The Vorticity Equation

Transcription

1 The Vorticit Eqation

2 Potential orticit Circlation theorem is reall good Circlation theorem imlies a consered qantit dp dt 0 P g 2

3 PV or barotroic lid General orm o Ertel s otential orticit: P g const Consider isentroic lo beteen to isentroic sraces For homogeneos incomressible lid, densit is constant So the horiontal area is inersel roortional to the lid deth A M const h h For a homogeneos, incomressible lid, P h const Where h ~ / 3

4 Westerl Flo Oer a Barrier Consider a esterl lo o air encontering a north-soth montain barrier. Ustream o the barrier, assme the lo is onal and niorm, ths = 0. P h const A otential temeratre srace near the grond aroimatel ollos the terrain. Pressre orces cased b the interaction o the lo ith the montain barrier case the ertical dislacement o an er-leel isentroic srace to be more sread ot horiontall (i.e., greater idth and smaller amlitde). Region 1: Colmn stretches; > 0. Region 2: Increase o latitde and redction in colmn deth; < 0. Region 3: Decrease o latitde and increase in colmn deth; > 0. Region 4: Sothard dislacement and decrease in colmn deth; < 0. The cclonic lo donstream o a montain barrier is knon as a lee-side montain trogh, or lee trogh. 4

5 Stationar (toograhicall orced) aes NCEP Reanalsis Z500 Janar mean 5

6 NCEP Reanalsis Z500 Deiations rom onal mean Janar mean 6

7 NCEP Reanalsis Srace eleation 7

8 What s ith orticit and aes

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22 The Vorticit Eqation Want to nderstand the rocesses that rodce changes in orticit. So derie an eression that incldes the time deriatie o orticit: Recall that the momentm eqations d dt d dt d dt d dt Ths e ill begin or deriation b taking Sm o orces in direction Sm o orces in direction d dt d d d d momentm eqation momentm eqation Eqialentl, e cold se the ector orm, and comte the crl o the ector momentm eqation d dv k dt dt 22

23 23 Vorticit eqation t 1 momentm eqation momentm eqation d d d d dt d t 1 We ill ork in Cartesian coordinates The addition terms or sherical earth come ot more natrall in ector orm t t 1 1

24 24 t t t 2 1 t 2 1 dt d t dt d Vorticit eqation (contined) Ths the orticit eqation,

25 25 Terms in orticit eqation dt d 2 1 A: Rate o change o absolte orticit olloing the lid motion B: Eect o horiontal elocit diergence on orticit C: Transer o orticit beteen horiontal and ertical comonents ( tisting term or tilting term ) D: Eects o baroclinicit ( solenoidal term ) A B C D For ressre coordinates, solenoidal term disaears d dt d V k V V In ractice, solenoidal qite term is small in height coordinates too

26 Absolte orticit (term A) Eanding, the rate o change o absolte orticit is ritten d dt t local tendenc o absolte orticit horiontal adection o absolte orticit ertical adection o absolte orticit = 2 sin. Being indeendent o and, e can rite d dt t Absolte orticit change de to (3d) adection o relatie orticit, and meridional adection o lanetar orticit 26

27 Vorticit diergence (term B) Eect o horiontal elocit diergence on orticit Diergence 0 Vorticit ill decrease i absolte orticit is ositie Vorticit ill increase i absolte orticit is negatie. Conergence 0 Vorticit ill increase i absolte orticit is ositie Vorticit ill decrease i absolte orticit is negatie. Consider orticit o a region being sqished into a smaller area. This mechanism is qite imortant or large-scale midlatitde 27

28 Tilting (term C) Transer o orticit beteen horiontal and ertical comonents Vertical shear in o ind is gies shear orticit abot an east-est ais. (consider addle rotating arond the solid orticit ector here is ero) 0 East-est ariations in the ertical elocit tilts the ector to be more ertical (dashed orticit ector). This ne ector has a ertical comonent, and ths is non ero. 0 d 0 0 dt 28

29 Solenoidal term (term D) Vorticit generation de to baroclinic strctre (densit nction o ressre and temeratre) 1 2 For a niorm ressre gradient, horiontal ariations densit means there is a non-niorm acceleration de to the ressre gradient orce d( ) PGF () = ( ) d Variations in acceleration rodce orticit d/dt ~ d(pgf )/d > 3 > 2 > > 3 > 2 > 1 lo) d 0; 0 0 This can occr in a baroclinic atmoshere, bt is absent in a barotroic atmoshere (or isentroic dt 29

30 What is a solenoid? From electrostatics, consider orce on a ire ith moing charge in a magnetic ield. Seciicall, the electron motion is de to the gradient in magnetic ield otential. For a lid, the same terminolog is sed to describe lo that is non-diergent (i.e., solenoidal). The elocit is de to a gradient in geootential ield. Forces can eist here there are densit dierences along the geootential srace 30

31 31 Vorticit eqation t 2 1 Change in relatie (ertical comonent o) orticit at a oint, Adection o relatie orticit Adection o lanetar orticit ( beta eect ) Conergence o absolte orticit Tilting o horiontal orte tbes Horiontal ariations in horiontal accelerations (baroclinic, or solenoidal term)

32 Field tri - assignment Measrements at East Bolder Commnit Park 11-12:15 Set at am. Some helers old be great! RTD 203 direct Broada to cams. Leaing Aro eer 13 mintes to cams (Broada Eclid) Bring: notebook to track take ield notes and log balloon theodolite angles. A atch ill be sel. 32

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35 55 th street. Soth rom Baseline Soccer ields Parking Meet here

36 Problems Holton