16. Rossby waves We have seen that the existence of potential vorticity gradients supports the prop-agation of a special class of waves known as Rossby waves. These waves are the principal means by which information is transmitted through quasi-balanced flows and it is therefore fitting to examine their properties in greater depth. We begin by looking at the classical problem of barotropic Rossby wave propagation on a sphere and continue with quasi-geostrophic Rossby waves in three dimensions. a. Barotropic Rossby waves on a sphere The vorticity equation for barotropic disturbances to fluid at rest on a rotating sphere is dη dt =0, (16.1) where η ≡ 2Ω sin ϕ + ζ. Here ζ is the relative vorticity in the z direction. Now the equation of mass conti-nuity for two-dimensional motion on a sphere may be written 1 ∂u ∂  + (v cos ϕ) =0, (16.2) a ∂λ ∂ϕ71where u and v are the eastward and northward velocity components, λ and ϕ are longitude and latitude, and a is the (mean) radius of the earth. Using (16.2) we may define a velocity streamfunction ψ such that 1 ∂ψ u = − , a ∂ϕand (16.3) 1 ∂ψ v = . a cos ϕ ∂λ The Eulerian expansion of (16.1) can be written ∂η u∂η v∂η + + =0,∂t a cos ϕ ∂λ a ∂ϕ or using (16.3), ∂η 1 ∂ψ ∂η ∂ψ ∂η  + − =0. (16.4)∂t a2 cos ϕ ∂λ ∂ϕ ∂ϕ ∂λ We next linearize (16.4) about the resting state (u = v =0), forwhich η =2Ω sin ϕ, giving ∂η 2Ω ∂ψ + 2 =0, (16.5)∂t a ∂λ where the primes denote departures from the basic state. 72   In spherical coordinates, η = ζ = kˆ·∇×V 1 ∂2ψ ∂ ∂ψ (16.6) = +cos ϕ cos ϕ . a2 cos2 ϕ ∂λ2 ∂ϕ ∂ϕ Let’s look for modal solutions of the form ψ =Ψ(ϕ)e im(λ−σt), where m is the zonal wavenumber and σ is an angular phase speed. Using this and (16.6) in (16.5) gives d2Ψ dΨ 2Ω m2  − tan ϕ − + Ψ=0. (16.7)dϕ2 dϕ σ cos2 ϕ This can be transformed into canonical form by transforming the independent vari-able using µ ≡ sin ϕ, yielding d2Ψ dΨ 2Ω m2  (1 − µ 2) − 2µ − + Ψ=0. (16.8)dµ2 dµ σ 1 − µ2 The only solutions of (16.8) that are bounded at the poles (µ = ±1) have the form Ψ= AP n , (16.9)m73Table 16.1. Meridional Structure of Pnm(ϕ) Rossby Waves on a Sphere m 0 1 2 3 1sin ϕ cos ϕ – – n 2 1 2 (3 sin2 ϕ − 1) −3sin ϕ cos ϕ 3cos2 ϕ – 3 3 2 sin ϕ(5 sin2 ϕ − 3) − 9 2 (5 sin2 ϕ − 1) cos ϕ 45 sin ϕ cos3 ϕ −45 cos3 ϕ where P nmis an associated Legendre function of degree n and order m,with n>m. The angular frequency must satisfy −2Ω σ = n(n +1) . (16.10) As in the case of barotropic Rossby waves in a fluid at rest on a β plane, spherical Rossby waves propagate westward. Their zonal phase speed is given by cos ϕ c = a cos ϕσ = −2Ωan(n +1) . (16.11) The first few associated Legendre functions are given in Table 16.1. The lowest order modes, for which m = 0, are zonally symmetric and have zero frequency. These are just east-west flows that do not perturb the background vorticity gradient and thus are not oscillatory. The lowest order wave mode, for which n = m = 1, has an angular frequency of −Ω and is therefore stationary relative to absolute space. This zonal wavenumber 1 mode has maximum amplitude on the equator and decays as cos ϕ toward the poles. Modes of greater values of n have increasingly fine meridional structure. 74MIT OpenCourseWarehttp://ocw.mit.edu 12.803 Quasi-Balanced Circulations in Oceans and Atmospheres Fall 2009 For information about citing these materials or our Terms of Use, visit: