Atmospheric Dynamics 1
Atmospheric Dynamics 1 ESCI 342
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ESCI 342 Atmospheric Dynamics I Lesson 9 Thermal Wind Reference An Introduction to Dynamic Meteorology 4ml edition Holton Reading Holton Chapter 3 section 34 THERMAL WIND The geostrophic wind in pressure coordinates is A 80 V8 k XTVpZ The difference in geostrophic wind between two levels is 73 471 2gtltvpz2 12gtltvpz1 2gtltvpz2 zl This shows that the difference between the geostrophic wind at two layer is parallel to the contours of thickness Using the hypsometric equation Z2 Z1 1n f 80 p2 the difference in geostrophic wind can also be shown to be parallel to the contours of layer average temperature 2 471 1n 12xvpf f p2 Since the difference in wind is parallel to the layer mean isotherms it is commonly referred to as the thermal wind and denoted as VT 7 12gtltvpz2 zl or 7 1n 12xvpf f p2 Rules for the thermal wind 0 The thermal wind is parallel to the thickness lines with low thickness to the left 0 The stronger the thickness gradient the stronger the thermal wind The rules for the thermal wind are analogous to those for the geostrophic wind except that thickness is substituted for geopotential height If you add the thermal wind to the geostrophic wind at the lower layer you will get the geostrophic wind at the upper layer Like the geostrophic wind the thermal wind is a de nition 0 The actual difference between the wind at two levels will equal the thermal wind only if the actual winds at the two levels are geostrophic However since the atmosphere is usually close to geostrophic balance the thermal wind is a good approximation to the actual difference in wind between two levels PHYSICAL EXPLANATION OF THERMAL WIND The phyml ban for the thermal wmd can be explamed a follow m y m hmmmm o n the average temperature m the layer 0 each uxface will be Mammal ng Vgl COLD H BACKING AND VEERING WINDS 0 v v V m 7 amendmg order ee example below V5 V2 V1 thh hexght V1 helght V1 V and temperature advecuon o Veexmgwmd mdxcate wanna advecuon o Backmg Wm mdxcate comm advecuon EXERCISES 1 The geostrophic wind is Vs IEx VPZ Take the partial derivative ofthis with 3 Rd A respect top and show that is rikXVPT 3p f p 2 The diagram below shows contours of 1000 r 500 mb thickness 3 o o J n 3 o J m 3 Assume the 1000 mb geostmphic wind is SW at 5 ms At the three black dots A A at ura 500 mb Use a latitude of45 N and d 175 km b l 39 front 3 Show that K rin 7 1Efo can be written as Ex AV 47 D p Dt 3 V2 Yam and that the gradient wind speed is fR 1 2 2 V3eTthf R 4fRVg iiiu 3V fRaiRJrZVgaiRJrZRig we LM 3p 7 3 JfZRZJrAfRVg 4 6 a Show that if the radius of curvature does not change with height then 8V3 i 1 8Vg a a p 1 i V p f R 3 Note In this formula the positive root applies to the regular cases regular high and regular low while the negative root applies to the anomalous cases anomalous high and anomalous low b For the regular high and the regular low is avap greater than or less than 8Vg 8p 7 Assume that the isotherms are parallel with the contours so that the radius of curvature doesn t change with height ESCI 342 Atmospheric Dynamis I Lemon 12 Vor city Reference An Introduction to Dynamic Meteorology 4 edition llolton nformal Introduction to Theoretical Fluid Mechonict Lighthill Reading llolton Chapter 4 section 42 VORTICITY The circulation of a fluid is defined as c E gsv oi From Stoke39 s theorem this is the same as CjVxl7dA 1 The quantity VXV is therefore also a measure of the rotation of the fluid and is called the vorticity o Vorticity is defined as at 2 WV I 1 TL respect to the vorticity vector I Circulation and vorticity are closely related 0 L theorem as szxvcjdei Vde n m u The components of the vorticity vector are a 5 Lt t iw u 1 at so do at J at at 54V r L plone to we ore ntort interested in the verticoz component of vorticity FrlIm now on p J Wquot ci vertical component n definition holds VORTEX STRETCHING O A vortex line is a line that is everywhere parallel to the vorticity vector m it 0 Vortex lines cannot begin or end in the interior of the uid They must terminate at a boundary of some so Vortex lines move with the uid 0 A vortex tube is a collection of vortex lines 0 Vortex tubes move with the uid and always consists of the same uid parcels 0 The circulation taken around a disk that is perpendicular to the vortex tube is equal to the average vorticity of the tube times the area of the tube C VdZIVxVdAjad 2m A O In a barotropic uid if a vortex tube is stretched the circulation doesn t change wever the crossrsectional area of the tube will decrease which means that the