This paper examines the implications for eddy parameterizations of expressi
ng them in terms of the quasi-Stokes velocity. Another definition of low-pa
ssed time-averaged mean density (the modified mean) must be used, which is
the inversion of the mean depth of a given isopycnal. This definition natur
ally yields lighter (denser) fluid at the surface (floor) than the Eulerian
mean since fluid with these densities occasionally occurs at these locatio
ns. The difference between the two means is second order in perturbation am
plitude, and so small, in the fluid interior (where formulas to connect the
two exist). Near horizontal boundaries, the differences become first order
, and so more severe. Existing formulas for quasi-Stokes velocities and str
eamfunction also break down here. It is shown that the low-passed time-mean
potential energy in a closed box is incorrectly computed from modified mea
n density, the error term involving averaged quadratic variability.
The layer in which the largest differences occur between the two mean densi
ties is the vertical excursion of a mean isopycnal across a deformation rad
ius, at most about 20 m thick. Most climate models would have difficulty in
resolving such a layer. It is shown here that extant parameterizations app
ear to reproduce the Eulerian, and not modified mean, density field and so
do not yield a narrow layer at surface and floor either. Both these feature
s make the quasi-Stokes streamfunction appear to be nonzero right up to rig
id boundaries. It is thus unclear whether more accurate results would be ob
tained by leaving the streamfunction nonzero on the boundary-which is smoot
h and resolvable-or by permitting a delta function in the horizontal quasi-
Stokes velocity by forcing the streamfunction to become zero exactly at the
boundary (which it formally must be), but at the cost of small and unresol
vable features in the solution.
This paper then uses linear stability theory and diagnosed values from eddy
-resolving models, to ask the question: if climate models cannot or do not
resolve the difference between Eulerian and modified mean density, what are
the relevant surface and floor quasi-Stokes streamfunction conditions and
what are their effects on the density fields?
The linear Eady problem is used as a special case to investigate this since
terms can be explicitly computed. A variety of eddy parameterizations is e
mployed for a channel problem, and the time-mean density is compared with t
hat from an eddy-resolving calculation. Curiously, although most of the par
ameterizations employed are formally valid only in terms of the modified de
nsity, they all reproduce only the Eulerian mean density successfully. This
is despite the existence of (numerical) delta functions near the surface.
The parameterizations were only successful if the vertical component of the
quasi-Stokes velocity was required to vanish at top and bottom. A simple p
arameterization of Eulerian density fluxes was, however, just as accurate a
nd avoids delta-function behavior completely.