Synoptic Meteorology Laboratory
Synoptic Meteorology Laboratory METR 4424
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Review of the Quasi Geostrophic System In synoptic lecture you have seen how scale analysis following set of consistent quasigeostrophic equations of the equations of motion leads to the Vortici Eg neglect friction 8 g Vpgg f7 fuvpv agg 39 80 df 7v V 7 v fa at g ng B 3 Vortici changes are caused by advection of relative plus earth s vorticity and by ty divergenceconvergence see Fig 67 in Holton Thermod amic Eg neglect diabatic term 7VgVprm at R Temperature changes are caused only by advection and adiabatic ascentdescent 80 Mass Contian Eg Vp VPT g 0 Examples of physical interpretation 1 850 lt 0 foi gt 0 horizontal divergence T 813 8 I 8 gt 0 VP lt 0 horizontal convergence 03ubsquot note gt 0 5 rising motion or J 8 50 gt 0 VP lt 0 horizontal convergence 1 ap 8a lt 0 8P VPV gt 0 horizontal divergence Z w and PTOZlt 0 39 sinking motion These 3 equations can be combined to help answer two of the primary questions of weather forecasting 1 Where are the highs and lows ridges and troughs going That is what are the height changes x Em 2 Where is the precipitation going to be ie What is the distribution of vertical motion as The answer to l is found by examining the height tendency equation fi 62 f3 6 R va p2xf VgVpggf egg 57 vgVpr The answer to 2 is contained in the mequation fi 62 NWTiggtvgyruflgv m Both of the equations were derived in synoptic lecture We shall rst provide a mathematical interpretation of these equations followed by a physical interpretation The mathematical interpretation is accomplished by the usual insideout approach First consider the vorticity advection term in the height tendency equation vb fblng39Vp gfJ We have seen earlier in this course that at point A Va Vpgg gt 0 so ivg Vp g flt 0 5 anticyclonic vorticity advection Thus the right hand side is negative and V x lt 0 Since VZ lt 0 means that Ngt 0 we see that AVA leads to g gt 0 or height rises Similarly at point B Va Vp glt 0 Vg39vp g fgt 0 cyclonic vorticity advection which leads to sz gt 0 and x lt 0 heightfalls 2 A quotEa 3 We now consider the di erential temperature advection term V3 77 Va VPT Here we need to estimate temperature advection at two or more levels For example suppose we have the following vertical distributions of temperature advection Thus TAlt 0 sz gt 0 and xlt 0 heightfalls 5 7Vg VpT Thus temperature advection increasing with height causes height falls Note that this can be caused by warm advection increasing with height or cold advection decreasing with height Similarly temperature advection decreasing with height causes height rises true for a warm advection decreasing with height or cold advection increasing with height see Fig 68 in Holton EN The mathematical interpretation of the wequation is as follows Consider the di erential vorticity advection term 8 vgm7EELVgVp afl Assume eg that vorticity advection increases with height true for both curves shown Then 83VAlt 0 but VA gt 0 and V20 gt 0 which implies that m lt 0 rising motion P o V Vng Therefore vorticity advection increasing With height leads to upward motion Similarly vorticity advection decreasing with height both curves shown leads to sinking motion See Fig 69 from Holton Consider the temperature advection term R V w7gv ngVPT This says 03 is large When gradients of temperature advection are large If We have warm advection then ivgVPT gt0 V2TAlt 0 Vin gt 0 Which implies m lt 0 or rising motion Similarly cold advection leads to 03gt 0 or sinking motion see Fig 610 from Holton Physical Interpretation of QuasiiGeostrophic Forcing Functions Consider the height tendency x equation and recall the vorticity advection term cLVgVpgg fJ If We have cyclonic vorticity advection CVA into aregion then the Q3 increases there 2W2 V1 vaorticity is m geostrophic ie 7 t VZZ then We have height decreases as Q increases Or in other Words larger vorticity values require stronger Winds Which require steeper pressure gradients which leads to height falls relative to the regions With Weaker or no cyclonic vorticity advection Similarly AVA requires height rises Consider the differential thermal advection term fi 8 R mi Vg39Wi Recall that we have shown that if temperature advection increases with height height falls temperature advection decreases