Frontogenesis driven by horizontally quadratic distributions of density

Abstract

Experiments by Simpson & Linden (1989) have shown that a horizontally nonlinear distribution of fluid density is necessary in order to produce frontogenesis. This paper considers the simplest case of such a nonlinear distribution, a quadratic density distribution in a channel. Two flow models are examined, porous media and Boussinesq. The evolution equations for both these flows can be reduced to one-dimensional systems. An exact solution is derived for porous-media flow with no molecular diffusion. Numerical solutions are shown for the other cases. The porous-media and inviscid/non-diffusive Boussinesq systems exhibit ‘classic’ frontogenesis behaviour: a rapid and intense steepening of the density gradient near the lower boundary while horizontal divergence reduces the upper-boundary density gradient to nearly zero. The viscous Boussinesq system exhibits a more complicated behaviour. In this system, boundary-layer effects force frontogenesis away from the lower boundary and at late times the steepest density gradients are close to mid-channel. One feature of these model systems is that they can exhibit blow-up in finite time. Proof of blow-up is given for the non-diffusive porous media and inviscid/nondiffusive Boussinesq cases. Numerical results indicate that blow-up also occurs for the diffusive porous-media case and that it may occur for the diffusive Boussinesq case. Despite the blow-up we believe that the model solutions can be applied to real situations. To support this a two-dimensional calculation has been made of Boussinesq frontogenesis in a long box. This calculation shows close agreement with the corresponding one-dimensional calculation up to times close to blow-up.