Baroclinic flows can also support Rossby wave propagation. This is most easily

Transcription

1 17. Quasi-geostrohic Rossby waves Baroclinic flows can also suort Rossby wave roagation. This is most easily described using quasi-geostrohic theory. We begin by looking at the behavior of small erturbations to a zonal background flow that varies only in the meridional and vertical directions. Beginning with the definition of seudo-otential vorticity (9.11), we let ϕ and q be reresented by zonally invariant background fields lus erturbations to them: ϕ = ϕ(y, )+ ϕ (x, y,, t) q = q(y, )+ q (x, y,, t) (17.1) We next linearize the adiabatic, frictionless form of the conservation equation for q (9.10) using (17.1): q q q + u g + v g =0, (17.) t x y where 1 ϕ u g =, f y 1 ϕ v g =. f x We also note from the definition of seudo-otential vorticity (9.11), that q 1 ϕ f 0 ϕ = + β 0 + y f 0 y S y u f0 u = β. y S (17.3) (17.4) Thus the meridional gradient of background seudo-otential vorticity deends on β, the meridional gradient of the vorticity of the zonal wind, and a measure of the curvature of the vertical rofile of the mean zonal wind. 75

2 Using the second of the geostrohic relations in (17.3) as well as the definition of seudo-otential vorticity, (9.11), the linearized conservation relation (17.) may be written ( )[ ] 1 f 0 ϕ 1 q ϕ + u g ϕ + + t x f 0 S f 0 y x =0. (17.5) We will examine solutions of (17.5) in the secial case that the stratification is constant, S = constant. We will also assume that the background zonal wind, u g, and the associated gradient of background seudo-otential vorticity given by (17.4) are slowly varying comared to the structures of erturbations to the flow. If this is the case, we can make the W.K.B. aroximation and reresent modal solutions to (17.5) as ϕ =Φe ik(x ct)+i y l(y,)dy +i m(y, )d, (17.6) where l(y, ρ) and m(y, ) are slowly varying functions of latitude and ressure. Substituting (17.6) into (17.5) gives a disersion relation: c = u g q / y. 0 k + l + f m S (17.7) Comaring this to the strictly barotroic disersion relation (10.8) shows the strong similarity between barotroic and baroclinic waves. The main differences are that in the baroclinic case, the meridional gradient of otential (rather than actual) vorticity serves as the refractive index for Rossby waves, and the vertical structure contributes to the disersion roerties of the waves. 76

3 The wave frequency, which remains invariant along the ray aths followed by the wave energy as long as the background flow is considered to be steady, is given by k( q / y) ω = kc = ku g. 0 k + l + f m Letting k i (k, l, m), the three comonents of the grou velocity are given by where (17.8) [ ( ( )) ] 0 ω q y f 0 klqy km f q S k l y c gi = = u g + m,,, (17.9) k 4 S 4 4 i r r r q q y, y r k + l 0 + m. S It is of some interest to comare these grou velocities to the hase seeds, which are given by [( ) ] ω q y k q y k q y c r = u g,, k i r [( l r ) m r ] k q y 1 r 1 S r (17.10) = c gx, l c gy, f0 m c g r 4 Thus, for quasi-geostrohic Rossby waves, the flow-relative grou velocity in the meridional and vertical directions is of oosite sign from the hase seeds in those directions. As quasi-geostrohic Rossby waves diserse in three dimensions, the associated wave numbers evolve following the vector grou velocity, according to the relationshi for the refraction of wave energy: f S dk i dt ω =, (17.11) x i 77

4 where the total derivative indicates the rate of change following the grou velocity: dk i k i k i = + c gj. dt t x j Using (17.8), the evolution of wavenumber (17.11) is [ ( ) ( )] dk i u g q / y u g q / y = 0,k +,k +. (17.1) dt y r ρ r The interaction between Rossby waves and the background flow is of great interest, because in quasi-balanced flows these waves are resonsible for conveying information from one lace to another. An elegant way of quantifying the interaction between quasi-geostrohic Rossby waves and the background state on which they are assumed to roagate is through the examination of Eliassen-Palm fluxes. The Eliassen-Palm theory is derived as follows. Since quasi-geostrohic flow on an f lane is nondivergent, we may write the conservation equation for seudo-otential vorticity, (9.10), in the form q = (u g q ) (v g q ). (17.13) t x y Consider now the time rate of change of zonal mean seudo-otential vorticity. First define a zonal average oerator { }, such that for any scalar A, 1 L {A} Adx, (17.14) L 0 where L is the distance around a latitude circle. Alying this oerator to (17.13) gives {q } = {v g q }. (17.15) t y 78

