Inertial Oscillations and Frontogenesis in a Zero Potential Vorticity Model

Geostrophic adjustment and frontogenesis are examined by means of a two-dimensional, inviscid, rotating and nonlinear fluid model that satisfies the condition of zero potential vorticity. The fluid is bounded top and bottom by level, rigid lids. The initial state is one of no motion, but an unbalanced horizontal temperature gradient is prescribed. The subsequent motion is represented as the sum of an inertial oscillation, with the frequency of the local Coriolis frequency f, and an evolving geostrophic flow. When a nondimensional parameter a, a Rossby number, satisfies a < 1, the gradient of the evolving geostrophic flow increases (frontogenesis) during the period 0 < t ⩽ π/f; the gradient decreases during the period π/f < t ⩽ 2π/f (frontolysis). When a ≥ 1, the relative vorticity of the evolving geostrophic flow becomes infinite: a discontinuity forms at the top and bottom boundaries during the period 0 < t ⩽ π/f. There is an equipartition of energy between the inertial oscillation and the geostrophic flow, and nonlinear interactions occur between them. An exact (Fourier) spectral representation of the solution on the bottom boundary is used to display the kinetic energy spectrum and the transfer of energy through the spectrum at the time that the discontinuity forms. Applications of the model to oceanic and to atmospheric frontogenesis and to restratification of the surface mixed layer, following a storm, are noted.