Instabilities of buoyancy-driven coastal currents and their nonlinear evolution in the two-layer rotating shallow-water model. Part 1. Passive lower layer

Abstract

Buoyancy-driven coastal currents, which are bounded by a coast and a surface density front, are ubiquitous and play essential role in the mesoscale variability of the ocean. Their highly unstable nature is well known from observations, laboratory and numerical experiments. In this paper, we revisit the linear stability problem for such currents in the simplest reduced-gravity model and study nonlinear evolution of the instability by direct numerical simulations. By using the collocation method, we benchmark the classical linear stability results on zero-potential-vorticity (PV) fronts, and generalize them to non-zero-PV fronts. In both cases, we find that the instabilities are due to the resonance of frontal and coastal waves trapped in the current, and identify the most unstable long-wave modes. We then study the nonlinear evolution of the unstable modes with the help of a new high-resolution well-balanced finite-volume numerical scheme for shallow-water equations. The simulations are initialized with the unstable modes obtained from the linear stability analysis. We found that the principal instability saturates in two stages. At the first stage, the Kelvin component of the unstable mode breaks, forming a Kelvin front and leading to the reorganization of the mean flow through dissipative and wave–mean flow interaction effects. At the second stage, a new, secondary unstable mode of the Rossby type develops on the background of the reorganized mean flow, and then breaks, forming coherent vortex structures. We investigate the sensitivity of this scenario to the along-current boundary and initial conditions. A study of the same problem in the framework of the fully baroclinic two-layer model will be presented in the companion paper.