Frontal Instabilities in a Two-Layer, Primitive Equation Ocean Model

Abstract

A linear stability analysis combined with an energy analysis is performed to discriminate between the various instabilities that may develop at upwelling fronts. In the present study, a two-active-layer model of finite depth is considered. Thus, the model includes a variable across-front bottom topography, a sloping interface, a surface elevation, and variable densities in the two layers. In addition, the energy analysis departs from earlier studies in that it makes use of the available gravitational energy to replace the conventional potential energy. The concept of available gravitational energy is akin to available potential energy, but avoids the constraint of considering a closed basin. Interestingly, the earlier findings of two preferred bands of unstable waves are retained in the present model. The first band (wavelengths of 10?30 km) is associated with the so-called frontal instability (frontal mode), and the second band (wavelengths of 60?70 km) is associated with a mixed barotropic?baroclinic instability (mixed mode). The growth rate of the frontal mode is typically in the range of one to two days, while the mixed mode is typically three to five days. Although the frontal mode dominates in most cases, an exception occurs when the horizontal shear (in terms of the jet speed divided by the frontal width) becomes large. Indeed, the frontal mode ceases to exist when the frontal width becomes small enough, depending on the horizontal viscosity. Another exception occurs when the frontal jet is caused by the sloping interface only (no upper-layer density front). In this case the frontal mode is cut off, lending further support to the theory that the smaller-scale waves found in the coastal transition zones of the world oceans indeed owe their presence to the existence of the upwelling front. When the vertical shear is increased, the present analysis reveals that the growth rates of all the unstable waves, in particular the waves associated with the frontal mode, are increased. Moreover, the mixed mode ceases to exist as a preferred band of unstable waves. A final case shows that the frontal mode is unaffected by a sloping bottom topography. This is in support of the suggestion that the frontal mode is trapped to the upper layer. Experiments with a numerical multilayer, primitive-equation ocean model support the findings of the linear stability analysis, both qualitatively and quantitatively. They also reveal a complicated nonlinear wave?wave interaction causing a transition from the well-organized linear instability wave pattern toward a new organized pattern of much longer scale, filament-type, structures.