with height height rises don t have to use terms Wmm or cold For example assume we have warm adveetion at all levels Mathematically we see we have a 75TAlt 0 x gt 0 rises 6 750A 0 x lt 0 falls 0 3912 What is the physical reason for this Recall that warm advection acts to warm each layer and will thicken each layer but will do so most where warm advection is the largest Now consider the old and new location of pressure surfaces a er the warm advection shown above has acted for a while pressure surfaces pushed upwards 7 height rises max thickening occurs here 39TA0 gt x0 pressure surfaces pushed downward 7 height falls Thus warm advection causes height rises due to thicker layers above the level of maximum advection and height falls below this level due to hydrostatic warming of the column Similarly cold advection produces both height rises beneath CA maximum due to increase in weight of the column and height falls above CA maximum due to accumulated effect of denser thinner A m x a ers W I vm n q H Now conslder the dxjferennalvartxcxty advecnan term ofthe aequanpn amp3 Vs Mano es du rentml vortaerty adveetaor eause vemcal motaorn Conslder vortrerty adveetror Why do mereasmg wrth herght 74 negleet temp adv If vortrerty o geostrophre get for euwe on rrght 7 large g merease gtlarge herght fall 7 small Egmcrease gt small helght fall Therefore we have athexmess decrease where VA dlfference ls greatest Usmg temperature equataor wth Thus the eolumr must em to keep atmosphere hydrostatae g lt 0 we see that upwardmotaor and adlaban eoolmg ls requlred a i m3 2 m lt 0 at R Srmrlarly we get smk mg mamm lfvomclty adveeta on decreases wth herght N m Although the eurve represerrts both posrtwe and rregatwe vortaerty adveetaorr we have sm n m both layers beeause adlabatl warmmg ls e motao requlred to explam the resultarrg thckness mereas MA 9 A Fmally we conslder the Iaplacmn athermaladveman term m the weequatror egv Vs VFT Ifwe have Wa1m advech39ori we know on lt 0 or rising motion Why Ifa layer experienc 39 39 s shown bel w This leads m a relative increase in advech39oriy is nckness will lricrease a o vorticity at the botmni and a decrease at the hip ofthe layer 3 g decreases w A a V g increases From the vorticity equation is 40D neglect vorticity adv 3 gt0 near the bottom gt Dw lt o convergence Thus and glto alo gt Dw gt o divergence e mass continuity equation we know convergence below and divergence above leads to rising mo on Lu c a j 39 39 Aquot J a f r e v W Similarly cold advection leads to subsidence TABLE 1 Idealized QG Description of a MidLatitude Baroclinic Wave Parameter A 500 mb B sfc low C 500 mb D sfc high trough ridge QgSOO from Increasing due Increasing due Decreasing due Decreasing due vorticity eq to convergence to CVA partly to divergence to AVA partly no VA cancelled by no VA cancelled by divergence convergence W500 from QG 3 Sinking due to Rising due to Rising due to Sinking due to eq CA no VA DIFF CVA TA WAX no VA DIFF AVA TA small small 7500 from QG X eq D 39 due D 39 due I due I due to to TA to CVA to TA AVA increasing with decreasing with height height H5001000 from D 39 due D due I due I due to temp eq to CA partly to adiabatic to WA partly adiabatic cancelled by ad cooling TA cancelled by ad warming TA Warming small Cooling small Qgsfc from Decreasing due Increasing due Increasing due Decreasing due vorticity eq to divergence VA small to convergence VA small to convergence VA small to divergence VA small assumes level of nondivergence is below 500 mb LEGEND VA TA CVA AVA CA WA vorticity advection temperature advection cyclonic vorticity advection anticyclonic vorticity advection cold advection warm advection To accompany Fig 611 in Holton p 141 Relationships between Vorticity Advection and Response ofx a Div and Thickness Fields 1 Vertical Pro le of Vorticity Change QG Height Change QG Vertical Divergence Thickness Response Vortic39 x Motion 0 Response 8 0 7 7 j 8P am 9g 0 t d I L H gt bu ecreasmg x lt 0 falls 0 gt 0 L 13ng Thickness increases m with height T wmng ag T g gt 0 and 1ncreaslng x lt 0 falls 0 lt 0 T QCOHVE Thickness decreases Av R A with height t cooling m x 0 d t H gt aquot Increasmg x lt 0 falls 3 lt 0 T 613ng Thickness decreases m with height T W mg ag T g lt 0 and 1ncreas1ng x gt 0 11585 m lt 0 T acorng Thickness decreases at s with height T mug ag 0 d T pl gt an Increasmg x