Synoptic Meteorology II: Introduction to Isentropic Potential Vorticity. 7-9 April 2015

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1 Synoic Meeorology II: Inroducion o Isenroic Poenial Voriciy 7-9 Aril 2015 Readings: Secions 4.1, 4.2, and o Midlaiude Synoic Meeorology. Inroducion We sar by casing he horizonal momenum equaion, neglecing ricion and resened wihou deriaion, ino isenroic coordinaes: D kˆ M (1) D In (1), he subscri o imlies ha he gradien oeraor is alied on an isenroic surace. M is he Mongomery sreamuncion. On an isenroic surace, he oal deriaie in (1) akes he orm: ( ) ( ) D D ( ) d ( ) (2) d where d/d is he diabaic heaing rae, which we reerred o as dq/d earlier his semeser. This erm saes ha cross-isenroic low occurs only when here is local diabaic heaing (d/d non-zero) conribuing o oenial emeraure no being consered ollowing he moion. This imlies erical moion beween isenroic suraces. (Noe, as we did in he las lecure, ha erical moion in isenroic coordinaes can also be along an isenroic surace, or dry adiabaic in naure.) Noe he similariy o (1) o is orm in isobaric coordinaes: D kˆ Φ (3) D In our las lecure, we demonsraed how he Mongomery sreamuncion on an isenroic surace is an analog o he geooenial heigh on an isobaric surace. Thereore, i makes sense ha he erm on he righ-hand side o he horizonal momenum equaion in isenroic coordinaes would be ormulaed based uon he Mongomery sreamuncion raher han he geooenial heigh. I we exand he oal deriaie in (1) by making use o (2), we obain: kˆ M (4) Inroducion o Isenroic Poenial Voriciy, Page 1

2 Inroducion o Isenroic Poenial Voriciy, Page 2 We now wish o obain he oriciy equaion alicable on isenroic suraces. Recall ha he oriciy equaion on an isobaric surace was obained by aking kˆ o he horizonal momenum equaion. In comonen orm, his is equialen o inding /x o he -momenum equaion and subracing /y o he u-momenum equaion rom i. Doing so, we obain: ( ) ( ) k ˆ (5) I we assume ha oenial emeraure is consered (urely dry adiabaic low), hen he diabaic heaing rae is zero. Thus, he hird erm on he le-hand side o (5) and he only erm on he righ-hand side o (5) are boh zero. Simliying, ( ) ( ) 0 (6) The irs erm o (6) can alernaely be wrien in erms o because does no change locally wih ime. As a resul, ( ) ( ) ( ) 0 (7) We can wrie he coninuiy equaion in isenroic coordinaes as: 0 (8) We now wish o combine (7) and (8) in such as way so as o eliminae he diergence erm rom he sysem o equaions. To do so, we mulily (7) by -g/ and add o i (8) mulilied by -g(). Doing so, we obain: ( ) ( ) ( ) ( ) 0 g (9) The inerse o he chain rule or arial deriaies may be used o combine he irs wo erms o (9) as well as he las wo erms o (9). This allows us o wrie: ( ) ( ) 0 g (10)

3 We deine P, he isenroic oenial oriciy, as: P gη (11) where η, he absolue oriciy. Because g is a consan, we can ake i ino he arial deriaies o (10), allowing us o subsiue (11) ino (10) and obain: P DP P 0 0 (12) D The Conseraion o Poenial Voriciy Equaion (12) saes ha he isenroic oenial oriciy is consered ollowing he moion along an isenroic surace (i.e., under dry adiabaic condiions). Gien ha we negleced ricion when we saed he horizonal momenum equaion in isenroic coordinaes, isenroic oenial oriciy is also consered only under ricionless condiions (i.e., no near he surace). The nonconseraion o isenroic oenial oriciy ollowing he moion on an isenroic surace hus allows us o iner where diabaic heaing is occurring and/or where ricion is imoran, boh oenially useul ieces o inormaion. The isenroic oenial oriciy P is a mulilicaie uncion o wo acors: Absolue oriciy η a roaional consrain Saic sabiliy -/ erical isenroe acking Isenroic oenial oriciy is a large osiie alue when cyclonic roaion is srong (η > 0) and/or where saic sabiliy is large, reresening isenroes ha are ighly acked in he erical (-/ >> 0). Normally, -/ > 0, such ha P < 0 only occurs where η < 0. Localized maxima in isenroic oenial oriciy are known as osiie oenial oriciy anomalies, whereas localized minima in isenroic oenial oriciy are known as negaie oenial oriciy anomalies. Because isenroic oenial oriciy is consered (does no change) ollowing he low, i saic sabiliy or absolue oriciy change in alue, he oher mus change in he inerse in order o kee he alue o he isenroic oenial oriciy consan. I he saic sabiliy increases, he absolue oriciy mus decrease. I he saic sabiliy decreases, he absolue oriciy mus increase. Inroducion o Isenroic Poenial Voriciy, Page 3

Boh o hese saemens hold rue in he inerse as well; changes in absolue oriciy mus be accomanied by changes in he saic sabiliy in order o kee he alue o he isenroic oenial oriciy consan.

