Geostrophic adjustment on the f-plane: Symmetric versus anti-symmetric initial height distributions

We study the fundamental process of geostrophic adjustment in infinitely long zonal and meridional channels of widths Ly and Lx, respectively, by deriving analytic solutions and simulating the linearized rotating shallow water equations (LRSWE). All LRSWE's variables are divided into time-independent (geostrophic) and time-dependent components. The latter includes Kelvin waves, Poincare waves, and inertial oscillations. Explicit expressions are derived for both components, which are confirmed numerically. Anti-symmetric and symmetric initial height distributions, η 0 ( x ), are considered, both of which introduce a length scale, D, into the problem. We show that for an anti-symmetric η 0 ( x ), (i) the rate of approach to geostrophy is D-independent; (ii) the decay rate of inertial oscillations is ∝ t − 1 / 2; and (iii) for D L y ≫ 1, the energy of the final state in any finite sub-domain of the channel exceeds that of the initial state, while for D L y ≪ 1 the energy in the final state is smaller than in the initial state. In contrast, for a symmetric η 0 ( x ): (i) the rate of approach to geostrophy increases with D; (ii) the decay rate of inertial oscillations is ∝ t − 3 / 2, and (iii) the energy of the final state is always smaller than that of the initial state. In meridional channels, the effect of boundaries is to (i) block the waves, propagation to infinity; (ii) alter the spatial structure of the geostrophic flow; and (iii) discretize the frequency spectrum, thus eliminating the inertial oscillations. The ratio of wave energy to initial energy decreases with Lx for anti-symmetric η 0 ( x ) and increases with Lx for symmetric η 0 ( x ).