very weak radiation and black with very strong radiation white areas will represent lots of water vapor aloft and black areas will represent little water vapor aloft So that s what s done Water vapor images are particularly useful for detecting patterns associated with uppertropospheric weather features such as troughs and jet streaks They are also important in ood forecasting since a wet upper troposphere makes convection more efficient at producing rainfall One thing you can t do with a water vapor image is detect lowlevel moisture or instability The moisture you see is too high in the atmosphere to give any direct information regarding the moisture distribution down in the boundary layer So don t try diagnosing surface dewpoints with a water vapor image In the example water vapor satellite image lighter colors correspond to larger amounts of humidity in the upper troposphere The driest air is off the coast of California Note that this is from the same time as the images shown in Section 61 which indicated low clouds southwest of California The humidity is large in the lower troposphere ATMO 251 Chapter 8 page 8 of 23 but is very dry in the middle and upper troposphere The other satellite imagery also showed very warm temperatures there and the combination of high temperatures and low humidity suggests that this air has been sinking A water vapor image There are numerous streaks and swirls in the water vapor image most of which are invisible in images taken from wavelengths not absorbed or emitted by water vapor Northeast of Hawaii there appears to be an upperlevel circulation of some sort and a Paci c cold front appears to be just reaching the northwestern comer of Oregon Since clouds also emit radiation at this wavelength you might see some very white areas with a much more jagged pattern than the rest of the image These are high clouds clouds high enough that there s not much water vapor above them to block the radiation These would be cirrus clouds or cumulonimbus clouds 64 Infrared Images ATMO 251 Chapter 8 page 9 of23 The water vapor images discussed in the previous section use radiation in the infrared range Other infrared wavelengths are absorbed by other gases or not at all Despite this generality the term infrared image is reserved for images at a wavelength that has almost no absorption and emission by the atmosphere such as 1112 microns In other words an infrared image looks right through the atmosphere Although there are slight variations in the ability of the Earth s surface to emit infrared radiation for the most part the Earth is a very ef cient emitter Thus variations in surface temperature are easy to detect as variations in infrared radiation intensity and they show up well on an infrared satellite image On a clear day or night you can tell just how warm or cold the surface of the Earth is from an infrared image On a cloudy day it s a different story Clouds absorb and emit infrared radiation too With the water vapor image the clouds weren t much of a problem because most of the clouds were hidden by the water vapor In a standard infrared image there s nothing to hide the clouds and in fact the clouds are hiding the surface of the Earth from the satellite This is a problem if you care about surface temperature but it s a good thing if you care about the clouds themselves As long as the clouds are a different temperature from the underlying surface it s possible to easily identify the edges of clouds in an infrared satellite image Not only can we distinguish clouds from Earth we can distinguish high clouds from low clouds The higher the cloud top generally the colder the cloud top temperature So an infrared image is useful for looking at different layers of clouds although it can only see the topmost layer at any given point The same grayscale plotting convention is often used as with water vapor images The weakest radiation is coming from the highest clouds and it is tagged as white Radiation consistent with warm surface temperatures is tagged black So even though you re really looking at different temperatures your brain tells you you re looking at clouds In the example infrared satellite image both high clouds and low clouds are visible over the Paci c Ocean The high clouds stand out because they are colored bright white in the image Low clouds are harder to see but there is quite extensive low cloud cover over the Paci c too The telltale sign is the rapid spatial variations from medium gray to dark gray the temperatures in the Paci c Ocean are much smoother and more uniform than that ATMO 251 Chapter 8 page 10 of 23 An infrared image 65 Visible Images Satellite images from the visible spectrum are unlike the other images discussed so far As noted before the Earth doesn t glow in the visible wavelength range so all the visible radiation detected by a satellite is scattered or re ected from the surface ofEarth or the surface ofclouds Since clouds are a very bright white compared to most surfaces they are easy to detect An advantage of visible images over infrared images is that visible images easily pick up the contrast between low clouds and the surrounding und as long as the ground is not covered with snow or is white for some other reason If the two surfaces are nearly the same temperature you may not be able to tell them apart with just an infrared image An obvious disadvantage of visible images is that because they rely on a radiation source external to the Earth they only function during daytime or during a full moon if you stare hard at the image There are times of year when visible satellite images are simply not available in polar regions ATMO 251 Chapter 8 page 11 of23 Of all the images it makes the least sense to talk about gradients in the context of visible satellite images Most features are true discontinuities here you have cloud here you don t A visible image In the example visible satellite image it s easy to see the low clouds because they are just as bright as the high clouds This ease is both an advantage and a disadvantage since it s harder to tell low clouds and high clouds apart in a visible satellite image than in an infrared satellite image You really need to look at both types of images at the same time to properly diagnose the cloud cover 66 Wind Pro lers and Radars RaWinsondes are one source of observations of Wind above the surface but it s a limited source The launches are typically twelve hours part and there s only one data point at each level from each sonde By contrast radar Wind profilers provide data With a frequency of about an ho d scanning Doppler radars provide data over a broad volume every ve minutes Plus unlike a ravvinsonde you can use a radar more than once ATMO 251 Chapter 8 page 12 of23 Pro lers and scanning radars detect wind in the same way The transmitter sends out a pulse of radiation The radiation travels at the speed oflight 30 X 108 msec Some ofthat radiation ies through the atmosphere and is never seen again but a lot is scattered by the air or stuff in it Some of the scattered radiation is actually scattered back toward the radar where it is detected The time lag between transmission and reception determines how far the radiation went before being scattered back Some of the radiation emitted by the radar encounters objects such as raindrops that cause the radiation to be scattered Some of the radiation is scattered back toward the radar where it is detected The radiation travels at the speed of light so the time interval between transmission and reception is recorded and is used to calculate the distance from the radar to the scatterer That covers the radar part how the spatial distribution of precipitation is determined by the radar Indeed the word radar is actually an acronym for RAdiation Detection And Ranging To determine wind it is necessary for each radar to transmit pulses of energy at a known frequency If the radiation is scattered back to the radar and still has the same frequency the scatterer was neither moving toward nor away from the radar If the radiation that returns to the radar has a longer wavelength the scatterer was moving away from the radar If a shorter wavelength the scatterer was moving toward the radar The amount of the wavelength shi is proportional to the speed of the scatterer ATMO 251 Chapter 8 page 13 of23 E Radiation is emiTTed by a radar TIME 1 Q q 39 4 Toward a raindrop If The I raindrop is moving away from I The radar each successive wave cresT from The radar encounTers The raindrop far 39 39 I Ther and farTher from the 39 radar As a result The wave B A cresTs scaTTered back TIME 2 E Q 9 To The radar by The I II raindrop end up being I further aparT Than The original wave cresTs This change in fre I A quency or wavelengTh is 39 I I The Doppler shifT and A by measuring E D C B The change Tune 3 E I I The moTion of l I The scaTTerer I Toward or I I away from The I radar can be t l A I I B I deTermmed E D C 393 I TIME 4 G l 39 39 I I I I I l I I a I I I A I B I I H G F E I C 3 A TIME 5 i II I 39 i I I I l I I I I l I A I I B C i I I I The scatterers of interest to weather radar are raindrops or snow akes Raindrops and to a lesser extent snow akes move through ATMO 251 Chapter 8 page 14 of 23 the air because they are being pulled downward by gravity However they are drifting along with the horizontal wind like a balloon so the horizontal velocity of the scatterers is an excellent estimate of the horizontal velocity of the air If the wind is blowing sideways rather than toward or away from the radar there won t be any change in the phase of the signal received by the radar so sideways velocities are undetectable Indeed if you think about velocity as a threedimensional vector and orient a coordinate system so that one component is parallel to the radar beam that s the only component of velocity that s measurable by a Doppler radar The other two components are perpendicular and invisible Despite this limitation there are many ways of determining all three wind components One simplesounding approach is to scan along the eastwest northsouth and vertical axes to measure all three components The catch is that the three scans being in three different directions measure winds in three different places Typically the wind varies by location so only close to the radar do you get an accurate wind estimate And if you wanted the wind close to the radar you could use an anemometer Winds aloft are much more interesting because no anemometer measures them As the radar scans in all directions its beam intercepts any scatterers located on an imaginary cone centered at the radar Objects that are farther from the radar are also farther above the ground Suppose the scanning radar points up at some angle and then scans a complete 360 degree circle Typically upward motion is much weaker than horizontal motion so if the scanning elevation angle is fairly shallow ATMO 251 Chapter 8 page 15 of 23 the vertical motion will hardly contribute to the radial velocity at all and can be neglected As long as the elevation angle is not zero each distance along the scan will correspond to a different height above ground All that must be done is determine the average horizontal wind components at each distance This is easily done assuming there s not much wind variation horizontally across the radar scan the wind will be blowing from the direction with the largest component toward the radar and will be blowing toward the direction with the largest component away from the radar The wind speed is given by those peak magnitudes Plvmouth State v ea her Center August A velociTy scan from The Columbia SC WSR BSD Doppler radar The color scale is on The lower lefT posiTive speeds are for moTion away from The radar and negaTive speeds are for moTion Toward The radar Range rings are 30 n mi aparT AT 1 range disTance from The radar of 20 n mi The sTrongesT speeds away are norTh of The radar and The sTrongesT speeds Towards are souTh of The radar Thus The wind is blowing from souTh To norTh The maximum wind speed away is 20 kT and The maximum wind speed Toward The radar mosT sTrongly negaTive is 16 kT So The average speed of The souTherly wind is 18 kT in The direcTion shown by The black arrow The heighT of The radar beam aT This range is 300 m above ground The blue arrow shows The maximum winds Toward and away from The radar aT a range of 60 n mi corresponding To a heighT above ground of 1300 m The wind is from TheSSW aT 3O kT ATMO 251 Chapter 8 page 16 of 23 One can estimate winds at different heights from a single tilted scan of the radar To keep the measurements close to the radar a higher angle scan can be used for higher altitudes The result is data describing the vertical distribution of winds Since this information is available for each volume scan of the radar a new set of observations comes in every ve minutes or so A wind pro ler takes a different approach Rather than scanning it uses three xed angles for its beams straight up fteen degrees to the east of straight up and fteen degrees to the north of straight up The beam pointing slightly to the east will measure a radial velocity that has contributions from the eastward wind component and the upward wind component Similarly the beam pointing slightly north of straight up will receive contributions from the northward and upward components But with the third beam pointing straight up the upward component is known exactly and vector arithmetic can be used to determine the other two components B A C Beam A points straight up measures W Beam B points 15 degrees north of vertical measures W and v Beam C points 15 degrees east of vertical measures w and u la l0 veIX39 Radial velocity measured by profilerX v velB velAcos15 sin 1 5 u velC veIAcos15 PROFILER 67 TimeHeight Sections of Wind The display of wind observations from a wind pro ler or VAD is usually done by means of a timeheight section The time section method takes advantage of the frequent pro les to display information about the time evolution of the wind Along each vertical column of the chart wind ATMO 251 Chapter 8 page 17 of23 barbs are plotted much as they are on a sounding diagram with north toward the top of the image There are two schools of thought regarding how to deal with the time axis The simple way is to have the time increase toward the right just as you might expect it to do The result is a simple gure that shows the evolution of wind at all measured levels in the atmosphere The other plotting style is to have time run backwards increasing to the le This approach is o en used for pro ler data with the plots typically covering 1224 hours The basis ofthis approach is timespace conversion In rnidlatitudes other than in summertime weather features typically move from west to east Weather features that pass a particular station before other weather features must have started out farther east than the others With time running backwards the timeheight section is like a vertical section running west to east The time section shows the relative positions of the weather features recorded as they pass the pro ler site ND ND MD MD ND m m up ND ND ND ND ND ML 2 N r 1 D D VD ND ND 4 39 in ma In the example of a VAD plot shown above from Columbia SC time in UTC runs from le to right and height is labeled in thousands of feet The last column of data is for the same time as the Doppler velocity scan shown earlier and the winds are similar to what was estimated from the velocity scan being moderate from the SSE at low levels and stronger ATMO 251 Chapter 8 page 18 of23 from the SSW at 40005000 about 12001500 In Wind speeds are remarkably uniform alo only exceeding 40 kt from the SW at the highest data point at the last time mx i Envuronmental Technology Laboratory Boundary Layer Wind Profiler Studies Data provided by the NOAA Envimnmental Technology Labovalcry I o g 5 5 7 3 E x 7 e 392 m 1 1 u 30 J r 7 U A an a a J i n 9 A 1 2 5 1 an 7 J 5 HemLt msl c 1 l Juiniai a m39 T117719 UTC In the example of a Wind pro ler plot above time plotted in Whole hours runs from right to left and covers a 24hour period To aid in reading the chart at a glance the Wind speeds are indicated both by barbs and by color The strongest Winds from the south occurred in the early morning hours and the Winds overnight on Sept 17 appear to be increasing just as they did on Sept 16 Some of the Winds at the lowest level and a lot of the light Winds above 15 km appear to be erroneous probably caused by weak signal strength in the returned radar beam or groundlevel interference ATMO 251 Chapter 8 page 19 of23 68 Other TimeHeight Sections Time sections can be used to display any meteorological information for which both vertical distribution and time evolution are important As with horizontal maps scalar variables are depicted on vertical time sections using contours or isopleths One such variable is temperature Temperature changes at different altitudes describe changes in temperature inversions and strati cation If the air column is close to convective instability temperature drops aloft imply decreasing stability All of these can be diagnosed from a vertical time section with temperature plotted Another aspect of the temperature distribution that is relevant for forecasting is the dry adiabatic lapse rate When the lapse rate is dry adiabatic the atmosphere is well mixed and neutral At low levels knowing where the atmosphere is neutral provides information about pollution distribution and maximum temperature alo it provides information about turbulence Jwvx mrW m Ago A M my ltlt39l nix 43mm 1 44 Mailigrwxrrrmm 4 ltrrrr 39 39MAQQLQJJMg s sWv k x2 z QMJJMQ i k w w M r r 71 7177 ma 4 WWWwa W rmm r fi f r n r M77775 trer mm s 4JJJ A J W JJX W quotquotimll gg gi AMJJJJ441 lll ll xuwwmwilimitaij l quotmeuiiw llllllllJJlUZJJJJJJJ s W ll In the example of a temperature timeheight section above isotherms are every 2 degrees C and winds are also shown The vertical axis is pressure The image is from a model forecast and covers three days with time running from le to right Notice how the warm ATMO 251 Chapter 8 page 20 of23 temperatures every a emoon are con ned to the lowest 100 mb or so above that level the temperature responds more to largescale changes than to the daily heatingcooling cycle The cooler temperatures on the last day are caused by a cold front passage note how the winds were from the south during the rst part of the forecast but have shi ed to NE by the end of the forecast More useful than temperature for diagnosing the dry adiabatic lapse rate is potential temperature Rather than looking for a particular vertical rate of variation of temperature one can look for where the potential temperature is constant with height And a time section of potential temperature shows all the other important information contained in temperature too Since potential temperature is proportional to temperature rises and falls of potential temperature say the same about temperature Variations in strati cation are even easier with potential temperature The greater the strati cation the larger the rate of increase of potential temperature with height The more tightly packed the isentropes potential temperature contours in the vertical the greater the strati cation vquot 1 Us i I I 7amp7 W M Aw k JNU W K UJJX I J quotmullWV 4 39 wf w M t wwmwmw f HJJW tirg yJM Ma zy Vzag iw k4 J 95 1 i V WMLMMAMWWJ quot 39 1 WW 7 7 W ffo f W I W W W Farr rm 39 MJ W44 WWW XXXLAELULUJI JJJ llllJJJJaLMQLU JJJ W ANNA lejllllJJ39JJ ll lJMlelUllLlJVJJJJJJ l W ll 57 mwnm cm1 in mg I L In the example for a potential temperature timeheight section for the same location as the temperature timeheight section potential temperature in K can be seen to increase upward compared to temperature which decreased upward Also the warmest times of the day ATMO 251 Chapter 8 page 21 of23 correspond to periods in which the potential temperature is constant in the lowest 100 mb or more This means that the lapse rate is dry adiabatic exactly what would be expected for the daytime planetary boundary layer In the model forecast a squall line is forecasted to pass the station about midway between 12 UTC on the l7Lh and 00 UTC on the 18m The gust front of the squall line shows up as lump of low potential temperatures at and near the ground Fronts and squall lines are usually easier to see in a potential temperature plot than a temperature plot because the high strati cation at the top of the pool of cold air corresponds to a cluster of potential temperature isopleths grabbing the viewer s attention 69 Vertical Sections The same plotting styles and conventions for vertical time sections apply to vertical sections in space Vertical sections can be constructed from gridded model output or from rawinsondes With gridded output the creation of a vertical section is straightforward However one will rarely nd rawinsondes lined up perfectly To create the section one determines the straight line representing the section and then assigns near rawinsondes to the points on the line closest to the rawinsonde stations Stations c use to The ver f cal perpendicular To The section line ATMO 251 Chapter 8 page 22 of23 Vertical sections are a superb method for visualizing the vertical structure of the atmosphere Unlike surface maps or constant pressure maps though it s not possible to designate standard vertical section locations Vertical sections are chosen based on the particular circumstances to for example be perpendicular to fronts or jet streams Thus vertical sections require the use of an interactive program images of vertical sections are rarely found on web sites When drawn to be perpendicular to a linear weather feature such as a front or jet a vertical section is known as a cross section Often the term cross section is applied to vertical sections in general but that use of the term is technically erroneous Questions 1 Obtain a radar scan at a nonhorizontal elevation angle Estimate the wind direction and speed every 100 m in the vertical Use trigonometry to convert horizontal distances on the radar scan to vertical elevations 2 Why is a typical water vapor image a lot fuzzierlooking than a typical infrared satellite image 3 Based on the known characteristics of visible and infrared satellite images describe clouds that would be hard to detect by satellite 4 Suppose the vertical velocity is zero and the horizontal wind is 20 msec from the southsoutheast 150 degrees What radial velocities will the three beams of a radar wind profiler record 5 Find four soundings that are almost located along a line Plot a crude vertical section of temperature by a setting up graph coordinates with the X coordinate being distance along the line and the y coordinate being pressure with the highest pressures near the bottom Examine the soundings and jot down the temperatures every 100 mb along each sounding at the appropriate locations on the vertical section Analyze the sounding by drawing isotherms every 10 C 6 Suppose the surface wind is 120 degrees at 10 knots the wind at 500 m is 180 degrees at 30 knots and the wind at 1000 m is 220 degrees at 20 knots Sketch what a radar radial velocity display would look like if the scan was at a high enough angle to see 1000 m above the ground ATMO 251 Chapter 8 Weather Observation and Analysis John NielsenGammon Course Notes These course notes are copyrighted If you are presently registered forATMO 251 at T exasAampM University permission is hereby granted to download and print these emrse notes for your personal use If you are not registered forATMO 251 you may view these course notes butyou may not download orprint them without the permission ofthe author Redistribution of these emrse notes whether done freely or for profit is explicitly prohibited without the written permission of the author Chapter 7 BITS OF VECTOR CALCULUS 71 Vector Magnitude and Direction Consider the vector shown in the diagram The vector is drawn pointing toward the upper right The origin of the vector is literally the origin on this xy plot Sup ose we want to know the magnitude of this vector In high school you probably learned about computing vector lengths by starting ATMO 251 Chapter 7 Vector Calculus page 1 of 21 with the magnitudes of the components of the vector computing the squares of the lengths summing them and then taking the square root Well we ll use that technique eventually but that s way too complicated for most weather analysis applications Instead let s keep things simple The length of the vector is proportional to its magnitude so once we know what a given vector length corresponds to we can just measure the vector and convert it to a magnitude Since the gure in this case has a grid background we ll start by asking what the magnitude of a vector would be if it were exactly one grid box side long Then we can see how many grid boxes the vector covers and convert that to a vector magnitude Enough hypotheticals let s do this for real Let s say the vector is the horizontal wind The magnitude of the wind is called the wind speed Now suppose each grid box corresponds to a wind speed of one meter per second 1 m s391 lfwe take a ruler to the page we nd that each grid box is halfaninch wide So a vector that s 2 inch long on this particular graph would have a magnitude of l m s39l A vector that s aninch long would be 2 m s39l a vector that s 1 2 inches long would be 3 m S4 and so forth If we measure the vector we nd that the vector is 16 inches long 1 So the wind speed is just a little bit more than 3 m s39 32 m s39l Speci cally it s r some of you it may be obvious where that answer came from If not this is like any conversion problem and solving conversion ATMO 251 Chapter 7 Vector Calculus page 2 of 21 problems is a necessary skill so let s work it out in detail There is one conversion here 1 m s391 05 graph inches In this case we know the length of the vector in graph inches Divide the conversion equation by the side with the units that have been measured 1 m s391 05 graph inches 05 graph inches 05 graph inches The right hand side is unity a number divided by itself Divide and simplify the left hand side so that its denominator is unity too 2 m s391 graph inch 1 Now as you know from arithmetic you can multiply any number by l and get the same number back Since the left hand side of the above equation is equal to 1 you can multiply your measurement by it to convert units So since the drawn vector is 16 graph inches we get vector magnitude 16 graph inches 2 m s391 graph inch 32 m s391 Okay that s reasonably quantitative for the wind speed so now what about the wind direction You might think the wind direction is from the southwest as the vector is drawn but that depends on which way north is If you place a piece of paper with the figure on the ground the vector might actually be pointing south or northwest or eastsoutheast To give a compass direction we must first know the orientation of the grid For cartesian coordinates such as these the conventional orientation has the x axis pointing toward the east and the y axis pointing toward the north This gives the