5 Now let v g = {v g } + v g, q = {q } + q, where v is the local, instantaneous dearture of v g from {v g }, but since g 1 ϕ v g =, f 0 x {v g } = 0. Thus (17.15) becomes t {q } = {v y g q }. (17.16) The time rate of change of zonal mean seudo-otential vorticity is equal to the convergence of the meridional eddy flux of seudo-otential vorticity. Using the definitions of q and v g, 1 f 0 ϕ = ϕ +, f0 S q 1 ϕ =, f 0 x v g where ϕ is the dearture of ϕ from its zonal average, we can write [ ] 1 ϕ ϕ ϕ ϕ ϕ 1 ϕ v g q = + + f0 x x [ x y x S ] 1 1 ( ϕ ) ( ϕ ϕ ) 1 ( ϕ ) = f0 + (17.17) x x y x y x y ( ) ( ϕ 1 ϕ 1 ϕ ) +. x S S x Taking the zonal average of this gives { } { } 1 ϕ ϕ ϕ 1 ϕ {v g q } = f0 +. (17.18) y x y x S 79

6 Using the geostrohic relations and the hydrostatic relation (15.8), this may be written { } f 0 {vgq } = {ugv g } vgθ (17.19) y Sπ F, where F is the Eliassen-Palm flux, given by { } F {u v f 0 g g }ĵ v g θ, ˆ (17.0) Sπ with ĵ and ˆ unit vectors in y and. Thus the northward comonent of the Eliassen- Palm flux is the geostrohic northward eddy flux of zonal (geostrohic) momentum, while the vertical comonent of the EP flux is the geostrohic northward eddy heat flux. The utility of the Eliassen-Palm flux lies in its role as a source of wave activity. This is a measure of the variance of seudo-otential vorticity, and is defined as 1 q A. (17.1) qy We can form an equation for wave activity by multilying (17.), modified to take into account dissiation, by q and taking the zonal average of the result: Now letting 1 {q } + q {v q } = {Dq }. (17.) t y q q y, y 80

7 and noting that q y is not a function of time or longitude, divide (17.) through by q y : { } 1 { q q } + {v q } = D, t q y qy or using (17.1) and (17.19), A + F = D, (17.3) t where { } D D q. q y In the absence of dissiation of seudo-otential vorticity, the rate of change of wave activity is roortional to the divergence of the Eliassen-Palm flux. Conversely, in a steady flow, creation or dissiation of wave activity is signified by a nonzero divergence of the Eliassen-Palm flux. In the case of lane waves, the Eliassen-Palm flux may be interreted as the flux of wave activity along wave ray aths, traveling at the grou velocity. This is shown as follows: First, using the definitions of wave activity (17.1) and seudo-otential vorticity (9.11) and the modal decomosition (17.6), we have {[ } y 1 1 A = (k + l )+ f ] 0 m Φ e ik(x ct)+i ldy +i md q y f0 S [ ] 1 1 f0 = Φ (k + l )+ m e i y ldy +i 4qy f 0 S md. (17.4) 81

8 On the other hand, using (17.6) in the definition of the Eliassen-Palm flux vector, (17.0), together with the usual geostrohic and hydrostatic relations, gives [ ] [ y y ] 1 F = klφ e i ldy +i md 1 mk ĵ + Φ e i ldy +i md. ˆ (17.5) f 0 S Now comaring (17.5) to (17.4), and using the grou velocity relations (17.9) together with the definition of wave activity, (17.5), shows that F i = c gi A. (17.6) Thus, for individual lane waves, the Eliassen-Palm flux is just the roduct of the Rossby wave grou velocity and the wave activity. This does not hold for disturbances consisting of more than one lane wave, or nonmodal disturbances, as will be discussed in Chater (?). Later on, we will find that the Eliassen-Palm flux is very useful for diagnosing the sources and sinks of Rossby waves from atmosheric observations. 8

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