Enstrophy and energy cascades in numerical models

John Thuburn
University of Exeter

John Thuburn (University of Exeter)
James Kent (University of Exeter)
Nigel Wood (Met Office)

Observations and diagnostic studies show that in the atmosphere on synoptic scales energy is transferred predominantly upscale while potential enstrophy is transferred predominantly downscale by turbulent dynamics, in qualitative agreement with idealized turbulence theory. Numerical models of the atmosphere require some scale-selective dissipation to absorb the downscale potential enstrophy cascade; it may be inherent in the numerics (e.g. in the interpolation used with semi-Lagrangian advection) or added as a separate term (e.g.
the del^4 dissipation typically used with spectral methods). This dissipation may be regarded as a subgrid model for the effects of unresolved eddies. This talk will address the question of whether it is an adequate subgrid model.

The barotropic vorticity equation is used as a model problem to study the effects of eddies smaller than a certain cut-off scale on the energy and enstrophy spectra at larger scales. Typically, enstrophy is transferred from scales near the cut-off to scales smaller than the cut-off, while energy is transferred from scales near the cut-off to the much larger energetically dominant scales. These effects are compared with those of typical numerical methods when the cut-off scale is the resolution limit.
Numerical methods can remove the required amount of enstrophy from small resolved scales, but tend to be insufficiently scale-selective. They also remove energy from resolved scales (again being insufficiently scale-selective), but do not replace it at the energetically dominant scales, leading to poor energy conservation over long times. These problems may be partly compensated by including a parameterization of energy `backscatter'.

Presentation (PDF File)

Back to Workshop II: Numerical Hierarchies for Climate Modeling