Vorticity. curvature: shear: fluid elements moving in a straight line but at different speeds. t 1 t 2. ATM60, Shu-Hua Chen

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1 Vorticity We hve previously discussed the ngulr velocity s mesure of rottion of body. This is suitble quntity for body tht retins its shpe but fluid cn distort nd we must consider two components to rottion: sher nd curvture. We introduce the term vorticity s mesure of rottion within fluid. sher: n t 1 t fluid elements moving in stright line but t different speeds. v Consider mrker in the fluid such s the dshed line crossing the stremlines. Initilly, the mrker might be norml to the stremlines, s t time t 1, but lter will be rotted to the position t t. The mrker line hs undergone n nticlockwise rottion (cyclonic in the N-hemisphere). We evlute the contribution to the dv vorticity s dn curvture: r v If, t rdil distnce r, the velocity is V, the curvtive component of vorticity is evluted s V/r in 1

2 the sme wy we define ngulr velocity. Convention: both components re ssessed the sme wy using the convention tht cyclonic (nticlockwise in the N- hemisphere) rottion is positive, nd nticyclonic rottion is negtive. The vorticity of volume of tmosphere, when evluted in this wy from the velocity of the ir reltive to the erth s surfce is clled reltive vorticity (Greek zet) Reltive vorticity v r dv dn If the tmosphere rotted like solid body, the two components would be equl nd thus the vorticity is equivlent to twice the ngulr velocity of solid object. To pply bsic lws of motion (such s the conservtion of ngulr momentum), we need to consider rottion in n bsolute frme of reference, nd must dd in the rottion due to tht of the erth itself. Twice the ngulr velocity of the erth bout locl verticl is, of course, equl to the Coriolis prmeter f sin, so we define bsolute vorticity s f On certin upper ir chrts (see, for exmple, the NGM 500 mb forecst mps), bsolute vorticity is plotted in units of -5 s -1. Thus, in mid ltitudes, where -1-5 s f, region of zero reltive vorticity would hve n bsolute vorticity of units. Centers of mximum vorticity re useful in identifying trough loctions, which re sometimes difficult to identify, otherwise, especilly since the sher component is difficult to ssess. As we ll see lter the reltionship between the flow field nd the vorticity pttern yields n importnt forecsting tool.

3 Constnt bsolute vorticity trjectories It hs been shown tht, to resonble pproximtion, the tmosphere moves in such mnner s to conserve its bsolute vorticity. Tht is, f consttnt Atmospheric trjectories with constnt bsolute vorticity execute sinusoidl-like pths round the hemisphere. Remember tht the Coriolis prmeter f (twice the ngulr velocity of the erth bout locl verticl) increses towrds the pole ( f sin ). If f increses s mss of ir moves northwrd, then must decrese, nd vice vers. The result is wvy trjectory round the hemisphere s shown below: N f decreses increses f increses decreses To understnd this, remember tht positive is cyclonic (nticlockwise) rottion, while negtive is nticyclonic (clockwise) motion, nd cycles between positive nd negtive vlues s the ir executes sine wve. 3

4 Rossby wves These lrge wve-like perturbtions observed in the mid-ltitude westerly flow re clled Rossby wves fter the Swedish meteorologist (who founded the first meteorology deprtment in the US t MIT in 198). He ws the first to recognize the importnce of such disturbnces to the globl circultion pttern. Anlysis of Rossby wves results in the following expression for the speed t which the wves propgte towrds the est: Rossby wve speed c v where V is the wind speed t mid-tropospheric levels (specificlly t the level of non-divergence ND), is the wvelength of the Rossby wve cos where 7.9 rdisus 6378 ltitude of km 5 ngle rd / s erth V wind speed C speed of propgtion of the wve

5 Since the lst term in the eqution is lwys positive, the wve speed C must be smller thn the wind speed. Thus, the ir psses through the troughs nd ridges t speed greter thn the pttern itself propgtes. Notice tht the shorter the wvelength, the lrger the wve speed C. The longer the wvelength, the smller the wve speed. ong wves seen on upper ir mps thus move slowly while shorter wves ripple through them. At ny one time, the globl circultion pttern is summtion of wves of different length in different stges of being in or out of phse. From the eqution, one cn see tht sufficiently lrge wve could hve negtive speed of propgtion (C < 0 if V ) nd the wve will retrogrde (move towrds the west). Exmple clcultions of wve speed C: Suppose V = 30 m/s, 5 o N cos ()(7.9 ( )cos 5 ) for wvelength = 000 km C = 30 (0.05 x -1 )( x 6 ) = = 3.6 m/s for wvelength = 8000 km C = 30 (0.05 x -1 )(8 x 6 ) = =.1 m/s Thus, the longer wve propgtes towrds the est t much lower rte thn the short wve. 5

6 Divergence nd convergence In our numerous discussions of current nd prognostic wether ptterns, we hve spoken of divergence nd convergence in the horizontl fields of motion t different levels within the troposphere. Anlysis of wve-like disturbnces (Rossby wves) shows tht the rte of chnge of vorticity reltes to fields of divergence nd convergence, s shown in the following digrm: V Tropopuse V C wve speed z C level of non-divergence mb 0 speed The curve lbeled V represents the profile of wind speed through the troposphere, reching mximum t the tropopuse (the jet strem). When V C s in the upper portion of the troposphere, convergence occurs in the horizontl wind field upwind of the trough line (west of the trough) nd divergence occurs down wind of the trough. In the lower prt of the troposphere ( V C ), the opposite is the cse. Compenstory verticl motions occur with scent hed of the trough (downwind) nd subsidence behind the trough, together with the cretion of res of low nd high pressure t the surfce, s shown. On wether mps which show upper ir flow ptterns, wy to identify divergence nd convergence is to exmine the reltionship between height contours (sy of 500 mb) nd clculted vorticity vlues (such s plotted on the NGM forecst 500 mb mps) 6

7 NVA X PVA 570 As the ir flows through the trough nd moves downstrem, it loses vorticity nd is forced to diverge horizontlly (recll tht this is the sme reltionship s dictted by the conservtion of ngulr momentum). Where the boxes formed by the crossing of the contours nd the vorticity isopleths re smllest, divergence (or convergence on the upwind side of the trough) is gretest. Alterntively, s ir moves downstrem hed of the trough, we cn sy tht positive vlues of vorticity re dvected by the ir prcels crrying the positive vorticity cquired t the trough line. The downstrem re is sid to be one of positive vorticity dvection (PVA), while the upstrem region is one of negtive vorticity dvection (NVA). PVA is ssocited with upper ir divergence, nd NVA is ssocited with upper ir convergence. 7

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