Unsteady similarity solutions and oscillating ocean gyres

The effect of time-dependent forcing on steady solutions representing basin-scale flows is investigated. Analytical and numerical solutions are considered separately and compared. We first use symmetry methods to show how any steady solution of the ideal thermocline equations can be used to generate a family of unsteady solutions, via an arbitrary function of time alpha(t). The resulting time-dependent solutions correspond to distortion of the isopycnal surfaces by a velocity field which varies linearly in the three coordinate directions. Although the displacements are linear, the fluctuations can lead to a form of nonlinear streaming wherever the function a appears nonlinearly in expressions for mass and heat fluxes. For an example steady solution, changes in internal energy caused by the time-dependence are associated with changes in thermocline depth and fluxes of energy from the western boundary, although it is unclear to what extent this behavior is specific to the example chosen. We also describe another symmetry of the time-dependent thermocline equations which generates wave-like solutions from arbitrary steady solutions. All the time-dependent solutions are special cases of a symmetry which applies to a general advection equation. Potential vorticity advection provides another special case. With the inclusion of convective and dissipative processes, a more realistic steady solution is found numerically in a flat-bottomed sector. If the surface forcing functions oscillate annually, the resulting flow resembles the analytical predictions. As the oscillation period increases, spatial variations in phase disrupt the agreement as first boundary and then diffusive effects become important. For decadal period oscillations, nonlinear streaming is found to significantly increase the meridional overturning.