ESCI 343 – Atmospheric Dynamics II Lesson 11 - Rossby Waves Reference: An Introduction to Dynamic Meteorology (4rd edition), J.R. Holton Atmosphere-Ocean Dynamics, A.E. Gill Fundamentals of Atmospheric Physics, M.L. Salby Reading: Holton, 7.7 and 12.3 BAROTROPIC ROSSBY WAVES Rossby waves owe their existence to the principle of conservation of potential vorticity. We first start with a barotropic fluid, for which the principle of conservation of barotropic potential vorticity states that 0=+hfDtDζ. Expanding this out yields the barotropic vorticity equation DtDhhfvVth=+∇•+∂∂βζζ. The linearized form of this equation with zonal mean flow only (0;0=≠vu) is ∂∂′+∂∂′+∂∂+∂∂+∂∂=′+∂′∂+∂′∂yHvxHuxHuxutHfvxutηηβζζ0. (1) If we assume that the mean depth of the fluid (H) is constant then equation (1) becomes ∂∂+∂∂=′+∂′∂+∂′∂xutHfvxutηηβζζ0. (2) Further assuming geostrophic balance we can write them in terms of the perturbation height as follows, xfgvfg∂∂=′∇=′ηηζ020 so that equation (2) becomes 2 22 20 02 20f fut c x c xηη η η η β   ∂ ∂ ∂∇ − + ∇ − + =   ∂ ∂ ∂    (3) where c is the phase speed of a shallow-water gravity wave. The waves supported by equation (3) are called Rossby waves. To find the dispersion relation for the waves supported by equation (3) we assume a perturbation of the form ()tlykxiAeωη−+= and substitute it into (3). This yields the following dispersion relation for Rossby waves, 2202cfkku+Κ−=βω. (4)2For waves that are short compared to the Rossby radius of deformation (given by c/f0), the wave number will be large compared to f0 /c. In this case the dispersion relation becomes 2Κ−=kkuβω,1 (4’) a result known as the shortwave approximation. DISPERSION PROPERTIES OF ROSSBY WAVES The phase velocity of Rossby waves with zero mean flow (0u=) is ( )2 2 2 2 20ˆ ˆkc K ki l jf cω β = = + Κ Κ Κ + , (5) and the group velocity is ()( ) ( )2 2 2 202 22 2 2 2 20 02ˆ ˆgk l f cklc i jf c f cββ− −= +Κ + Κ +. (6) The plots on the next page show the dispersion properties for Rossby waves with zero mean flow, using a value of β = 10−11 m−1 s−1 and Rossby radius of deformation of 3000 km. Some things to note: • The frequency is always negative, and becomes larger in magnitude for the longer wavelengths (smaller wave numbers). • The zonal phase speed is always negative in the absence of mean flow. • The zonal group speed may be either positive or negative, depending on the horizontal wave number. o Long waves propagate energy westward in the same direction as the phase speed. o Shortwaves propagate energy eastward, opposite to the phase speed. • The meridional phase speed is negative, but the meridional group speed is positive. o The meridional energy propagation is opposite to the phase speed. If there were a non-zero mean flow it would simply be added to the phase speeds and group speeds. 1 In most meteorological textbooks equation (4’) is the dispersion relation that is given for Rossby waves, and is derived directly by ignoring the right-hand-side of equation (2). Physically this implies ignoring the vertical stretching of the fluid column. Equation (4) is the more general form of the dispersion relation.3 Angular frequency. Zonal phase speed.4 Zonal group speed. Meridional phase speed.VERTICALLY PROPAGATING ROSSBY WAVES In a stratified fluid it is possible to have Rossby waves that have a vertical component of propagation. To study thesegeostrophic potential vorticity (in place of the barotropic potential vorticity used in the previous discussions). In the absence of diabatic heating, quasivorticity is conserved and therefDtDgwhere the static-stability parameter is given asConverting this to height coordinates yieldsDtDgWe can write this in terms of the so that 5 Meridional group speed. VERTICALLY PROPAGATING ROSSBY WAVES In a stratified fluid it is possible to have Rossby waves that have a vertical component of propagation. To study these waves we have to use the concept of quasigeostrophic potential vorticity (in place of the barotropic potential vorticity used in the previous discussions). In the absence of diabatic heating, quasi-geostrophic potential vorticity is conserved and therefore the following equation holds (see Lesson 4) 011020=∂Φ∂∂∂++Φ∇ppfffσ stability parameter is given as dpθθασ∂−=. Converting this to height coordinates yields 012020=∂Φ∂∂∂++Φ∇zzNfffρρ. We can write this in terms of the streamfunction, 0fΦ≡ψ, In a stratified fluid it is possible to have Rossby waves that have a vertical waves we have to use the concept of quasi-geostrophic potential vorticity (in place of the barotropic potential vorticity used in the geostrophic potential ore the following equation holds (see Lesson 4)602202=∂∂∂∂++∇zzNffDtDgψρρψ. (7) The linearized form of equation (7), with mean flow in the zonal direction only, is 02202=∂′∂+∂′∂∂∂++′∇∂∂+∂∂xzzNffxutψβψρρψ. (8) Since we are interested in waves which may propagate vertically great distances, we need to use a slightly modified form of the sinusoidal solution for the streamfunction, ( )tmzlykxieAωρψ−++=′, (9) where ρ is a function of height. To simplify things we also assume and isothermal, anelastic atmosphere, so that HgdzdgN =−=ρρ2. Putting equation (9) into equation (8) yields the dispersion relation 242 202 24HkukfNmN gβω= −  Κ + +    . (10) Rossby waves are often forced by topography. To find out when such stationary waves (with frequency zero) can propagate vertically we rearrange equation (10) for vertical wave number to get 2 222 202 204HN fNmf u gβ  = − Κ −    . (11) Vertical