On the energetics of stratified turbulent mixing, irreversible thermodynamics, Boussinesq models, and the ocean heat engine controversy

Abstract

A key issue in stratified turbulence theory concerns the nature of the link between D(APE), the dissipation rate of available potential energy APE, and W_{r,turbulent}, the turbulent rate of change of background gravitational potential energy GPE_r, which are both controlled by molecular diffusion. For Boussinesq fluids with a linear equation of state, this link is simply W_{r,turbulent}=D(APE), widely interpreted as implying that GPE_r increases at the expense of APE, in contrast with the laminar case where GPE_r increases at the expense of internal energy (IE). This idea is revisited here by regarding IE as the sum of three distinct subcomponents: available internal energy (AIE), exergy (IE_{exergy}), and dead internal energy (IE_0). In this new view, D(APE) is the dissipation rate of APE into IE_0, while both W_{r,laminar} and W_{r,turbulent} convert IE_{exergy} into GPE_r. The equality W_{r,turbulent}=D(APE) thus states that IE_{exergy} is converted into GPE_r at the same rate as APE is dissipated into IE_0. For non-Boussinesq fluids, the equality D(APE)=W_{r,turbulent} is at best a good approximation, for W_{r,turbulent} is generally smaller than D(APE), and sometimes even negative for a strongly nonlinear equation of state. In a second step, the link between stirring and mixing is examined for a wind-and buoyancy-driven thermally stratified ocean to determine whether these constrain the mechanical sources of stirring, as recently advocated. It is established that the coupling between stirring and mixing cannot refute the traditional buoyancy-driven view of the so-called meridional overturning circulation, in contrast to recent claims. In fact, the buoyancy forcing appears to be as important as the mechanical forcing in stirring and driving the large-scale ocean circulation.