ATMS 310 The Vorticity Equation. The Vorticity Equation describes the factors that can alter the magnitude of the absolute vorticity with time.

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1 ATMS 30 The Vorici Eqaion The Vorici Eqaion describes he acors ha can aler he magnide o he absole orici ih ime. Vorici Eqaion in Caresian Coordinaes The (,,,) orm is deried rom he rimiie horional eqaions o moion: F r = α () F r = α (2) (a) (b) (c) (d) (e) here: a) local rae o change o he meridional or onal eloci comonens b) adecion erms c) ressre gradien orce d) Coriolis orce e) Fricion orce The deriaion ill no be carried o here. The mehod is as ollos: ) Take o Eqaion () 2) Take o Eqaion (2) 3) Sbrac he resl o se 2 rom he resl o se Aer se o he rodc rle, simliicaions, and cancellaions, he resl is he orici eqaion in (,,,) coordinaes: ( ) = (3) (a) (b) (c) (d) (e) () (g) F F r r 2 (h) (i) The erms o he orici eqaion are deined as ollos:

2 a) Rae o change o relaie orici a a grid oin b) Zonal adecion o relaie orici c) Meridional adecion o relaie orici Noe ha i erms b) and c) increase he orici, e call ha Posiie Vorici Adecion or PVA. I he ork o decrease he orici, i is called Negaie Vorici Adecion or NVA. d) Verical ranser o relaie orici e) Adecion o lanear orici Noe ha lanear orici is adeced b he meridional comonen o he ind, since lanear orici onl changes in he meridional direcion (ncion o laide) Air moing norhard rom he Eqaor ill enconer increasing lanear orici, casing a cre o he righ or more ani-cclonic moion. This ill decrease he local rae o change o orici since ani-cclonic sin is negaie orici. Sohard moing air ill enconer decreasing lanear orici, casing he air o moe more o he le (cclonicall, increasing he rae o change o orici). ) Diergence Term I he horional ind diergence > 0, he change in orici ill be donard. I here is horional ind conergence, ha ill imar more osiie orici. g) Tiling Terms Esseniall his erm reresens he change in erical eloci in he horional direcion. I air is rising aser in one area han anoher, his imars a iling eec, hich creaes sin in he amoshere h) Fricion erms i) Solenoidal Terms Reresens he eecs o he ressre gradien acceleraion. I he ressre gradien aries in sch a a o rodce clockise roaion, his means a negaie change in he relaie orici (and ice ersa). Holon combines he arial erms in Eq. (3), does no consider he ricion erms, and sbsies or he (d/d) erm o ield Eq. 4.7 on. 0: D D ( ) = ( ) 2 (4)

3 Vorici Eqaion in Isobaric Coordinaes The ses o derie he orici eqaion in isobaric coordinaes are he same as Caresian, ece one begins ih he isobaric eqaions o moion. The resling eqaion is: = ω ω ω F r r ( ) All erms remain he same EXCEPT he solenoidal erms dro o. Recall ha he solenoidal erms reresened he eecs o he ressre gradien orce on he orici. For an isobaric srace, here are no ressre gradiens. Ths, here canno be a solenoidal erm. Alicaion o he Vorici Theorem In 988, he olloing aer as blished in Monhl Weaher Reie, a roessional blicaion o b he American Meeorological Socie: MacDonald, B.C. and E.R. Reier, 988: Elosie cclogenesis oer Easern U.S., Monhl Weaher Reie, Vol. 6, The eamined 20 eraroical cclones 0 deeloed normall, and 0 ere caegoried as raidl deeening (aka BOMBS ), deined as a srace ressre all o a leas 24 mb oer 24 hors. The calclaed he erms o he orici heorem o r o ind he imoran conribors o he raidl deeening sorms. Reglar cclones had: ) Diergence erm rodced signiican conribions o he orici onl in he lanear bondar laer (high conergence -> more orici) 2) Verical orici adecion no imoran 3) PVA hrogho mos o he rooshere 4) Lile change in orici a all leels as sorm moes rom inciien sage o he mare sage. Bombs had: ) High osiie and negaie ales o he diergence erm hrogho he enire rooshere, ih a shar reersal o sign near he 500 mb leel 2) Verical orici adecion osiie (ard) hrogho he rooshere 3) PVA large onl in he er rooshere 4) A dramaic increase in orici a all leels as sorm moed rom is inciien o mare hase F

4 The conclde ha bomb generaion deends criicall on he generaion o osiie orici b he conergen lo in he loer rooshere. Sch a rocess necessiaes he reeisence o amosheric olmes conaining signiican amons o osiie orici hich can be dran ino he region o inciien cclogenesis. Scale Analsis o he Vorici Eqaion We can eamine he relaie magnides o each erm in he orici eqaion hrogh a scale analsis. Firs, i can be shon ha he relaie orici ( ) is oen small comared o he absole orici: / 0 = s m s m L U 4 0 s Ths, ma be negleced in he diergence erm o he orici eqaion: ( ) Noe ha near he cener o inense cclonic sorms, aroaches 0-4 and hs e canno no make his simliicaion since i is o similar magnide o. Similar scale analses can be erormed on he oher erms o he orici eqaion sing ical mid-laide snoic-scale ales gien on. 04 o Holon: = All nis o hese erms are s -2. I e reain onl he erms ha are 0-0 are greaer, e ge a orm o he orici eqaion alid or snoic-scale moions: ( ) = D D (5)

5 For inense cclonic sorms, he relaie orici ms be added back in o he diergence erm on he righ hand side o Eq. (5). Wha does Eqaion 5 sa in ords? Basicall, The change in absole orici olloing he horional moion on he snoic scale is gien b he concenraion or dilion o lanear orici cased b he conergence or diergence o he horional lo. Diergence along he lo ill loer he absole orici o he lo. Conergence along he lo ill increase he absole orici o he lo. Eqaion 5 also hels o elain h cclones are mch more inense han ani-cclones. I here is conergence, his ill increase, hich ill increase he relaie orici, hich leads o een more cclonic orici. I here is diergence, his ill decrease. Eenall, he relaie orici ill become negaie and eacl oosie o he lanear orici (). From hen on, he absole orici ill no change, no maer ho mch more diergence occrs in he lo.

Vorticity equation 2. Why did Charney call it PV?

Vorticity equation 2. Why did Charney call it PV? Vorici eqaion Wh i Charne call i PV? The Vorici Eqaion Wan o nersan he rocesses ha roce changes in orici. So erie an eression ha incles he ime eriaie o orici: Sm o orces in irecion Recall ha he momenm