vorticity must increase 0 Stretching a vortex tube causes it to spin faster 2 1 111 23942 0 This phenomenon is known as vortex stretching o Vottex sttetehing explains Why a WhiIlpool foIms oveI youI bathtub dIainl As the voIteX tube moves oveI the dIain it becomes sttetehed causing it to spin mote I i l The sense of the Iotation is detetmined by the oIiginal Iotation of the voIteX tube o befote it moved ovet the am 0 Vottex sttetehing also helps explain the fotmation of mesoeyelones and toma oesas 39 39 Hex quot RELATIVE VERSUS ABSOLUTE VORTICITY tieity also depends on Whethet it is measuted in an absolute ota g flame 0 As With eiIeulation voI Iefetenee flame o1 in a I t39 The minty 39 L l P 39 quot A W my and is gi enthe symbol 0 o The uuieit 39 39 39 39 39 w oi y and is given the symbol 41 o The vottieity of the sutfaee the Cotiolis patametet f see e 0 Absolute Ielative and planetaty vottieity ate elated via 0 0 In the atmosphete Ielative vottieity is usually much less than the planetaty vorticity Thetefote the absolute vottieity is usually a positive value 39 39 quotquot WhenWe quot L 39 39 inLessonlOWe saw that thete WeIe two anomalous cases of balanced ow an anomalous high A 39 10W 1 ML ok m nh lmi n Iquot h an of the Earth is called planetaty vottieityl It is equal to Xeteise 1 o 3 absolute vorticity which may be one reason why they aren t observed frequently on the synoptic scale since it is difficult to think of a process in the atmosphere that would generate negative absolute vorticity over a large area CURVATURE VERSUS SHEAR 0 Vorticity may be Visualized by imagining a paddle wheel moving with the uid ow 0 If the paddle wheel is rotating clockwise then there is negative or anticyclonic relative vorticity 0 If the paddle wheel is rotating counter clockwise then there is positive or cyclonic relative vorticity O The relative vorticity may be due either to 0 Curvature 0 Shear 0 This is best visualized in natural coordinates where the vorticity can be written as 4 2 LV Z an R A B where Term A is the vorticity due to shear and Term B is vorticity due to curvature 0 You can t necessarily tell the sense of vorticity by just looking at the streamlines 0 It is possible to have cyclonic curvature with anticyclonic shear or vice versa In most cases V XV must be calculated to find the sign of the relative vorticity 0 Relative vorticity in the atmosphere is usually on the order of 10 5 tolO4 s71 O The picture below shows the 500 mb geopotential height and the 500 mb absolute vorticity units are s71 X 105 8 amp TUE 392 29 600 Ar Apr 29 2003 GEOSTROPHIC VORTICITY The vorticity due to the geostrophic wind is called the geostrophic vorticity 42 Since the geostrophic wind can be given in terms of a streamfunction ill 91 g a y a 3 3x a the geostrophic vorticity is equal to the LaPlacian of the streamfunction s VZII 0 Since the streamfunction is related to the geopotential field via VI fJICD fo lgo Z where f 2 f0 constant then the geostrophic vorticity can be calculated directly from the geopotential heights On the synoptic scale we often approximate the actual wind by the geostrophic wind In the same vein we often approximate the actual relative vorticity by the geostrophic vorticity 0 This is convenient since we can calculate vorticity directly from the geopotential heights and don t need the actual wind observations Remember the geostrophic vorticity like the geostrophic wind is a definition It is close to but not necessarily equal to the actual vorticity g E g EXERCISES 1 a Show that the circulation of a at disk of radius r in solid body rotation is 2 A vertically oriented vortex tube is in your bathtub The tube is circular with a radius C 27rPr2 where P is the component of angular velocity perpendicular to the disk b Show that the component of vorticity perpendicular to the disk is just 2P c Use your result to show that the vorticity of a point on the Earth s surface is 29 sin 4 and is therefore equal to the Coriolis parameter of 5 cm The tube is rotating clockwise as viewed from above with a tangential velocity of 05 cms a Calculate the average vorticity of the tube b As the tube moves over the drain it is stretched and its radius shrinks to 1 cm What is the new average vorticity 3 Calculate the vorticity of the following ows at point xy 1m 2m 14 u a oxy 1402m 1 71110 1 s 1 v voy u u b 0y 140 2 s71 v0 1s 1 v vox u u c 021402ms 1101m 1s 1 v vox u 140 1 71 71 71 d 140 st vo lms k 21ml 09m v v0 coskx sinly 4 Show that if f and 0 are assumed constant the geostrophic vorticity on a constant altitude surface is 1 4 inp g f p 5 a Show that if f is allowed to vary with latitude