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

Abstract

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, Poincaré waves, and inertial oscillations. Explicit expressions are derived for both components, which are confirmed numerically. Anti-symmetric and symmetric initial height distributions, η0(x), are considered, both of which introduce a length scale, D, into the problem. We show that for an anti-symmetric η0(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 DLy≫1, the energy of the final state in any finite sub-domain of the channel exceeds that of the initial state, while for DLy≪1 the energy in the final state is smaller than in the initial state. In contrast, for a symmetric η0(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 η0(x) and increases with Lx for symmetric η0(x).

Geostrophic adjustment on the mid-latitude β-plane

“…The calculations of the energy evolution, discussed in details in §5.4, reveal that the fraction of the initial (potential) energy converted to Poincaré wave energy during the adjustment process decreases with the length of the initial perturbation, 2D. This is consistent with the results found on the f -plane in Yacoby et al (2021). Hence, to increase the energy of Poincaré waves, we decrease the channel length, L x , to 12 so D = L x /12 = 1 instead of 10.…”

“…Since Gill's seminal study (Gill, 1976) it became common practice to analyze the energetics of the adjustment problem by calculating the energy conversion ratio [see, e.g., Grimshaw et al (1998), Fang and Wu (2002), Yacoby et al (2021)]:…”

“…Figure 9 shows the geostrophic steady state on the f -plane (b = 0) in wide (L y = 60, left column) and narrow (L y = 4, right column) channels. The steady solutions were calculated using the analytic expressions developed in subsection 5 of the Appendix in Yacoby et al (2021) for long channels (L x ≫ D). In both channels the steady solutions vary with y only near the boundaries so their variation with y is less pronounced in wide channels as the solution there remains (nearly) independent of y The temporal development of η on the β-plane (b = 0.005) in wide and narrow channels was simulated with L x = 120 so D = L x /12 in equation ( 11) was set to 10 (simulation C).…”

“…The geostrophic adjustment theory was first studied in the 1930s when Carl-Gustaf Rossby published the pioneering works on the subject (Rossby, 1937(Rossby, , 1938). Rossby's work was extended in several papers, e.g., Blumen (1972) and Yacoby et al (2021). However, as in Rossby's original studies, nearly all of these earlier studies addressed particular aspects (e.g.…”

Geostrophic adjustment on the midlatitude β plane

“…The geostrophic adjustment theory was first studied in the 1930s, when Carl-Gustaf Rossby published pioneering works on the subject (Rossby, 1937(Rossby, , 1938). Rossby's work was extended in several papers, e.g., Blumen (1972) and Yacoby et al (2021). However, as in Rossby's original studies, nearly all of these earlier studies addressed particular aspects (e.g., initial or boundary conditions) of the adjustment on the f plane; thus, their applicability to the real ocean is limited.…”