lt 0 falls 3 lt 0 T 613ng Thickness decreases in with height T Coolmg ag T a gt 0 and 1ncreas1ng x lt 0 falls m lt 0 T acorng Thickness decreases in With height cooling Relationship between Temp Adv And Response ofHeight a Div and C Fields Vertical Profile of Temperature QG Height Change QG Vertical Divergence Vorticity Response Temp Advection Change Motion Response 6g f D 8 0 7 X 0C u I W TA decreasing with T N height rise 03 lt 0 T eDiva 8 K lt 0 at S max warming max m lt 0 T T gtConvlt quotM TA increasing with fall in lt 0 T T hei ht 3 warning increases P with height fall to lt 0 T T We gtconv lt 8C gt 0 m 0 at mm cooling decreases fall 0 gt 0 T L a r with height 200 max warming fall 03 ltlt 0 T T m gtconv lt 8C gt 0 TA increases with T T at 5 height t gtconvlt 8Q Dm Small warming fall m lt 0 T t g gt 0 a Relationships between Temp Adv And Response ofHeight co Div and C Fields cont Vertical Pro le of Temperature QG Height Change QG Vertical Divergence Vorticity Response Temp Advection Change Motion Response 6g f D OC u iv 8 03 7le at K 8P am TA increases with PL height falls 03 gt 0 L gtConvlt 9i gt 0 8t 5M large cooling 3 gtgt 0 L L d 39th 39 gt 0 t 8 g lt 0 NW TA ecreases W1 uses In eDlv at o height L Q cooling L x x rises to gt 0 eDiv Temp advection L at m decreases With m 0 L R quota g gt 0 hei ht ms uses 0 lt 0 T a o 4 warmmg T Large cooling rises 03 gtgt 0 L ace 8 H Tem d lt 0 p a v e a at 5N decreases With m gt 0 L L helght Div ema ag uses 0 N 0 L L 5 ltlt 0 REVIEW OF HYDROSTATIC CONCEPTS Recall the hydrostatic equation gas law and de nition of Virtual temperature 1 dpdz pg p pRTV TV 161w To derive an expression for obtaining heights from temperature and moisture data we rst separate variables dzRTV dp g P and integrate over an atmospheric layer 22 pz 22 pz 21 P1 I dz E I Tvdlnp 21 g P1 Let TV be the mean Virtual temperature between p1 and p2 Then 22 21 jpz dlnp RTV ln p2p1 g p g or z 21 RTV ln p1pz 2 g which is the hypsometric eguation If we de ne z 7 21 as the thickness h then the thickness equation is h E In p1pz g For a xed lower pressure p1 and upper pressure pz 7 eg for the 1000500 mb layer IUg ln p1pz is a constant K Thus h Ki 3 which emphasizes the point that the thickness of a layer is solely due to its mean virtual temperature We can use equation 3 to estimate expected thickness errors due to radiosonde temperature errors First take differentials 5h 7 Ksiv If we assume a 1 C systematic error in the radiosonde thermistor we can construct the following table for the corresponding thickness errors Layer 5h 1m 1000 7 850 mb 5 1000 7 700 mb 11 1000 7 500 mb 20 1000 7 300 mb 35 1000 7 200 mb 47 1000 7 100 mb 68 You can allow for about 50 of these values as the margin of error in the reported height values when adjusting your contours for smoothness and obeying geostrophic wind spacing You should be able to prove to yourself the following applications of hydrostatic concepts i troughs lows tilt toward cold air ii ridges highs tilt toward warm air iii cold lows intensify with height iv warm lows weaken with height v warm highs intensify with height vi cold highs weaken with height You should use the above rules to develop vertical consistency between contour analysis at different levels Why Does Vorticity Advection Increasing With Height Cause The Air To Go Up Consider vorticity equation 7VvpltzoitVV 1 Let n F 8 V V and assume steady state 8 motion gt recall a 7C V Where C de Cy t is the speed of the trolgh or ridge Rewrite 1 as iaVnVVp fifg or viaw 7t 2 Now assume vorticity advection increases with height 7 this usually requires Vto increase With height as shown Note that the level of nondivergence LND exists where C V Assume WgtE motion consider Xcomponent of 2 only u7 a fit Now below LND c gt u ucx lt 0 and since 2 17lt0 LHS is gt 039 fg gt 0 means g lt 0 or convergence below LND X Similarly above LND u gt cx 5 g gt 0 or divergence above LND By mass continuity we must have upward motion in this region Aw gt T CV gtgtO i m gtt K cowxv f Q 39gt I I 1 Pg 67 Schemauc 500 mb geopohenual eld showmg regxons of posmve and negame advecuons ofrelame and planemry vomclty 6 2 DEVELOPMENT OF THE QUASLGEOSTROPHIC SYSTEM Fxg of Lanperature advecuon w the upper level hexght Lmdencxes A and B desxgnate respectwely reglons ofcold advectxon and mm advecuon m the lowemoposphee u K Fig 69 Schematic 500 mb