A Simple Box Model of Stochastically Forced Thermohaline Flow

A modified Stommel two-box model is considered as a minimal representation of the buoyancy-driven ocean circulation. In the limit of fast temperature relaxation only the salinity evolves in time while the temperature is clamped to the prescribed ambient value. The box model has no intrinsic variability: just two linearly stable and one unstable equilibria. A finite perturbation is needed to shift the system from one stable equilibrium to the other. The minimum amplitude and duration in time of the perturbation are calculated. A stochastic component of the freshwater flux forcing is then added to model the effect of changes in the global hydrological cycle due to the “weather.” The stochastic forcing is a source of extrinsic time dependence. The salinity gradient obeys an equation analogous to the trajectory of a viscous particle in a double-welled potential, subject to Brownian agitation. If the amplitude of the stochastic driving is above a certain threshold, then there is a finite probability of switching from one stable equilibrium to the other. The threshold variance and the average residence time in each equilibrium are calculated. For timescales on the order of the average residence time or longer, the box model behaves like a random telegraph process. The stochastic driving also induces a “rattling” around each steady equilibrium whose frequency is proportional to the curvature of the potential well at each equilibrium. The probability of being in each well can be calculated and, within each equilibrium, the box model behaves like an Ornstein-Uhlenbeck process. Finally the spectrum of the salinity gradient is calculated analytically using standard approximations in stochastic processes. The approximate analytical results are in excellent agreement with those obtained by direct computation.