Boundary conditions on quasiStokes velocities in parameterisations
Killworth, P.D.. 2001 Boundary conditions on quasiStokes velocities in parameterisations. Journal of Physical Oceanography, 31 (4). 11321155.
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Abstract/Summary
This paper examines the implications for eddy parameterisations of expressing them in terms of the quasiStokes velocity. Another definition of lowpassed 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 secondorder 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 quasiStokes velocities and streamfunction also break down here. It is shown that the lowpassed 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 quasiStokes streamfunction appear to be nonzero right up to rigid boundaries. It is thus unclear whether more accurate results would be obtained by leaving the streamfunction nonzero on the boundary – which is smooth and resolvable – or by permitting a deltafunction in the horizontal quasiStokes 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 eddyresolving 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 quasiStokes 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 timemean density is compared with that from an eddyresolving 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) deltafunctions near the surface. The parameterisations were only successful if the vertical component of the quasiStokes velocity was required to vanish at top and bottom. A simple parameterisation of Eulerian density fluxes was, however, just as accurate and avoids deltafunction behaviour completely.
Item Type:  Publication  Article 

Additional Keywords:  WOCE, boundary conditions 
Date made live:  25 Nov 2003 +0 (UTC) 
URI:  https://nora.nerc.ac.uk/id/eprint/100246 
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