that the geostrophic vorticity on a constant pressure surface is 1 81 3 ivch 27 f f 3x b Use scale analysis arguments to see if it is reasonable to ignore the second term in the above expression on the synoptic scale so that we can still use 1 Eivch 41 f even though f is not constant with latitude Hint f 1 BCD Ex is of the same order of magnitude as the geostrophic wind ESCI 342 Atmospheric Dynamics I Lesson 6 Scale Analysis Reference An Introduction to Dynamic Meteorology 3ml edition Holton Reading Holton Chapter 2 section 24 SCALE ANALYSIS OF THE HORIZONTAL MOMENTUM EQUATIONS 0 Not all of the terms in the momentum equations are significant If a term is much smaller than the others then it is reasonable to ignore it under certain circumstances variables and parameters in the equations 0 For scale analysis we don t assign exact numbers just order of magnitude The orders of magnitude are assigned for specific scales of motion For instance they To assess which terms can be neglected we assign an order of magnitude to all the would be quite different for the study of tornadoes than they would be for the study of hurricanes 0 The following parameters are also used TZLU For synoptic scales the following orders of magnitude are appropriate 0 Using these scales and parameters the terms in the u momentum equation have the following orders of magnitude Bu Bu V Bu W Bu uvtancj MW 1 3p 29quot Sim 29WCOS P 32 32 32 i i i i 7 i if v 7 7 V7 a Bx By dz 0 a p Bx 3x2 ayz azz U L U L WUH U a UWa tsp01L 29 U sin 45 29W cos 45 vULZ vUHZ 104 104 10 5 10 5 10quot 10 3 10 3 105 10 16 10 12 ms2 ms2 ms2 ms2 ms2 ms2 ms2 ms2 ms2 ms2 1 The time scale 139 LU is called the advective time scale It is the time it would take for a parcel of uid to travel the entire horizontal length of the ow O A similar analysis for the v momentum equation is a awa We 2W m 41p 49 a1 a1 Bt Bx By Bz a a p By V 3x2 ayz azz U L U L WUH UVa UWa Lapp L 29 U sin 45 vULZ vUHZ 104 104 105 105 10quot 103 103 1016 1012 ms2 ms2 ms2 ms2 ms2 ms2 ms2 ms2 ms2 Many of the terms are very small compared to others and can therefore be ignored without much loss of accuracy We can therefore ignore the curvature terms the viscous terms and the Coriolis term that involves the vertical velocity 0 Ignoring these terms yields a much simpler version of the horizontal equations of motions 31 31 31 Biz igl2 2vsin Bt Bx Vay Bz pr av av av wal lal 2ousin Bt Bx By Bz 0 By Note We could have also ignored the vertical advection terms but it is not too much of an inconvenience to keep them By defining the Coriolis parameter as f E ZQsingi the horizontal momentum 0 equations assume the familiar form D 1 B in i7p fv Dt 0 Bx Lip Dt 0 By 0 In vector form the horizontal momentum equation is mi 1 A a 7 7V k x V Dt p HP f H where the subscript H indicates we are taking only the horizontal components of I7 and V VH u i v v in i j Bx By 0 In this form of the momentum equations the total derivatives are known as the inertial terms The terms on the right hand side are the pressure gradient and Coriolis terms respectively THE ROSSBY NUMBER AND GEOSTROPHIC CYCLOSTROPHIC AND GRADIENT BALANCE O The horizontal momentum equation in the form lvp 12 x f 7 Dr 0 can be further analyzed using scale analysis If we divide it through by if V we get T A 7 k XL M Di plfvl lf Vl Using the representative scales the order of magnitude of these terms are L 6P fL pr The dimensionless combination Uf L is defined as the Rossby number named for Gustav Rossby R0 E U f L When the Rossby number is much less than unity R0 ltlt 1 then the acceleration term can be ignored and the only two terms left are the pressure gradient term and the Coriolis term which must be nearly in balance 0 This is known as geostrophic balance and the velocity in this case is known as the geostrophic wind 0 The momentum equation in this case reduces to 12 x N lvp which is solved for the geostrophic wind to yield a 1 A V 7k XV g f p p When the Rossby number is much greater than unity R0 gtgt 1 then the Coriolis term can be ignored In this instance the only terms that are left are the acceleration and the pressure gradient terms and so the acceleration is a direct result of the pressure gradient force DV 1 Dt VP 0 This type of balance is called cyclostrophic If the Rossby number is of the order of unity R0 1 then all three terms must be retained This is known as gradient balance and the wind in this case is known as the gradient wind The following table summarized these results erms ltlt gtgt For large scale synoptic scale motion the Rossby number is of the order 0 7E10ff S6 01 10 s 10 m which shows that on these