graph the same orientation as a map if you normally look at a map with north at the top With this orientation of the graph specified the vector is indeed pointing from the southwest But hey southwest isn t good enough We determined the wind speed to a precision of 01 m s39 surely we owe our estimation ofthe direction similar care So if you take out your protractor that is if you have a protractor or remember what one is and measure the angle relative to north the y axis you will find that the wind vector is pointing toward a compass heading of about 58 degrees see Chapter 3 for the correspondence between cardinal directions and degrees For most vectors that might be a good enough answer but remember again from Chapter 3 that wind is an exception Meteorologists express wind direction as the direction the wind is coming from not going towards So we must add 180 degrees to get the compass ATMO 251 Chapter 7 Vector Calculus page 3 of 21 heading on the opposite side of the compass dial this Wind direction is 238 degrees Suppose you don t have a protractor With a calculator you can still compute the orientation With just a little trigonometry Look at the next gure Which shows the triangle formed by the vector relative to the axis representing north The tangent of the angle we are looking for is equal to ba And the values ofb and a are easily seen from the gure b 4 ATMO 251 Chapter 7 Vector Calculus page 4 of 21 the distance ofthe endpoint ofthe vector from they axis is 28 grid boxes 14 m s39l using the conversion we worked out earlier and a the distance ofthe endpoint ofthe vector from the x axis is 18 grid boxes 09 m s39l Plugging these values into a calculator the arctangent of 2818 or 156 is 57 degrees mighty close to the previous estimate Add 180 to that and we get a wind direction of 237 which is equal to 238 to within the level of accuracy of our estimate Without a ruler we could also have used simple trigonometry to determine the magnitude of the wind By the Pythagorean Theorem a2 b2 02 the square ofthe length ofthe vector A little more calculator work tells us that 09 m s39lZ 14 m s39lZ 277 m2 s39Z so c is the square root of277 mZ s39Z or168 m s39l This too is pretty close to our ruler estimate To summarize so far we have determined the approximate value ofthis wind vector to be 16 m s391 from 238 degrees based on its length and orientation To do that we needed to know the length scale and orientation of the graph The vector we worked with began at the origin of the graph But neither magnitude nor direction depends on the location of the vector on the graph Any vector drawn on the graph with that magnitude and direction will have the same length and the same orientation as the one already drawn there 7 we would say that the two vectors are equal Each of These vee ror39s is equal to each of fhe vectors ATMO 251 Chapter 7 Vector Calculus page 5 of21 72 Vector Addition and Unit Vectors any ordinary mathematical operations that are normally applied to numbers can be applied to vectors as well The simplest such operation is addition You can t add a number to a vector because any two 39ngs to be added together must be similar in form and a number is just a number while a vector has both magnitude and direction You may object that you add numbers to vectors all the time For example you might say that if you re driving at 55 miles per hour toward the north a vector and add 5 miles per hour a scalar you re then going 60 miles per hour toward the north True but what you added wasn t really a scalar it was a vector because that extra 5 miles per hour were directed toward the north It had a direction as well as a magnitude Any two vectors can easily be added graphically To do so rst draw the second vector so that it begins where the rst vector ended Then a vector drawn from the beginning of the rst vector to the end of the second vector is the sum of the two vectors A visual example is shown here To add the blue vector to the green vector just move or redraw the blue vector so that it starts where the green vector 1e off The sum of the two vectors is shown in purple it starts at the beginning of the green vector and ends at the end of the blue vector To add The green and blue vecTors on The lefT draw The second To STar T where The first ends The vecTor sum in purple begins aT The beginning of The firsT vecTor39 and ends uT The end of The second vecTor Another example can be drawn from the previous section Remember the components of the vector We ve redrawn it here Suppose that the triangle side labeled a was really a vector that started at the origin and ended 18 grid boxes in the y direction Also suppose that the triangle side labeled b was really a vector that started at the end of a ATMO 251 Chapter 7 Vector Calculus page 6 of 21 and ended 28 grid points over in the x direction Then the vector labeled 6 would be the sum ofvectors a and b One last vector addition example is the car example from earlier in this section It s kind of hard to draw a vector from the beginning of one to the end of the other When both vectors point in the same direction So the diagram shows the sum vector off to one side Also so that the ATMO 251 Chapter 7 Vector Calculus page 7 of 21 vectors will t on the page the scale is different from the previous examples A vector drawing is not much use without a scale such a drawing shows direction but not magnitude 73 Vector Multiplication and Components Vectors oriented along the x and y axes come up so frequently that there s a simpler way to describe them using the concept of unit vectors A unit vector is simply a vector with a magnitude of 1 in whatever units are currently popular Three unit vectors have special names unit vectors i j and k are the unit vectors oriented in the positive x y and 2 directions respectively The i and j unit vectors are shown here ex While you can t add a number to a vector there s nothing wrong with multiplying a vector by a number When you multiply a vector by some number you simply multiply the magnitude of that vector by the number without altering the vector s direction unless the number is negative in which case the new vector will point in the opposite direction What happens if you multiply a unit vector by a number The general result is a vector with the magnitude of the number and the direction of the unit vector Unless the number is negative in which case you get the direction opposite the unit vector For example let s say you multiply the unit vector j by the number 09 m s39l Multiplication by a positive number doesn t affect the direction of the vector so it s still pointing toward the north Its magnitude is 1 x 09 m s39l or simply 09 m s391 This is vector u from earlier ATMO 251 Chapter 7 Vector Calculus page 8 of 21 Vectors are often written in terms of components Our vector a is 09 m s391 j This is precisely the multiplication in the previous paragraph When 09 m s391 is multiplied by j you get vector a We would say that the j component of vector a is 09 m s391 and that the i and k components are zero If an arbitrary vector does not happen to be parallel to one of the standard unit vectors it will have more than one component Vector c from above has components of 14 m s391 i and 09 m s391 j Describing vector c as 14 m s391 i and 09 m s391 j or equivalently 14 m s391 i 09 m s391 j conveys precisely the same amount of information as saying the vector c is 17 m s391 from 238 degrees A componentbased description of vectors is common in many circumstances and is especially useful for mathematical manipulations of vectors Any vector in two or n dimensional space can be written as the sum of two or n component vectors oriented parallel to the axes Indeed it is so common when considering the wind vector that special variable names are applied to the magnitudes of the components of wind in the x y and 2 directions u v and w In the case of our favorite wind vector c we would say that u 14 m s391 and v 09 m s39l The simple nongraphical way of adding two vectors is to add the two vector components Thus if we had a wind vector of u 25 m s391 and v 35 m s391 and we wanted to add to it a wind vector ofu 15 m s391 and v 1 m s39l we could do it in our heads The answer is a wind vector ofu25ms39115ms3914ms391andv35ms39171ms39125ms391 Much easier than graphing the vectors adding them that way and then measuring the magnitude and direction of the sum Another very practical reason for expressing vectors using their components has to do with what happens when we remember that the universe we sense has three spatial dimensions not just two We have expressed the wind direction with respect to compass headings but air is free to move up and down as well as sideways Spatially speaking the velocity of the air is a threedimensional vector It s still possible to specify a direction in three dimensions by giving the compass heading as well as the elevation angle the angle that the vector differs from horizontal Astronomers specify angles this way all the time as do radar scientists But with component notation it s really easy to include another dimension just add the third component Heck if you want to you can keep adding components until you get eight or ten dimensions Don t laugh 7 physicists do this all the time Aside from being convenient for mathematics and more than two dimensions component notation for wind is of practical forecasting value ATMO 251 Chapter 7 Vector Calculus page 9 of 21 Meteorologists care quite a bit about the sign and magnitude of w because it is intimately related to the formation or lack thereof of clouds and precipitation Speci cally for forecasting the horizontal components mselves have much less individual value When working with wind in a forecasting environment a hybrid strategy is most o en used specifying the horizontal part of the wind as a speed and direction and using w to specify the vertical wind 74 Vector Subtraction Vector subtraction always seems to give some students the heebie jeebies I m not sure why The act of subtracting two vectors using components is just as easy as adding two vectors Graphically there are two equally good ways to subtract vectors while adding vectors only has one good way Perhaps that s the problem there are too many ways of getting the right answer so students get confused To subtract vectors using components just add subtract the components of the second vector from the components of the rst vector So keeping with our previous example if we want a 7 b we can compute apof25ms39 715ms391 ms391 andavof35ms3917l ms39l45m s A A Objective ve v Lns cTor b from vec ror a A u One graphical way to subtract vectors is to recognize that a 7 b a b So just draw b backwards and add it to a The other graphical way is just as simple just draw both vectors starting at the origin then draw a ATMO 251 Chapter 7 Vector Calculus page 10 of21 third vector starting at the endpoint of b and ending at the endpoint of a This third vector is a 7 b Notice from the pictures that both techniques yield the same answer CD U1 v ElS u One wa To subtract vector b from vector u is To add b To a l 0 f u mls Another way 3 To subtract A vecTor b from 7 vecror a is To l E draw b and a a gt from The same 1 s rar ring pain and than draw 0 a vector from The and of b To The Iand ofla 1 mls An important application of vector subtraction is wind shear Frequently it is important to know the difference in wind between one level and another Determining the difference between the vector wind at one level and that at another requires vector subtraction ATMO 251 Chapter 7 Vector Calculus page 11 of21 75 The Dot Product There are two other vector operations that are common in meteorology as well as anywhere else that uses vectors the dot and cross product The use of the term produc seems to imply multiplication and indeed the computation of dot and cross products using components involves multiplication However I personally don t think that multiplication is the right way to think about these operations in a meteorological setting Take the dot product Let s write the u and v wind components of a vector u as Ma and Va Expressed as components the dot product of horizontal wind vectors a and b is uaub vavb If you have three or more components to your vectors just keep adding the products of the individual components until you run out of components Expressed as magnitudes and directions the dot product is lul lbl cos 6 where lul and lbl are the magnitudes of the two vectors and 6 is the angle between Note that the dot product is a single number a scalar rather than two components or a magnitude and direction a vector so the dot product is often referred to as the scalar product of 1 onto the y axis That makes a The y of So far what we ve said about the dot product is merely annoying fodder for memorization The dot product takes on physical signi cance though if we keep in mind that the dot product of some vector with a unit ATMO 251 Chapter 7 Vector Calculus page 12 of 21 vector is the projection of that vector onto the axis represented by the unit vector Imagine that there was a light off way in the distance to the right ofthe rst gure in section 71 The line labeled a would be the shadow cast by the vector c onto the y axis We say that a is the projection of c onto y The length of this line is computed as and is equivalent to the dot product c 0 j In components we have 14 m s39 0 09 m s391 1 09 m s39l This is exactly equal to the v component of c Similarly if we compute the dot product using magnitudes and directions we get 16 m s391 1 cos 58 09 m s39l In summary the dot product is a measure of the magnitudes of two vectors and the smallness of the angle between them The smallness is measured by the cosine of the angle if the two vectors are at right angles the cosine is 0 and so is the dot product The dot product can be thought of as the projection of one vector onto another multiplied by the magnitude of that second vector 76 Advection as a Dot Product What does it take to get a large dot product For starters it helps if the two vectors being multiplied together are large rather than small Also the angle between the two vectors should be as small as possible if the two vectors are parallel the dot product is largest Does that sound familiar Remember advection advection is strongest if the wind is strong and the gradient of the thing being advected is also strong Furthermore the wind should be close to perpendicular to the isopleths of the advectee that is close to parallel to the gradient vector of the advectee Those are the same principles as for a dot product so it should not be surprising that advection actually is a dot product The advection of some scalar quantity T such as temperature is written as V0VhT This is a new equation so first let s check to see if it makes sense The gradient vector VhT points across isopleths isotherms toward higher values of temperature If the wind is blowing in the same direction the dot product of the two vectors will be largest and meanwhile the wind will cause the temperatures to drop This is cold advection and the minus sign in the de nition of advection indeed produces a negative value in this example ATMO 251 Chapter 7 Vector Calculus page 13 of 21 Let s explore the equation a bit more While gradient notation is useful for writing things concisely it takes a while to get used to the meaning so let s gure out how the de nition of advection works when the vectors are written in component notation First the gradient VhT The components of this vector are de ned as VhTE i i j 8x 8y Each component of the gradient vector is equal to the rate at which T changes in the corresponding direction For example the x component is equal to the rate at which T changes in the x direction Analysis pressure mb tens and units digits There are several ways of estimating gradients from a map with isopleths Here these methods are illustrated using a sample pressure analysis By convention the isobars are labeled with only the tens and units digits the leading 10 in this case is dropped All methods require knowing the horizontal scale of the map Beyond this you can either estimate the magnitude and direction of the gradient directly and then determine the components of the gradient vector as the projections on X and y or estimate the two components of the gradient vector and then compute the magnitude of the gradient using the Pythagorean Theorem You could also compute the direction of the gradient using trig but an exact direction is rarely needed in practice ATMO 251 Chapter 7 Vector Calculus page 14 of 21 Me rhod 1 Defermine The smalles r spacing be rween confours on eiTher side of The point of interest The direction of This line Toward higher values shown in green is The direction of The gradient vector vp 4 mb 115 km 0035 mblkm vp 0035 mblkm 280 degrees ApAx Vp dot i ApAx 0035mbkm 00528090 ApAx 0035mbkm 0985 Ale 0034 mblkm ApAy 0035mbkm 0052800 ApAy 0035 mblkm 0174 ApAy 0006 mblkm Okay say we ve computed the horizontal gradient of temperature using one of the three methods For computing advection we also need the horizontal wind v We already know the components of v ui vj To get the advection multiply the corresponding components together and throw in a minus sign ivthT 7u2 T7v2 T X y We ve seen something like that equation before Suppose we choose to orient the coordinate system so that the x axis is parallel to the ATMO 251 Chapter 7 Vector Calculus page 15 of 21 Method 2 Determine the horizontal derivatives by estimating the spacing of adjacent isopleths in the x cm y directions works as long as the isopleths are not too far apart ApAx 4 mbl120 km ApAx 0033 mbkm ApAy 4 mb340km ApAy 0012mblkm Vp 0033mblkm i 0012mbkmj vp 00332 00122 mblkm pl Ik V 035 mb m wind Then v is zero and the second term on the righthand side vanishes What s 1e is the same equation that was introduced in Chapter 3 Back then we stated that the equation only worked if x is chosen to be parallel to the wind That s true because only then is v zero One could compute the advection using that equation but when you re eyeballing advection on a map it s much easier to use alternative way of writing the dot product iai ibi cos 9 For advection this is 7 v0 VhT ilvlthTlcos8 Just take the wind speed multiply by the temperature gradient and nally multiply by the cosine of the difference in angle between the wind vector and temperature gradient vector ATMO 251 Chapter 7 Vector Calculus page 16 of 21 Method 339 Determine The horizontal derivatives by estimating The values of The analyzed field at equal shor39f distances an ei ther side of fhe point of interest Ax Ay 200 km 2 grid boxes ApAx 10108mb 10178mbl200km A Ax 0035mblkm AplAy 10156mb 10140mb200km ApAy 0008mblkm Vp 0035mblkm i 0008mblkmj Vp D0352 00082 mblkm Vp 036 mbkm An example is shown in the gure on the next page It doesn t hurt to have a few key values of cosine memorized cos0 1 cos60 05 cos90 0 cos120 05 and cos180 1 77 The Cross Product The cross product is like the dot product a measure of the magnitudes of two vectors and the largeness of the angle between them The largeness is measured by the sine of the angle if the two vectors are parallel the sine is 0 and so is the cross product So the cross product is sort of the opposite of the dot product ATMO 251 Chapter 7 Vector Calculus page 17 of 21 Analysis temperature Celsius Temperature gradient from 45 deg shown in green at about 0016 Ckm Wind from 360 deg a about 10 ms Advec tion 0016 Ckrn10 ms cos360 450001kmm Advec tion 11x104 Cs To ge l more convenient units multiply by 11x104 53 hrs To get Advec on 12 63 hrs One thing that makes the cross product a bit more complicated than the dot product is that the answer is a vector rather than a scalar The mathematical techniques for computing the complete cross product are suf ciently involved that we won t go into them here Fortunately for us the magnitude of the cross product vector is fairly simple As you might imagine from the above description the magnitude is lal lbl sin 9 For future reference we ll note one property of this magnitude computation here If one of the vectors is a unit vector oriented perpendicular to the other vector the magnitude of the cross product is simply the magnitude of the other vector The direction of the cross product vector is a bit stranger The cross product vector is perpendicular to the plane on which the two vectors being crossed lie But you don t really need to know that now What ATMO 251 Chapter 7 Vector Calculus page 18 of 21 you do need to know is the rule for graphically determining the orientation of the cross product of two vectors At most schools they use something called the righthand rule Here in Aggieland we use the Gig em rule Begin by making the Gig em signal with your right hand Next point your st in the same direction as the rst vector Then without changing the direction of your forearm rotate your thumb so that the palm of your hand is facing the direction toward which the second vector points The direction of the cross product is now given as the orientation of your right thumb Arm points toward green vector palm faces purple vector Thumb gives orientation of cross product of green and purple vectors The orientation of the cross product is shown in maroon I bet you always thought that the Gig em signal was just a pleasant but meaningless way of expressing Aggie spirit Now you know the truth It s really our way of saying to the rest of the world Hah We know how to determine a cross product Take pride in your mathematics Unlike the dot product you may be able to survive a meteorology curriculum without fully understanding the cross product 78 Divergence and Curl There are a couple of calculus operations that share the notations of dot and cross products For reference we list them here but we won t make use of them for a few more chapters ATMO 251 Chapter 7 Vector Calculus page 19 of 21 Although these operations can be applied to any vector most common meteorological applications of them involve the horizontal wind So we ll always work with wind here The horizontal divergence is de ned as A related quantity the threedimensional divergence is Bu 6v 6w VOVE 6x By 62 The vertical component of the curl of the horizontal wind is called the relative vorticity It is de ned as k0VXVEg Questions 1 Sketch arrows with the following direction and magnitude on a piece of paper Use a consistent horizontal scale for each The directions are provided using the wind convention the direction the vectors are pointingfrom a 270 deg 5 ms b 330 deg 8 ms c 196 deg 1 ms d 48 deg 12 ms e 92 deg 2 ms 2 Compute the components of the vectors in Question 1 3 Compute vectors ab bc cd de and ea from Question 1 4 Repeat question 3 using purely graphical techniques and determine the accuracy of your answers 5 Compute vectors ab bc cd de and ea from Question 1 6 Repeat question 5 using purely graphical techniques and determine the accuracy of your answers 7 Compute the dot products of aampb bampc campd dampe and eampa from Question 1 ATMO 251 Chapter 7 Vector Calculus page 20 of 21 8 If the horizontal temperature gradient is oriented toward the SE at 2 C 100 km compute the temperature advection with winds a through e from question 1 9 List all the possible directions that the cross product of two horizontal vectors can point ATMO 251 Chapter 7 Vector Calculus page 21 of21 Weather Observation and Analysis John NielsenGammon Course Notes These course notes are copyrighted If you are presently registered for ATMO 251 at Texas AampM University permission is hereby granted to download and print these course notes for your personal use If you are not registered for ATMO 251 you may view these course notes but you may not download or print them without the permission of the author Redistribution of these course notes whether done freely or for profit is explicitly prohibited without the written permission of the author Chapter 14 VORTICITY 14 1 Curl Like divergence and gradient curl involves derivatives of the components of a vector Like gradient curl is a vector The mathematical way of writing the curl of some vector 7 is V X 7 Even if the vector 7 is entirely horizontal the curl is a fully three dimensional vector The de nition of curl in three dimensions is most clearly written in the form of a determinant as follows i j k vw3 3 a 6x 6y 62 u v w Here we have explicitly assumed that the vector in question is the velocity so the three velocity components appear on the bottom row If you know what a determinant is great If you don t know what a determinant is or how to compute in three dimensions don t worry You can get by with knowing what the three components of the curl are a 6w 6v 6u 6w 6v 6u va 1 3 k 6y 62 62 6x 6x 6y Why do they call it the curl Because it measures the tendency of the vector eld in this case the velocity to rotate Consider for ATMO 251 Chapter 14 page 1 of 16 example the vertical component of curl the last term in the preceding equation For an entirely twodimensional world this is the only component of curl that is nonzero Suppose now that you are looking down on a low pressure center with its associated cyclonic counterclockwise circulation If the wind is in geostrophic balance we know that the divergence is zero but what about the curl The vertical component of vorticity is V mp nent positive in the case of counterclockwise circulation around a low centerx u component 39 negative easterly I negative I l positive northerly southerly I l westerly AulAy lt 0 Alex AuIAy gt 0 Alex gt 0 ATMO 251 Chapter 14 page 2 of 16 To proceed let s check out the sign ofthe vertical component of curl East of the circulation center the wind ought to be from south to north a positive v while west of the circulation center the wind ought to be from north to south a negative v So v changes in the x direction speci cally it increases with increasing x That means that the derivative ofv with respect to x is positive North of the circulation center the wind should be blowing from east to west while south of the circulation center the wind should be blowing from west to east So to the south u is positive and to the north u is negative Thus the derivative of u with respect to y is negative since u decreases with increasing y Taking stock we have the vertical component of curl equal to a positive term minus a negative term Since minus a minus is a plus the vertical component of curl must be positive overall You can con rm the same thing for the other two components Imagine that you re looking at the x 0 plane from a position on the positive x side Suppose you see counterclockwise rotation of the wind If you work out the signs of the various derivatives you ll nd that the x component of curl is positive Repeat the procedure with counterclockwise rotation in the y plane and you get a positive y component of cur This wind field has a posi rive x component of cur l lt Y 5 X To remember all this go back to the gig em rule Point the thumb of your right hand in the direction of the component of curl The vector pattern that produces a cur1 component