Time-Dependent Adjustment in a Simple Model of the Mid-Depth Meridional Overturning Cell

Abstract

Abstract A recently proposed reduced-gravity model of the warm-water branch of the middepth meridional overturning circulation in a rectangular basin with a circumpolar connection is extended to include time dependence. The model describes the balance between gain of warm water through northward Ekman advection across the circumpolar current, loss of warm water through eddy fluxes southward across the current, net gain or loss of warm water through diabatic processes north of the current, and changes in the thickness of the warm-water layer. The steady solutions are the same as those found previously, when the previous parameterization of diabatic fluxes is used. Time-dependent solutions are considered for the approach of the solution to a new equilibrium when the forcing or parameters are abruptly changed and then held fixed. An initial adjustment occurs through a combination of boundary and equatorial adjustment, followed by planetary wave propagation. The longer-term adjustment to equilibrium consists primarily of the slow change in eastern boundary thickness of the warm layer, which controls the mean depth of the entire layer. An approximate analytical solution of the time-dependent equations yields an explicit expression for the intrinsic time scale of the long-term adjustment, which depends on the eddy and diabatic flux parameters and on the equilibrium solution toward which the time-dependent solution adjusts. Numerical solutions are also considered with a second, advective–diffusive diabatic flux parameterization. These solutions differ quantitatively but not qualitatively from those with the original parameterization. For the range of parameter values considered, the adjustment time scale has dimensional values of several decades to more than a century, but the meridional flux of warm water may respond to changes in external parameters or forcing much more rapidly than this time scale for equilibration of the eastern boundary thickness and thermocline structure.