scales the atmosphere is close to being in geostrophic balance Hence the actual wind should be close to the geostrophic wind 5 MORE ON THE GEOSTROPHIC WIND The geostrophic wind is a definition 0n the synoptic scale the actual wind should be close to the geostrophic wind because R0 ltlt 1 but will rarely be exactly equal to the geostrophic wind The components of the geostrophic wind are L LP 3 f 0 3y L LP V3 f 1 9x The geostrophic wind is parallel to the isobars with lower pressure to the left in the Northern Hemisphere The geostrophic wind speed is directly proportional to the pressure gradient In pressure coordinates the geostrophic wind and components are 7 i12gtltvltrgt gi xvz f f LELALZ g f 3y f 3y i8ltlgtg0 BZ V 7 if g f 3x f 3x Therefore on a constant pressure surface 0 The geostrophic wind is parallel to the isohypses with lower heights to the left in the Northern Hemisphere o The geostrophic wind speed is directly proportional to the geopotential height gradient Another important feature of the geostrophic wind is that it is non divergent V 0 78 0 iffis constant The geostrophic wind is sometimes written in terms of the streamfunction 1 defined as 11 pfp in pressure coordinates 11 cpf where f and 0 are constant In this case the geostrophic wind is 73wa 31 a al 3 3x THE AGEOSTROPHIC WIND o The thttetehte between the actual wlnd and the geomophlc wlnd l called the ageostmphlc Wm L L typltally mall m companxon to the geomophlc wmd motion in the atmosphere A O ageomophlc wlnd V Therefore even thought the ugeomphtt wian small t7 139 very important SCALE ANALYS S OF THE VERT CAL MOMENTUM EQUAT ON thl caxe P l the Vemcal Vanatlon ln pxexxuxe whlch l 1000 mb 31 may 731 20W 1 Bl Br Bx By Hz a 9 Hz 3 By2 Hz UWL UWL w H U a amppH 29 U m 45o g VWE tut112 10 10 10 10 10 10 10 loquot9 loquot m2 m2 m2 m2 m2 m2 mz m2 m2 o Thl analyxl how that the preuure gradient and gravltytexm ate dommaht 0 Th etote ah the Synoptic tale the atmoxphexe can be axxumed to be m hydmtatlt balance and the vettltal momentum equatlon lmpllfle to 3p 7 pg at A more ngomu analym we Mexuxrtzle Meteut39uluglmlMudelmg by Plelke Show that the h mm39 rehf n 39 m 39m 39 the ver 39mll 39 en hxmle tumh malle hurt the ham 071ml len h mlt H ltlt L Thl condltlon cemmly r 1 applle 0n the Synoptic male EXERCISES 1 P Squot 4 Show that 12Xngv Perform a scale analysis of the horizontal momentum equations in component form for the whirlpool formed as your bathroom sink drains Which terms are important in this case Water has a density of 1000 kgm3 and a kinematic Viscosity of V 18X10 6 m2 s71 The horizontal pressure difference across the whirlpool is 10 Pa Use a reasonable estimate for the horizontal velocity based on your own experiences What is the Rossby number for a tornado Does the Coriolis force effect a tornado Expand the horizontal momentum equation lvp 12 x f 7 Dr 0 to show that in component form it is ial 5 ialfu Dr 0 3x Dr 0 By and therefore yields the two component equations D in ial fv Dr 0 3x D 1 a 1 1 f Dr 0 By A7 5 At the four pomt hown m the ptctuxe below emmate the mamtude ofthe geoxtmphlc wmd Axxume a demtty of123 kgma and a lautude of 45 The Rabat ate labeled h mb t t I km 6 out what term can be lgioxed ESCI 342 Atmospheric Dynamics I Lesson 14 P0tential Vorticity Reference An Introduction to Dynamic Meteorology 4ml edition Holton AtmosphereOcean Dynamics Gill Reading Holton Chapter 4 sections 43 45 and 46 THE BAROTROPIC QUASIGEOSTROPHIC VORTICITY EQUATION 0 Using the incompressible continuity equation we can write the quasi geostrophic vorticity equation as a 3w 3 Vg 0VH g vg fg 0 If we integrate this equation from the surface to the top of the atmosphere we have zT zT aw J a Vg VH g 16de Ifgdz fwzrwzol O In the case of a barotropic uid the velocity and hence vorticity is independent of height so we get Zo 3 e f f Dz Dz gVIV viw w 7 T 0 1 at g Hig km 0 ll Di 0 If we assume the height of the top of the atmosphere is fixed so that M is constant and also recognizing that DZ 7 V8 39VZO this becomes a a f c a Vg 0VH g vg ZV80VHZO 2 A variant of equation 2 was used by the first operational numerical weather prediction models the so called barotropic models 0 If the terrain is at then equation 2 becomes a g a a Vg 0VH g vg 0 0 The barotropic model was useful for predicting the vorticity at the level of nondivergence 500 mb 0 A barotropic model cannot predict development of a circulation other than that in uenced by terrain because the only divergence allowed is due to terrain effects POTENTIAL VORTICITY IN A BAROTROPIC FLUID 0 Equation 1 can be written as 8ng a th Vg39VH g vgZE 0 If We include the