Nonlinear dynamics in Langmuir circulations and in thermosolutal convection

Abstract

Two-dimensional motions generated by Langmuir circulation instability of stratified layers of water of finite depth are studied under a simplifying assumption making it mathematically analogous to double-diffusive thermosolutal convection with constant solute concentration and constant heat flux at the boundaries. The nature of possible motions is mapped over a significant region in (S, R) parameter space, where S and R are parameters measuring, respectively, the stabilizing and destabilizing agencies in the problem. In the Langmuir circulation problem R measures the effects of wind and surface wave action, and S measures the stabilizing effect of buoyancy: in the thermosolutal problem, R measures the destabilizing effects of heating, while S measures the stabilizing effect of solute concentration. Effects of lateral boundary or symmetry conditions are found to be crucial in determining the qualitative behaviour. Complex temporal behaviour, including intermittently chaotic flows, are found under suitably constrained (no flux) lateral conditions but are unstable and not realized when these constraints are relaxed and replaced by periodic lateral conditions. Multiple steady states also arise, with those found under constrained lateral conditions losing stability either to travelling waves, or to other steady states when the lateral boundary conditions are relaxed. In some regions of the parameter space, multiple stable nonlinear motions have been found under periodic boundary conditions. The multiple stable states may either be coexisting travelling waves and steady states (different from those found under the constrained lateral conditions). The existence of robust travelling waves may explain some field observations of laterally drifting windrows associated with Langmuir circulations.