Thermohaline loops, Stommel box models, and the Sandström theorem

The Stommel two-box, two flow-regime box model is kinematically and dynamically equivalent to the flow in a onedimensional fluid loop, although one having awkward and extreme mixing coefficients. More generally, such a loop, when heated and cooled at the same geopotential, provides a simple example of the working of the Sandström theorem, with flow intensity capable of increasing or decreasing with growing diffusion. Stress dominates real oceanic flows, and its introduction into the purely thermally driven loop generates oscillations, multiple states, and instabilities at low diffusivity. When, within the Boussinesq approximation, salinity forcing and mixed boundary conditions are further introduced, an intricate pattern of response appears, dependent upon at least five non-dimensional parameters, including the time of onset of salinity forcing. The ability, in a one-dimensional loop, to produce such a rich array of dynamical behaviors, dependent in detail upon the problem parameters, suggests that in the absence of any general results relating one-to three-dimensional fluid flows, identification of the time-dependent behavior of a GCM with that of the onedimensional loop Stommel models should be regarded as still primarily speculation.