relative vorticity in the divergence term remember We ignored it earlier because it is much smaller thanf then this equaiton can be Written as a c fDh is V o v V 37 at 3 H 3 3 h Dt which is the same as D D 31an f Elnh 2 0 Dt h 39 o The quantity 3 fh is conserved following the uid parcel in a barotropic uid 0 This quantity is called the barotropic potential vorticity POTENTIAL VORTICITY IN A BAROCLINIC FLUID 0 The circulation theorem for a baroclinic uid is DC d 7 4 l 3 Dt p 0 The equation of state for the atmosphere can be Written as Pa 11 p 7 RJ Rd 6 p 39 Rd GEL Po 0 On a surface of constant potential temperature the density is a function of pressure only which makes the rightehandeside of 3 equal to zero Thus 0 On an isentropic surface the circulation is constant Imagine a section of a stream tube of crossesectional area 54 lying between two isentropic surfaces function of onlyp and d The circulation on the isentropic surface is given by A The mass contained within the stream tube is m pd l s so that the circulation per unit mass is C a m p 39s The length of the stream tube is 6s 860 80 so that the circulation per unit mass is C a 80 m 600 as o If the ow is adiabatic then the ends of the stream tube will always lie on the isentropic surfaces Thus m is constant as is 6 Since the circulation C is also constant this means that a 80 if const as On the synoptic scale we can assume that isentropic surfaces are nearly horizontal so that the stream tube is oriented along the z axis Therefore we can write 19 const 0 a and in terms of pressure this is Iii Lg Zip const 0 8p it If the atmosphere is hydrostatic then this becomes a 0 gm 7 const 3p or g const 8 The quantity Pg gf is called Ertel s potential vorticity and is the form of potential vorticity appropriate to the atmosphere 0 P is conserved following an air parcel in adiabatic ow and is therefore a good tracer of air parcels under conditions where diabatic heating latent heat of condensation radiation etc can be neglected POTENTIAL VORTICITY AND THE GENERATION OF LEESIDE TROUGHS Conservation of potential vorticity can be used to explain the formation of lee side troughs in westerly ow over a mountain barrier see Holton section 43 3 EXERCISES 1 Show that WT me DhDt 2 Expand D2 Tf0 to show that it gives 88it70Vh v fDh t t 3 a Show that 1131 41 Vzo t b Show that 78 0 Vzo 10 zo so that the quasi geostrophic barotropic V01ticity equation can be written as a 81 f 7V2 J V2 72 7 at II 14 II ax h we ESCI 342 Atmospheric Dynamics I Lesson 8 Geostrophic and Gradient Balance Reference An Introduction to Dynamic Meteorology 3m1 edition Holton Reading Holton Chapter 3 sections 32 and 33 NATURAL COORDINATES 0 In the absence of friction the Wind blows parallel to the isobars with low pressure to the left of the ow in the Northern Hemisphe e 0 If A f39 quot 39 39 u the ind direction and pointing toward the left of the ow and s to be a coordinam that is in the direction of the Wind then the balance of forces in the s and n directions is Eiizg Diniif ilal th pan 0 dinalt2 is the centripetal acceleration of the parcel so it isjust din 7 V2 alt2 R W ere R is the radius of curvature defined as positive for cyclonic curvature and negative for anticyclonic curvature 0 When the ow is straigh R m 0 In natural coordinams the equations of motion are then i if 1 Dt p as lip V2 R7fV7pan 2 If the ow is parallel to the isobars or height contours on a constant pressure surface then apas 0 and the speed of the ow is constant GEOSTROPHIC FLOW If the isobars are straight then R 2 co and 2 gives us the geostrophic wind speed 7iP g f 1 an o This always gives us a positive value for speed since apan lt 0 in the Northern Hemisphere o The formula also works in the Southern Hemisphere because there although apan gt 0 f lt0 and so it still gives a positive value for speed INERTIAL FLOW If the pressure gradient is zero then equation 2 shows that the acceleration is caused solely by the Coriolis force and we have Vm fR This type of balance is called inertial balance In inertial balance the ow is circular with radius of R Vm f 0 Since by definition Vis always positive then R must be negative and so inertial ow is anticyclonic The period of the inertial ow is found by dividing the circumference of the inertial circle by the speed 27rR 27139 7 7 7 Vm f o The inertial period is shorter at higher latitudes and is infinity at the Equator Pure inertial ow is rarely important for the atmosphere but does occur in the oceans CYCLOSTROPHIC FLOW If the rotation of the Earth is unimportant large Rossby number then the Coriolis term can be ignored and equation 2 shows that V