contours solid lines and 1000 mb contours dashed lines indicating regions of strong motion due to differential vorticity advection I Fig 610 Schematic 500 mb contours thin solid lines 1000 mb contours dashed lines and surface fronts heavy lines indicating regions of strong vertical motion due to temperature advection Na min Pain m idly mmquot reefon Unaon 9 U 11 l I 750 cm mm V oduum mum A a C Fig 611 Secondary circulation associated With a developing baroclinic Wave top schematic contour solid line 1000 mb contours dashed lines and surface fronts bottom vertical profile through the line 11 indicating the divergent and vertical motion fields Western Region Technical Attachment July 19 1966 No 20 66 The Physical Relationship Between Vorticity Advection and Vertical Motion An occasional review of the physics underlying the use of routine analyses and0r forecasting procedures is a good practice to follow It emphasizes the proper use of a technique by bringing back to each forecaster s attention the strong and weak points of the technique plus any restrictions regarding its use This Technical Attachment reviews the relationship between positive vorticity advection at 500 mb as indicated on facsimile charts and vertical motion Discussion of vorticity may seem to be out of date in the light of comments in previous Attachments These have suggested that the use of vorticity charts has reached its zenith and will probably be on the decline in the future as NMC improves primitive equation model NWP forecasts which do not involve vorticity While this is still considered true vorticity charts based on the barotropic model are expected to be with us for several years to come The vorticity lines on the current facsimile 500 mb analyses and prognoses are isopleths of absolute vorticity 7 ie relative vorticity plus the Coriolis parameter 7 and are labeled in units of 10395 per second Vorticity isopleths have two important uses in operational forecasting 1 They add detail to the 500 mb contours by clearly locating shortwave troughs and ridges when the associated curvature of the contours is small or missing see Figure l 2 They make available a useful tool indicated vorticity advection 7 for forecasting clouds and precipitaion With regard to the latter the use of vorticity advection is an intermediate step taken to indicate the sign of vertical motion If we were given the vertical motion directly and we expect that PE forecast verticalmotion charts will soon be transmitted on facsimile there would be no need to use the vorticity advection as a verticalmotion indicator The relationship between positive vorticity advection at 500 mb and upward vertical motion is not unique One of the best ways of seeing this is to examine the simplified vorticity equation 1 V C Vn KDavpV V is the wind at a given point on the constantpressure surface C is the translation speed of the synoptic system e g trough 0r ridge movement and VT is the gradient of the vorticity isopleths at the given point under consideration on the constantpressure surface K is a constant for purposes 0fthis discussion VVn is the vector notation for vorticity advection CVn is the local change of vorticity at a point due to the movement of troughs and0r ridges assuming no change of shape and intensity of the system as it moves What this equation says is that the indicated vorticity advection on a smntrpressure surface and the local change lhdlcaled by the speed of the trough or ndge assoclated Wth ths advecuon delelmlhes whelhec dlvergence or convagmce IS laklhg place h the lndlcatedvomclty advectlon I Figure 1 36HR soc m PMNOSIS ISLEVEL KDEL VT 00002 July 13 1966 Vorticity Isopleths Contours If we assume as is usually done that the level of nondivergence is close to the 500 mb surface then VcVn 0 or saying this in another way the vorticity advection determines the motion ofthe associated trough or ridge ie vVn cVn But this doesn t tell us anything about the sign ofthe vertical motion Thprpfnre n it advection can 39 39 with upward as downward vertical motion It is only when we assume that the troughs and Wk Lhdtand nu s p ie n Vn doe that the wind component perpendicular to the vorticity lines increases with height that we can use positive vorticity advection at 500 mb as an indicator of upward vertical motion To demonstrate this let us apply the vorticity equation