A Generalized Mathematical Model for Geostrophic Adjustment and Frontogenesis

Fronts, or regions with strong horizontal density gradients, are ubiquitous and dynamically important features in the atmosphere and ocean. The process of frontal development, whereby an initial horizontal density gradient is amplified, often into a near-discontinuity, is known as frontogenesis. Here, we will describe a new generalised mathematical model of frontogenesis for flows with arbitrary rotation (Rossby number) and stratification (Froude number). Historically, frontogenesis driven by a convergent strain field has been studied in the low Rossby number regime (e.g. Hoskins & Bretherton, 1972). In contrast, Blumen (2000) and colleagues have dealt with frontogenesis occurring spontaneously during the geostrophic adjustment of an initially unbalanced flow with a large Rossby number. Our model provides a unifying framework applicable at arbitrary Rossby number, and limiting to the Hoskins & Bretherton and Blumen solutions in the limit of small and large Rossby number, respectively. The model describes the transient adjustment of a stratified flow with uniform potential vorticity between rigid lids. In the absence of strain, the solution is dynamically split into a balanced steady state and unbalanced transient part composed of inertia gravity waves. A key result is that the existence of a steady state does not imply attainability of that state; a frontal discontinuity, induced by the transient waves, may prevent the steady state from being reached. The criteria for the occurrence of a frontal discontinuity is determined in terms of the Rossby and Froude numbers, and compared with the criteria for the existence of a steady state. Our model is also applicable for arbitrary values of strain. With a small amplitude strain, the solution is approximately split into a slow-time part matching the Hoskins and Bretherton solution, and a fast-time part composed of propagating inertia-gravity waves. For large strain fields, this timescale separation breaks down, and the strain and wave fields interact. In this limit, large amplitude inertia gravity waves are generated even for geostrophically balanced initial conditions.