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 LyCombining double low line4Rd (narrow channel, where Rd is the radius of deformation) and LyCombining double low line60Rd (wide channel) for the particular initial conditions of a resting fluid with a step-like height distribution, η0. In the one-dimensional case, where η0Combining double low lineη0(y), we find that (i) β affects the geostrophic state (determined from the conservation of the meridional vorticity gradient) only when bCombining double low linecot(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., bCombining double low line0.005); and (iv) for bCombining double low line0.005, harmonic waves approximate the waves in narrow channels, and trapped waves approximate the waves in wide channels. In the two-dimensional case, where η0Combining double low lineη0(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.
Bibliographical noteFunding Information:
This research has been supported by the ISF-NSFC Joint Research Program (grant no. 2547/17).