Figure 1. Schemaic o a column o absolue oriciy beween wo isenroic suraces a an iniial (le) and a subsequen (righ) ime aer he oriciy column has been sreched erically.

4 Boh o hese saemens hold rue in he inerse as well; changes in absolue oriciy mus be accomanied by changes in he saic sabiliy in order o kee he alue o he isenroic oenial oriciy consan. We can demonsrae his grahically using a erical column o absolue oriciy beween wo isenroic suraces, as deiced in Figure 1. Figure 1. Schemaic o a column o absolue oriciy beween wo isenroic suraces a an iniial (le) and a subsequen (righ) ime aer he oriciy column has been sreched erically. The rae o roaion a he iniial ime, as measured by he absolue oriciy, is gien by η 0. The wo isenroic suraces, reresened by and, are searaed by a known incremen. Likewise, he dierence in ressure beween hese wo suraces, is also known. I his column o oriciy is sreched erically, he absolue oriciy increases. This can be shown using he oriciy equaion, wheher in is ull or quasi-geosrohic orm. The new absolue oriciy is gien by η 1, where η 1 > η 0. The searaion beween he wo isenroic suraces, under dry adiabaic condiions, remains. Howeer, by irue o sreching he column erically, he dierence in ressure beween hese wo isenroic suraces has increased. Thus, since is larger, / is smaller. Thus, as absolue oriciy increased, he saic sabiliy decreased o consere isenroic oenial oriciy. On he synoic-scale, isenroic oenial oriciy anomalies eole hrough a combinaion o ranslaion (moion/adecion), roaion, and deormaion by he synoic-scale wind ield. For hese rocesses, isenroic oenial oriciy is consered ollowing he moion. Howeer, ricion and/or diabaic rocesses can also imac he isenroic oenial oriciy. When hese rocesses are imoran, isenroic oenial oriciy is no consered ollowing he moion. We will examine his conce in greaer deail in a laer lecure. Inroducion o Isenroic Poenial Voriciy, Page 4

5 Tyical Values o Isenroic Poenial Voriciy and he Dynamic Trooause The unis o P are relaiely comlex: m K m s P 2 kg s s Pa s s kg 2 1 K m m K s For yical mid-laiude, synoic-scale low, g 10 m s -2, η 1 x 10-4 s -1, and / -10 K er 100 hpa (-10 K er Pa). I should be noed, howeer, ha he acual alue o he saic sabiliy is yically less han his scale alue exce in he resence o srong emeraure inersions. I we lug hese alues ino (11), we obain a characerisic alue o P 1 x 10-6 m 2 K s -1 kg -1. For simliciy, we erm his alue o be equal o 1 PVU, where PVU sands or oenial oriciy uni. In he rooshere, P is yically less han or equal o 1.5 PVU. Exceions are yically conined o smaller-scale henomena associaed wih srong cyclonic roaion, such as roical cyclones. In he sraoshere, where he saic sabiliy is ery large as oenial emeraure raidly increases wih heigh, P is yically greaer han 2.0 PVU. The rooause reresens he ransiion region beween he lower roosheric alues and higher sraosheric alues o isenroic oenial oriciy. This gies rise o he consruc o he dynamic rooause, which is mos commonly reresened by he 1.5 PVU or 2.0 PVU surace. The dynamic rooause is he reresenaion o he rooause by a surace o consan oenial oriciy. Noe ha as he isenroic oenial oriciy is consered ollowing he low on an isenroic surace, so oo is oenial emeraure consered ollowing he low on a surace o consan isenroic oenial oriciy. Consequenly, where oenial emeraure changes ollowing he low on he dynamic rooause, one can iner ha diabaic rocesses (e.g., laen hea release) are ongoing and (oenially) imoran o he eoluion o he synoic-scale aern. Since oenial emeraure generally increases wih heigh, analyses o oenial emeraure on he dynamic rooause can be used o iner he relaie heigh o he rooause. Where oenial emeraure is relaiely warm on he dynamic rooause, he rooause isel is a a relaiely high aliude, inerring an uer roosheric ridge. Conersely, where oenial emeraure is relaiely cold on he dynamic rooause, he rooause isel is a a relaiely low aliude, inerring an uer roosheric rough. This is demonsraed by Figure 2 below. Relaiely warm oenial emeraure on he dynamic rooause is ound across he easern hal o he Unied Saes. Conersely, relaiely cold oenial emeraure on he dynamic rooause is ound across he wesern Unied Saes. The horizonal wind ield is anicyclonically-cured across he easern Unied Saes whereas i is cyclonically-cured across he wesern Unied Saes, in agreemen wih wha we would exec Inroducion o Isenroic Poenial Voriciy, Page 5

gien he heory aboe. As one migh exec, he sronges winds on he dynamic rooause are ound in conjuncion wih he sronges horizonal gradien o oenial emeraure along he dynamic rooause.