in that direction is given by the ngers of your right hand if you are forming the shape of a gig em The vectors are oriented or the wind is blowing along your ngers pointing from the hand along the curve of the fmgers to the tips of your ngernails ATMO 251 Chapter 14 page 3 of 16 142 Vorticity So far we ve used the term curl exclusively In meteorology or in uid mechanics in general the curl of the wind or ow eld that is the curl of the velocity gets a special name the vorticity Meteorologists are actually a little bit lax about the term When they say Vor city they are generally referring only to the vertical component of the curl of the wind When the full threedimensional curl is meant meteorologists generally refer to the Vorticity vector We will follow this convention here For the rest of this chapter the vorticity means the vertical component of the curl of the velocity Vorticity has a crucial physical interpretation The vorticity is a measure of the spin of the wind about a vertical axis with counterclockwise spin being positive Imagine you stick a fourpronged propeller into the air with the spin axis of the propeller vertical We ll take the Cartesian coordinate system as being oriented so that the four prongs are sticking directly north south east and west respectively Think about the two prongs sticking east and west For the wind to try to cause them to spin counterclockwise you d want the northward component of wind to be larger on the east prong than on the west prong In other words the derivative of v in the x direction would need to be positive and the larger the derivative the stronger the turning impulse As for the two prongs sticking north and south to get them to turn counterclockwise you d want the eastward component of wind to be larger on the south prong than the north prong So the derivative of u in the y direction would need to be negative and the more negative the better Thus the larger the vorticity the greater the tendency of the wind to spin a propeller ATMO 251 Chapter 14 Positive curvature vorticity causes counterclockwise spin l7 page 4 of 16 There are lots of different wind patterns that would cause a propeller to spin and that therefore have vorticity We ve considered only one example so far that of wind rotating in a circle Vorticity that results from horizontal variations in the direction of the wind is called curvature vorticity For positive curvature vorticity the wind must be curving counterclockwise or toward the le facing downwind Negative curvature vorticity would have the wind curving in the other direction Negative curvature vorticity causes clockwise spin The other type of vorticity is called shear vorticity Here the shear refers to horizontal shear rather than the vertical shear we discussed in the last chapter Imagine a perfectly straight wind eld and let s say it s blowing from west to east If the wind speed changes in the northsouth direction so that for example the westerly winds to the north are weaker Note that it39s the net effect on the imaginary propeller that matters air nee e hitting all arms in the same way as long as there s a stronger push in one direction than the other Positive shear vorticity causes counterclockwise spin ATMO 251 Chapter 14 page 5 of 16 than the westerly winds to the south we have horizontal shear But notice that this situation corresponds to one of the propeller cases so that the winds on the north and south prongs of the propeller would act to cause the propeller to turn The northsouth wind being zero everywhere has no effect so we have a counterclockwiserotating propeller and therefore we must have positive vorticity The basic rule is that if you are facing downwind and the winds are weaker to your left and stronger to your right you have positive shear vorticity If winds are weaker than your right the shear vorticity is negative Negative shear vorticity causes clockwise spin even Though no air is going in a circle The total vertical component of vorticity is the sum of the shear vorticity and the curvature vorticity Sometimes you have both simultaneously sometimes they add together and sometimes they cancel e atmosphere even if the wind is not blowing is rotating counterclockwise in the Northern Hemisphere That s because the Earth th it itself is rotating and the atmosphere is rotating w1 Including the effect of the Earth s rotation gives us the absolute vorticity The absolute vorticity has two contributing terms the vorticity associated with the wind and the vorticity associated with the spin of the Earth We call these two contributing terms the relative vorticity and the planetary vorticity The planetary vorticity is exactly equal to the Coriolis parameter f The relative vorticity is usually represented as the Greek letter Q So the absolute vorticity is f ATMO 251 Chapter 14 page 6 of 16 the Earth itself is spinning 143 Vorticity and Convergence There are several reasons to care about vorticity One simple reason is that high vorticity implies strong cyclonic circulation centers or strong troughs If cyclonic vorticity is concentrated along a line it marks the windshift associated with a front Fronts always have cyclonic vorticity But perhaps the most important reason is related to the relationship between vorticity convergence and vertical motion which lets us diagnose the intensi cation of cyclonic circulation centers or troughs and their converse anticyclonic circulation centers or ridges It would be great if we could diagnose changes in pressure directly Since pressure is proportional to the weight of air in the overlying air column one might think there s some hope of diagnosing temperatures and thereby diagnosing pressures But that s not very workable because horizontal temperature advection and vertical motion tend to act to oppose each other and the net temperature change is dif th to determine ATMO 251 Chapter 14 Planetary vorticity the air is spinning cyclonically even when it39s standing stillquot because page 7 of 16 We can also diagnose the forces acting on the wind and thereby diagnose changes in the wind But to do that we would need to be able to predict changes in the pressure gradient force so we have the same problem as if we re trying to diagnose pressure Vorticity however is fairly easy to diagnose There are two basic techniques the first involving convergence and vertical motion the second involving something called potential vorticity First we ll talk about the connection between vorticity convergence and vertical motion In Chapter 9 the connection between convergence divergence and vertical motion was first discussed and they go hand in hand A brief review if there s largescale upward motion there must be convergence in the lower troposphere and divergence in the upper troposphere Conversely if there s largescale downward motion there must be divergence in the lower troposphere and convergence in the upper troposphere Now for the connection between all this and vorticity D3 f QVXVh 0 7 other stuff This equation states that the rate of change of vorticity following an air parcel is proportional to the absolute vorticity times the negative of the divergence ie the convergence plus some other stuff The other stuff would be rather ugly if I wrote it all out but fortunately the larger the horizontal scale the smaller the other stuff Once we get to weather systems larger than a few hundred kilometers or so the other stuff is completely negligible and the divergence and convergence controls the vorticity Consider changes to vorticity in the lower troposphere If there s a low pressure center it will necessarily have cyclonic relative vorticity which is positive If the low intensifies the vorticity must increase The above equation states that for the vorticity to increase there must be convergence and therefore aloft there must be upward motion If instead the low weakens there must be divergence present and downward motion aloft High pressure centers have anticyclonic relative vorticity This is negative in the Northern Hemisphere but the absolute vorticity the sum of the relative vorticity if and the planetary vorticity f will almost always be positive in the Northern Hemisphere So for the high pressure center to intensify the vorticity must decrease Hence there must be divergence present and therefore downward motion present aloft Notice that if the issue is the intensification or weakening of a cyclonic disturbance a trough in the upper troposphere you still need convergence but since that convergence would be taking place in the ATMO 251 Chapter 14 upper troposphere it would mean downward motion in the middle troposphere The basic rule is as follows midtropospheric upward motion implies that the relative vorticity will become more positive or less negative at low levels and less positive or more negative aloft and mid tropospheric downward motion implies that the relative vorticity will become more negative or less positive at low levels and less negative or more positive aloft 144 Predicting Vertical Motion Using the principles described in the previous paragraph we can use vertical motion to diagnose intensification and weakening of weather systems All that s left is to predict vertical motion In past chapters we ve encountered several indicators of vertical motion One is friction friction produces crossisobar ow and convergence within lows while the crossisobar ow is outward and divergent around highs However frictionallyinduced vertical motion is the one kind of vertical motion that doesn t lead directly to intensification of largescale weather systems because the friction that causes the vertical motion also slows down the wind That s one of the things hidden in the other stuff Another is the effect of curvature and ageostrophic winds around upperlevel troughs and ridges Upward motion is expected downstream of troughs and upstream of ridges while downward motion is expected downstream of ridges and upstream of troughs So for example if a low level cyclone is located downstream of a trough one would expect to find upward motion and thus lowlevel convergence and intensification of the cyclone Another is the effect of accelerations within a jet streak Upward motion is expected beneath the righthand side of the entrance region and beneath the lefthand side of the exit region while downward motion is expected beneath the lefthand side of the entrance region and beneath the righthand side of the exit region Another is the effect of intensification or weakening of fronts If a front is becoming stronger there should be upward motion on its warm side while if a front is weakening there should be upward motion on its cold side Another is the effect of topography with upslope and downslope ow Except here is where the simple rule about lowlevel convergence being associated with upward motion doesn t apply The effect of topographic upward and downward motion will be much easier to explain in the context of potential vorticity later in this chapter ATMO 251 Chapter 14 page 9 of 16 Rather than going through a laundry list of possible mechanisms it would be nice to have a single equation that relates vertical motion to other straightforward easily observable aspects of the atmospheric state Such an equation does exist It s called the omega equation a er the symbol for vertical motion in pressure coordinates There are various forms of the equation involving different ways of arranging the terms different simpli cations and cancellations We won t go into the details here but the physics behind it is a threedimensional analog to the vertical motion associated with frontogenesis Basically anything that changes the strength or orientation of the vertical wind shear or horizontal temperature gradient experienced by an air parcel requires vertical motion to keep everything in thermal wind balance Until it s time to study the omega equation it will be suf cient to utilize the list of patterns and processes associated with vertical motion 145 Potential Vorticity The close relationship between vorticity changes and divergenceconvergence suggests that the vorticity of a particular air parcel depends entirely on its history of divergence and convergence If we now imagine that air parcel as being threedimensional horizontal convergence would squish it horizontally and stretch it vertically Similarly horizontal divergence would stretch it out horizontally and squish it vertically In principle if you had a way of determining how tall an air parcel is you could tell how much vorticity it had A J HorizonTal convergence Horizontal divergence requires vertical stretching requires vertical squishing There is such a way Remember that potential temperature is conserved by air parcels and increases upward Thus the potential temperature at the top and bottom of an air parcel stays the same even as the air parcel stretches and squishes Changes in the vertical spacing of isentropic surfaces surfaces of constant potential temperature corresponds to changes in the vertical extent of air parcels ATMO 251 Chapter 14 page 10 of 16 Since air parcels conserve potential Temperature vertical stretching or squishing of air parcels corresponds To increasing or decreasing The vertical separation between isen rropic surfaces The vertical spacing of isentropic surfaces has a name the strati cation If the isentropic surfaces are close together the vertical derivative of potential temperature with respect to height or pressure is large and the strati cation is high If the isentropic surfaces are far apart the vertical derivative of potential temperature is small and the strati cation is weak So if an air parcel undergoes horizontal convergence its absolute vorticity will increase and its strati cation will decrease Conversely if an air parcel undergoes horizontal divergence its absolute vorticity will decrease and its strati cation will increase It takes a fairly simple bit of mathematics to show that the product of vorticity and strati cation stays the same they go up and down proportionally so that their product is constant This product is known as the potential vorticity which we will write as P a gr 4 a 0 Dr Dz 6p The potential vorticity is conserved under the following conditions 1 no iction39 2 no diabatic heating such as latent heating Above the ground there s no friction and if there s no cloud either there s no signi cant diabatic heating SoP is conserved most of the time Also strictly speaking the vertical vorticity inP should be computed from the derivatives of wind on the isentropic surfaces rather than on constant pressure or height surfaces One simple application of potential vorticity is in ow over topography Picture air owing over a mountain range As air goes up ATMO 251 Chapter 14 page 11 of 16 and over so do the isentropic surfaces Since potential vorticity is constant the horizontal variations of vertical spacing of isentropic surfaces can be used to infer the places where the absolute vorticity is large or small Over the mountains the isentropic surfaces are necessarily closer together and the absolute vorticity is smaller On the lee downwind side the air parcels stretch and the absolute vorticity increases Most of the time there s a cyclone in the lee of large mountain ranges for this reason g gt Low Vorticity S High Vonicity j Low Vorticity 315 K J High Vorticitv 320 K 310 K High Vorticity 305 K As air flows over mounTains air parcels geT squished and Their vorTiciTy decreases As iT descends The lee side of The mounTains The air parcels are sTreTched verTically and Their vorTiciTy increases Thus cyclones weaken as They pass over mounTains and inTensify again on The lee slopes One thing to be careful of diagnosing vorticity directly this way only works within a single isentropic layer For instance in the example shown above if the 315320 K isentropic layer was shallower the vorticity in that layer wouldn t necessarily be lower All we can say is that within a given isentropic layer the vorticity decreases as air goes up the mountain range and increases as the air comes back down 146 Potential Vorticity and Vertical Motion Potential vorticity isn t just useful for understanding lee cyclogenesis More generally it s the most straightforward wa of understanding the dynamics of the atmosphere The principles of potential vorticity dynamics are listed here 1 Potential vorticity is mostly conserved You knew that one already 2 Anywhere the potential vorticity is larger than most of what surrounds it at the same isentropic level is a positive potential vorticity anomaly If the potential vorticity is smaller than what surrounds it at the same isentropic level it s a negative potential vorticity anomaly ATMO 251 Chapter 14 page 12 of 16 3 If the winds are balanced a single positive potential vorticity anomaly is at the center of a cyclonic circulation a negative anomaly is at the center of an anticyclonic circulation The winds weaken away from the anomaly in all three dimensions 4 If there are multiple anomalies the effects of all the anomalies can be added together to get the overall wind pattern 5 The stratosphere being very stably strati ed has much higher potential vorticity than the troposphere 6 Anomalies of potential temperature near the ground have the same effect as potential vorticity anomalies in the interior of the troposphere X In this vertical section isentropic surfaces are in red The tropopause is brown and a positive potential vorticity anomaly at the tropopause is in purple The full balanced state of the atmosphere is determined by this configuration There is a cyclonic circulation around the anomaly so the wind is into the page to the east and out of the page to the west with the vector symbols located where the winds are strongest Because of thermal wind balance there must be a horizontal temperature gradient above and below the wind maxima as indicated by the slopes of the isentropes Potential vorticity within the troposphere is uniform where the vorticity is cyclonic the isentrope spacing is larger In the stratosphere the potential vorticity is also uniform In item 3 balanced means geostrophic and hydrostatic balance and therefore thermal wind balance as well Winds pressures an temperatures are all consistent with each other A given potential vorticity distribution constrains the winds pressures and temperatures to have a ATMO 251 Chapter 14 page 13 of 16 particular pattern as well Even the isentropic surfaces are constrained to have a particular pattern by the particular arrangement of potential vorticity And you can t get a cyclone or anticyclone a trough or a ridge without a potential vorticity or nearground potential temperature anomaly at its core We ll now apply these principles to diagnosing vertical motion The key is the potential temperature surfaces Since potential temperature is conserved air that starts out on a particular isentropic surface can never leave it Since the isentropic surface positions are determined by the potential vorticity distribution air must move up or down along isentropic surfaces if they are sloped Suppose there is an upperlevel trough Since a trough is cyclonic there must be a positive potential vorticity anomaly associated with it That anomaly is generally located at the tropopause as shown in the previous gure With just the one potential vorticity anomaly in an otherwise undisturbed environment there would be no vertical motion Even though many of the isentropic surfaces have slopes the air is just circulating around the potential vorticity anomaly without going up or down The situation changes if the potential vorticity anomaly is embedded in a sheared environment Usually the temperature decreases M In midlatitudes with a horizontal temperature gradient the westerly winds will generally be strongest at the tropopause Here the wind is drawn in a frame of reference relative to the potential vorticity anomaly Ahead of the anomaly in the troposphere air is lifted upward and behind it the air sinks At the same time the air39s vortioity increases as it moves beneath the anomaly and decreases as it leaves ATMO 251 Chapter 14 page 14 of 16 toward the pole in rnidlatitudes so this is the ordinary situation Because there s a horizontal temperature gradient there s also vertical wind shear as shown in the gure on the previous page Essentially the tropopause potential vorticity anomaly is zooming eastward past most of the rest of the atmosphere As the anomaly approaches air ahead of it in the troposphere must be lifted upward because the isentropic surfaces slope upward there As the anomaly goes by the air descends again This process has been called the yacuum cleaner effect because it s as though the potential vorticity anomaly is sucking the isentropic surfaces and the air toward i So far we ve described the vertical motion caused by the interaction between the largescale shear and the bumps in the isentropic surfaces due to the anomaly There s also vertical motion caused by the interaction between the largescale sloping isentropic surfaces and the winds due to the anomaly as follows since the isentropic surfaces in the troposphere must slope upward toward the north because of the horizontal temperature gradient northerly winds are associated with downward motion and southerly winds are associated with upward motion With a cyclonic potential vorticity anomaly the air ahead to the east of the anomaly is moving northward and ascending and the air behind the anomaly is moving southward and descending WiTh a cyclonic circulaTion on a sloping isenTropic surface The air ascends on The easT side as iT glides up The sloping surface and descends on The wesT side To summarize once the relative motion of the air and the isentropic surfaces are known it is easy to determine where the air is gliding up and where it is gliding down Notice that both the convergencedivergence technique and the potential vorticity technique predict that there should be upward motion ahead of an upperlevel trough and downward motion behind an upper level trough This is to be expected if the two techniques gave ATMO 251 Chapter 14 page 15 of 16 contradictory answers one of them would be wrong In practice meteorologists use whatever approach is easiest and least ambiguous Questions 1 Suppose u and w are both zero in a particular area but v increases toward the south at the rate of 1 ms per 10 km and increases with height at the rate of 10 ms per km What is the curl of the velocity Express the answer as a vector 2 Sketch a vertical section for which the y component of the curl of the velocity eld is negative 3 Sketch wind fields with the following characteristics a Anticyclonic shear vorticity with the u component of horizontal wind zero everywhere b Cyclonic curvature vorticity and anticyclonic shear vorticity 4 At 50 N the convergence is 15 X 10391 s391 and the relative vorticity is 32 X 10395 s39l Is the relative vorticity increasing or decreasing How rapidly 5 One PVU potential vorticity unit equals 1 X 10396 m2 K s391 kg39l Show that the potential vorticity in the troposphere should be somewhere near 1 PVU 6 Sketch a vertical section with isentropes and winds consistent with a lowlevel positive potential temperature anomaly 7 What part of the atmosphere in the vicinity of a negative potential vorticity anomaly is a warm core anticyclone ATMO 251 Chapter 14 page 16 of 16 Weather Observation and Analysis John NielsenGammon Course Notes These course notes are copyrighted If you are presently registered for ATMO 251 at Texas AampM University permission is hereby granted to download and print these course notes for your personal use If you are not registered for ATMO 251 you may view these course notes but you may not download or print them without the permission of the author Redistribution of these course notes whether done freely or for profit is explicitly prohibited without the written permission of the author Chapter 4 CREATING A HAND ANALYSIS 1 Introduction An analysis is a depiction of the state of the atmosphere as determined from observations Within this de nition lie the secrets of creating a good subjective analysis Why learn how to create a handmade analysis Aside from the fact that you may need that skill someday the most important reason is that it teaches you how the atmosphere ts together how to take snapshots of pieces of the atmosphere and assemble them in your mind to form a coherent whole Creating a good hand analysis requires that you understand both the observations and the atmosphere 42 Observations The most important word in this de nition is the word observations There are in nite possibilities for an analysis depending on the person or computer algorithm performing the analysis and the purposes of the analysis itself There is no single right analysis But the easiest way to produce a atout wrong analysis is to miss an observation An analysis could be gorgeous and sophisticated but if it doesn t agree appropriately with the observations it s worthless ATMO 251 Chapter 4 page 1 of 22 So the analyst s rst responsibility is to consider all the available data or at least as much as can be considered in the available time This is not as hard as it sounds In the course of drawing the analysis you will cover the map with lines and will necessarily look at most of the data in the process Three tips First when confronted with a map with nothing but data on it take a good hard look and get a sense of the basic patterns in the data Second when you are drawing contours look beyond the data on either side of the contour and make sure that you re considering all the data in the area Third when you think you have nished scan the map and make sure all the data ts You will nd that when you first see the map with data it s a big jumble of numbers When you nish the analysis and scan it for the last time you will see all the patterns in the data and most of it will make sense In fact that s the most important reason to do an analysis in the rst place to discover what the data is telling you and to understand what s really going on in the atmosphere 43 Types of Analysis Errors Individual teachers of hand analysis may have a different list or a different ranking but in my opinion there are seven different types of errors that can be made on an analysis In order of severity with the worst rst they are 1 Analysis unclear Whatever your analysis right or wrong it has to be clear enough for someone to read it Known mistakes should be cleanly erased and labels should be clearly written and placed with suf cient frequency The amount of expected clarity depends on the amount of time available to perfect the analysis but in all cases it must be possible to determine what the analyst had in mind 2 Impossible analysis It is quite possible to create an analysis depiction that is utterly and physically impossible One example of an impossible analysis feature is two contours with a different value on each contour crossing each other There are others some of which will be discussed below 3 Disagrees with the data Some disagreement with the data is OK in the interest of