We have found a new way to express the solutions of the RSM (Reynolds Stress
Model) equations that allows us to present the turbulent diffusivities for heat, salt and
momentum in a way that is considerably simpler and thus easier to implement than in
previous work. The RSM provides the dimensionless mixing efficiencies ?? (? stands for
heat, salt and momentum). However, to compute the diffusivities, one needs additional
information, specifically, the dissipation ?. Since a dynamic equation for the latter that
includes the physical processes relevant to the ocean is still not available, one must resort
to different sources of information outside the RSM to obtain a complete Mixing Scheme
usable in OGCMs.
As for the RSM results, we show that the ??’s are functions of both Ri and R?
(Richardson number and density ratio representing double diffusion, DD); the ?? are
different for heat, salt and momentum; in the case of heat, the traditional value ?h = 0.2
is valid only in the presence of strong shear (when DD is inoperative) while when shear
subsides, NATRE data show that ?h can be three times as large, a result that we
reproduce. The salt ?s is given in terms of ?h. The momentum ?m has thus far been
guessed with different prescriptions while the RSM provides a well defined expression
for ?m (Ri, R?). Having tested ?h, we then test the momentum ?m by showing that the
turbulent Prandtl number ?m/?h vs. Ri reproduces the available data quite well.
As for the dissipation ?, we use different representations, one for the mixed layer
(ML), one for the thermocline and one for the ocean’s bottom. For the ML, we adopt a
procedure analogous to the one successfully used in PB (planetary boundary layer)
studies; for the thermocline, we employ an expression for the variable ?N-2 from studies
of the internal gravity waves spectra which includes a latitude dependence; for the ocean
bottom, we adopt the enhanced bottom diffusivity expression used by previous authors
but with a state of the art internal tidal energy formulation and replace the fixed ?? = 0.2
with the RSM result that brings into the problem the Ri,R? dependence of the ??; the
unresolved bottom drag, which has thus far been either ignored or modeled with heuristic
relations, is modeled using a formalism we previously developed and tested in PBL
We carried out several tests without an OGCM. Prandtl and flux Richardson
numbers vs. Ri. The RSM model reproduces both types of data satisfactorily. DD and
Mixing efficiency ?h (Ri, R?). The RSM model reproduces well the NATRE data.
Bimodal ?-distribution. NATRE data show that ? (Ri<1) ?10?(Ri>1), which our model
reproduces. Heat to salt flux ratio. In the Ri>>1 regime, the RSM predictions reproduce
the data satisfactorily. NATRE mass diffusivity. The z-profile of the mass diffusivity
reproduces well the measurements at NATRE. The local form of the mixing scheme is
algebraic with one cubic equation to solve.