6 gien he heory aboe. As one migh exec, he sronges winds on he dynamic rooause are ound in conjuncion wih he sronges horizonal gradien o oenial emeraure along he dynamic rooause. We can conirm hese insighs by comaring Figure 2 o Figure 3. A 300 hpa, a rough o low ressure is ound across he wesern Unied Saes whereas a ridge o high ressure is ound across he easern hal o he Unied Saes. A any gien laiude, air emeraures are relaiely cold in he iciniy o he uer-roosheric rough and relaiely warm in he iciniy o he uer-roosheric ridge. Since ressure is consan on an isobaric surace, his imlies relaiely cold uer roosheric oenial emeraure across he wesern Unied Saes and relaiely warm uer roosheric oenial emeraure across he easern Unied Saes. The sronges winds are ound along he erihery o he rough and ridge, where he horizonal heigh and oenial emeraure gradiens are a heir larges. Beore roceeding, i is worh saing a cauionary noe. Air emeraure yically increases or a leas remains consan across he rooause. As a resul, air emeraure in he hear o an uer roosheric rough on an isobaric surace inersecing he rooause may aear warm comared o is surroundings, in conras o he insigh drawn aboe. Thereore, exreme care mus be aken when analyzing eaures on isobaric suraces inersecing he rooause. The same does no hold rue on he dynamic rooause, howeer; here, cold (warm) oenial emeraure always corresonds o a lower (higher) rooause heigh. Figure 2. Poenial emeraure (shaded; unis: K) and wind (barbs; hal: 5 k, ull: 10 k; ennan: 50 k) on he dynamic rooause, as reresened by he 1.5 PVU surace, a 1200 UTC 10 Noember Also deiced is he hpa layer mean relaie oriciy (conours; eery 5 x 10-5 s -1 ). Image couresy Heaher Archambaul. Inroducion o Isenroic Poenial Voriciy, Page 6

7 Figure 3. Sreamlines (black conours), horizonal winds (shaded; unis: k), and uer air obseraions (barbs: wind in k, red numbers: air emeraure in C, green numbers: dewoin emeraure in C) a 1200 UTC 10 Noember Image couresy Sorm Predicion Cener. Alicaion o Poenial Voriciy o Isobaric Suraces I should be noed ha isenroic oenial oriciy is consered ollowing he low only on isenroic suraces. I is no consered on isobaric suraces, or een on consan heigh suraces. Gien ha mos meeorological daa is obained, dislayed, and inerreed on isobaric suraces, i is air o ask wheher here is a similar quaniy ha is consered on isobaric suraces. The Erel oenial oriciy, or EPV, is consered ollowing he ull hree-dimensional low on isobaric suraces so long as he low is dry adiabaic and ricionless. Mahemaically, he EPV can be exressed as: ( ˆ k ) EPV g (13) In (13), subscris o 3 reer o he gradien being ealuaed in all hree dimensions x, y, and. I we comlee he ecor oeraions in (13) o exand he EPV ino is comonens, we obain: 3 3 EPV ω ω u g η y x x y (14) I we resume ha he horizonal oriciy he erms inoling ω and he arial deriaies o u and wih resec o is relaiely small, we can simliy (14). This resumion is equialen o saing he erical moion and is horizonal gradiens are relaiely small on he Inroducion o Isenroic Poenial Voriciy, Page 7

