Geostrophic adjustment on the midlatitude beta plane

Analytical and numerical solutions of the linearized rotating shallow water equations are combined to study the geostrophic adjustment on the midlatitude beta plane. The adjustment is examined in zonal periodic channels of width L-y = 4R(d) (narrow channel, where R-d is the radius of deformation) and L-y = 60R(d) (wide channel) for the particular initial conditions of a resting fluid with a step-like height distribution, eta(0). In the one-dimensional case, where eta(0) = eta(0)(y), we find that (i) beta affects the geostrophic state (determined from the conservation of the meridional vorticity gradient) only when b = cot(phi(0))R-d/R >= 0:5 (where phi(0) 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, beta 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 eta(0) = eta 0(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 beta 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.