separately at 850 500 and 300 mb and assume the wind increases with height with no large gt 10 degrees changes in wind direction a b blind 5901 gt but gain gt Figure 2 Schematic drawings of Vertical wind shear and Vertical distribution of divergence At 850 mb vltc then in the equation K VC Vn 7 WV the term VC is negative and this requires that DWV be negative ie convergence At 500 mb V c then DWV 0 At 300 mb v gt c then VC is positive which indicates divergence This is the case when wind blows through the contour pattern These results are plotted schematically in Figure 2b assuming a linear change of divergence with height Thus ahead of a trough convergence is indicated below 500 mb and divergence above and this calls for upward motion with a maximum at 500 mb This is the model we assume when we relate positive vorticity advection at 500 mb to upward vertical motion andor to convergenttype synoptic patterns at or near the surface Note that if there is little or no vertical wind shear positive vorticity advection at 500 mb does not indicate largescale upward motion Also there are occasions when troughs move faster that the indicated vorticity advection Usually they are large amplitude troughs associated with old occluded systems When this occurs downward motion is associated with positve vorticity advection and upward motion with negative vorticity advection ie upward motion behind the trough instead of ahead of it French and Johannessen in their article on the association of indicated 300 mb positve vorticity advection and cirrus clouds 2 found this reverse situation to occur 14 of the time in the data they studied Figure 3 is an example of this situation taken from their article Pilot reports of overcast cirrus ie assumed upward vertical motion along the ight path are indicated by the black stripe and broken cirrus by the hatched stripe in the figure 100 In I 1 39tICuv I39ll w o 10 rl Iv quotplnu Figure 3 300 mb chart 0300Z February 1 1953 Dashed lines are isopleths of ZZ at lOOfoot intervals these can be considered isopleths of relative vorticity Solid lines are contours at 200 foot intervals The track of the observing aircraft is indicated and its position at 0300Z located by the arrow Other limitations in using positive vorticity advection as an indicator of upward vertical motion are 1 errors in the barotropic forecast 2 vertical motions related to temperature advection below 500 mb the theoretical relationship which Sutcliffe pointed out in 1947 3 A good discussion of this second point and others is given in the NAWAC Manual 4 The relationship between clouds and precipitation and upward vertical motion is not simple The amount of moisture available is an important item in this relationship also the magnitude of the vertical motion is involved In general positive vorticity advection areas are usually much larger than the related cloud shield In our Region forecasters have found empirically that when positive vorticity advection areas are associated with precipitation the absolute value of the vorticity usually exceeds 12x10395 per second In summary the simple relationship between positive vorticity advection and cloudiness while very useful has important limitations Therefore this relationship is only one of several indicators that the forecaster must consider in preparing cloud and precipitation forecasts For example moisture distribution temperature advection airmass stability and surface heating or orographic lifting must also be taken into account References 1 Cressman GP An Approximate Method of Divergence Measurement Journal of Meteorology April 1954 Pages 8384 2 French J E and Johannessen K R Forecasting High Clouds from HighLevel ConstantPressure Charts Proceedings of the Toronto Meteorological Conference 1953 Pages 160171 3 Sutcliffe R C A Contribution to the Problem of Development Quarterly Journal of Royal Meteorological Society October 1957 Pages 370383 V 4 NAWAC Manual Part II November 1961 Pages 914 V Thermal Vorlicity Interpretation march g r i 2 EV zrrzmgv h 1 Vonnclty advecnon rrrcreasrrrg wnh height 8 B B B 2 B 2 3h gt 7 gt0 ZZL gt0 hgt0 lt0 aw am am a alw JV at Thrckness decreases rf neglect TA only 3 lt 0 T car keep Thrckness change hydrostatic 2 Tempaature advecuon W A cg decrease cgmcrease im igr lt0 8 8t cu decrease r Increase Drv aloft conv Below warm advecuon gt nslng moaon