Comments on Vertical Vorticity Advection

Transcription

1 Comments on Vertical Vorticity Advection It shold be fairly intitive that ositive maima in vertical vorticity are associated with cyclones, and ths ositive cyclonic vorticity advection might be a sefl forecast tool. NOTE however that the evoltion of synotic-scale meteorological systems is governed by more than jst vertical vorticity advection. ather, its the change in the vertical shear of the horizontal wind associated (i.e., forced by) with differential [i.e., f(z)] vorticity advection that drives an ageostrohic vertical circlation which adiabatically adjsts the horizontal temeratre gradients in order to maintain thermal wind balance. The divergence/convergence associated with the vertical circlation imacts the vorticity distribtion throghot the trooshere (throgh stretching) -- not jst aloft. The vertical circlation that reslts from QG differential CVA tends to be an order of magnitde larger than that indced via bondary layer circlations (and ths we neglect the Ekman generated vertical motion). Derivation of the QG height tendency eqation GOAL: We want to eliminate ω in the QGVE and QGTE Note that if we can do this, then as Holton says, we can determine the characteristics (i.e. the evolving height field) withot any knowledge of ω! FIST: ewrite or QG vorticity and thermodynamic eqations so that they are a fnction of Φ and ω only. We IGNOE the diabatic heating term in the thermo eqation (althogh it may be imortant!). From or QG thermo eqn... T T t g T g σ ω y - Q c neglect QGTE Using the following in the QGTE χ Φ --, and Φ -- t T - T -- Φ --

2 We have: -- T T t g T g σω y -- t g g y -- Φ -- σω and, t g g Φ -- y σω Φ -- Vg Φ -- t σω χ * What haened to the ressre, i.e. why no derivatives of? ewriting the eqation above we have χ Vg Φ -- σω 1 So how/why does the vertical derivative of the geootential tendency change? Or how does the thickness change? [ecall that χ ( Φ t) t( Φ ) ] What is term 1? Here are a cole of ways to look at it with the oint being that yo sholdn t lose site of the basic relationshi between thickness and temeratre! First consider a cold advection attern (-z section): We can always go back to temeratre here to show the same thing, i.e. χ at t Φ+ Φ Φ g Φ -- g > Φ -- ( ) g Φ Φ COLD (smaller thickness) lower heights -- Φ Vg Φ -- Φ -- t ( T) Vg Φ -- t > < (large -) < (small -) -- Φ Φ [-- ( + ) -- ( ) ] < from above time Φ -- ( + ) ( T) t + + WAM (greater thickness) higher heights term 1 > χ > at Φ -- T < if CAA T n + 1 T n 1 t + HOT > (T n-1 > T n+1 ) n-1 n n+1

3 Now consider a region of warm advection (where I have jst reversed the temeratre/thickness gradient given in the CAA eamle above), this time let s look at it in (-y) lane χ Φ -- t Vg Φ -- ** From the figre below we see that at the bo in the center of the domain, we are increasing the thickness (between two ressre srfaces) and therefore the trend (or tendency) in Φ Φ( 1 ) Φ( ) is to become more negative in time! It s like shing aart the ressre srfaces, e.g. the figre below. y large thickness WAM g small thickness COLD Φ -- gz Φ warm advection increasing thickness time 1 In the eqation (**) above, we note that for warm adv. the wind has a comonent in the direction oosite thickness (temeratre) gradient. Vg Φ V T g -- T g > T >< t WAA Note that the thickness gradient is oosite Φ -- becase Φ -- ( T)!!!!!!! What is term? The second term, which involves ω, yields χ >, if ω < (rising motion), and χ <, if ω > (sinking motion) Hence, rising motion is associated with decreasing thickness (cooling), while sinking motion is associated with increasing thickness (warming). As yo will see later, this resonse (i.e. the verti-

4 cal motion is considered a resonse to the forcing ) is eactly what is needed to maintain hydrostatic balance (the QG atmoshere is hydrostatic) in the resence of height rises/falls. ecall we set ot to eliminate ω in the QGVE and QGTE, we have a modified eqation for the QGTE above (*). We can do something similar for the QGVE Alying t g g y 1 ζ g --- Φ f ζ g ω f - βv g We have... t g g y f Φ ω f - βv g earranging Φ -- t f V 1 g f --- Φ ω f βv g + f - χ 1 Bt term above is jst f V g f, therefore χ f V 1 g --- Φ + f f + f ω - This eqation says that the Lalacian of the geootential tendency is eqal to the advection of absolte (geostrohic) vorticity by the geostrohic wind ls the vorticity generation de to the stretching (divergence) of the ageostrohic wind. However we still have 3 nknowns (χ, Φ, ω and two eqations...so still want to get rid one of the variables (ω)! Note that ω (re geostrohic motion) is a soltion to Holton eqations 6.1 and 6. for the secial cases of barotroic flow (no -deendence) zonally symmetric flow (no deendence) Ine of these two conditions are not satisfied, setting ω for 6.1 and 6. yields two indeendent eqations for χχ(φ) which is overdetermined (i.e. yo d get two searate soltions for χ!).

5 Conseqently the vertical motion lays a critical role in or QG system by coling the two eqations. We re almost there now... We now mltily Holton ~6.13b by and then differentiate w.r.t.? (Pressre - shold be obvios since we are trying to eliminate ω!) σ χ ( σ) Φ Vg -- σω --- χ σ ---Vg σ Φ -- f oω And then differentiating w.r.t. ressre... f --- oχ σ ---Vg Φ -- f σ oω f o σ --- χ f ---Vg o Φ -- ω f σ o - Now we can eliminate ω by combining (adding) or two new eqations (hybrid QGVE and a differentiated form of the QGTE), f --- oχ + σ χ f ---Vg o Φ -- ω f σ o - f V 1 g f --- Φ + f + f ω - We do this tye of thing all the time - and have to - in order to eliminate nknowns in many or systems of eqations (we then tyically go back and solve for the eliminated qantities once other qantities are known - hint!). Anyway, rearranging things above, we have... Geootential Tendency Eqation A B f --- o + χ σ f absolte vorticity advection ---Vg o Φ -- σ f V 1 g f --- Φ + f C differential thickness advection