lag pan 0 This type of balance is called cyclostrophic Cyclostrophic ow can be either cyclonic or anticyclonic Either way the speed is positive because 0 for anticyclonic ow R lt 0 and apan gt 0 o for cyclonic ow R gt 0 and apan lt 0 The ows in tornadoes and in your bathtub drain are examples of cyclostrophic balance GRADIENT FLOW 0 If no terms can be neglected then equation 2 remains as is It is quadratic in V and can also be written as V2fRV537po pan The solutions of this equation are the gradient wind fR l 2 2 Rap V 7r7 R 77 3 8 2 2W pan U 0 In order for the wind speed to be real and not have an imaginary component the following condition must hold f2R2 47Ra7pzo 0 an 0 As long as R and a p an are of opposite signs this isn t an issue 0 For those cases were R and a p an are of the same sign then there is an upper limit placed on the pressure gradient such that 2 ELI S f 0 R an 4 This explains why on a synoptic scale weather map we often see tightly wound lows with large pressure gradients right down to the center while we never see large pressure gradients in the center of high pressures For the high pressure case there is a physical limit as to how large the pressure gradient can be near the center 0 There are several distinct combinations for the signs of R apan and the root of the equation These are applies only if R and a p an have the same sign O Sign of Sign of Root Restriction Type of circulation R a p an on Radical 3p fzp Physically impossible because PGF and Coriolis are a S TlRl opposite the needed centripetal acceleration none Anomalous low 7 Rotates anticyclonically PGF and Coriolis in same direction none Physically impossible because Vg can t be negative none Regular low 7 Rotates cyclonically PGF and Coriolis opposed none Physically impossible because Vg can t be negative 8p fzp Anomalous high 7 Rotates anticyclonically PGF S 4 W and Coriolis opposed Faster than regular high ap fzp Regular high 7 Rotates anticyclonically PGF and S 4 W Coriolis opposed Slower than anomalous high 0 If we exclude the physically impossible solution we are left with four possibilities Sign of Sign of Root Restriction Type of circulation a p an on Radical none Anomalous low 7 Rotates anticyclonically PGF and Coriolis in same direction none Regular low 7 Rotates cyclonically PGF and Coriolis opposed 3p fzp Anomalous high 7 Rotates anticyclonically PGF a S TlRl and Coriolis opposed Faster than regular high 3p fzp Regular high 7 Rotates anticyclonically PGF and a S Coriolis opposed Slower than anomalous high For many years the anomalous high and low solution though physically possible were thought not to occur in the atmosphere or ocean However there have been enough observations of cases of anomalous gradient winds to show that anomalous ow sometimes occurs though not over large areas For more information on the anomalous solutions refer to Fultz 1991 Quantitative nondimensional properties of the gradient wind J Atmos Sci 48 869 875 Chew F and MH Bushnesll 1990 The half inertial ow in the Eastern Equatorial Pacific A case study J Phys Ocean 20 1124 1133 Mogil HM and RL Holle 1972 Anomalous gradient winds Existence and implications Mon Wea Rev 100 709 716 Alaka MA 1961 The occurrence of anomalous winds and their significance Mon Wea Rev 89 482 494 Note that the anomalous high exhibits the curious behavior that the wind speed actually increases as the pressure gradient force decreases O The anomalous high and anomalous low both become inertial ow as the pressure gradient goes to zero or as radius goes to infinity 0 The gradient wind equation can also be written in terms of the geostrophic wind speed Vg as see exercises V will my gr 2 fR g TRAJECTORIES VS STREAMLINES O A trajectory is a curve tracing the successive points of the particles position in time 0 A streamline is a line that is tangent to the velocity at a point at a given instant I Trajectories and streamlines only coincide if the uid motion is steady O O O O O The local rate of change of wind direction is a S VSLJ at RT RS where RT and RS are the radii of curvature for the trajectory and the streamline respectively EXERCISES 1 Suppose the magnitude of the pressure gradient for a circular regular low is given as ap Acos RSL an L ap02RgtL an where A 25X10 3 Pa m l L 250 km and R is the distance from the center of the low Using a computer program such as Excel IDL Matlab or other software of your choice make graphs of the following a The geostrophic wind as a function of R for 0 S R S L b The gradient wind as a function of R for 0 S R S L c The ageostrophic wind as a function of R for 0 S R S L Use values for density