A note on the numerical representation of surface dynamics in quasigeostrophic turbulence: Application to the nonlinear eady model

The quasigeostrophic equations consist of the advection of linearized potential vorticity coupled with advection of temperature at the bounding upper and lower surfaces. Numerical models of quasigeostrophic flow often employ greater (scaled) resolution in the horizontal than in the vertical (the two-layer model is an extreme example). In the interior, this has the effect of suppressing interactions between layers at horizontal scales that are small compared to Nδz/f (where δz is the vertical resolution, N the buoyancy frequency, and f the Coriolis parameter). The nature of the turbulent cascade in the interior is, however, not fundamentally altered because the downscale cascade of potential enstrophy in quasigeostrophic turbulence and the downscale cascade of enstrophy in two-dimensional turbulence (occurring layerwise) both yield energy spectra with slopes of -3. It is shown here that a similar restriction on the vertical resolution applies to the representation of horizontal motions at the surfaces, but the penalty for under-resolving in the vertical is complete suppression of the surface temperature cascade at small scales and a corresponding artificial steepening of the surface energy spectrum. This effect is demonstrated in the nonlinear Eady model, using a finite-difference representation in comparison with a model that explicitly advects temperature at the upper and lower surfaces. Theoretical predictions for the spectrum of turbulence in the nonlinear Eady model are reviewed and compared to the simulated flows, showing that the latter model yields an accurate representation of the cascade dynamics. To accurately represent dynamics at horizontal wavenumber K in the vertically finite-differenced model, it is found that the vertical grid spacing must satisfy δz ≤0.3f/(NK); at wavenumbers K > 0.3f/(Nδz), the spectrum of temperature variance rolls off rapidly.