Does the nonlinearity of the equation of state impose an upper bound on the buoyancy frequency?

Abstract

Mixing in the ocean is usually accompanied by a net reduction in volume caused by the nonlinear nature of the equation of state. This contraction-on-mixing at a certain depth implies that the whole water column above this depth slumps a little and so suffers a reduction in gravitational potential energy. Under certain circumstances the gravitational potential energy of the entire water column can decrease as a consequence of mixing activity at a certain depth. We examine Fofonoff's hypothesis that in these circumstances the net reduction of gravitational potential energy of the whole water column causes a local increase in the turbulent mixing activity at the location of the original mixing. Fofonoff proposed that this increased local mixing diffuses the local property gradients until the criterion for positive feedback is no longer satisfied, so providing an upper bound for the vertical stratification in the ocean. Bearing in mind the relatively inefficient nature of turbulent mixing at causing diapycnal fluxes (the majority of the turbulent kinetic energy goes directly into internal energy), we find that the criterion for positive feedback is a factor of approximately seven more difficult to achieve than has been realized to date. An examination of oceanic data shows that while Fofonoff's original criterion for positive feedback is often exceeded, the more appropriate criterion is almost never approached. The positive feedback hypothesis assumes that the reduction in the gravitational potential energy of the whole water column appears at the location of the original mixing as an increase in the turbulent mixing activity. We show that this very focused oceanic response is extremely difficult to justify. For example there is no such feedback in a strictly one-dimensional water column; rather all of the reduction in gravitational potential energy appears as an increase in internal energy at the depth of the original mixing and there is no possibility of any positive feedback to increase the turbulent mixing. As the positive feedback hypothesis is lacking a convincing theoretical basis and is not supported by oceanic data, we do not believe that it acts as an effective upper bound on oceanic stratification.