and Coriolis parameter of 0 123 kg m 3 and f 1041 s l 2 For the same values of pressure gradient Coriolis parameter and radius of curvature the ow around a regular low is slower than the ow around a regular high Give a physical explanation of this fact Drawing force diagrams should be very helpful 5quot P If p0 is the pressure in the center of a circular high pressure find the radial pressure profile pr that will give the maximum allowable pressure gradient at each radius r from the center of the high Fquot Using the pressure distribution you calculated from part a sketch the gradient wind speeds for the regular and anomalous high as a function of distance from the center of the circulation Use 0 123 kg m 3 and f 1041 s l O How does the answer to part b compare to pure inertial ow 4 The geostrophic wind speed in natural coordinates is given as V g f 1 an Use this to show that the gradient wind equation can be written as V wept my gr 2 fR g ESCI 342 7 Atmospheric Dynamics l Lesson 10 7 Vertical Motion Reference AVL Itrrmdrtcrrmr ra Dymzmlc Metem39ulugy 4 edition llolton Reading llolton chapter 3 sections 31 35 and 36 PRESSURE COORDINATES 0 Pressure is o en a convenient vertical coordinate to use in place of altitude altitude is given by the hydrostatic equation 3P a 7 fps Ln helght coordinates the vertical velocity is d eflned as W DzDt npiessuie coordinates the vertical velocity is de ned as E E and is commonly called slmply umegtz o The units of mate pas o en mbS is also used While upoxl39live omega mean humming motion 0 w and mate related as follows a Biz l DD l Lz Dr Dr B W Dr By W Dr Hp W Br W vtwand it 2 i 1 k gt 3 m By pg 9 it actually perpendicular to the direction ofw isperpendicularto a surface a also be written 0 The dlrectlon of a a r e of constant pressure whlle of constant z see diagram w th t 7m Hz and Hz 9 wt 93 m the term mpg The term which represents the vertical movement of a constant pressure surface is also small compared to the 0 pg term Therefore we can use a E 0 g w to convert between wand w on the synoptic scale The total derivative in pressure coordinates is 7 LAME Dt at 3x 3y 3p 0 The conversion of a height derivative to a pressure derivative is accomplished as follows aaza aa 3p 3p dz 8 dz 39 In pressure coordinates the directions of the unit vectors 5 j and I are the same as in height coordinates The x and y axes are still horizontal and not oriented along the constant pressure surface The vertical axis is still vertical perpendicular to x and y o The 14 and v components of the wind are the same in both height and pressure coordinates MOMENTUM EQUATIONS IN PRESSURE COORDINATES O In pressure coordinates the horizontal momentum equation is Dr A 7quot VpltIgt k X f VH Horizontal momentum equation t O The hydrostatic equation in pressure coordinates is a CD a7 0 Hydrostatic equation P CONTINUITY EQUATION IN PRESSURE COORDINATES O The continuity equation is pressure coordinates is derived as follows Dm D 7 7 6 6 6 0 Dt Dt 390 x y Z If the atmosphere is in hydrostatic balance then p6z 6 p g so we write the expression above as D 7 6 6 6 0 Dt x y p This expands out as D D D D 7666 6676 6676 6676 0 Dt x y p y th x x th y x yDt p which can also be written as 726x 6th From the fact that L2 6y Dt l D 6y67pE6p0 D D D 7 6x 6u 7 6 6v 7 6 60 Dt Dt y Dt p we get 614 6v 660 7 7 0 6x 6y 6p which in the limit as the parcel becomes infinitesimally small becomes 91 a La 3x a y a p The full continuity equation in pressure coordinates looks very much like the incompressible continuity equation This is one of the advantages of using pressure coordinates Continuity equation THERMODYNAMIC ENERGY EQUATION IN PRESSURE COORDINATES O The thermodynamic energy equation in pressure coordinates is DT D p c 7 a 7 J p Dt Dt which expanded out and using the definition of 0 becomes alrvxvpp 911 L at 3p cp cp A B C D E In this form the terms represent Term A 7 Local temperature tendency Term B 7 Horizontal thermal advection Term C 7 Vertical thermal advection Term D 7 Adiabatic expansioncompression due to vertical motion Term E 7 Diabatic heating radiation latent heat etc 0 Terms C and D can be combined and written as 8T 0 ol 80 p 8p cp 0 a p R d a and defining the staticstability parameter 039 as a 80 0 ap we get the following form of the thermodynamic energy equation in pressure coordinates a7T 7 7V1T ai 3 Rd 0p Thermodynamic energy equation A B C D Note that this definition of the static stability parameter is different than Holton s definition of the static stability parameter SF on pg 59 They are related by 0 Sde P 0 Ln thl form ofthe equatlon the Veltlcal advectlon and