Spurious multiple equilibria introduced by convective adjustment

Abstract

The application of bifurcation analysis to ocean climate models is substantially hampered by difficulties associated with the use of convective adjustment, i.e. a parameterisation of convection in which the vertical diffusion of heat and salt is greatly enhanced whenever the water column becomes statically unstable. When tracking steady solutions of these models in parameter space, problems arise due to the occurrence of a multitude of saddle-node bifurcations, each of which is related to a minor reorganisation of convection. In this paper, we analyse the origin of the multiple steady states in detail using a one-dimensional column model. By analytical evaluation of the eigenvalue problem conditions are formulated under which bifurcations may occur. Using numerical continuation methods a regime of multistability is identified and it is shown that the number of coexisting steady states increases with increasing resolution, while the extent of parameter space in which they occur decreases. A comparison of the numerical results to corresponding analytical solutions reveals that the multiple equilibria are inherent to the discretisation and hence artificial. Apparently, successful application of convective adjustment requires knowledge of subgrid-scale tracer fields. Two alternative convective adjustment schemes are proposed that (partially) overcome these problems. Results from a fully implicit model based on the two-dimensional primitive equations indicate that the physically relevant phenomena due to the nonlinear character of large-scale oceanic flow, notably the bifurcations related to the salt advection feedback, are preserved under the alternative formulations.