The system discussed by Stommel (1987) and Welander (1982), in which heating and evaporation at the surface of the ocean are balanced by vertical turbulent mixing, is studied analytically and numerically for mixing laws appropriate to salt fingers, rather than mechanical turbulence. Stommel and Welander found for mechanically-driven turbulent mixing that a limit cycle of T and S exists (that is, T and S oscillate) in the presence of steady forcing. We find that the usual salt finger parameterizations, in which salinity flux coefficient and buoyancy flux ratio decrease with increasing density ratio, do not allow a limit cycle. This result holds whether the flux parameterization is for an interface using the “4/3 power law” laboratory relationships or in terms of vertical gradients. Rather, all initial conditions either evolve to a steady balance or lead to the upper layer becoming denser than the lower layer and overturning. In addition, we find that commonly used mechanical turbulence parameterizations for eddy diffusivity vs. Richardson number do not vary rapidly enough to allow a limit cycle in the Stommel/Welander model, although recent observations of equatorial turbulence do. Hence the possible existence of a limit oscillation in evaporatively-driven areas of the ocean depends critically on the type of vertical mixing which occurs, and on the precise form of its parameterization.
Ruddick, Barry, and Lingqi Zhang. 1989. "The mythical thermohaline oscillator?." Journal of Marine Research 47, (4). https://elischolar.library.yale.edu/journal_of_marine_research/1948