smoothness or ignoring obvious data errors as will be seen below but other times an analysis will clearly disagree with the data because the analyst didn t notice the data An analyst should learn to read all the available data whenever possible 4 Disagrees with indirect data Some meteorological elds contain information about other fields For example pressure and wind ATMO 251 Chapter 4 page 2 of 22 elds are closely related Ignoring the information from all available elds on a surface map is a serious error 5 Elements missing The lack of a contour perhaps because of not noticing an observation in the comer of the map is an error of omission that is almost as serious as placing a contour in the wrong place 6 Wrong smoothness The concept of smoothness in an analysis will be discussed below Frequently this error is one of agreeing too closely with the data 7 Wrong application of conceptual model Much of the art of analysis is recognizing weather features based on limited amounts of data If you fail to recognize a feature or put it in the wrong place your analysis of that feature is next to worthless 44 Determination The act of creating an analysis is not merely a graphical exercise The analysis represents the analyst s best judgment about what lies between the observations and in particular which characteristics of the atmosphere between the observations deserve to be depicted in the analysis If the analysis were simply an alternate view of the data it wouldn t be worth drawing Its value lies in the information that the analyst adds to the map while creating the analysis The idea that there are judgments to be made and not simply single correct solutions to be divined is often difficult for the novice analysis to grasp Even more difficult is the concept that the best possible analysis does NOT coincide precisely with the data This is not to say that the analyst ignores some of the data but rather that the good analyst understands that the data itself is inherently imprecise and doesn t pretend otherwise The situation is similar to the scientist conducting an experiment and seeking to determine the relationship say between the amount of heat added to a beaker of water and the temperature of that water This scientist understands or at least expects that the relationship is linear as long as the water doesn t get hot enough to boil and wants to determine the proportionality constant the heat capacity So heat is applied measurements are taken and the measured temperatures are plotted on a graph as a function of the integrated amount of heat The data tends to lie along a line but with some scatter The scientist determines the best fitting straight line perhaps through linear regression and the resulting line is the best possible estimate given the data of the relationship between the added heat and the resulting temperature ATMO 251 Chapter 4 page 3 of 22 not complete the graph by stmply connecttng the dots The better approach was to recogntze that the measurem often ew ntl tmprectse Maybe thermometerwastnawatmp othebeakerdun gone meas andtnacoolerpartof beakerdurtngano er Maybeahotbubbleof nstng waterpassed across the mom era ettrne emeasur ent was taken Maybe thethermometer ltselfwas tmprectse orenattc In any case the sctenttst had to see past ose errors to dtscem the underlytng patta n and the resulttng bestrflt ltne ts not wrong because tt doem t pass through every dot on In a real sense tt s RIGHT because tt doem tpassLhrough may do onthe gaph But you may say what tf measurements of temperature tn the beaker were all perfectly correct and represmtatlve7 Our sctenttst may have m ssed a real tnteresttng phenomenon by hts or or a som ought to be If the scatter from a stmtght ltne does not constst ofmndom errors ofapproprtate magnttude then a stmtght ltne lm t the nght analysts Constder now measurements ofternperature at surface weather stattons The same types of errors tn the beaker found tn the atmosphere m b eryt st es are tn locattons that are the or colderthan most ofthelr surroundtngs 3 ch a stte ts satd to be unrepresentatty e and an analysts that tntends to represent the typtcal warmer ATMO 251 chapter 4 page 4 of 22 conditions in the surroundings shouldn t deviate to include an unrepresentative point measurement Similarly sometimes the weather at a particular spot happens to be unusual A good example of that would be an isolated thunderstorm that nails a weather station That station will be cold and cloudy but that single observation may be the only one for dozens of miles Unless that thunderstorm has significance on a larger scale it shouldn t intrude in the analysis of the largerscale temperature pattern Finally some measurements are just plain wrong If an analyst assumes that the measurements are correct the analysis becomes wrong The meteorological analyst must know how large the representativeness siting and instrument errors ought to be Apparent disagreements between the analysis and the observations that are small enough to be within the noise level of the data are perfectly acceptable and in fact desirable if they lead to a simpler cleaner analysis If a data point s value is far from what the analyst expects the analyst should at least attempt to determine why it is so different What is the source of error Or does the analysis have the wrong expectation 45 State of the Atmosphere Subjective analysis would be really hard if all we were analyzing were a bunch of numbers distributed on a map Indeed the first few analyses you perform the data WILL all seem like just a bunch of numbers You have my sympathies but fear not you will outgrow this state Soon if not already you will know enough about the atmosphere to recognize that the data are evidence of recognizable weather patterns present in the atmosphere That s good news for it greatly reduces the number ofpossible analyses that might be consistent with the data It reduces it from a large infinity to a small in nity Whatever analysis you produce has to be something that the atmosphere is capable of producing too If the pattern you analyze can t happen then it must be wrong Beyond the impossible analyses are the unlikely analyses With multiple ways of analyzing the data some of these possible atmospheric states are much more likely to happen than others Since you don t have data at every single point and the data wouldn t be perfectly correct even if it did exist everywhere you have to decide which possible atmospheric state seems to make the most sense given the data that s available within the constraints of what the analysis is allowed to show An analysis isn t supposed to show everything After all each tiny bit of air has its own temperature which is slightly different than the next ATMO 251 Chapter 4 page 5 of 22 my be eraquot and by the lame yuu39ve gene a hundred yards yeu39ve s edavmllh erwnalelllly A guud rule emumle ls llala synepue Dr mesusmle l l es ele shewseeemres duwnluthesmle represented by me lyplml elesemuee sfa ng but nu farmer If eala ls espenally wane Lhaanalysls shuuld sell anempne represem mudures malmgmeepresmz eeme sale era cmplehundredkllumelers Dr se mes are wenal mses They arelung leunlee Furthe puxpuses enhe aealysls us the lung elmeeslee Lhalmaners It39s akale eeawa very eelallee temperature fallem alung a fmnt as lung as that fallem stretches a cunsldemble elslaeee almg me 39un39 Ifan aealysls depmds un wlal almesphene slates are pessllele and Whattheanalysls lsalluwedlu Shaw fulluwsthanhe me ealamglel lad tn dl amt best analyses m dl amt nrcums39ances Let39s leak at seme Examples emes alppese we have a smug enempelalwe masur nen mg an astrwest llee alppese 39huse mseremmls lr pleuee eea mph lunkllkethe gure shUthere Syn x e emajs m ezm oMm m midemail I39mv TEMPERATU RE What39s me prupa39 analy5ls7 What39s Lhe m1quotw1ue er lempelalwe belwem Lhasa eala pmnts397 What llne Ur eew s me lempesalerev Tu tell we need in knuw39he relauve aeeelaey fths eala pmnls and me heezeelal sale ervanaeee llal sheele be resulved Let39s wmy abuultheaccumcy em Suppusewewa39elu aeea lempelalwe sale in Lhele slee enhe magam New me mg say Luuk at 39huse lama large vanaeees m lempelalwe eangmg fmmth lewsns in mm Thusetempemmre ATMO 251 Chap39a39A fage em pm memt emymulme wally m earl Ital am am 1 m4 TEMPERATURE F vanataehs arepretty hrg they musthe rea1 13911 drame analysxs su aste he cunslstent wrth all the dataquot what dues sueh an analysts luuk hkev Isrtthrs yama gammamt Lt mIIMaxm we M put ma Am 0147 L ra nulJwr k TEMPERATURE F Orthe the cm the next page 7 analyses uLhmugh every datapmnt Andxfsv temp mmr But estarhatahg temperature rs net the ule Dry The analysis muSL represent areahzahle hkely arsmhutaeh eftherea1terhperatures Hewhkelyrs rt that the hehzehtal pamal aehvatave eftemperature er Lhs gaph the ATMO 251 Chapter4 page 7 ufzz pm nmmaIQMIJ umymulmu 11441 M Max ml 4m Mi a mum st TEMPERATURE F slupe1s cunstantbetwem stauurrs and suddenlyjumps te anewva1ue at eaeh stauurn What atmuspheue prueess wuuld eause the temperature te reaeh pumty peaks and truughs rather than varyrrrg smuuthly wr pusruurn Ifyuu nk abuut rt yuu wru reauze thatthe secund gmph rs a pussrme reahsh usmbuuur uftemperature and e rst graphrs a hrgrlyrmplausrble une Set as ameteumlugmal aralysrs 2 stheWmner As a general rule unless yuu have a guud reasur fur malyzmg arseuuurrurues sueh as m 1 dun39t du rt what xfwe had a m erent temperature 52137 cVam Aswaij Mlmmjwh aM am 4 m4 cam 11ml 4410 will a TEMPERATURE F ATMO 251 Chapter4 page a ufzz ln thrsparueular ease thetemperatureyanauuns are sh small that ey are well WALhm the range ufuncenamty ufa typlcal temperature measurement Thtsmeansthat eapparentwarm muts anueulusputs can39t he trusted and the hest analysts ls slmply a stratgnt lme The eutuffhetween areal temperature dlfference and une llkely tu he due tu randum errurs depends un the eld bang analyzed Fur temperature METARmeasurements shuuldbe deemedtu he aeeuratetu uffb mueh mrllrhar and wrnu arreeuun ean easl y he u y twenty degrees hr sh mure lfwmds are lrght and there ls a lut uflucal tenatn m the way We39ve seen the extreme eases regarulng the relauye amplltudes quhe uhseryea yanahrlrty and the uhseryauun errur Nuw eunsruerthe lssue hf matral yanauuns and what shuuld he represented un an analysls Suppuse we eunstaer the fulluwmg dlsmbunun hf an eparlnnglutswu dbe ew erareas Butrt enure state ufTexas suehaleyel ufuetal ls unreasunahle n ea small eulleeuun ufuhseryauuns shuuld he surt uflumpeatugethermtu a general pattern Sueh an analysls ls muleatea hy the gaph heluw 1 nmmafmw rlmytmmw Mm aw PanWW 621 110mm 7 TEMPERATURE F Suppuse we have the uppusrte pruhlem whreh ls unfurtunately nurte eummun nut enuugh data Ifthe datals yery sparse yuu must use the extra n wh mng hetween stauuns Fur Example suppuse the data represents temperatures aeruss an ldeallzed yersmn quhe westem Umtea States ATMO 251 Chapter4 page 9 ufzz Havmg guud gengaphlcal knuwledge yuu need rt m meteurulugy fur ths uw that ehangesrn eleyauun e hayehrgrmpaets un temperature and tln tunemaurmumtan ranges are lueated as shuwn un the next a E i 3 a i a E e e yam gammahm dmtmst have um cal Ila 119 tux4 ummu 39 alummin TEMPERATURE F Nuuee that the euuler temperatures tend te he assuerated wrtln the muuntann ranges Thlsmakes sense lfyuu knuwtlnattemperaturetendstu deereasewrth eleyauun Thus there39s apattemrn the data and that pattem has a suund physreal basls 5 we ean make use quhat pattern m uur analysrste help uuthetween stauuns Three quhe muuntann ranges haye temperature datapeunts assuuated wrtln them hut the fuunh seeund hum the nght dues nut Based un the pattern we uhserye elsewhere we mfer that there uught te he euul temperatures alung that muuntann range Furtlnenmure the uthergaps m the data are nut alung muuntann ranges su temperatures shuuld he relauyely watm there My analysls ls shuwn m the next gure See huw the same data purnts ean yreld many dffermt analyses eaeh ufwhlch ls euneet under e lsnu aa ulumately yuur analysrs ls meant te he ufhenedt and value hutln te yuurselfandtu utlners Furyuurself pat ufthehenedt eumesrn the eunstrueuun quhe analyss and the resulung unayeudahle need te study e data and make sense ufrt Furuthers the analysrsrs fundamentally a ATMO 251 Chapter4 pagelnufzz enmmumeatanns deytee lt ts therefnre tmnnrtant that the analysts ennyey the enneet tnfnrmatann tn aytsually stratgntfnrward and dear manner 4524M naeWanna aImy vn mw nwzy m cal41ml am 111M man111m TEMPERATURE F 45 Cnntnur Spacing and Gradients lt has ntten heen sad that a gnnd analysts luuks llke awurk nfart Unless ttme ts an nyemdtng faetnr the analysls shnuld shnw as mueh neatness and attentann tn detatl aspusslble lndeed neatness wlll he nne nfthe measures by whteh ynur analyses wlll he graded That sad 1 shnuld wam ynu that fnrthenumnses quhese an ys euhtsmtsnntan apprnnnate style nfattn emulate The data nntnts are dts ete and tsnlated hutthe analysls musthe the nnnnstte smnnth and uwmg Remember that ynu are nnttust analynng the values ufsume sealar eld sueh as temperature analysts alsn deptets the hnnznntal yanannns are fundamentally tmnnrtant fnrnredtenng the eynlutann quhe weather The nyemdtng styhsne rule tn drawtng enntnurs ts tn nay attentann tn the enntnurspaetng Dn nntput enmers nr atp curves m ynur enntnurs unless tt39s meetdeally ealled fur tn the data nr hy the weather phenumenun hetng an yze and dn nnthe sn earelesstn ynuranalysts that enntnurs enme elnse tn tnuehtng fur nn apparentreasnn lust as the sealar eld sueh as temperature shnuld he assumed tn yary smnnthly a mssthemap unlessthe data and weather dtetate ntherwtse sn the ATMO 251 Chapter4 pagell ufZZ contour spacing should also vary smoothly across the map Appropriate smoothness is a difficult technique to learn but it is so important that it will be the second of the three factors by which your analysis will be graded Cmaps are among all possible ways of displaying two dimensional information the best and easiest type of map for estimating spatial derivatives Visually they give an immediate sense of the strength and direction of derivatives Quantatively it is possible at selected points to make precise as accurate as the analysis itself computations of spatial derivatives We ll start with quantitative methods then move on to the quicker qualitative ones Recall the technique in Chapter 3 for computing a horizontal derivative from a contour map picking two points on either side of the point of interest estimating the value of the field at those points subtracting one value from the other and dividing the difference by the distance between the points It works But there s a much easier way The derivative of temperature with respect to x is the change in temperature over some distance in x divided by that distance with the distance interval taken as small as possible On a map with isotherms the isotherms themselves define intervals of temperature If the isotherms are five degrees apart each time you cross another isotherm the temperature has changed five degrees Here s how you can use that information to estimate a derivative Draw your line through your point of interest parallel to x Now go both ways along the line from your point and find where the first isotherms cross the line Then estimate the distance between those crossing points The difference in temperature between those two isotherms divided by the distance between them is the average derivative of temperature with respect to x over that interval This technique is similar to the earlier one but instead of choosing a fixed distance and estimating the temperature change you find a known temperature change and estimate the distance It amounts to the same thing It s simpler and more accurate because instead of having to estimate two temperatures you only need to estimate one distance The quantitative technique suggests what the qualitative method would be The closer together the isotherms as long as they are not all the same isotherm the more rapidly temperature changes with distance and the larger the derivative If the isotherms cross your x line at points that ATMO 251 Chapter 4 page 12 of 22 are rarapan the nennhnmamnn yuur denmuve cnhnpmaunn vmuld he very big and the denmuve llselfmust h e small 5 10nmi ATAX 5F12nmi 042 Fnmi n the m1 wudd yuu den genemlly Ere ah uuthuw nmckly lempEmlure changes m the x direcuun a1nne yuu care ahnmuney nhecnnn mu on mum hkely yuuwuuld Ere ahnm whch anecnnn lempEmlure changes mm s mp1dlyand exacuy huw mpldly n changes m that anecnnn Hmrn the magnmne n r change and the direcuun m whch that change uncurs A magnmne and a direcuun shunns a 1m hke a vecmn duesn39l m The name ufthxs vecmns the gradam The anecnnn nnhe gradient vecmns the anecnnn m whmh the nuanmy m quesuun increases meg mpldly The magnmne nnhe gamenus the denmuve ufthat nuanmy m that direcuun ATMO 251 ChapletA page 13 cm As with all vectors the gradient vector can be expressed as components Those components are simply the partial derivatives with r r already Here we see that two of them are the components ofthe gradient vector So ifyou know how to compute a spatial derivative or two you lmow how to compute a gradient Mathematically the gradient oftemperature is written as W The is read as grad 391quot since we have three spatial dimensions the gradient I I I I I I I U I class we will concern ourselves only with the twodimensional vector ambiguity these notes will use the subscript z to identify the gradient as twodimensional only V27 with the two gradient components one could use trigonometry to gradient from a weather map you may as well do it the easy way Magnitude of temperature gradient 5F20mi 025Frni Direction toward 155 degrees SSE ATMO 251 Chapter 4 On a weather map with contours the direction of the gradient is everywhere perpendicular to the contours themselves pointing in the direction of higher contour values The magnitude of the gradient is everywhere equal to the spacing between the contours divided into the contour interval That s even simpler than derivatives To see why this works imagine a gradient vector Orient your cartesian coordinate system so that one axis is in the direction of the gradient Now think about the components of that vector in that coordinate system The component along that axis is equal to the whole magnitude while the component along the other axis being perpendicular to the vector is exactly zero This zero is the derivative of the quantity in question along that direction And if the value of that quantity doesn t change along that direction that direction must be parallel to the contours there Remember the contours are lines along which the value of that quantity doesn t change And so the gradient must be perpendicular to the contours The relationship between contours and gradients is so fundamental so organic to the depiction with contours that it s easier to eyeball the gradient of a quantity than it is to eyeball that quantity itself Think about it if you want to know where that quantity is largest you have to scan the map read some contour labels and nd the high spot If you want to know where the gradient is largest you just look for where the contour lines are closest together as long as they re different lines After all if a 60 degree isotherm is close to another 60 degree isotherm the gradient between those isotherms must average out to zero It s only where the contours show rapidly changing values that the gradient is largest In calculus you ll work with gradients a lot But you may never deal with them in the context of contour maps In meteorology contour maps are how we see gradients As we will continue to see gradients are fundamentally important to our understanding of how the atmosphere behaves and evolves So now think about your responsibility when doing an analysis drawing your own contour map You re not just drawing isopleths you re creating a graphical representation of the gradients That s why you must always be conscious of the orientation and especially the space between the contours This also calls to mind another reason for drawing smooth contours if the contours are erratic and one contour doesn t look anything like the next one it is impossible for someone to look at the space between the contours and infer the magnitude and direction of the gradient It sounds wrong but a smooth map actually provides more information than ATMO 251 Chapter 4 page 15 of 22 a jagged map because it clearly shows both the quantity being analyzed and its gradients 47 Creating Your Depiction To get a good artistic analysis you should learn to follow a basic procedure The procedure I recommend is as follows 1 Identify yourself Write your name and the current date in the lower right comer of the map 2 Examine the map Spend a few seconds examining the map to get a sense of the overall pattern and to decide where to start the analysis Once you learn frontal analysis you will use this step to sketch in the initial positions of your fronts 3 Start simple Select a portion of the map where you think it will be easy to place contours For height or pressure this would be a place where the winds are relatively strong or uniform For other scalar fields find an area where the gradient is large Regions of light winds or nearly uniform fields are tricky save them for later 4 Select your contours anal contour interval Some fields such as height have standard contour intervals at each level For most others you are on your own For temperature for example every 2 5 or 10 degrees might make sense depending on the situation You should pick an interval that lets you draw enough lines to show the structure of the field throughout the map As for the specific contour values the hard and fast rule is that every contour value should be evenly divisible by the contour interval So for example if you are analyzing every 10 degrees a 50 degree contour is okay but a 54 degree contour is not 5 Draw lightly Your initial contours should be drawn very lightly in graphite pencil and be constantly refined for smoothness Sketch in a few labels as you go so that you don t confuse contours When you have all the contours where you want them add permanent dark contour labels enough so that each line can be identified without hiring a detective and then retrace the contours darkly and smoothly If you have drawn your initial contours lightly enough it will not be necessary to erase them but go back and clean up any sloppy squiggles 6 Draw with lower values to the left As you get more proficient in your analysis technique you can ignore this rule but it s very important for beginners Remember that each contour is representing not only the values of a scalar but also its gradient If the gradient is directed toward one side of the contour at one point it can t suddenly switch to the other ATMO 251 Chapter 4 page 16 of 22 side The way to ensure that your gradients are consistent is to draw with lower values to the left It will become obvious why left is preferred when you begin analyzing pressure and heights To do this start a contour on the edge of the map or somewhere in the middle and draw so that the lower values are to the left of the direction of motion of the pencil If you have to stop the contour for some reason mark the end that you finished with so that you can pick it up again from there Always go in the same direction sort of like combing or brushing your hair 7 Draw alternating contours Once you have your rst contour drawn do not then draw the contour right next to it if the gradients are strong Instead pick the second contour away or the fourth and fill in the remaining contours later This technique has two benefits it lets you see what the overall pattern will look like at an early stage and it makes it easier to determine the proper contour spacing You will nd that if you already have the 60 and 70 degree contours the 65 degree contour will be a piece of cake 8 Visualize the field Think of the contours you draw as contours on a topographic map and visualize the resulting terrain If you can do this successfully you will avoid impossible or pathological analyses 9 Erase As you add contours you will discover that your rst few contours probably aren t in the right place Do not hesitate to erase parts of them and adjust their position To continue the art analogy the analysis is like a sculpture that only reveals its true shape after every square inch of rock has been refined 10 Add intermediate contours If there s a large gap between contours over a portion of the map such that it s pretty much impossible to eyeball the gradient there you need to add one or more intermediate contours Intermediate contours are distinguished from primary contours by making them dashed rather than solid lines The intermediate contour should be a value halfway between the primary contours or halfway beyond if the intermediate contour is to be the last contour within a high or low In exceptional circumstances you can add intermediate contours between the intermediate contours at 1A the primary contour interval 11 Label In addition to adding contour labels step 5 above you must also label the extrema that is maxima and minima Pet peeve please remember that the singular is maximum or minimum and the plural is maxima or minima The conventional labels are H and L for highs and lows of pressure or height W and K for extrema of temperature they stand for the German words for warm and cold and X and N for maXima and miNima of everything else Beneath the extremum label write the analyzed extreme value of the eld and underline it The extreme value ATMO 251 Chapter 4 page 17 of 22 vnu gmemlly be duse m but nut nuuusmy equal tu me nagan Dr luwest data pmm m me am 12 thzxarmmz the map Take unu mare luuk at me analysts u make sure the mfun39muun 15 presmled darly and that mare are nu xmpussxble Dr cunfusmg mtuurs u Cnmmn Ermxs We cundude Lhs nhap39a39 wnn Examples ur cumm n ermrs and ups un huwtu avmd LhEm 55 58 63 e 6 56 58 62 58 72 7 E 76 74 68 Mar 1 Unmoath comm 1n Lhs analysis mu cuntuur lmes are Jagged and dun39l gve any sense ufa u haempatlem Thaexs nu attempt m deplde tanpaature mum pr pa39ly and 5 me analysts adds nuunng n he data that39s alrady mare An examplexsthehump mLh mule unnu n dame cumuurm mu cams unnunup There39s nu reasun futhB cmluurlu mmd farthest nunh mare mnu araly rally behaved that me lempemmre Was a m Warmm mu cams unnu map that mfm39rmuun shuuld ve bean uunvuyuu m Lhe7EI dame cunmurluu Instadthe7 dame cuntuur snuws nu hml uruns bump The n dame mtuur un mungm seamstu nuvuuuen parfunmnmlly drawn u pink uer39 me 58 degree uusumuun WALhuul any ATMO 251 Chap39a394 page 18 um cunsda aum ufwhat mudurethebeluw armgqlhave A m smp magmlly unnamed mm m the edge enhe map is an uhhkely pauem 55 58 63 enas mueh Ere w the wanng belwem Eun39 urs The Eun39 urs gel duse lugmher and repan wnhuul any ned Suppu m the underlymg data 1139 s as thugh the aralyit has wkatl E11 data phubxaquot the cumuurs seemm haveheeh drawnas raras pessmle fmmLhe ta Thare39snu physiml msm that the miraknusphere eehmurs39 shuuld knewwhae the gauehs are andawudlhem 