 = q17. Quasi-geostrophic Rossby waves Baroclinic flows can also support Rossby wave propagation. This is most easily described using quasi-geostrophic theory. We begin by looking at the behavior of small perturbations to a zonal background flow that varies only in the meridional and vertical directions. Beginning with the definition of pseudo-potential vorticity (9.11), we let ϕ and qp be represented by zonally invariant background fields plus perturbations to them: ϕ = ϕ(y, p)+ ϕ(x, y, p, t) (17.1) qpp(y, p)+ qp (x, y, p, t) We next linearize the adiabatic, frictionless form of the conservation equation for qp (9.10) using (17.1): ∂qp  ∂qp∂qp  + ug + vg =0, (17.2)∂t ∂x ∂y where 1 ∂ϕ ug = − ,f ∂y (17.3)1 ∂ϕ .v = g f ∂x We also note from the definition of pseudo-potential vorticity (9.11), that ∂qp 1 ∂ϕ ∂f0 ∂ ∂ϕ = ∇2 + β0 + ∂y f0 ∂y ∂p S ∂p ∂y (17.4)∂2u ∂f2 ∂u = β − − 0 . ∂y2 ∂p S ∂p Thus the meridional gradient of background pseudo-potential vorticity depends on β, the meridional gradient of the vorticity of the zonal wind, and a measure of the curvature of the vertical profile of the mean zonal wind. 75     Using the second of the geostrophic relations in (17.3) as well as the definition of pseudo-potential vorticity, (9.11), the linearized conservation relation (17.2) may be written ∂ ∂ 1 ∂ f0 ∂ϕ 1 ∂q∂ϕ + ug ∇2ϕ + + p =0. (17.5)∂t ∂x f0 ∂p S ∂p f0 ∂y ∂x We will examine solutions of (17.5) in the special case that the stratification is constant, S = constant. We will also assume that the background zonal wind, ug, and the associated gradient of background pseudo-potential vorticity given by (17.4) are slowly varying compared to the structures of perturbations to the flow. If this is the case, we can make the W.K.B. approximation and represent modal solutions to (17.5) as ϕ =Φe ik(x−ct)+i y l(y � ,p)dy� +i p m(y,p �)dp� , (17.6) where l(y, ρ)and m(y, p) are slowly varying functions of latitude and pressure. Substituting (17.6) into (17.5) gives a dispersion relation: c = ug − ∂qp/∂y (17.7). 20k2 + l2 + f2mS Comparing this to the strictly barotropic dispersion relation (10.8) shows the strong similarity between barotropic and baroclinic waves. The main differences are that in the baroclinic case, the meridional gradient of potential (rather than actual) vorticity serves as the refractive index for Rossby waves, and the vertical structure contributes to the dispersion properties of the waves. 76        The wave frequency, which remains invariant along the ray paths followed by the wave energy as long as the background flow is considered to be steady, is given by k(∂qp/∂y) (17.8) ω = kc = kug − . 20k2 + l2 + f2mS Letting ki ≡ (k, l, m), the three components of the group velocity are given by 20∂ω  qpy  f02 2  2klqpy 2kmfq Sk2 − l2 − py , (17.9)= = ug + m cgi , , 4 4 4S ∂ki r r r where ∂qpqpy ≡ ,∂y f2 r 2 ≡ k2 + l2 + 0 m 2 . S It is of some interest to compare these group velocities to the phase speeds, which are given by ωqpy kqpy kqpycr ≡ = ug − , − , − ki r2 l r2 m r2 k2qpy 1 r2 1 S r2 (17.10) = cgx − 2 r4 , −2 l2 cgy , −2 f02 m2 cgp Thus, for quasi-geostrophic Rossby waves, the flow-relative group velocity in the meridional and vertical directions is of opposite sign from the phase speeds in those directions. As quasi-geostrophic Rossby waves disperse in three dimensions, the associated wave numbers evolve following the vector group velocity, according to the relation-ship for the refraction of wave energy: dki ∂ω = − , (17.11)dt ∂xi 77where the total derivative indicates the rate of change following the group velocity: dki ∂ki ∂ki = + cgj . dt ∂t ∂xj Using (17.8), the evolution of wavenumber (17.11) is     dki ∂ug ∂2qp/∂y2 ∂ug ∂2qp/∂y∂p = 0,k − + ,k − + . (17.12)dt ∂y r2 ∂ρ r2 The interaction between Rossby waves and the background flow is of great interest, because in quasi-balanced flows these waves are responsible for conveying information from one place to another. An elegant way of quantifying the interaction between quasi-geostrophic Rossby waves and the background state on which they are assumed to propagate is through the examination of Eliassen-Palm fluxes. The Eliassen-Palm theory is derived as follows. Since quasi-geostrophic flow on an f plane is nondivergent, we may write the conservation equation for pseudo-potential vorticity, (9.10), in the form ∂qp ∂ ∂ = − (ugqp) − (vgqp). (17.13)∂t ∂x ∂yConsider now the time rate of change of zonal mean pseudo-potential vorticity. First define a zonal average operator {}, such that for any scalar A, 1  L {A}≡ Adx, (17.14)L 0 where L is the distance around a latitude circle. Applying this operator to (17.13) gives ∂ ∂ {qp} = −{vgqp}. (17.15)∂t ∂y78           Now let vg = {vg} + vg , qp = {qp} + qp , where v is the local, instantaneous departure of vg from {vg }, but since g 1 ∂ϕ vg = ,f0 ∂x{vg} = 0. Thus (17.15) becomes ∂ ∂ ∂t{qp} = − ∂y{vg qp }. (17.16) The time rate of change of zonal mean pseudo-potential vorticity is equal to the convergence of the meridional eddy flux of pseudo-potential vorticity. Using the definitions of qp and vg , 1 ∂f0 ∂ϕ qp  = f0 ∇2ϕ + ∂p S ∂p , 1 ∂ϕ vg  = ,f0 ∂x where ϕ is the departure of ϕ from its zonal average, we can write 1 ∂ϕ ∂2ϕ ∂ϕ ∂2ϕ ∂ϕ ∂ 1 ∂ϕ v  q  = ++g p f2 ∂x ∂x2 ∂x ∂y2 ∂x ∂p S ∂p0 1 1 ∂  ∂ϕ 2 ∂  ∂ϕ ∂ϕ  1 ∂  ∂ϕ 2 = f02 2 ∂x ∂x + ∂y ∂x ∂y − 2 ∂x ∂y (17.17) ∂ ∂ϕ 1 ∂ϕ 1 ∂ ∂ϕ 2 + − . ∂p ∂x S ∂p 2S ∂x ∂p Taking the zonal average of this gives 1 ∂ ∂ϕ ∂ϕ ∂ ∂ϕ 1 ∂ϕ g p{v  q  } = f02 ∂y ∂x ∂y + ∂p ∂x S ∂p . (17.18) 79    Using the geostrophic relations and the hydrostatic relation (15.8), this may be written   ∂   ∂f0 {v q } = −{u v }−