adlabatlc expannonoornpremon are comblned lnto one term Terrn c 0 mt negatlve nurnber fol an umtable atmoxphele VERTICAL MOTION o rt arrn rnr Tbrtr ltmu tb tntn A from other meaxuled quantltlex o Kinematic methodon calcuhting Vertical motion 0 One method of calculatlng the vertlcal motlon l the kmemm method whlch above the ulface to get p My mm Vt Vdp o f the ulface n at and the ulface plexxule tendency n zero then Wr 0 and we have t ap VF Ivdp p and lntegrated dlvelgence gwer downward rnotlon wlnd Ln fact the ageoxtlophlc wlnd l of the older ofthe enol ln the wlnd obxelvatlon themxelvex Thl mean that dlvelgence calculated from the obxelved wlnd rnayhave large erro 0 Though lt lm t much uxe fol calculatlng actual Value ofvertlcal rnotlon the and lt relatlon to vertlcal rnotlon mall at the top of the tropoxphere a graph ofthe Veltlcal motlon wlth herght would look omethlng llke that hown below 11 vVlt0 0 Since awap must disappear at some level the divergence also disappears at that level This leads to the conclusion that I There is some level in the atmosphere at which there is no horizontal divergence This level is known as the level of nondivergence or LND 0 Observations indicate that the level of non divergence usually occurs at around 600 mb However since 600 mb is not a standard pressure level for reporting traditionally meteorologists consider 500 mb to be the level of non divergence O Adiabatic method for calculating vertical motion 0 Another method for calculating vertical motion uses the thermodynamic energy equation solved for wand assuming adiabatic conditions so that w alX7V T 039 p at p o A drawback to this method is that temperature tendency must also be known PRESSURE TENDENCY EQUATION 0 An equation for the change in pressure at a fixed point in the atmosphere can be derived as follows 0 Differentiate the hydrostatic equation with respect to time to get a a p 80 37 E g 37 Substituting from the continuity equation gives mm gvg pv gaizmw which when integrated from some level z to the top of the atmosphere yields ap ap giVHIpl7dz g0w at at 1 Since BpBt at the top of the atmosphere is zero the equation for the pressure tendency at level z is a m gIVHIpVdzgpw This equation can be expanded as a M a M a affrig39IJVH 39degIV39VHJngJW Using the ideal gas law we can show that VHp Rive PT R T72 TVHP PVHT so the equation becomes g pVH Dde Ri l7DVHpdz l7VHTdzg0w A B C D E O The physical interpretation of the pressure tendency equation is as follows 0 Term A represents the local pressure tendency 5 Term B represents the vertically integrated divergence above the level of interest I Integrated divergence above the layer leads to lower pressure I Integrated convergence above the layer leads to higher pressure Term C represents integrated advection of pressure I If the winds are in geostrophic or gradient balance this term will be zero Term D represents the integrated temperature advection l Advection of warm air lowers the pressure Term E represents advection of mass across the layer l Upward vertical velocity leads to increased pressure as the mass is moved above the level since it is the mass above the level that determines the pressure in a hydrostatic atmosphere I At the surface of the Earth if the surface if level then Term E would be zero EXERCISES 1 Show that the hydrostatic equation in pressure coordinates is BCDap a Hint Start with Bpaz pg and use the chain rule and the definition of geopotential al1 31 L 8p cp 03p Rd Hint Take aap of T 0pp0K 2 Show that 3 If horizontal advection and diabatic heating are negligible then the local temperature tendency from the thermodynamic energy equation is l Q a R d This equation says that if the atmosphere is stable then downward motion will result in an increase in temperature at a fixed level while if the atmosphere is unstable then downward motion will result in a decrease in temperature at a fixed level Give a physical explanation as to why this occurs 4 Use the adiabatic method to estimate the 500 mb vertical velocity 0 for the following situation The temperature tendency is zero The temperature at 600 mb is 13 C at 500 mb it is 19 C The wind at 500 mb is from the SW at 20 ms and the temperature at 500 mb increases toward the West at 1 C100 km 5 For a typical tropical cyclone which is a warm core circulation in gradient balance explain whether each term in the pressure tendency equation contributes to surface development lower pressures at the surface or to weakening higher pressures at the surface Which terms do you think are most important for surface development
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