5 s analysis lSVerg Awndmg S39auuns wnh Eunmurs almust always results m gamst wheh are mane and way mu mng Mar 5 Impomla contauv The mlpussbleanalysxs aheve has a 1m gumg rum The Eunmurs aremceand mnemhm sun eehsxs39ehm39h the data They have the me em lugetha39 cumparedm the rest enhe data as a guudldalu du whamas dune hare and hat at ehsemheh as a cullecnve 58 Dr 58 wnh a hmt LhathelEmpaature mtrases m 39hewest ATMO 251 Chap39a394 page 19 em Onme gm rand sme ermermp 15 wheretheanalyslhas geuen mm kuuble A nemee mm mm see nmhmg Wmn wnh Lhs aralysls at m game me n dame xsu39herm ms mcely belwem temperatures m me S sandlempemmresmthe s Eulluukmure dusely asyuu fulluw e cunmuratmss memap nmle m n31Lpan enhe hme annular anymturesare unnsle andpa ermeumemeyare units gut The 15 Impassble Amma39mdlmhun uftmublexs than fewslzum mrs suchas39he 55 and a2 EInt Wham side eugme haves n dame mluur Mar 4 Gunan Tmsanalyslsrmgqlbea guudm39ezprmzhm enhe 6313me data was all curred a m are data pmnl 15 darly an mums n mms em that me 27 152 lypugaph em mLhB 72 Lhalappmred enme Ether maps The analystthuug1kas beneved me 27 even Lhuugu n duem l make any sense gvm me surmundlng data Data Dr ebsemuens whmh me analysts beneves are mmeuus shuuld be mked mmugh wnh an X un me analysis 5 that n 15 Elmr at me mam has cunsdaed and dimegarded the dam ATMO 251 Chap39a394 page 2n er22 M74 AnathEr drdedu walk 15 me an degee cumuur alung me smmem edge drme rmp Thare39 s m day an duemly suppmme tamaature bang gala man an degees mere but we sun dr cunsxs39enl wnh me surmundlng lanpaalure gamem Cmtuurs nmr me edge drme map are ajudgnmt an Ifwuhavea gnud1um able msm fur mpulahng a ednmur m the edge drme map at pumng a new une mane gd ahad But mandmg a cumuurjust m ll up due map is nut a suf nemly guud msun a alters blank seendn den 2 cumpleualy wrung seem u Cnmlmim Lmng m analyze wen ukes pmedee qunately becummg a guudaralysus nmmaelyan andmnself Tu aralyzewellyuuhavetu 1mm undemend dammdemandh data dmd em d huw atmnwhere rks Tmsxs best daneby gumg dueeuym the dam andlddqngam dekmgaums um Inlmmgm draw e atmnsphere du39u 1mm see me anuwhere m all 115 mm my and mlme auunshlps E E a Qumims ATMO 251 Chap39a394 page 21mm 1 Create an imaginary set of observations then create five separate analyses each of which is an example of errors 1 2 3 5 and 6 respectively from Section 42 2 Take a segment of a topographic map and extract observations of height at random locations Stop when you think you have enough observations to show the basic topographic pattern Exchange your observations with a partner and then try analyzing them to see how much of the real topography you can infer from the observations Finally assess your analysis To the extent that your analysis does not agree with the real topography does the fault lie with the limited number of observations or with the analyst 3 On a contour map identify the location with the strongest gradient Estimate the magnitude and direction of the gradient there 4 On a contour map identify all locations where the gradient is exactly zero based on the analysis alone 5 Take an existing analysis Draw a horizontal line through it Construct a graph of the analyzed eld as a function of distance along the line Your distance units should correspond to those of the map and your graph should agree with the location of every contour line as it crosses your horizontal line Discuss whether the information in the analysis is sufficient to enable you to draw a fairly precise graph ATMO 251 Chapter 4 page 22 of 22 Weather Observation and Analysis John Nielsen Gammon Course Notes hese course notes are copyrighted If you are presently registered forATMO 251 at TexasAampM University permission is hereby granted to download and print these course notes for your personal use If you are not registered forATMO 251 you may View these course notes 5 39 39 39 im them without 39 39 1 14 um author Redistribution of these course notes whether done eely or for p 39 quot 39 39 I quotquot 39wiu39tom the 39 39 39 me a Chapter 1 WAYS OF SEEING 11 Introduction Firstyear college students in a physics course spend lots of time studying objects such as sliding blocks point masses and pe ect springs The hysics concepts are sometimes difficult to grasp but the underlying principles appl to the real Wor too The i ealized treatment in physics class helps to isolate individual concepts and separate them out from other complicatmg factors The physics principles are easy to visualize For example here is a diagram illustrating an elastic rebound The object hits the Wall and bounces off ofit If there s no energy loss in the collision We know the speed of the object after the bounce it s the same as it Was before the A39IMO 251 Chapter 1 page 1 of 17 bounce and the direction of motion of the object after the bounce its angle away from the wall is equal to its previous angle toward the wal 39r39L r1 ifwe hitting another molecule and bouncing off of it Now both objects are moving but the motions are still possible to figure out ifwe make some quot quot 39 1 s 39thenamreof the collision Already though the true nature of the reallife interaction e two molecules which involves electromagnetic forces the motion of atoms within the molecules and various ways of exchanging energy is ignored when the collision is assumed to be ideal Now take that more complicated situation one real molecule colliding with another and multiply it by a thousand Imagine how a ow multiply that by a million Multiply the result by a million Multiply the result by a million a ain We re up to 102 molecules now not even a mole but still incomprehensibly large Now consider a billion billion billion of L L C1021 1 mm molecules A er all that your re still not even close to the total number of molecules in the Earth s atmosphere It gets worse The atmosphere is not composed ofjust one type of molecule Indeed it has things in it besides gas molecules such as oxygen nitrogen and water vapor The impurities take the form of cloud droplets rain and snow and aerosol particles with typical sizes ofmicrons 10396 m quotquot 10393m quot r 39 rea i m takeplar e in the atmosphere do so when individual molecules interact Meanwhile the A39IMO 251 Chapter 1 page 2 of 17 primary circulations systems of the troposphere involve simultaneous motions across much of the Earth planetary waves for example can easily be 10000 km 107 m long With important stuff happening on the molecular scale the planetary scale and every scale in between it seems impossible that computers will ever be able to simulate everything that s going on in the atmosphere simultaneously And if the atmosphere is too complex for computers what hope do humans have Anyone who drives must track multiple objects cars as they move in two dimensions a multilane road and predict their future shortterm evolution Now imagine if cars could y and were not constrained to stay at individual ight levels Now imagine if there were so many cars that the sky was nearly lled with them Then what if the cars moved almost randomly changing directions when necessary to avoid collisions Add in some trucks which have different behavioral characteristics Now if each car represented a molecule imagine how many cars would there be to correspond to the atmosphere And how would you possibly depict what is going on at any given time To see and understand the atmosphere we must make drastic simplifications Fortunately because of the way the atmosphere works such simplifications are possible We don t need to know what each individual molecule is doing to know what the sea breeze does for example Instead we think of the atmosphere as a continuous constantly moving uid Fluids don t bounce they ow and swirl and wiggle in three dimensions Even though all of this motion frts Newton s laws well we no longer have the luxury except in very special circumstances of thinking of the atmosphere like we think of an object If one part of the atmosphere moves other parts move as well and it s that entire assemblage of motion that we must understand if we are to know how the weather behaves Even the uid ow of the atmosphere takes place on a wide range of scales Fortunately further simplifications are possible with respect to individual atmospheric phenomena Generally that s good but it introduces its own complexity the simplest laws governing a jet stream for example are different from those that are most important for a thunderstorm So intuition developed for one set of phenomena at one scale of motion in general does not apply on other scales of motion Learning how the atmosphere works is often a matter of unleaming knowledge that s been recklessly extrapolated from one phenomenon to another ATMO 251 Chapter 1 page 3 of 17 H72 a my ow e M MAW 7 t mt1m m4 do In addmon to srmph eataon on thebasrs of d1xngsquot we are stuck h ea r1y vrsuahnng the a re m tvvo 64m s the type of that ean be drawn a any dAmEnsmns three spat pornt m spaee and tame the pressure ve1oerty temperature and eoneentrataons o m ene ns uen r e o s ay tvvo spataal dAmEnsxons and on me dAmE e mos here are staue Thxs leaves us m a srtuataon akm to the blmd men eneountenng the e ean pteees othe atmosphere butrtrs very mmmltto get asense ofthe whole the rest ofthrs seetaonvve3911eneounter seven fundamentany differentways ofseemg the atmosphere Eaeh lets us understand the atmosphere m apartaeular vvay None rs a substxmtefor any ofthe others ATMO 251 chapter 1 page 4 of17 Different people Will be able to see in some Ways more easily than others but a meteorologist must be able to use them all 12 Graphs st common scienti c image for conveying information is the graph A meteorological graph typically displays the value of one or more parameters as a function of one of the four dimensions For example a meteogram displays various parameters such as temperature dew point Wind speed pressure etc as a inction of time The other three dimensions are st tr e 39 n t ove ar The m t 39s however relatively useless unless the specific station or its location is identified W1 ati 0 tion each point along the ra hs in the meteogram corresponds to a particular value at a single point in space and time NDRRNDSCDDPS Hind peed EmitsDir 37mm Morgans Pam m drum awe95129 amazes21 A E 39A E g e i e i e e l E 3 E e 7 nEEa BaaEB aBen nEEi Mei Mei EEEe Emma mama 15mm Emma mama 15mm Emma DateTime mm speed g memen RRNDSCDDP Esrememe pressure Plot weer Morgans rem m drum 29969529 29959521 mm miem mm i w 319129 1 19199 nEEa BaaEB aBen nEEi Mei Mei EEEe Ema mama 15mm Emma mama 15mm Emma DateTime EMT newneser pressure Coastal Ocean Observing Program station at Morgan s Point Texas 31 me snows winds and pressures that undergo a repeating daily diurnal cycle A39IMO 251 Chapter 1 page 5 of 17 By connecting the dots the data points a meteogram emphasizes the continuously changing nature of the weather as it passes over a particular location or changes over the course of the day By plotting graphs of multiple parameters on a single meteogram the relationships among the parameters particularly as they change simultaneously or not are emphasized Shown here is a meteogram for Morgan s Point Texas The meteogram covers a 48hour period Look first at the pressure graph on the bottom Notice the two peaks in pressure at 1600 GMT or 11AM local time Those peaks are common and can be seen in pressure observations worldwide as long as the pressure isn t otherwise falling or rising too rapidly Look closer and you ll see there tend to be secondary peaks around 0400 GMT or 11PM These too are common worldwide With two peaks 12 hours apart this feature is known as the semidiurnal tide The diurnal part means daily The semi part means half meaning there is a new peak every halfday The tide refers to widespread change in the mass of the atmosphere much like the widespread change in sea level associated with the ocean tides This atmospheric tide however is caused by the sun heating the atmosphere and causing a globalscale wave to set up Ordinarily you might see falling pressure on a barometer and fear the onset of bad weather With this meteogram you can see that if the pressure fall takes place between 1600 GMT and 2200 GMT there is nothing to fear The other graph shows wind speed and direction The two lines on the graph are the average wind speed and the peak wind gusts Naturally the peak winds will always be stronger than the average winds in any particular period The wind direction is indicated by gray x s with the direction scale on the right You will see how to interpret wind directions in the next chapter but for now note that the wind direction here too is undergoing a fairly regular diurnal cycle with the wind direction shifting from large values nearly 360 degrees to small values nearly 0 degrees at about 1200 UTC on both days What other repeating patterns can you see Another common meteorological graph is the sounding diagram A plotted sounding typically shows graphs of temperature and dew point as a function of height or pressure The sounding diagram throws convention on its head or more precisely on its side by swapping the ordinate and abscissa Instead of height increasing along the x axis it increases along the y axis Other liberties are taken with the diagram such as tilting or even bending one of the coordinate axes Nonetheless as with a normal graph once the location and time of the sounding are given each point of plotted data on a sounding diagram corresponds to a particular value at a single point in space and time ATMO 251 Chapter 1 page 6 of 17 A suundlng 39nm Tupeka Kansas ls shuwn here Yuu Wlll leam huw te lntel39pret suundlng magmas m Chapter 5 Fur new nete that the data 39nm the lewest levels ufthe atmusphere ls atthe buttum and temperature ls lndlcated by the dlagunal blue lmes whlch are drawn at als Thus the temp stature at the guund accurdlng te the suundlng abuut zuc wt M r er l n mt mw m 5mm W ATMO 251 Chapter 45m page7ufl7 13 Isopleths While a graph shows a parameter as a function of a single space or time dimension isopleths show a parameter as a function of two dimensions Usually but not necessarily those two dimensions are X and y making the gure an isopleth map When analyzing a weather map the first step is usually to generate an isopleth map Isopleth maps have historically been so fundamental to meteorological seeing that they have developed their own vocabulary For example isopleths depicting pressure are called isobars isopleths depicting temperature are called isotherms isopleths depicting height are called contours isopleths depicting changes in dewpoint over time are called isallodrosotherms et cetera Impress your friends casually work the word isallodrosotherm into your next conversation preferably without accidentally spitting Two different types of isopleth maps are shown here the standard isopleth map and the colorbanded map Both contain exactly the same information but the different ways of depicting the information emphasize different aspects of the temperature field ususzmzunvuuu 2n m7 usnazinzuuvuuu 2n m7 Isopleth maps are subject to a variety of interpretations The simplest is based on the concept that each isopleth represents a particular value of the contoured f1eld Along the 20C isotherm for example the temperature is everywhere 20C So isopleths can be thought of as simply connecting all the places on the map where the isoplethed field has a particular value But this interpretation of isopleths neglects most of the map the space between the isopleths The color flll version of an isopleth map emphasizes those spaces So for example dark blue on the color flll map indicates all temperatures on the map between 19C and 20C By comparing the two maps you will see that the southern margin of the 19C 20C blue band corresponds to the 20C isotherm and the northern margin of the l9C20C blue band corresponds to the 19C isotherm So the same information is in both maps just depicted differently ATMO 251 Chapter 1 page 8 of 17 HUWever Lhae39smuremfumaum that mnbemfared 39nman xsuplethmapLhanjustvmaemetmlpmmrexsexamly190 armour snmewherembelwem M 252 32pmquot ume uf39anpa alure lets me arhxtmry mam ufthexsuplethed eldatany eslwaym do mmsm damdethe mewa39 sumaua me lEmpa39ature at any alluws me mm m 29mm me wine exy dusetu me znc lsulh mju une subsegnenlawaym fad wevmuld m ure39henelube 20me subsegnenllempemmre z W mm Wm M quot4 L Wow 1 mm mw M i W aa mm quot2quot W M circle mac 99 prunedures 15 equivalennu me mffa39enceb m Erang gzphby dmwmg mm hnes belwem me data pmms and Crating a gaph by dmwmg a smnu39h curve 39hruugh me data pmms melsuplethmaps Shawn hare are runhe same hmeas me suundlng Remanber39halthelempemmreal Tupekaalthalumewas znc ATMO 251 Chap39er 1 page 9 Dr 17 Topeka is located at the exact center of the contour Does the temperature indicated on th contour maps agree with the temperature that was measured at Topeka at that time by the radiosonde saunding 14 Plotted Data Generally an analysis starts with numerically and graphically depicted t39 e map A ap 39 d t d t necessarily be isoplethed though There are circumstanceswhen it s best to leave the data alone and let it speak for itself One virtue of aweather map with plotted data is that everything you see is an actual obserwtion with an isopleth map the accuracy of t 1 1 t t t or p a ma i in analysis with a plotted weather map there s no analysis so nothing is made u or imagined everything you see is real or as real as a possibly inaccurate observation can e Plotted weather maps typically display several variables simultaneously Most o en such weather maps show temperature dew point wind speed and direction sea level pressure cloud cover and present weather On the map shown below you can find Topeka s weather six hours a er the time of the sounding shown earlier More comprehensive weather maps also show pressure changes cloud types past weather station identifier and other information There are other standard sets of information for use in plotted upper air charts Such maps have evolved over time to convey the maximum amount of information a uta ap t iLhe ateofh h 39h39 ATNIO 251 chapter 1 page 10 of 17 Other plotted maps only show a single parameter This may be preferred over a contour map when the observations are insuf cient to indicate the proper shape of the contours One such case that often arises is precipitation data Particularly in summertime rainfall is very erratic One place might get nailed by a thunderstorm but nary a drop a few miles away Since observations are typically far apart compared the horizontal scale of precipitation variations an attempt at analysis of precipitation would be expected to get the value of precipitation between observations wrong Often it s better to simply plot the data on a map perhaps using colors to represent different precipitation values and not attempt to imagine where contour lines would go the map below each observation is colorcoded by the amount of precipitation recorded there Notice the irregular precipitation distribution across Kansas Any attempt to infer how much rain fell between any stations in Kansas would probably be inaccurate Precipiloiion in 8142006 8202006 Generuled 3212005 ul HPRCC usinq provisiunul data MOM Ramonal Climate Centers Sometimes plotted data and isopleths are combined on one map This can be useful if additional information is plotted or even if it s the same data so that the viewer can see the data coverage and spacing on which the isopleths are based Where there s plenty of data the viewer ATMO 251 Chapter 1 page 11 of 17 knuws that me aralysrs rs hkely m be rename where there39s hale data the analysrs may be We mum than a guess 15 Images Images are must u en used with radar and satellite data We cuntuurmzpsthe aretwurdxmensmnalpluts Typrea11ymereare nut e eh Sputun me 1mages rs bright Dr dark dependng un huwmuch hgut reaehed the satellite nmthatpan quhe Earth 3 33 Q s a Wave Nains niggered by winds passing aver the eastern capes of Nova Sauna The reseduddr sp acmg betwem datapmnts uf an urdmzry msrme satellitexngexs 1 km msrs cyprea1 s aerr ufsurfaceweather sranuns such as base shuwn cm are e a apparen n e a T e mteremng wavelrke cluuds are uver was but tuget er they cuver an area as small asthetypxcal ubservanun mung uverlznd ATMOZSI Chapter pagelZ d 17 Images and color band maps both show colorcoded data in two dimensions so what is the difference between them The main difference is that an image is not the product of an analysis No attempt has been made to determine the value of a meteorological eld at a particular point constrained to only show whatever detail is discernable from the often widely spaced observations a ine what an isopleth map of the precipitation observations plotted earlier would look like Compare that to an image o r estimated precipitation for a similar period zoomed in on Kansas nsas 7Dav Dbserued Precipitation Valid 5222006 12m uTc air qiun ueumey V 7 397 V 7 39 linrnli r i libiuuul gimme Beninre 7 7 Mummy 7 mllnshulu ummum 6Mdlnn cnlhv 39 l fly me puma 7 gm an Gumsi ciiv Jutn39mquot Judge 2in FEMquot EB Iiyssus warmeimgs JIDEIEII Mumus i v mmyulle 39 r i 7 mi 4i See how the image shows the rainfall from showers on a scale that s so small that the available observations can t possibly detect many of them An analysis of observations necessarily ignores many things that happen between the observations If you only care about the largerscale weather ignoring the smallscale stuff is a good thing If you need to know the smallerscale stuff too the image is essential Not all detail is good Because images are derived from raw observations they also contain some amount of noise Not all the pixel topixel variations are real39 some are caused by the instrument or by the vagaries of the observation process Isopleths are usually designed to eliminate or minimize noise For an uneducated observer who is not able ATMO 251 Chapter 1 page 13 of l 7 to recognize noise an isopleth analysis may be a safer way to depict the state of the atmosphere 16 Visualizations As used by meteorologists visualizations are representations of data that retain visual cues for the threedimensional variation of the data eld Usually the visualization will be drawn in perspective and inherently include looping or the ability to change perspective interactively Visualizations can be powerful conceptual aids but the limitation that the visualization has to be depicted in a twodimensional plane hinders its ability to convey quantitative information Sometimes the visualizations are simply twodimensional images or contour maps that can be rotated or moved about Other times the visualizations depict isosurfaces solid or transparent surfaces along which a parameter attains a particular value the spatial equivalent of an isopleth One good way of imagining an isosurface is as an eggshell The egg occupies the space on one side ofthe eggshell and the rest of the world occupies the space on the other side With an isosurface of say 80F on one side of the isosurface is all air that has a tempearture greater than 80F and on the other side is the rest of the world Perhaps the most useful visualizations are those that depict the wind field Because wind vectors inherently represent things moving a visualization can show this motion directly Often individual pieces of air are tagged with balls or ribbons and the visualization depicts the motion of these balls or ribbons as the wind carries them around The three dimensional nature of the motion can be conveyed by shading or other visual cues or changes of perspective 17 Equations An equation conveys information about the atmosphere but instead of depicting the value of individual parameters it describes in a mathematically precise way how the different parameters relate to one another or evolve through time Many students think of equations as a string of symbols that can be manipulated according to magical mathematical rules but equations in meteorology have another important use the visualization of the behavior of the atmosphere To see how an equation can be used for seeing think about the relatively straightforward dynamical equation Fma How would you visualize this equation Hopefully not as some gigantic threedimensional letters resting on a field somewhere Instead your visualization should be ATMO 251 Chapter 1 page 14 of 17 some hing hke a hand pushlng a block along a 1a mled surface You would yield a shongex accelemuon 0 how he same fmce on a more how he diffexmt vanames are mtenelatedr as one changes another mus change oo 77w amo 4ou anle F rna The nea hmg about an equauon as a means ofvlsuallzmg he mosphexe s ha s timeless 1 apphes a a a ny ame s e an unless as he pxo c ofa esmcuve appxoxlmahon apphes a any gwm pom wuhm he amosphexe oo Jus s he o he ways ofseemg he atmosphexe are mcomple e so oo me he equauons descnhmg how he atmosphexe woxks Bu Ifyou ve go a descnpuon of he s a e of he aunosphexe aheady and you comhme wi h he equauons ha govem he mosphexe you ve go some hang quue powerful a means ofpxedlcung he mosphexe Any me you mcountex an equauon m hs course 0 any othex hehavmg he way he equauon says u behaves ATMO 251 chap ex 1 page 15 of17 18 CuncepmalMudels Having the complete state of the atrnosphae as well as the equations that govem its evolution is geat ifyoulie a supercomputer but l ch itswaytoomu Inform ion oranor mary uman To dealwltthls n on a meteorologists use ldeallzed conceptual model takes the current weather situation and ms wellunderstand weather systems to it This conceptual model oi the weather depicts how walm and cold fronts move and evnlve Meteorologists a infoimation For examplea q me ofsurface observatlons mi op ntem eratuie andachange in Wi lso use conceptual models lo deal wlth Loo mile s ue andwlnd dlrectlons atthe slngle station Ultimately leamlng aboutthe atmosphere largely consists of conceptual modelrbulldlng with education and expalmce is dlffamt what can itpossibly mean to recogilze somethlng7 ltmean that the situation fits the same conceptual model as somepast situation If ATMO 251 chapterl page 16 of17 that the atmosphere will evolve in a similar fashion as the last time the same conceptual model applied A good forecaster can build a conceptual model if necessary for every situation If a particular conceptual model doesn t seem to apply exactly the forecaster should know how those differences between the conceptual model and the real atmosphere will affect what the atmosphere does Chapter 1 Questions 1 The atmosphere gets less and less dense with altitude and extends over 100 km above the Earth s surface If it were all the same density it would only be about 10 km thick Using that fact the approximate density of the atmosphere 1 kgm3 the known size of the Earth and Avogadro s number estimate the total number of molecules in the atmosphere 2 What are some benefits to adding isotherms to a map depicting surface temperature observations 3 A ship has measured sea surface temperatures every hour as it crossed the Atlantic Ocean Describe three possible ways of depicting those