Statistical hydrodynamics of the thermohaline circulation in a two-dimensional model

We study a simple conceptual model of the climate-mean ocean thermohaline circulation, the two-dimensional Boussinesq model of Thual and McWilliams, generalized to include a random surface salinity flux, representing synoptic variability in freshwater sources. In a small aspect ratio limit previously considered by Cessi and Young for the deterministic model, we show that the dynamics reduces to a stochastically-driven one-dimensional Cahn-Hilliard equation (with non-conserving boundary conditions). If the random force is space-time white noise, a non-linear fluctuation-dissipation relation permits an exact determination of the statistical equilibrium distribution. The effect of the spatial distribution of the systematic (non-random) salinity flux on the stability of multiple equilibria is studied in detail. We determine a flux distribution for which the model caricatures the circulation states in the Atlantic during the last glacial period. For this case, we determine the multimodal equilibrium distributions associated to the multiple states of circulation. We also study the dynamics of random state-transitions by a simulation of the stochastic Cahn-Hilliard equation. Finally, with a weak, periodic modulation in the freshwater forcing of the correct frequency, the model is shown to exhibit stochastic resonance, i.e. the transitions synchronize with the periodic force. The stochastic resonant frequency is calculated analytically in the model by the Langer formula for the mean residence time and compared with numerical solutions.