The stability of a zonally averaged thermohaline circulation model

A combination of analytical and numerical techniques are used to efficiently determine the qualitative and quantitative behaviour of a one-basin zonally averaged thermohaline circulation ocean model. In contrast to earlier studies which use time stepping to find the steady solutions, the steady state equations are first solved directly to obtain the multiple equilibria under identical mixed boundary conditions. This approach is based on the differentiability of the governing equations and especially the convection scheme. A linear stability analysis is then performed, in which the normal modes and corresponding eigenvalues are found for the various equilibrium states. Resonant periodic solutions superimposed on these states are predicted for various types of forcing. The results are used to gain insight into the solutions obtained by Mysak, Stocker and Huang in a previous numerical study in which the eddy diffusivities were varied in a randomly forced one-basin zonally averaged model. Resonant stable oscillations with century scale periods are predicted with structures that compare favorably with those found in the previous study. In cases with large horizontal diffusivities, the two-cell pattern is also stable, which parallels cases in the previous study where large vacillations were seen between the three stable steady states. Further, in cases with large horizontal {\it and} large vertical diffusivities, no one-cell pattern can be realised and the only asymptotic behaviour found is the two-cell pattern.