temperature observations and discuss the advantages and disadvantages of each 4 Using plotted upper air maps at several levels extract the temperatures at some particular location Plot those extracted temperatures on a sounding diagram Use pressure as your yaxis with pressure decreasing upward 5 Take a radarestimated precipitation image and estimate how many random point observations would be sufficient to adequately discern the spatial pattern of precipitation if the radar image wasn t available Describe the basis for your estimate Then tape the radar image to a dartboard and using darts generate the appropriate number of points for observations Copy down the point observations on a second map How well do they show the rainfall pattern 6 Describe your ideal tool for visualizing the atmosphere ATMO 251 Chapter 1 page 17 of 17 Weather Observation and Analysis John NielsenGammon Course Notes These course notes are copyrighted If you are presently registered for ATMO 251 at Texas AampM University permission is hereby granted to download and print these course notes for your personal use If you are not registered for ATMO 251 you may view these course notes but you may not download or print them without the permission of the author Redistribution of these course notes whether done freely or for profit is explicitly prohibited without the written permission of the author Chapter 3 SPACE TIME AND MOTION 31 Wind Observations We can divide weather elements into scalars and vectors Anything that can be represented as a single value is a scalar Almost all observed atmospheric quantities are scalars Temperature dewpoint pressure rainfall infrared radiation concentration all of these are things whose value is speci ed by a single number The one important exception to the ubiquity of scalars is wind Wind like all vectors has a magnitude and a direction The wind is de ned as the motion of the air at a particular location averaged over some period such as two or ten minutes According to normal convention the wind is a twodimensional vector Strictly speaking air moves in three dimensions eastwest northsouth and updown and sometimes the velocity vector is taken to be the full threedimensional wind but it is more common in normal use to work with the vector horizontal wind and treat the vertical component of air motion as a separate scalar Thus an air parcel might be said to have a wind vector of 16 ms from 130 degrees and also be ascending at 24 cms Remember that wind directions are expressed as the direction the wind is coming from not the direction it is going toward As with all vectors the wind can be described in terms of its components as well as a speed and direction There s no law that requires it but you are probably accustomed to the three cartesian coordinates being designated as x y and z By similar inviolate convention the three components of air motion toward the x y and 2 directions respectively are represented as u v and w When the air motion was described as 16 ms from 130 degrees and ascending at 24 cms that 24 cms was the w ATMO 251 Chapter 3 page 1 of 29 edmpdhehT drThe an vetdeuy The edmpdhmz drmendh m The pdsmvez dueendh Yuu edmd mmpule The d and v edmpmehTs drmdum Tmm The hdhztheT weed and daremmn usmg mgunumeky ThaT Teehmdue wdT he reviewed m Chap39a39 12 Whth yuu ehddse Te Express The hmzm39zl wmd as sepamTe Bump Ts r252 weed and daremunyuu heede use Wm hurhhes Thus as yuu mm lmagne e masur nent drThe wmd gmemlly d Th sTedmmdhanemdmeTensaeup an mar The eup Tape ensures apartmular dmemmn arm The speed ufrmzum drThe eups is retarded and cunvmed Th 2 wmd speed usmg a prsvmus mlxbmum The thd me is Smpla a gmemlly at plane ThaT mans xtselfwnh The wmd m MM quotWM WWWMM a 7 WW 71mm 44w mu TMTM awny an sdmedmes The Wm thdThsTmmstare edmhhed mm are wnh The wmd weed msn39umenl mduhTed m The ladmg edge drThe wmd me sThse The thd we pths Ward The wmd The anan mElEr En he a ATMO 251 chapTerz pagez um prUpEUa39 Fmpe ers gana39ally swund menu ehanges m wmd weed man eups bemusetharmumenlarmxs walla 11 MMu mms M mom Man A w I 4W cam H mm aiw 5 enWM ea me e2 wnh Wm pmpe la s meumea at new anges yuu dm l need a wmd Vane at all When me pmpe la xs pmmed mwem me West and me utha39pmnled mwem me suuLh me pmpella s meeuy masure me Wm eempenems enhe hnnzun39zl wmd A we mgummen39y men gves wu thewmd weed and lineman Even pmpella s39zkeum m m up Whanfyuu Wanllu msure sk is a sum an Sn 393 m ach eemp nml 1y sunng required fees suund pulse ermued by unetxansducerm be recavedby armlhertmn ducer s und Waves are affeded by wmdjust uke am are wnh a mi me suund Waves muve ras39a wnh a headwmd mey mnve sluwa Yuu rmyknuwthatthe weed erseunms alsu a ededby lanpaalure Tu eliminate any cumphmung 353515 the masummmhs repE39Bd wnh me mmua39 mwme receiver The duffermce m Lmnsll mes lspr pumuml m mewmd weed th tranmtlerrecava39 pans unenled alung mree axes me mree eempenems erme an veleeuy an be masured Tms e ufmsn39ument mnbemnrembust 39hannurmal amnmmea39s bemuse therearenu muvmg fans ATMO 251 chap39az fags em rc lawmanJ 4392 m WWW muu A M s 4 mm m M Wm H W WM 4 user r u mmx mu WWW mm 31 Pln jlgWin lsan l Ovhzrtha On a Wther map sm ars are u en pluueu srrrrply as numbErs Whue Wm numbErs By speed and ureeuurreuu1u supply be wrmen un me map m represmt me Wmd 212 pammlar pan u39s mueh mure use m use a yaphml represmmum A Wmd u mmthalmxmlvesa urre pamuel m mewmu urreeuurr Enables me eye m immediately pereeve me urreeuurr m wrueh arr rs ueveurrg wnhuuz ramug m md numbErs and gure um campass umumgs mu types ufwmd uepreuurr are eummurr vemnrs and barbs a eyeusn me nummml wruu speeu um a veemr plat and me me veaur sale rray nut be appmpnale fur me me ranges ufwmd speeds up may appmr un me me rrap sxmuhaneumly Wmu Earbs are almnst exetusrvely useu rur Wmth rraps The Wmd tarp migrated as an armw cemered cm the uusemuurr lumuun The armwhad Was m me urreeuurr me wruu was bluwmg inward and me fathers urme anqu un me Ether srue urme mum ATMO 251 chap39erz fage4 um W Original o gt Intermediatek Modern 3 57 mm 946 mg mg dmeebm me wmd Was bluwmg hm Rama men have me laugh sbmebne gm me bngm ma resemths eed m me by 512 ufthe armwrep smt me wmd weed ufusng39hemmba ufbarbsunlhefa39herlurep Wm w FunhEr re nements fulluwed barbs was svenmally planed mly un me smebnbean Ether bu39hsxdes d2 mndardazedcudmg sys39em Was duped 5 at 2 shun barb represents veh1utsalung barb 3931 lentils and a pamanl y k nuts The appmpnate ebmbmebm deletmmss the ma speed much bke rurran numaals are cumpused br muus ebmbmebms ufleners Chap39a39 3 fags 5 any ATMO 251 wyndm anmnn 01M 7M 0 kt 12 kt 37 kl 812 kt 1317 kt 3337 kl GeLesv 4852 M 5862 kl 9397 kt 168172 kl Jvii 842 kt In Southern Hemisphere er eunvenuun the barbs are un apameular srde ufthe SuuLhEm Hemsphere us unless the wrnu rs reauy screwy relauve tn the pressure eld the wrnu barbs are un the luwpressure srue quhe anuw shaa One lag srrnnhseauun makes Lhmgs just abxt cumphcated fur the meteumlugm Onee yuu39ve gut the baek pan quhe anuw shaa and the barbs fur wnu speed the furwzrd nan quhe arruw sha xs cumpletely redundant The breeuun quhe wrnu rs already umquely sneerseu by the baek nan quhe arruw sha urawrn the 39nnt panjust utters up the we a39mzp Sn n rs that the symbuls represenung Wm breeuun evuweu su Lhatthsy are un the she quhe sauun where the wnu rs bluwmg am mtherthan the sue the wrnu rs bluwmg m Tu keep yuur ATMO 251 Chapter page a ufZB directions straight remember that the barbs are like feathers and imagine that the station itself is the arrowhead A surface map typically includes much more than just the winds Allweather 39 u quot 39 the station location This plotting convention is followed rigorously so that ev one knows immediately which numbers correspond to which weather elements The most common variables are listed below Some are less common than others in part because many types of observations don t record them The temperature is plotted just to the northwest of the station location The pressure is plottedjust to the northeast ofthe station location r qr yetens To save space and confuse novice meteorologists only the tens unit and tenths digits are plotted and the decimal point is excluded Thus a pressure of 10133 mb is plotted as 133 3 N 39SKY COVER SKC D O No clouds FEW 1 CD1 One tenth or less bul not Zen 2 G Ywnitenms no tnree emns SCT 3 Ch FouMemtIa 4 G 39 Fivetennis 5 E iv BKN 6 Q Sevenlenlhs to eighttenths 7 0 Nineten1hsor overcast wlth u eni YIth OVC B Complewlypvemasttenlambs Wnnn 9 Skyunscured r y mm a wmw m 39 39 w ml 4 Win WW4 quotWMm M mm mh ATMO 251 Chapter 3 page 7 of 29 The dewpoint is plottedjust to the southeast ofthe station location The cloud cover is plotted within a small circle marking the station location Unique symbols correspond to different degrees of cloudiness with more and more of the circle lled in as the sky gets udier The present weather condition is plotted just to the west of the station location between the temperature and the dewpoint Unique WW a cum Huvelnpmnl NEW magma w Mm nan aring pm mm mm lag mm wmm mm by am mmmuam mm mm tvaxinm mm a was a uhuewa wn Mlermmnm a in u mjlllenl cn unuaus raln Ila in mm m quot fr e az39w g 39 mar Vraazinnl NOT mum illnhl an a um al mama a 111m hanwy 1 ma 0 ubsewallan uhnruliwn a Dhl lll l buswnllcn Inlulmmam Inn at Mllkaa heavy me I bagp murmurM Inquot nu mu m u mum mnwllnksi ullnm m Imam uhlurvl u an Mugema nr humy IIIWMI M lllll m1 mm mm Vlolanl mm mm quotadmin n ill W mm snum ram shaw erm rm shawlIll an ATMO 251 Chapter 3 page 8 of 29 s a 7 a 9 wmunmaaudu39rlquot I DU 11 a I I AM W I d v l i D I I v V apld um Burn 1 00 an frvgg39 digg min1w wlnd I am mum gnn mmmv wn a a m n v Q um ur ubuni In pallhum lullFrztnm nl an usurynan H quoton mquot n a g V placiplunm wm lepnaum wun 39 mm 3 quotm Mm in m mm Irmn am ff o bnnu 5Wquot nf ifil anraun E Brwn r nu anl nmm mm 11 an MDT m up m station muglugnl v minimum 5 Ilnn nun 39 39 s u now Ehwilirsulhnlher quot YMMNIIDIM snawm oi mm a 1 llln m6 a nun and mquot Fan m m mm m wlrhwt 39 mm am now 3mm mum Isl A a m m hknuu 31m E019 El lucipimnhan 1 v but LIT nulmn D V quotmm a quota I v W quota mum n MI a unm mafu mgr u Wum lun um nl39 nbm nuilwulan H N IHII39II n W inn lvlllo SavI dui mm am n ma Imur sllnm m madmu dlilll g www um willy m i Ihm a m H n vy drllllng n n n w naninlly m snarl nl mu a xa n mm snow gal Hy hlgh mum n a m H a n v v hlwlnv l n u w wanmam man Fun 1 legquot hm sky NOT nimle bl nu anpm able chmvau an my mm mm FD M ii l a m39mnnnlaWnaquot 53mm or 35mm lhlckev during pm quotWIN Fun a ma sky nor scam me In blaun m hecamu lhlpksr urirw was nuur Fug dawnsllinn mus sky dimmi a E avn uulna Fag wmn m Nm uls nemlhla cumrnwun dllzzla Manama or mm 39qum and mm allanl mum m unnw pauuu NOT Ileaxlnlh mm m I in a min i I IIth at mm or N uman 3935 39 N Ileezlng emu i madman 57 new chmwnllan r r Cunllnll l l B l 39 39 REM Ill 11 luv an lllr ll L Mariamtl nr lulu I 5 5quot muquot m new 7quot mm ur HR n mus a N 51 quot 39 Wquot39quot N quotagsz 39 My Mm OMEmlIFn I I l u Cnnlll lubus 395 quot UK W quotn Bland i H 1 mum m 5 w M w me my 3 an n w 3 quot 39 39 m quot M39 quot h quot quot g mm M wllhnul A 1 xime at an 3 an wlch nut my in mm mg for sum mm mm m Mumm ur haivly mm an170 ltIlgtv sugnl minim of mm Donny a Ice pnllBl Mllrar wllvwl min or rain and swig mind A We v Marianne or my mnwarts m V l m requot m HIV and wow mlm sum hnwarm m mm th or alm ummn r lnInd quotnew man Ml aesdwamd mn Ihundar sugm a mammal munazumm wnn dul HIII hul wm min In mm DI ubuwu n r 51 HA lgt m ur mod damnrm NEH a M av nh nnmlm Sllg ma I H a n y mund Ilovm quotWWIll l m wlInialn and nr wow 31 lnln al ahaMullah f2 Thundolslaym cum Mneg mm mm x anrm nr sln v 1mm 1 mus or analwanna symbols correspond to weather conditions too although not every possible weather condition has its own symbol weather is occurring simultaneously the largest numbered one takes precedence The rst few present weather symbols are not drawn as a separate circle but instead are tick marks on the station location circle and are generally not plotted at all on surface maps The two digit weather code for each weather element is the sum of the number at the beginning of the row and the number at the top of the colunm The full chart has been cut down the middle into two pieces so that it would t in this book An example Slight or moderate thunderstorm with hail at time of observation is found in the bottom row of the second half Referring to the first half the bottom row is 90 The column heading for this weather ATMO 251 en more than one type 0 Chapter 3 l Ha vy mun norm mm m m m 01 vhlmva nan page 9 of 29 is 6 Thus such a thunderstorm Would be coded as 96 in a synoptic or ship report METAR reports do not have onetoone equivalences With these Weather elements so Whichever one most closely matches the METAR report is used The pressure tendency is plotted just to the east of the station location The tendency is given in tenths of mb and is preceded by a symbol Which indicates the nature of the pressure variation over the past three hours rising then steady for example PRESSURE cm Dada TENDENCY No a Rising then fall 4 sleady same as 1n9same as or f a h D high r man 3 am 390 Falllng than rlsingsama as u Risin th n 5 lower an 3 steady a ristng hours ago 1 then rlslng mom WW E Falllgg hag H Momma 6 5195 y p a ng H 51 d1 pressure Ihen alllng move 2 Egan 39y39 w 3 0ng mane iv slowly i i u 59 Fallin slaadil m Emmemc Falling ar steady 7 unstegdity y39 ngvfsiiarvever 3 tll39uen infl ng gr ham 3 r am 5 quotSH ashtray a B falling theri ailing more ram dly The visibility in statute miles is plotted just to the West of the present Weather The past Weather Weather that Was signi cant but had ended by the time of the observation and is not incorporated in the present Weather symbol is plotted just to the southeast of the station location There is an abbreviated set of symbols for past Weather The loW middle and high cloud types are represented by symbols placed just south of the station location just north of the station location and Well north of the station location respectively The sea surface temperature is plotted south of the loW cloud type for those stations on the Water such as buoys and ships ATMO 251 Chapter 3 page 10 of29 Cod n W1 W2 PAST WEATHER 0 Clear ur lav clouds scattered or quot01 vallabla sky Emarep Clnudy broken cur overcast 5 Sandslorm pr dusRslorm or arming or blowing snuw 1 g Fug Ice fog 1hIck haze or thick smoke 39 39 Dlizlla Haln 39 Show or rain and snuw mixed nr ice petrels 39X39 V Showerfs Thunderstorm wim or wilhoul 39 precipilallun 39 30041361an kn e um ohms bygpr adhj gm ar 39 1 am so norfarm by spvemng uui cl Cu with ax dlf39ersrll levels runawngs enemy Mucus amirovm lap o en anwlanapedtw q mmm eu Sp 5 or send p ATMO 251 Chapter 3 page 11 of 29 Nu Description Amidwad Fmquot lmemnllqnal Canal 1 4 Thin A lmosl of cloud layer semilransnarent Thick As grealer parl sulllcienlly dense lo tilde sun r mean at N3 2 42 3 W Thin Ac mostly semllransparanl claud elements not Thi AG in patches CIOMH elemevils cnnlinually changing andDr umum ng a mule Ihan one level changing much and at a smgle level 4 0 5 1 Thin A m bands or in a layer gradually Spraadlng hale ave sky and usually thickenlng as a w 5 w A formd by me spreading mil of Cu 0 Ch Dnuhlelayered Ac or a wink layer 01 Au not my crsasmg in Al With As andl nr N5 76s all 99 Code C No H Description imam Flam mammal cm A in the farm a Gushapeu luits ur Ac with turrels Ac 0 a chaull c 5kquot usually al uillaranl levelsl malches ol dense Cl are usually presenl also Fllamenls ul CM 1 quotmales lamaquot scallered and rm 1 neleasing Dense Cl in palches or Wlsleu sheaves usually no 2 4 increasing sonlmimes Ilka remains al ea or aware or lulls Dense cl allen anvrlrshaped derlved lmm m assoJ 3 739 cialea mix CD 4 Ci ollen haokrshaped gradually spreading over quot12 sky and usually imckcnmg as a Whole Ci n n in managing bands ur Cs alone 5 2 enerally averspleadin and growing aenser the 9 nonllnuous layer nol leachlng 45 alulude Ci and Cs alien In converglng hands or OS aione 6 generally ovalspleadmg and gmwmg densel ma commuaus layer exoaeomg 45 altitude 7 g 5 Vall of 05 covering 18 enllre sky 8 A Cs ncl lnciaaslng and not calming enlila sky 9 CC alone 01 CC Wllh aume Cl or C5 hul Ihe Cl being lhe mam alrrilulm cloud A39IMO 251 Chapter 3 page 12 of29 rThe weedand dmemmn ersth mmmmsplmledlu the ESL er me SE surfaee tEmpmmre An Examplexspmmded Ufa planed Shp ubsewauun The Shp ubsemauumsme une al was denuded m Chapter Smee me Shp am nulrepu duudlypesmu duud typesare shewn 315 980 3 T1 20 27 R an n V7HCS 33 Derivatives inTm am 3 c eeznmea Lhauhemle er ufameleumlugml 1 me ankle mmed wnh ehelp era gzph ufthat qLEnmy gemluthegaphandmasure nr Elly E 3 a E have an awed van same Sn ms manhe angmneehmque measures rewedlu ume uf almelenrulugml qLEnu Ifthemeleumlugml quanmyxs 39ana39alure I fur exampleme demanve wnhrespedlu hmexswnuan am Ifyuu havm39t a any Blemusavmdmemt Encym assume matanh enms sumesun ufmable mm mmpm me d39 s and me slash as part Ufa s gle cumplEx symbul Lhatmmns the rate ufnhange armpaamemm rewed m mequot Just as gaphs En alsu be 11an almg 21131 mrecnuns 5 tan En demanves be eumpmed WALhrespedtu the wand dire mns gm wand I yuukave a gaph ersume quanmymthex y Lhm wu En esuma39e me ammuvemmrespemu x21 any g the gaph by drawmg me Bngmt hue and cumpuhng me wnh ewedluums As furmtauunthe pmm am slupe ju as vmuld be dune demanve enempaaaxe wnh awed m x is wnuen dTak ATMO 251 Chapter page 13 em 07Z D4Z 3 haurs OS OSL 30L TIME um 39 39l3900 ATMO 251 Chapter 3 page 14 of 29 So we ve covered the cases of no calculus and some calculus If you ve had lots of calculus you know that each of these derivatives should be something called partial derivatives since atmospheric variables typically change in all three directions as well as in time Rather than using dd for derivatives partial derivatives are written with 6 6 instead In math the partial derivatives are enclosed in parentheses and subscripts are used to indicate which variables are being held constant When dealing with stuff that varies in space and time meteorologists usually don t bother with the subscripts and it s always just assumed that the remaining spatial andor time dimensions are being held constant What are typical magnitudes of these things In a normal morning the temperature is rising at the rate of several degrees per hour To write the derivative in mks we need Ks Temperatures aren t normally reported in K but since K is C 27315 a change of one degree C corresponds exactly to a change of one degree K An hour is 3600 seconds if the temperature goes up 18 K in that time the rate of change of temperature would be 5 x 10394 Ks For horizontal derivatives imagine that the temperature decreases 5 K for every 500 km of distance Then the derivative of temperature with respect to x dTdx would be 1 x 10395 Km 34 Advection Wind affects the weather in many ways Most obviously if the wind is strong the weather is windy Wind also by moving air from place to place causes the weather to change If the air being blown in has different characteristics than the air being blown out the weather will evolve This easytounderstand effect of the wind is an important meteorological process known as advectz39on Example 1 Suppose the air 30 nautical miles upstream of Savannah Georgia at 2PM has a carbon monoxide CO concentration of 80 parts per billion ppb This means that out of one billion molecules of air 80 of them are carbon monoxide molecules Suppose that the wind speed is 15 kt 15 nautical miles per hour Suppose finally that there are no sources or sinks of CO between there and Savannah At what time will the concentration of CO in Savannah GA be 80 ppb Answer The 80 ppb air must travel 30 n mi to reach Savannah It is moving at 15 n mi hr 15 kt Divide the distance by the speed to get the time required 30 n mi 15 n mi hr 2 hr Since it was 2 PM when the air was 30 n mi away from Savannah it will be 4 PM when that air reaches Savannah Example 2 Suppose the CO concentration in Savannah at 2PM was 60 ppb How rapidly will the CO concentration change Answer The rate of change of CO concentration has units of concentration divided by time To compute the rate of change we can ATMO 251 Chapter 3 CO 80 ppb Savannah dlvlde the total change by the length oftlme over whlch that change occurred The changewas 80 ppb 7 60 ppb 20 ppb The length oftlme us 2 hr10 ppb perhour or 3 x 104 ppbsec Now let s look atthe algebra We wlll wnte Lhe Wlnd speed as M m w t t l m Savannah The drstahee lnterval Ax ls the drfferehee rh posrtroh h ths e30nml SLO olntln the downwlnd drreotroh T e speed o d d b our notatroh that ls v Thls deflnltlon should be famlllar to you To determrhe the tune lnlerval ohe rearranges the equatroh ahd solves rt ATMO 251 chapter 3 page 16 of 29 30nmi IUI 15nmihr Now we ll write the change in concentration at Savannah as ACOX The subscript x indicates that the location Savannah doesn t change we re talking about a change in time at a fixed point We computed that change as Moi At One could easily do the whole problem at once by combining results Thus MCOI z 39639 mm At Ax Finally it would be useful to be able to estimate the rate of change of CO without actually having to explicitly determine what the future CO value would be We can do this because in this case the amount by which CO goes up in two hours in Savannah is equal to the amount by which CO decreases between the upstream point and Savannah at 2PM MCOI M001 If you sit still in Savannah you start out in 60 ppb CO air and end up two hours later in 80 ppb CO air If you y a highspeed jet from 30 n mi upstream of Savannah to Savannah itself at 2PM you start out in 80 ppb CO air and end up in 60 ppb CO air Two hours of waiting corresponds to thirty miles of distance because it takes the air itself two hours to go thirty miles Bottom line we can estimate the rate of change of CO over time by looking at how much it varies over distance as long as we know how fast the air is moving In equation form MCOI z 39639 M001 At Ax This is the basic principle and the importance of advection By knowing the transport of air with different properties we can predict how those properties ought to change at any given location I m sure you ve used this idea many times For example if the wind is blowing from the north and temperatures are lower to the north you d expect it to get colder And unless something else is going on to cause temperatures to change it will get colder 35 The One Dimensional Advection Equation ATMO 251 Chapter 3 page 17 of 29 Ready for some calculus You probably can sort of see what s coming The equations in section 34 are something called nite dz39 rence equations They involve computing the change difference in some quantity over some nite measurable interval of time or space The tool of calculus allows us to take it to the limit to write down the corresponding equation for the rate of change of the CO This equation is 6CO u6CO 6t ax Unless you ve had several semesters of calculus this equation will look rather unfamiliar so I ll talk you through it First of all the funny 6 symbols play the same role as the d s in ordinary calculus they indicate differentiation In this case the term on the left hand side of the equation is the derivative of CO with respect to time t The reason we use 6 rather than d is that CO is not just a function of time it s a function of the spatial dimensions as well in this case there s only one spatial dimension x taken into account This type of derivative is called a partial derivative in a sense the derivative with respect to time or one dimension only represents one part of the variable s variations The parentheses with the subscript tell you what s not supposed to change The lefthandside term for example is the change of CO in time at a fixed location The x location of the measurement doesn t change it s Savannah in our example Now you know enough to be able to decipher the righthand side of the equation The term u is the component of wind in the positive x direction remember And the derivative term is the rate of change of variation of CO in the x direction holding time constant in our example the time is 2PM Unless explicitly stated otherwise in meteorology all partial derivatives are assumed to be with respect to space and time so we can drop the parentheses If you see a BBt know that x y and z or whatever the vertical coordinate happens to be are to be held constant if BBx know that y z and t are to be held constant etc It is not sufficient for you to memorize an equation such as this one or even to explain what each of the terms mean In meteorology as in most other fields you have to be able to apply the equation to real situations and to understand really understand how it works So let s look at an example that s similar to our 30mile blob example Consider a contour map of CO concentrations with winds overlaid Suppose you want to apply the onedimensional advection equation and estimate the rate of change of CO at Savannah According ATMO 251 Chapter 3 page 18 of 29 partial derivative of co with respect to x quotquot 9 8 7 x Savannah V X x 15 I7 mi We must rst choose the orientation of our coordinate system so 39 39 39 A 39 4 That s easy 39 39 39 so the conventional x direction will do Next we need 11 That too is easy since I have set up the example so that u is 15 kt everywhere Finally we need an estimate ofthe partial derivative ofCO with If this were ph you d probably have no problem estimating the magnitude o the derivative you d 39 a couple ofpoints on the graph one on each side ofthe location ofinterest and divide the the size ofthe interv mar a u m al The result in the example shown is 20 ppb 30 n mi or 0 557 ppb mi n You apply the same basic technique for estimating derivatives from a twodimensional contour map Pick two points aligned parallel to 39 39L 394 epartialquot 39 39 l J t m d a the point on the decreasing side of Savannah dividedby the distance ATMO 251 Chapter 3 page 19 on9 between the points The result in ma case is 20 wk 30 ml or 41557 ppbn ma ATMO 251 Chapter 3 page 20 on9 Now there s nothing left but to plug the numbers into the advection equation M u M 15nmihr 0667ppbnmz39 lOppbhr it 6x 2 So the CO at Savannah should be increasing at the rate of 10 ppbhr due to advection This is the same numerical answer as in the example in Section 34 That s no coincidence The wind speed is the same in both cases and if you look closely at the map in the present example you ll nd that the CO 30 n mi upstream of Savannah is 80 ppb just as before And if you go back and review the procedures in the two sections you ll nd that there s really no difference between them in both cases we re estimating how rapidly CO will change at a point due to the wind blowing in different values of CO We needed to know how fast the wind was blowing and how much CO varied with distance in that direction One final but important point We ve been working with CO concentrations because there s not a lot that changes CO besides J and 39 