8 synoic-scale and ha he rooshere is aroximaely baroroic such ha he horizonal wind seed and direcion are nearly consan wih heigh. This simliied orm o (14) is hus gien by: EPV gη (15) Equaion (15) is idenical o (11), exce alied on an isobaric surace and saed exlicily as an aroximaion. The basic inerreaion o mid-laiude, synoic-scale weaher henomena is idenical wheher isenroic or Erel oenial oriciy are uilized. Howeer, care should be aken o remember ha he isenroic oenial oriciy is only consered ollowing he moion on isenroic suraces, while Erel oenial oriciy is only consered ollowing he moion on isobaric suraces. (Boh are consered only or ricionless, dry adiabaic low.) Likewise, i should be reieraed ha he aroximae EPV gien in (15) aboe is jus ha an aroximaion o he ull EPV. For greaes accuracy, he isenroic oenial oriciy should be comued and inerreed exclusiely on isenroic suraces while he ull, raher han aroximae, Erel oenial oriciy should be comued and inerreed exclusiely on isobaric suraces. Neerheless, he orm o oenial oriciy gien by he aroximae Erel oenial oriciy in (15) is ha which is mos oen considered in synoic-scale meeorology, largely because i is simler o comue han boh is ull orm on isobaric suraces and is isenroic oenial oriciy counerar on isenroic suraces. Alicaion o Isenroic Poenial Voriciy o Synoic Analysis Earlier, we saed ha lower alues o oenial emeraure on he dynamic rooause imly lower rooause heighs and uer roosheric roughing. Meanwhile, higher alues o oenial emeraure on he dynamic rooause imly higher rooause heighs and uer roosheric ridging. This is demonsraed grahically in Figure o Bluesein (Vol. II), reroduced below as Figure 4. I we ake he dynamic rooause o be he 1.5 PVU surace, we ind ha i is ound a relaiely low aliudes and on relaiely cold isenroic suraces near he oles. Conersely, i is ound a relaiely high aliudes and on relaiely warm isenroic suraces near he equaor. Inroducion o Isenroic Poenial Voriciy, Page 8

Figure 4. Verical cross-secions o zonally-aeraged, climaological mean oenial oriciy (dashed lines; unis: PVU) and oenial emeraure (solid lines; unis: K) or (a) January and (b) July.

9 Figure 4. Verical cross-secions o zonally-aeraged, climaological mean oenial oriciy (dashed lines; unis: PVU) and oenial emeraure (solid lines; unis: K) or (a) January and (b) July. Reroduced rom Synoic-Dynamic Meeorology in Midlaiudes (Vol. II) by H. Bluesein, heir Figure These conces can be iewed in ligh o hickness argumens. Aeraged oer he course o a year, inciden solar radiaion is greaes a he equaor and lowes a he oles. Corresondingly, he annually-aeraged, layer-mean roosheric emeraure is greaes a he equaor and lowes a he oles. This imlies relaiely low hicknesses a higher laiudes and relaiely high hicknesses a lower laiudes. Consider he case where he lower surace is aken o be he ground, which is a a consan aliude, and he uer surace is aken o be he rooause. Lower hicknesses hus imly a lower rooause heigh, as seen near he oles, while higher hicknesses imly a higher rooause heigh, as seen near he equaor. Figure also demonsraes how one isenroic surace can be in he sraoshere a one locaion and in he rooshere a anoher. Take, or examle, he 350 K isenroic surace. I is ound beween hpa a nearly all laiudes, aricularly in anel (a). In he roics, his is locaed in he rooshere, beneah he dynamic rooause. Howeer, he 350 K isenroic surace inersecs he dynamic rooause a around 30 N/30 S laiude, and a higher laiudes i is siuaed wihin he sraoshere. In Figures 2 and 3, we demonsraed he link beween oenial emeraure anomalies on he dynamic rooause and roughs and ridges on uer roosheric isobaric chars. Now, we demonsrae he link beween oenial oriciy on an isenroic surace and roughs and ridges Inroducion o Isenroic Poenial Voriciy, Page 9

Poleward exensions o he dynamic rooause relec uer roosheric ridges, he resence o anicyclonically-cured low, and relaiely warm emeraures. Figure 5.

10 on uer roosheric isobaric chars. This is illusraed by Figures 5 and 6. Equaorward exensions o he dynamic rooause relec uer roosheric roughs, he resence o cyclonically-cured low and relaiely cold emeraures. Poleward exensions o he dynamic rooause relec uer roosheric ridges, he resence o anicyclonically-cured low, and relaiely warm emeraures. Figure 5. Isenroic oenial oriciy (conours; eery 1 PVU) on he 325 K isenroic surace alid a (a) 1200 UTC 16 May 1989 and (b) 1200 UTC 17 May Reroduced rom Synoic-Dynamic Meeorology in Midlaiudes (Vol. II) by H. Bluesein, heir Figure Inroducion o Isenroic Poenial Voriciy, Page 10

11 Figure hpa heigh (solid conours eery 60 m), emeraure (dashed conours eery 5 C), and uer air obseraions (saion los o emeraure in C, dew oin deression in C, wind in k, and heigh in dam) a (a) 1200 UTC 16 May 1989 and (b) 1200 UTC 17 May Reroduced rom Synoic-Dynamic Meeorology in Midlaiudes (Vol. II) by H. Bluesein, heir Figure Inroducion o Isenroic Poenial Voriciy, Page 11

Lecture 2: Telegrapher Equations For Transmission Lines. Power Flow.

Lecture 2: Telegrapher Equations For Transmission Lines. Power Flow. Whies, EE 481/581 Lecure 2 Page 1 of 13 Lecure 2: Telegraher Equaions For Transmission Lines. Power Flow. Microsri is one mehod for making elecrical connecions in a microwae circui. I is consruced wih