Vorticity equation 2. Why did Charney call it PV?

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1 Vorici eqaion Wh i Charne call i PV?

4 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 eqaions Sm o orces in irecion Ths e ill begin or eriaion b aking ( momenm eqaion) ( momenm eqaion) Eqialenl, e col se he ecor orm, an come he crl o he ecor momenm eqaion V k

5 Vorici eqaion ( ) ( ) momenm eqaion momenm eqaion We ill ork in Caresian coorinaes The aiion erms or sherical earh come o more narall in ecor orm ( ) ( ) - Vorici eqaion (conine) Ths he orici eqaion,

6 Phsical iniion? ( ) ( ) Terms in orici eqaion ( ) ( ) A: Rae o change o absole orici olloing he li moion B: Eec o horional eloci iergence on orici C: Transer o orici beeen horional an erical comonens ( ising erm or iling erm ) D: Eecs o baroclinici ( solenoial erm ) A B C D For ressre coorinaes, solenoial erm isaears ( ) ( ) ω ω V k V V In racice, solenoial qie erm is small in heigh coorinaes oo

7 ( ) ( ) ( ) ( ) ( ) Absole orici (erm A) Eaning, he rae o change o absole orici is rien local enenc o absole orici horional aecion o absole orici erical aecion o absole orici Ω sinφ. Being ineenen o an, e can rie ( ) ( ) Absole orici change e o (3) aecion o relaie orici, an meriional aecion o lanear orici

Eec o horional eloci iergence on orici Diergence Sreching/iergence (erm B) > 0 ( ) Vorici ill ecrease i absole orici is osiie Vorici ill increase i absole orici is negaie.

8 Eec o horional eloci iergence on orici Diergence Sreching/iergence (erm B) > 0 ( ) Vorici ill ecrease i absole orici is osiie Vorici ill increase i absole orici is negaie. Conergence < 0 Vorici ill increase i absole orici is osiie Vorici ill ecrease i absole orici is negaie. Consier orici o a region being sqishe ino a smaller area. This mechanism is qie imoran or large-scale milaie ssems. Tiling (erm C) Transer o orici beeen horional an erical comonens Verical shear in o in is gies shear orici abo an eas-es ais. (consier ale roaing aron he soli orici ecor here is ero) > 0 Eas-es ariaions in he erical eloci ils he ecor o be more erical (ashe orici ecor). This ne ecor has a erical comonen, an hs is non ero. < 0 ( ) < 0 > 0

9 Vorici generaion e o baroclinic srcre (ensi ncion o ressre an emerare) > 3 > > Solenoial erm (erm D) 4 > 3 > > 0 ; 0 ( < < ) > 0 For a niorm ressre graien, horional ariaions ensi means here is a non-niorm acceleraion e o he ressre graien orce. ( ) PGF () ( ) Variaions in acceleraion roce orici / ~ (PGF )/. This can occr in a baroclinic amoshere, b is absen in a baroroic amoshere (or isenroic lo) For r aiabaic lo Thogh eerimens The solenoi erm isaears in ressre coorinaes. Wh? Pressre ariaions along ressre sraces is ero The solinoial AND iling erm isaears in isenoic coorinaes. Wh? Aiabaic coniion, means no erical moion so cerainl no sheer in i! Here he erical eloci is no θ θ & θ

10 Recall orici aecion Aecion o relaie orici: ku(kl) A cosk cosl Aecion o lanear orici: -bk A cosk cosl The aecion o relaie orici ominaes or shor aes Aecion o lanear orici ominaes or small aes (hs conirming ha as seen in he rojec, small aes moe sloer hen here is onl aecion) Gien a esar mean lo, eec, shor aes (iniial los, ec) en o moe easar, hile longer aes en o moe esar (in roacie long ae linke o oograh an lan se conras, o en no o moe er mch a all) This can be all seen ih he barooic moel

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

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