Problem Meto611 Set #9 Midlatitude planetary waves 1) (borrowed from Peter Rhines) The equation for The quasigeostrophic streamfunction for a barotropic slowly moving fluid is given by: 2 + =0 t x (1.1)
Assume a solution of the form: ( x, y, t ) = real (exp(ikx + ily it ) . a) Show that as the
frequency 0 with constant wavenumber ( k 2 + l 2 ), the meridional wavenumber becomes much greater than the zonal wavenumber. Sketch the crests in this case. b) Show that in this limit the energy in the midlatitude Rossby Waves propagates westward. b) In this limit show that the advection terms neglected above are in fact unimportant so that low frequency forcing tends to form zonally oriented jets (think of Jupiter). 2) Find the eigenfrequencies and eigenfunctions (a.k.a. normal modes) for midlatitude Rossby waves in a square basin of length A and constant depth H*. 3) Consider weak quasigeostrophic motion in a stratified uniformly ( N ( z ) = N o ) layer of fluid (0<z<H). For simplicity assume a rigid lid at z=H. The linearized Q.G. potential vorticity equation is given by:
2 2 f o2 2+ 2+ ( 2 ) + =0 x t z N z x y
(1.1)
Assume the motion is separable in the vertical (assume a solution of the form = 0 ( x, y, t ) ( z ) ). Find the dispersion relation for horizontally propagating waves (solve 1.1 assuming o ( x, y, t ) = exp(i (kx + ly t )) and find a relationship between frequency and wavenumber).
*hint: first, go back and resolve the problem of the normal modes of shallow water gravity waves in a square basin. Then try to apply the same methodology here. The one complication is going to be that Rossby Waves, whether they are propagating energy eastward or westward, always have westward phase velocity. Thus, the normal modes will appear to have crests and troughs propagating westward!