Boundary conditions on quasi-Stokes velocities in parameterisations

Abstract

This paper examines the implications for eddy parameterisations of expressing them in terms of the quasi-Stokes velocity. Another definition of low-passed time averaged mean density (the modified mean) must be used, which is the inversion of the mean depth of a given isopycnal. This definition naturally yields lighter (denser) fluid at the surface (floor) than the Eulerian mean, since fluid with these densities occasionally occurs at these locations. The difference between the two means is second-order in perturbation amplitude, and so small, in the fluid interior (where formulae to connect the two exist). Near horizontal boundaries, the differences become first order, and so more severe. Existing formulae for quasi-Stokes velocities and streamfunction also break down here. It is shown that the low-passed time mean potential energy in a closed box is incorrectly computed from modified mean density, the error term involving averaged quadratic variability. The layer in which the largest differences occur between the two mean densities is the vertical excursion of a mean isopycnal across a deformation radius, at most about 20 m thick. Most climate models would have difficulty in resolving such a layer. We show here that extant parameterisations appear 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 features make the quasi-Stokes streamfunction appear to be non-zero right up to rigid boundaries. It is thus unclear whether more accurate results would be obtained by leaving the streamfunction non-zero on the boundary – which is smooth 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 unresolvable 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 parameterisations is employed for a channel problem, and the time-mean density is compared with that from an eddy-resolving calculation. Curiously, although most of the parameterisations employed are formally valid only in terms of the modified density, they all reproduce only the Eulerian mean density successfully. This is despite the existence of (numerical) delta-functions near the surface. The parameterisations were only successful if the vertical component of the quasi-Stokes velocity was required to vanish at top and bottom. A simple parameterisation of Eulerian density fluxes was, however, just as accurate and avoids delta-function behaviour completely.

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