Numerical simulation of the neutral boundary layer: a comparison of enstrophy conserving with momentum conserving finite difference schemes

The application of integral constraints to a nonlinear system of dynamical equations is a method by which one can insure that a chaotic flow maintains its integrity over long-term integrations. Such constraints have been a mainstay of climate models, largely out of necessity to eliminate unrealistic long term trends in such quantities as enstrophy, moisture and so on. It is interesting that although large eddy simulations (LESs) deal with a similar long-term integration problem (i.e. simulations over many eddy lifetimes), there has been little attention paid to the integrity of the conservation properties of the underlying dynamics schemes. This may in part be due to the strong emphasis that has been placed upon the subgrid scale diffusion schemes that are typically used to complement the dynamics schemes and in effect compensate for many of their flaws, in addition to acting as a physical representation of subgrid scale turbulence.

Simulation of a classic Ekman boundary layer is ideal for studying how an LES conserves momentum, enstrophy, etc., because internal flow dynamics are completely inertial and do not involve the effect of stable stratification. Andren et al. (1994) compared the results (i.e. profiles of flux, variance, etc.) of four LES codes each simulating a classic Ekman boundary layer which is formed by surface friction and constant geostrophic wind. The classic Ekman solution occurs in a completely neutral atmosphere, involves a balance between simple local down-gradient subgrid scale transport, pressure gradient force, and Coriolis effect, and results in a spiral on a hodograph. We note that the classic Ekman PBL is a theoretical BL and never occurs exactly as such in the real atmosphere.

In this paper we study the performance of four advection schemes having varying degrees of imposed integral constraints on the simulation of the classic Ekman boundary layer. The 3-D LES of an Ekman boundary layer explicitly resolves eddies which may be responsible for nonlocal transport and thus may not necessarily result in a solution matching the classic Ekman spiral. The match may be closer depending on how much of the simulation is dominated by diffusion, be it physically defendable, or acting to compensate for numerical deficiencies.

The four schemes we compare emphasize (1) momentum conservation, (2) simple enstrophy conservation for nondivergent flow, (i.e. Sadourny, 1974), and (3) enstrophy conservation for divergent flow (i.e. Arakawa and Lamb 1981) and (4) a simple second order finite difference form of the equations in advective form. Because of the three dimensional nature of turbulent flow, both enstrophy conserving schemes are applied separately in all three dimensions, rather than only in the vertical as in the original climate model applications.

The results suggest that there is a strong impact from the imposition of integral constraints on the structure of the resulting boundary layer. In fact, only as the integral constraints are relaxed does the classic Ekman solution emerges, perhaps driven by the increased need for diffusion to combat the numerical noise. On the other hand, explicit enstrophy conservation tends to preserve the small grid scales forced by the bulk friction scheme, requiring explicit assistance from subgrid scale turbulence to smear large eddies resolvable sizes. Statistics, spectra and animations of the results will be given at the oral presentation.