39 But A quot is a process that affects all spatiallyvarying atmospheric variables The most common advection to consider is temperature advection The equation that describes the effect on local temperature of temperature advection in the x direction is simply dropping the parentheses like I warned you I would 6T 6T u 6t 6x You apply it like you would any other advection equation Take a map with isotherms and winds estimate the advection and thereby determine how quickly temperature should be rising or falling due to advection 36 Equations 0n the Brain The advection equation is our first complicated equation Whenever you re reading something and encounter an equation you should stop a moment to understand it Play around with it in your mind try out simple examples check if it makes sense to you in all situations that you understand This is how you see the atmosphere with equations First state the equation in words Let s use the onedimensional temperature advection equation The equation states that the rate of change of temperature with respect to time at a particular location is equal to the negative of the wind speed times the rate of variation of temperature ATMO 251 Chapter 3 page 21 of 29 in the direction toward which the wind is blowing Until you understand the equation in both appearance and in words go no farther xt see ifthe equation makes sense ifvanous terms are equal to zero waind speedis zero no air is being transported so the Wind s not What if tarac is zero What situation does that conespondto Ifthe 39 39 39 39 i hm temperature is uniform along hex axis so the air being blown in would be the same temperature as the airbeing blown out Again temperature should not change so it makes sense that tarac wouldbe 3 Before moving on let s add a complication Suppose the 39 39 uni 39 an union uni inure 39 39 colder uu eventually drop even though the equation says that dTdt should be zero Indeed it will but the equation only accounts for the change in emperature being caused by the wind at that instant To apply it over a much longer time you dhave to averageu and 67quot or over amuch larger distance not just measure or compute them locally x 3 Savannah 15nmi L L ATMO 251 chapter 3 page 22 of29 Okay we ve checked the zero situations Now let s see if the sign makes sense If x points toward where the wind is blowing it must always be positive so that term s easy The sign of the temperature change with time must therefore be determined by the sign of the downwind variation of temperature If BTBx is positive that means if you graphed T versus x the slope would be positive In other words if you moved a short distance downwind in the positive x direction you d find warmer temperatures Conversely if you move upwind in the negative x direction you d nd colder temperatures And since the wind is coming from the negative x direction those colder temperatures are blowing in and the temperature at your location should be dropping Sure enough with u positive if BTBx is positive too the minus sign makes the whole right hand side negative so BTBt would be negative according to the advection equation It works We know the temperature should be dropping and the equation says it would be dropping One last check of magnitudes Using that last example we would imagine that a stronger wind would make the temperature drop faster and the equation agrees Also if horizontal temperature gradients are weak and the temperature doesn t change all that much upstream the temperature should drop more slowly and the equation agrees Don t take my word for it try working it out with a few simple arithmetic examples The end result of all of this is to convince yourself that the equation makes sense It s not enough to believe the equation or take my word for it that it is true you have to make sure it s consistent with your own mental model of how the atmosphere works We ve tried all sorts of different cases no wind strong wind horizontal temperature variations no horizontal temperature variations weak horizontal temperature variations and the equation indicated that the right sort of thing should happen every time The only thing we haven t done with the temperature advection equation is check out the numbers To do that let s make up an example Suppose the wind is blowing at 5 ms and the temperature gets colder upstream 5 K colder after only 100 km First try solving it as a word problem We know that when that 100 km upstream air gets here the temperature will be 5 K colder So how long does it take The definition of velocity is distance covered divided by the time it takes so rearranging the time it takes must be the distance covered divided by the velocity So 1x105 m divided by 5 ms gives us the time 2x104 seconds 20000 seconds or about six hours So the temperature ought to decrease 5 K every 20000 seconds the rate of change is 5 K 2x104 s or 25x10394 Ks ATMO 251 Chapter 3 page 23 of 29 Now let s do the math of the advection equation We already know u To measure BTBx we ve got the information that the temperature decreases 5 K for every 100 km in the negative x direction Turn it around the temperature increases 5 K per 100 km in the positive x direction That s BT x 5 K 1x105 m or 5x10395 Km Now for the calculation BTBt u BTBx 5 ms 5x10395 Km or 725x10394 Ks Same answer So the equation works correctly in this example Try another example It s not as good as a mathematical proof but you ll learn to trust the equation as well as trust your ability to understand it Follow the same procedure with all equations that are not instantly obvious to you 1 Say it in words 2 See if it makes sense if each variable or derivative in turn is zero 3 See ifthe signs make sense 4 See if what happens makes sense if certain variables get larger or smaller 5 Make up a concrete example and work it through We ve done all these things together for the advection equation For the remaining equations in this book and the other equations you encounter in other courses doing all these things is your responsibility 37 Space Time Conversion No this is not science fiction No black holes here Look again at the advection equation and see what happens if the wind speed u is temporally and spatially uniform In other words the wind speed doesn t change with time and it s the same speed everywhere Specifically let s make u a constant negative value the same everywhere or at least as far out as our graphs are going to extend Then the advection equation says quite simply that BTBt is proportional to BTBx They are identical except for a proportionality constant which is the negative of the u component of wind ATMO 251 Chapter 3 page 24 of 29 Think about the implications for a moment or more than a moment The quantity BTBt tells us the slope of the graph of temperature vs time Starting at one particular time and value of temperature you could draw a line along the graph that has the proper slope BTBt everywhere and end up with the correct graph of temperature vs time Or instead of doing it graphically you could do it mathematically by integrating the known values of BTBt with respect to time The constant of integration is precisely the same thing as the starting value of temperature you would use to draw the graph Knowing 6T6t gives us the shape of the graph everywhere Now think about a graph of temperature as it varies in x rather than in time According to the advection equation with uniform negative u the variations in x are exactly proportional to the variations in time on our earlier graph The shape of the graph is identical the only thing that changes is the ordinate of the graph it s in different units and one meter is not the same thing as one second so if the graph is plotted it might look stretched out or scrunched up We can ensure that the graphs really are identical by designing the ordinate in x to correspond exactly to the ordinate in I How Think about the equivalence of time and space here Look at the temperature some distance upstream say 1 km How long will it take that air to get to you We ve done the math already and it depends on the wind speed thusly t xu So ifu 10 ms t 100 s Similarly ifwe know the time delay we can compute the distance using x u I Now suppose we take the graph of temperature vs time and for every time label on the ordinate we multiply it by iu Now instead of seconds the ordinate is labeled in meters and we have the graph of temperature vs distance without changing the graph at all just changing the label And this is the distribution of temperature in x that must eXist if the wind is steady and uniform and there s nothing else affecting the temperature Under those conditions whatever changes in temperature occur at a given location must correspond to variations in the upstream temperature pattern And you can look at the horizontal temperature variations upstream and know that you can directly predict how the temperature is going to change at your station Is this a useful forecasting technique How realistic is this special case that has the wind being steady and nothing but advection affecting temperature The steady wind is not too much of a problem If the wind speed changes it would just change the timing of the temperature changes but fundamentally there s not much difference in what is happening As ATMO 251 Chapter 3 page 25 of 29 Temperature F lDistqnc km m 3 Temperature F Tlma st before or after ocal noon ATMO 251 Chapta39K page 25 ufZB for temperature there are lots of other things that affect it The most obvious is the regular rise and fall of temperature that occurs during the day and night But that happens everywhere nearby at the same time If you know the spatial distribution you d just have to add the expected rise and fall from time of day to the temperature changes due to advection to get a forecast taking both effects into account 38 Phase Speed In the case of advection described above the variations in the temperature pattern were moving at the speed of the wind But suppose something is not moving at the speed of the wind then what There are many examples in the atmosphere of phenomena that move along at a steady speed with little change in structure One good example is a cold front A cold front almost by definition has a significant wind shift across it so it s just about the opposite situation from the uniform wind case Yet cold fronts tend to move at a fairly steady speed and they intensify or weaken very slowly The only things that often cause a rapid change in cold front structure are mountains and convection The spacetime conversion concepts in Section 37 can be applied to any atmospheric feature that is basically just moving along and it doesn t matter what s causing it to move As long as you can tell how fast it s moving that speed can be used in the advection equation in place of the actual wind speed This speed of motion is called the phase speed It is so named from wave theory and it refers to the motion of the point along a wave that has a particular phase Although the physical cause of motion is different the application of the phase speed is the same as in the case of simple advection Some feature is some distance away and if it s moving at some particular speed you know how long it will take to get to you Closer parts of the feature moving at the same speed will arrive sooner and more distant parts will arrive later The time record of observed meteorological variables at a particular space can be converted directly into a depiction of the horizontal structure of the phenomenon The technique of taking observations in time and using them to discern the spatial structure of the atmosphere is a common one in mesoscale analysis When observations are far apart perhaps even farther apart than the size of the phenomenon the hopefully frequent observations in time may be the only information available regarding the detailed spatial structure of the phenomenon ATMO 251 Chapter 3 page 27 of 29 A good example ofmis goes backto cold fronts You may have surface observations mat say a particular cold front is between two m quot where because there s no data in the space between the two stations But t L L L d1 station A DODGE B CITY ATMO 251 Chapterii page 28 on9 With clear welldefined fronts with a known phase speed time space conversion lets you pin down almost the exact the location of the front at any given time Questions 1 Write down and explain the onedimensional advection equation for water vapor mixing ratio 2 Suppose the CO is going up 5 ppb per hour at your location because of advection and is now 110 ppb a If present trends continue what will the CO be 2 hours from now b If the wind is blowing from the north at 20 kt what is the current value of CO 10 n mi north of your location c If instead the CO 10 n mi north of you is 110 ppb and the CO 10 n mi east of you is 112 ppb what is the current wind direction and speed 3 Describe something that moves from one place to another without any material actually traveling from one place to the other How does it move 4 Examine a surface map with winds and temperatures Pick a few locations and estimate the magnitude and sign of the temperature advection at those locations Express your answer in units of degrees Celsius per hour degrees Fahrenheit per day and Kelvins per second 5 You re driving quickly down a road in Montana You find that for the first 50 miles the temperature increases rapidly but there s no wind For the next 50 miles the temperature is steady but there s a strong tailwind For the next 50 miles the temperature increases rapidly again but there s no wind For the next 50 miles the temperature is steady but again there is a strong tailwind Considering the 200mile stretch as a whole is therewarm J cold orzero J ATMO 251 Chapter 3 page 29 of 29 Weather Observation and Analysis John NielsenGammon Course Notes These course notes are copyrighted If you are presently registered for ATMO 251 at Texas AampM University permission is hereby granted to download and print these course notes for your personal use If you are not registered for ATMO 251 you may view these course notes but you may not download or print them without the permission of the author Redistribution of these course notes whether done freely or for profit is explicitly prohibited without the written permission of the author Chapter 12 ATMOSPHERIC STRUCTURE 121 Structure of the Troposphere and Stratosphere The very names troposphere and stratosphere refer to the contrast between turbulence and vertical mixing in the troposphere and the stable layered stratosphere This chapter examines the basic structure of the troposphere and stratosphere the two lowest layers of the atmosphere and considers the importance of the interface between them the tropopause in the structure and behavior of weather systems Without the Sun the Earth would be a cold boring ball Sunlight drives the temperatures and winds of the atmosphere How much of the typical state of the atmosphere is due directly to the effect of Sun and how much is due to adjustments and reactions taking place in the atmosphere In the stratosphere the basic picture given by the direct effects of the sun is rather close to reality Quite simply the top of the stratosphere is warm because a relatively large amount of solar radiation gets absorbed there One might think that if solar radiation is absorbed at the top of the stratosphere there ought to be a whole lot more solar radiation absorbed at the bottom of the stratosphere or in the troposphere simply because the air is a lot denser there Well sure if the same amount of radiation that reaches the top of the stratosphere made it down into the troposphere there d be a lot of absorption down low But only the most energetic wavelengths of radiation are subject to absorption by the gases in the atmosphere and once you get past the top of the stratosphere most of those wavelengths have already been absorbed Because of this ATMO 251 Chapter 12 page 1 of 23 absorption the amount of energy still to be absorbed farther down in the stratosphere or troposphere gets smaller and smaller as the radiation penetrates lower and lower into the atmosphere The stratosphere is very stably stratified cold below warm above so the air motion in the stratosphere is mostly flat The troposphere is only weakly stable so there is much less resistance to vertical motion J WARM STRATOSPHERE g 3 TROPOSFH ERE 9 9 COLD 3d WARM If absorption of solar radiation by the atmosphere were the only important process heating the air the temperature would be quite cold at ground level because there would be essentially no energy le that the air could absorb As it is the warmest part of the stratosphere is at its top and the temperature decreases downward toward the troposphere becoming quite cold as low as 15 km above ground level But lower down there s something else that absorbs solar radiation the ground Remember that the air at low levels is heated by coming in contact with the ground This heating is responsible for the relatively warm temperatures we observe at ground level Temperatures at the ground are typically 50 K or more warmer than temperatures at the tropopause So there are two main places where heat is added to the atmosphere at the top of the stratosphere and at the bottom of the troposphere The differences in where heat is added are responsible for the fundamental differences between the troposphere and stratosphere In the troposphere the heating makes the atmosphere less stable As you warm ATMO 251 Chapter 12 page 2 of 23 lfraviole t solar radiation Thick pur Ie rays is absorbed qui ly in The w eleng39fhs other colors mostly pa 3 T rough The atmosphere where The hn be reflectedsca e ed by clous or absorbed or reflecTeSca t tere by e ground or ocean W en The grand absorbs radiafion if TemperaT re goes up and it heats 139 e air abo e if up a low layer the rate at which the temperature decreases with height becomes larger and larger If left unchecked eventually the lapse rate would exceed the dry adiabatic lapse rate and unstable overturning of the affected air would result In contrast adding heat to the top of the ATMO 251 Chapter 12 page 3 of 23 stratosphere helps to generate and perpetuate a deep temperature inversion with warmer temperatures aloft making the stratosphere very stable On a sounding the tropopause is usually detectable as a sudden transition from temperature rapidly cooling with height to temperature staying constant or warming with height In order to cover all possible circumstances the World Meteorological Organization has arbitrarily defined the tropopause as the base of the first deep layer above 500 mb where the temperature cools at less than 2Ckm 122 The Real Tropospheric Lapse Rate The simple description in the previous section has one critical inconsistency If the vertical temperature pro le in the troposphere is determined by air becoming unstable and mixing vertically the lapse rate in the troposphere should be dry adiabatic Instead we observe it to be much stabler than dry adiabatic There must be some additional process heretofore not discussed that causes the middle and upper troposphere to be warmer than we would otherwise expect One such additional process is called latent heating If air containing water vapor rises past its lifting condensation level the water vapor will condense and the air will not cool as rapidly anymore as it ascends While dry vertical instability would drive the atmosphere toward the dry adiabatic lapse rate moist convection drives the atmosphere toward the moist adiabatic lapse rate Indeed in the tropics the observed lapse rate is very close to the moist adiabatic lapse rate throughout the troposphere Convection in the tropics is the primary driving factor controlling the overall vertical temperature structure of the troposphere In midlatitudes particularly during wintertime we don t observe moist convection very often Yet the observed lapse rate is still fairly close to the moist adiabatic lapse rate This turns out to be a consequence of winter storms Such storms tend to be f1ercer if the stratification in the troposphere is weak The net effect of winter storms is to carry warm air northward and upward and cold air downward and southward Preferentially causing cold air to sink and warm air to rise quite apart from any upright instability has the net effect of decreasing the lapse rate and increasing the stability So while moist convection rules in the tropics midlatitude cyclones low pressure systems not tornadoes rule closer to the pole Both effects give us a lapse rate in the lower troposphere that is close to 5 Kkm rather than 10 Kkm So while this is stable for upward ATMO 251 Chapter 12 page 4 of 23 and downward motion it s not very stable Furthermore vertical motions are fundamental to the processes controlling the lapse rate so such vertical motions must be common features of the wind eld in the troposphere All this makes the troposphere quite unlike the stratosphere 123 Response to Vertical Motion We ve already noted that vigorous vertical motion must be more common in the troposphere than the stratosphere Now let s utilize the concepts of the previous chapter to see what vertical motion does to the troposphere and stratosphere First a review The change in temperature at a particular level due to upward or downward motion is a combination of two effects vertical temperature advection and adiabatic expansion or compression The quantity known as potential temperature nicely wraps both of these effects into a single package With potential temperature the change in temperature at a given level due to vertical motion is directly proportional to the vertical derivative of potential temperature and the amount of vertical displacement wTt96t962 If potential temperature is uniform with height upward and downward motion don t change the temperature pattern at all If potential temperature increases with height downward motion will raise the temperature at a particular level and upward motion will lower the temperature Graphically it s easy to imagine potential temperature Think about contours of potential temperature in a vertical section The contours predominantly run horizontally The closer together the contours the stabler the atmosphere and the bigger the effect of vertical motion Since potential temperature is conserved it is perfectly okay to imagine that the potential temperature lines or isentropes are actually being carried along by the air as it moves up or down The opposite situation would be air that is weakly stable or neutral In that situation the isentropes would be far apart and there could be little change of potential temperature or temperature for that matter caused by any vertical motion So the only other thing you need to know is because the stratosphere is stable its isentropes are tightly packed in the vertical In the troposphere the isentropes are far apart and the air is much less stable Now imagine that there is some ascent In the troposphere that ascent will cause some cooling The pressure at the base ofthe ascent would rise a bit gradually cutting off the horizontal convergence that must be present at low levels beneath the ascent In the stratosphere the ascent would cause a much greater and more rapid cooling The pressure would ATMO 251 Chapter 12 page 5 of 23 rise quickly and the convergence and associated ascent would be easily sti ed One could say that the stratosphere is much more resistant to upward motion You d have the same problem with descent In the stratosphere the downward motion would cause rapid warming lowering the pressure at the base of the descending air and cutting off the horizontal divergence that would otherwise be present Large s rrafifica rion verTical mo rion causes large changes in po ren rial Tempera rure and Tempera rur e e causes changes in po ren al TemperaTure and Tempera rure Ajf J Individual air parcels too suffer the same fate Imagine an unstable air parcel ascending through the upper troposphere At altitudes corresponding to the upper troposphere there s not much water vapor left in any air parcel so the lapse rate would have to pretty close to dry adiabatic for that air parcel to still be unstable and continue rising But once it hits the stratosphere by de nition the surrounding air is stable and a deep temperature inversion is probably present Any ascending air parcel would quickly become much cooler than the surrounding stratospheric air and would sink back down ATMO 251 Chapter 12 page 6 of 23 These processes have an obvious effect during moist convection A classic deep thunderstorm has a broad anvil alo and an overshooting top The anvil is typically near the tropopause level because that s where the air ceases to be unstable and reaches its level of neutral buoyancy Above the anvil over the updra core a few knobs of higher cloud can o en be seen This is the overshooting top and it is caused by air hitting the stratosphere while ascending fairly rapidly As the air rises farther it becomes much cooler than its surroundings and proceeds to sink back down 435 H The presence of an overshooting top is an excellent marker for vigorous convection The lower stratosphere tends to be pretty close to isothermal no matter where you look so the size of an overshooting top will almost entirely depend on the strength of the updraft beneath it and the upward momentum of the air as it reaches the stratosphere 124 Convergence at the Tropopause Not only is the tropopause the interface between the troposphere and stratosphere it s also a place where the tropospheric jet stream tends ATMO 251 Chapter 12 page 7 of 23 to be strongest In earlier chapters we noted that variations in the height pattern along the jet stream are associated with accelerations of the jet stream winds and that those accelerations in turn imply ageostrophic winds and certain patterns of convergence and divergence We will now consider the vertical motion response to jetlevel convergence and divergence For brevity we will only consider divergence If you reverse all the verbs and adjectives the same arguments apply to convergence If we think about divergence at the tropopause as being associated with jet accelerations there are certain basic areas that are associated with divergence as discussed in Chapter 11 One is the region between a trough and a downstream ridge Another is to the right of the entrance to a jet streak or to the left of the exit of a jet streak If there s horizontal divergence at the tropopause there must be some combination of downward motion in the lower stratosphere and upward motion in the upper troposphere Since the stratosphere is resistant to vertical motion in reality most of the vertical motion must take place in the troposphere So divergence in the upper troposphere is almost always associated with upward motion beneath it Again because of the vertical structure of the atmosphere convergence at near the tropopause is mostly associated with downward motion in the troposphere An even greater generalization is possible Because the wind patterns present at jet stream level tend