The Temporal-Residual-Mean Velocity. Part II: Isopycnal Interpretation and the Tracer and Momentum Equations

Abstract

Mesoscale eddies mix fluid parcels in a way that is highly constrained by the stratified nature of the fluid. The temporal-residual-mean (TRM) theory provides the link between the different views that are apparent from temporally averaging these turbulent flow fields in height coordinates and in density coordinates. Here the original TRM theory is modified so that it applies to unsteady flows. This requires a modification not only to the streamfunction (and hence the velocity vector) but also a specific interpretation of the density field; it is not the Eulerian-mean density. The TRM theory reduces the problem of parameterizing the eddy flux from three dimensions to two dimensions. The three-dimensional TRM velocity is shown to be the same as is obtained by averaging with respect to instantaneous density surfaces and the averaged conservation equations in height coordinates and in density coordinates are the same except for a nondivergent flux that is identified and explained. The TRM theory demonstrates that the tracers (such as salinity and potential temperature) that are carried by an eddyless ocean model must be interpreted as the thickness-weighted tracers that result from averaging in density coordinates. The extra streamfunction of the temporal-residual-mean flow, termed the quasi-Stokes streamfunction, has a simple interpretation that proves valuable in developing plausible boundary conditions for this streamfunction: at any height z, the quasi-Stokes streamfunction is the contribution of temporal perturbations to the horizontal transport of water that is more dense than the density of the surface having time-mean height z. Importantly, the extra three-dimensional velocity derived from the quasi-Stokes streamfunction is not the bolus transport that arises when averaging in density coordinates. Therefore the Gent and McWilliams eddy parameterization scheme is not a parameterization of the bolus velocity but rather of the quasi-Stokes velocity of the temporal-residual-mean circulation. The physical interpretation of the quasi-Stokes streamfunction implies that it must be tapered smoothly to zero at the top and bottom of the ocean rather than having delta functions of velocity against these boundaries. The common assumption of downgradient flux of potential vorticity along isopycnals is discussed and it is shown that this does not sufficiently constrain the three-dimensional quasi-Stokes advection because only the vertical derivative of the quasi-Stokes streamfunction is specified. Near-boundary uncertainty in the potential vorticity fluxes translates into uncertainty in the depth-averaged heat flux. The horizontal TRM momentum equation is derived and leads to an alternative method for including the effects of eddies in eddyless models.