Geostrophic Adjustment in a Rotating Channel: Symmetric and Anti-symmetric Initial Height Distributions

Abstract:

Large scale flows in the ocean and atmosphere are in near geostrophic balance as is evident from the smallness of the associated Rossby number. A fundamental theory in geophysical fluid dynamics is the transition of an initial unbalanced state to a geostrophically balanced final state, referred to as geostrophic adjustment. The present study extends the classical geostrophic adjustment theory of a step-like initial height distribution, η₀(x), in an infinitely long zonal channel to symmetric and anti-symmetric η₀(x). These two η₀(x) introduce a length scale, D, to the problem in addition to the channel width, L. The extension combines analytic considerations with numerical integration of the Linearized Rotating Shallow Water Equations (LRSWE). All variables of the LRSWE are divided into a time-independent (geostrophic) part and transients described by Kelvin and Poincaré waves (including inertial waves at infinitely long zonal wavenumber). Explicit analytic expressions are derived for both parts and these expressions are confirmed by numerical simulations. For the anti-symmetric η₀(x) we show that: (i) the rate of approach to geostrophy is independent of D; (ii) the decay rate of inertial oscillations is proportional to 1/√t , and (iii) for large D⁄L the energy of the final state exceeds that of the initial state, while for small D⁄L the energy of the final state is smaller than that of the initial state (see left panel in the figure below). In contrast, for the symmetric η₀(x) we show that: (i) the rate of approach to geostrophy increases with D; (ii) the decay rate of inertial oscillations is proportional to (1/√t )³, and (iii) the energy of the final state is always smaller than that of the initial state (see right panel of the figure below). The figure below shows the difference in energy per unit width between the initial and final states as a function of D for several values of L.