to extend down at least to the middle troposphere we might expect that the vertical motion associated with convergence and divergence diagnosed at the tropopause might actually extend through most of the troposphere And indeed it does Most largescale vertical motion begins close to the ground and ends close to the tropopause Across the spectrum of vertical motion from an ascending air parcel to largescale ascent the tropopause acts like a squishy lid to vertical motion To a first approximation the tropopause can be considered to be a rigid lid a solid barrier to upward and downward motion All the weather action takes place in the troposphere 125 Veltical Wind Shear Is it windy in the basket of a hot air balloon The answer is no A balloon has so little mass relative to its size that it drifts along with the wind Someone riding with the balloon feels almost no wind whatsoever This is true even if the wind is blowing 60 ATMO 251 Chapter 12 page 8 of 23 miles per hour The landing might be a tad rough but while in the air the balloon is racing along with the wind and if you were riding with the balloon and closed your eyes you wouldn t even know a breeze was blowing IT39S windy but since The balloon moves with The wind nothing is felt and The flag hangs limply small helium balloon to a string and attach the string to your car s antenna then take it out on the highway at 60 mph The string of the balloon will be nearly horizontal and the balloon will be dangling behind the antenna If there s a mouse hanging on to the balloon that mouse is feeling a strong wind Of course it s not Vvind that the mouse is feeling the actual wind might be calm Instead it s the air rushing past the car But the balloon doesn t know the difference It s behaving just as if the car were parked but pointing into a 60 mph wind Wi rh The car moving or 60 mph the helium balloon Trails behind as air rushes past Ready for the third situation Imagine that the small helium balloon is now tied to the top ofthe large hot air balloon Will the balloon string be pointed straight up or will the small balloon be trailing behind the big balloon just as it trailed behind the car s antenna The answer is that more than likely the small balloon will be straight above the big balloon Even though the big balloon is moving at ATMO 251 Chapter 12 page 9 of 23 60 mph so s the air around the big balloon If the small balloon feels that same 60 mph wind it will dri along at the same speed as the big balloon and there will be nothing to keep it from pointing straight up t Everything j sT drifts along The key to all these examples is the relative motion of the air In the case ofthe car the balloon is tiedto something that s moving at a speed 60 mph different from the air around it whether we consider the moving car example or the parked car example In the case of the hot air balloon the helium balloon is tied to something that s moving at the same speed as the air around it so there s no extra wind pushing on any side of the balloon to keep it from pointing straight up Or is it Actually we have only assumed that the wind at the level of the small balloon is identical to the wind at the level of the big balloon In the real atmosphere there could be a difference The wind at one height is not necessarily going to be equal to the wind at another height When there s a difference when the wind is blowing at a different speed or a different direction or both at one level compared to another level that phenomenon is called vertical wind shear or wind shear for short Suppose the wind at small balloon level is weaker than the wind at big balloon level That situation is sort of like the car example the small balloon is being pulled faster than the air in which it is embedded so it will lag behind the big balloon Conversely if the wind at small balloon level is stronger the small balloon will be blown out in front of the big ATMO 251 Chapter 12 page 10 of 23 balloon If the speed stays the same but the wind changes direction with height the small balloon will be blown to one side or the other Anytime the small balloon would not be pointing straight up we have wind shear e 4 Because af The verTical wind shear The small balloon is being dragged Through The air by The large balloon 126 C ompu ng Wind Shear from Wind Vectors Wind shear can be thought about in two equally valid but different and therefore contradictory ways The rst way is probably the simplest wind shear is just the difference between the winds at two different levels So if the big balloon was at a level where the wind was 60 kt and the small balloon above was where the wind was only 50 kt the magnitude of the wind shear between those two levels is 10 kt We ll call this quantity the di erence shear just to give it a name Mathematically if we consider some low level a and some higher level b the wind shear is the vector wind difference the wind vector at level b minus the wind vector at level a Shear being a vector has both magnitude and direction So for example if the wind at big balloon height level a is 60 kt from the west and at small balloon height it is 70 ATMO 251 Chapter 12 page 11 of 23 kt from the west the wind shear would be a vector of 10 kt pointing from the west Conversely if the wind at small balloon height is 50 kt from the west the wind shear would be a vector of 10 kt pointing toward the west rather than from it More complex examples might require doing the subtraction graphically or with a calculator Sometimes it s easiest with wind components Suppose for example there s mainly a change in direction At big balloon level say the wind is from the west at 65 mph That means u 65 mph and v 0 Meanwhile at small balloon level the wind is from the westsouthwest at 65 mph with u 60 mph and v 25 mph To compute the wind shear subtract the components Doing so we get u 5 mph and v 25 mph making a vector that is oriented toward the north northwest Using the same terminology as for wind itself we would say that the vertical shear is southsoutheasterly So while big balloon is drifting toward the east small balloon is leaning off toward the north northwest The other and more fundamental meaning of vertical wind shear is as the vertical derivative of the horizontal wind Rather than the amount the wind changes between two particular altitudes it is the rate at which the wind changes with height We ll call this quantity the derivative shear If you re still with me you may be wondering how do you take the derivative of a vector The answer is you do it one component at a time The x component of the vertical shear is the vertical derivative of the u component of the wind and the y component of the vertical shear is the vertical derivative of the y component of the wind So 651 6u 6v l 62 62 62 The subscript h signifies that only the horizontal wind is included in wind shear A little bit strangesounding I know the vertical shear only involves the horizontal wind In pressure coordinates there s a similar definition of the wind shear we ll add a minus sign so that the shear vector corresponds to the change in wind in the upward direction whatever the coordinate system 6u 6v l J 6p 6p 6p ATMO 251 Chapter 12 page 12 of 23 To evaluate the derivative form of vertical shear with real data whether in height coordinates or pressure coordinates you d have to determine the values of u and v at two different levels subtract the higher from the lower and then divide by the vertical distance between the two levels to get the x and y components of the shear But notice that our example above with the 65 mph wind does all that except divide by the vertical distance This then is the difference between the first de nition of vertical shear and the second de nition whether or not you divide by the vertical interval This brings up a related point When expressing the shear as the difference in wind between two levels the answer is relatively meaningless unless everyone knows which two levels are being used in the computation On the other hand the derivative version of shear is valid at a single level just like a horizontal derivative is valid at a single point on a graph So it makes sense to talk about the 1000 to 500 mb vertical shear on one hand and to talk about the vertical shear at 500 mb on the other The former would be the difference shear and the latter would be the derivative shear 127 The Mathematics of Thermal Wind The concept of thermal wind is perhaps the most alien of concepts considered in this course It requires sophisticated spatial reasoning advanced mathematical reasoning or both Some are better at spatial reasoning than others but the necessary spatial powers can be acquired through repeated critical examination of weather maps Similarly extensive instruction in calculus supplies the relevant mathematical background but most will be able to follow the logical reasoning immediately Either way thermal wind is the key to understanding the atmosphere as a complete unit rather than a set of discrete seemingly independent levels It is key to looking at one map and instantly knowing what must be happening at other levels It is key to looking at a sounding from one location and knowing what will happen there in the next 24 hours Finally it is a reality check if how you imagine the atmosphere to work is inconsistent with thermal wind your imagination is wrong Despite its name thermal wind while a vector is not a true wind That s why I ve put it within quotation marks occasionally Instead it is a vertical wind shear representing the change of horizontal wind with height More specifically it is the geostrophic vertical wind shear the rate of change of the imaginary geostrophic wind with height As we ve seen on normal weathermap scales the wind is close to geostrophic balance so the true vertical wind shear will be close to the geostrophic vertical wind shear too So the thermal wind is not a wind that blows Instead it ATMO 251 Chapter 12 page 13 of 23 is a wind shear the sort of thing that makes a small helium balloon tilt away from vertical when suspended from a large hotair balloon The thermal wind being just the geostrophic vertical wind shear can be written in pressure coordinates as 66 Bu 3 339 avg The thing about the thermal wind is not what it is but what it relates to Geostrophic wind of course is directly related to the horizontal pressure gradient by de nition as g a ampkxvhz f To get the vertical shear of the geostrophic wind we ll differentiate both sides with respect to pressure and throw in our minus sign at iampikxvhzampkxvh6zj 6p f 6p f 6p Okay so the geostrophic wind shear is proportional to the vertical variation of the horizontal pressure gradient force or its equivalent in height coordinates So It turns out and this is key that the right hand side of that equation can be simpli ed into something much more meaningful I have taken the liberty of rearranging the order of differentiation on the right hand side of the previous equation to relate it directly to the hypsometric equation which relates thickness to temperature 6ZR 6p pg Now follow this If the vertical variation of height with respect to pressure is proportional to temperature then the vertical variation of the ATMO 251 Chapter 12 page 14 of 23 horizontal height gradient is proportional to the horizontal temperature gradient Let s say that again If Vertical variation of height is proportional to temperature Then Vertical variation of height gradient is proportional to temperature gradient Taking the horizontal gradient of both sides of the previous equation Vh R4th 6p pg and kcrossing that equation gives us 62 R 6p kah dkgtltVhT Pg And since according to what was said a few paragraphs ago the vertical variation of the horizontal pressure gradient is proportional to the geostrophic vertical wind shear that must mean that the horizontal temperature gradient must also be proportional to the geostrophic wind shear One more time the horizontal temperature gradient is proportional to the geostrophic vertical wind shear Combining our altered hypsometric equation with the equation for geostrophic wind shear gives 66 3RdeVhT 6p pf So for one more time the geostrophic vertical wind shear is proportional to the horizontal temperature gradient Now that you ve made it this far go back and read this section again Slowly And make sure that you understand each step ATMO 251 Chapter 12 page 15 of 23 128 Interpreting Thermal Wind ne nice thing about the thermal Wind equation is it is similar to the geostrophic Wind equation so the same techniques for estimating geostrophic Wind from height contours apply to estimating geostrophic Wind shear thermal Wind from temperature contours isotherms Compare the equation for thermal Wind with the equation for geostrophic Wind avg 7R4 kahT alnp pf cg g Jkavhz Aside from a few constants they are identical geostrophic Wind shear is to horizontal temperature gradients as the geostrophic Wind itself is to horizontal height gradients Consider the example below Geostrophic winds are directly related to the height pattern 144 9 147 gt 153 gt just as geostrophic wind shear is directly related to the temperaeture pattern 9 a so x 7 ATMO 251 Chapter 12 page 16 of 23 This gure is a horizontal map so north is to the top of the page and east is to the right If the contours are height contours the vectors would represent the geostrophic wind If the contours are temperature contours the vectors would represent the thermal wind that is the geostrophic vertical wind shear Either way the relationship between the contours and the direction and magnitude of the vectors is the same Where the contours are close together the gradient is strong so the wind if they re height contours or wind shear if they re temperature contures must be strong too The implications of the concept of thermal wind are substantial Because pressure is strongly tied to largescale winds through geostrophic balance and pressure is also strongly tied to temperature variations through hydrostatic balance the winds and temperature must also be strongly tied Speci cally they re connected through the thermal wind Wherever there s a strong temperature gradient the geostrophic wind vector must be changing rapidly with height Take the typical midlatitude wintertime situation of weak winds near the ground and strong westerlies at jet stream level The vertical wind shear is the strong westerly vector minus the weak vector so the wind shear is also westerly and strong Just as westerly geostrophic winds imply lower heights to the north westerly geostrophic vertical wind shear implies lower temperatures to the north Indeed if you have weak winds near the ground and strong winds at jet stream the strong winds alo will always be nearly parallel to the midtropospheric isotherms with colder temperatures to the left if you are facing downwind N Cold to the left of the thermal wind vector Lowlevel wind Thermal wind wind shear Upperlevel wind Warm to the right of the thermal wind vector ATMO 251 Chapter 12 page 17 of 23 Note that this technique for inferring the temperature distribution only works so easily if the winds aloft are really strong or the winds near the ground are really weak Otherwise you d have to be careful and subtract the two vectors to determine the magnitude and direction of the vertical wind shear through the troposphere and therefore the magnitude and direction of the mean temperature gradient through the troposphere 129 Warm and Cold Cores Another useful application of thermal wind is in understanding the vertical structure of high and low pressure systems For simplicity let s consider only situations in which the wind direction is constant with height so there is no directional shear only speed shear In other words the shear vector is parallel to the wind vector at both levels Suppose you have a low According to thermal wind the colder temperatures should be to the left of the wind shear If the wind speed increases with height the wind shear is in the same direction as the wind itself so the colder temperatures must be to the left of the wind that is near the center of the low Conversely if the wind speed decreases with height the wind shear is pointed in the direction opposite the wind itself so colder temperatures would be to the right of the wind and warmer temperatures would be to the left near the center of the low The first ofthese is called a coldcore low and the second is a warmcore low Atypical upperlevel cutoff low pressure system is a coldcore low while a hurricane is a warmcore low We see from this that the fact that hurricanes are strongest near the ground is directly related to the fact that hurricanes are warmest in the eye Conversely coldcore lows are strongest aloft since as we noted wind speeds are stronger aloft The same principles apply to highs An anticyclonic circulation that increases in strength upward must have cold temperatures to the left of the wind outside the anticyclone and warm temperatures to the right of the wind inside the anticyclone Conversely an anticyclone that weakens with height has cold temperatures in its core Warm and cold highs and lows are best visualized in vertical sections The next diagram shows the height of various pressure surfaces in a slice through a warmcore low Since this is a vertical section the strength of the wind is emphatically not related to the spacing of the pressure surface Instead the slope of the pressure surfaces is proportional to the strength of the geostrophic wind The spacing between the pressure surfaces is the thickness and a larger spacing corresponds to warmer temperatures So in this figure we have warmer temperatures near the center and therefore the slope of the pressure surfaces and the strength of the geostrophic wind decreases upward ATMO 251 Chapter 12 page 18 of 23 The 250 mb map would have a weak low with heights barely lower inside the low than outside The 500 mb map would have a somewhat stronger low and would also feature warm temperature within the low The 850 mb map would have the strongest low and strongest winds of all 250 mb Z 500 mb V700 mb V850 mb ax In the center of this diagram are the lowest heights on pressure surfaces The thickness or vertical spacing of pressure surfaces is largest in the center too so this low is warm core As a consequence of the thicknesses the strength of the low decreases with height By now you re probably so indoctrinated to geostrophic wind that you re looking at the diagram above and envisioning geostrophic winds going from west to east If so you re wrong Geostrophic winds are only parallel to height contours an a constant pressure map and this is a vertical section Instead in a vertical section the spacing of pressure surfaces is directly related to the temperature distribution Now you can infer geostrophic winds from this vertical section but perhaps not in the way you might expect Remember that the geostrophic wind is proportional to the height gradient or the slope of constant pressure surfaces The vertical section above shows a part of that slope the component of the height gradient in the direction of x According to the geostrophic wind relations the stronger slope corresponds to stronger geostrophic wind39 one of the stronger winds in the preceding figure is directly above the label x What about direction Since the geostrophic wind in the northern hemisphere blows with higher heights to the right we can look at the diagram and see that to the right of the low the geostrophic wind must be ATMO 251 Chapter 12 page 19 of 23 blowing into the page and to the left of the low the geostrophic wind must be blowing out of the page You can use diagrams such as this to determine the temperature pattern associated with any arbitrary height pattern Start with a sketch of a couple of constant pressure surfaces Slope them strongly where you want strong winds slope them gradually where you want weak winds Now look at how far apart your two lines are Where they are close together the thickness and therefore the temperature is low where they are far apart the thickness and therefore the temperature is high 1210 Temperature Advection from Sounding and Profiler Data We ve already looked at soundings and wind profiler output as providing essential information about the vertical structure of the atmosphere There s also some horizontal structure information in there too if the wind is close to geostrophic balance one can look at the winds and determine which way the pressure gradient or height gradient points Now with thermal wind we can look at the vertical variation of wind in a sounding or profiler plot and determine the wind shear and therefore the horizontal temperature gradient If westerly winds increase with height for example there must be colder temperatures to the north and warmer temperatures to the south Suppose we take the case of directional shear Suppose at one level say 850 mb the wind is from the northwest at 30 kt and at a higher level say 700 mb the wind is from the southwest at 30 kt Whip out a pencil and paper to verify that the 850 to 700 mb wind shear is a vector from the south at 42 kt If the wind is in approximate geostrophic balance the colder temperatures must be to the west and the warmer temperature must be to the east ATMO 251 Chapter 12 page 20 of 23 700 mb 850 mb vertical wind shear event WARM 850 mb wind We also must have cold advection Note that at both 850 mb and 700 mb the wind has a component from the west from colder air toward warmer air Temperature advection is proportional to the temperature gradient multiplied by the component of wind parallel to that temperature gradient So the stronger the westerly component of wind and the stronger the vertical wind shear the stronger the temperature advection n this particular example the wind backed with height That is with increasing altitude the wind vector rotated counterclockwise Note that we re not saying anything about the horizontal wind pattern just the vertical variation of the wind vector itself It turns out that whenever the wind backs with height to the extent that geostrophic balance holds there must be cold advection and whenever the wind veers with height rotates clockwise there must be warm advection You can verify this for yourself by sketching a few examples This was a prime forecasting tool in the days before the widespread realtime exchange of weather observations The most up to date information a forecaster has is the observation that forecaster makes 1f the forecaster checks the clouds and notices a signi cant difference between the wind direction at ground level and the wind direction alo beyond what friction alone would produce he or she can predict with con dence an impending change in temperature And although it s a bit offtopic I must interject a few words about severe weather Meteorologists know that supercells and tornadoes require lowlevel wind shear On TV the conditions favoring such supercells are o en referred to as a clash of hot and cold air masses or some such verbiage Well it s true that you need warm and cold air masses next to each other to get a strong temperature gradient and thus strong wind shear But the two air masses don t need to be battling each other39 the mere fact of them sitting next to each other is good enough ATMO 251 Chapter 12 page 21 of 23 12 11 Recap The vertical wind shear is the vertical variation of the horizontal wind It can be expressed in units of wind itself by subtracting the wind vector at a higher level from the wind vector at a lower level or in units of wind speed per unit height or unit pressure obtained by computing the derivatives of each of the wind components The thermal wind despite its name is not properly a wind Rather it is de ned as the geostrophic vertical wind shear The thermal part of the name comes in because the geostrophic vertical wind shear magnitude is proportional to the horizontal temperature gradient and the shear vector is oriented parallel to the isotherms with colder temperatures 90 degrees to the left of the direction toward which the shear vector points Wherever the wind is nearly in geostrophic balance the thermal wind will nearly correspond to the actual vertical wind shear From these basic facts one can conclude that beneath the jet stream should tend to be a strong temperature gradient since the vertical wind shear tends to be strongest there The colder temperatures will be below and to the left of the jet stream while the warmer temperatures will be below and to the right of the jet stream At the jet stream level itself temperature gradients will be weak because the wind is neither increasing with height that happens below jet stream level nor decreasing with height that happens above jet stream level Closed circulations whose intensity varies with height imply temperature variations as well since the intensity changes mean there must be vertical wind shear Specifically a coldcore circulation will be stronger aloft if it s a low and be weaker aloft if it s a high and a warm core circulation will be stronger aloft if it s a high and be weaker aloft if it s a low A very cold core might have a high at the ground and a low aloft The vertical wind shear can also be used to diagnose temperature advection Cold advection implies winds backing with height while warm advection implies winds veering with height Questions 1 On several sounding diagrams identify the location of the tropopause 2 Suppose the lowest 3 km of the atmosphere has convergence at the rate of 3XlO395 s39l Compute the rate of change of temperature at heights of 1 km 2 km and 3 km due to the resulting vertical motion if the ATMO 251 Chapter 12 page 22 of 23 atmosphere is a weakly stratified BTBz 8 Ckm and b strongly stratified BTBz 0 Ckm 3 Given two weather maps one with winds on the 500 mb surface and one with winds on the 850 mb surface sketch a map with vectors depicting the difference shear between the two levels The difference shear in this case is defined as the 500 mb wind minus the 850 mb wind 4 Given a map of temperatures at 700 mb sketch the vector field that represents the geostrophic vertical wind shear at that level 5 If your answers to 3 and 4 are for the same time compare your two vector elds How well do the patterns of direction and relative magnitudes agree What reasons are there for them to not match exactly 6 Using a map of j et stream winds at 250 mb and the knowledge that winds at the ground are likely to be much weaker determine the areas where the troposphere is relatively wa1m and where it is relatively cool Look at some maps of temperature within the troposphere and evaluate how well the 250 mb wind field allowed you to estimate the overall temperature pattern 7 Sketch imaginary horizontal maps of 850 mb height and 500 mb height that would be associated with a a cold core high and b a cold core low 8 A sounding shows winds from the east in the lower troposphere and winds from the west in the upper troposphere What is the temperature advection Explain ATMO 251 Chapter 12 page 23 of 23

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