Geostrophic adjustment on the midlatitude <i>β</i> plane

Abstract

Analytical and numerical solutions of the linearized rotating shallow water equations are combined to study the geostrophic adjustment on the midlatitude β plane. The adjustment is examined in zonal periodic channels of width Ly=4Rd (narrow channel, where Rd is the radius of deformation) and Ly=60Rd (wide channel) for the particular initial conditions of a resting fluid with a step-like height distribution, η0. In the one-dimensional case, where η0=η0(y), we find that (i) β affects the geostrophic state (determined from the conservation of the meridional vorticity gradient) only when b=cot(ϕ0)RdR≥0.5 (where ϕ0 is the channel's central latitude, and R is Earth's radius); (ii) the energy conversion ratio varies by less than 10 % when b increases from 0 to 1; (iii) in wide channels, β affects the waves significantly, even for small b (e.g., b=0.005); and (iv) for b=0.005, harmonic waves approximate the waves in narrow channels, and trapped waves approximate the waves in wide channels. In the two-dimensional case, where η0=η0(x), we find that (i) at short times the spatial structure of the steady solution is similar to that on the f plane, while at long times the steady state drifts westward at the speed of Rossby waves (harmonic Rossby waves in narrow channels and trapped Rossby waves in wide channels); (ii) in wide channels, trapped-wave dispersion causes the equatorward segment of the wavefront to move faster than the northern segment; (iii) the energy of Rossby waves on the β plane approaches that of the steady state on the f plane; and (iv) the results outlined in (iii) and (iv) of the one-dimensional case also hold in the two-dimensional case.