Lateral Friction in Reduced-Gravity Models: Parameterizations Consistent with Energy Dissipation and Conservation of Angular Momentum

The f-plane reduced-gravity model has been extended with the parameterization of lateral friction in the momentum equations. The parameterization should preferably fulfill some requisites. One of them is that in the absence of external torques the change in angular momentum should be determined by boundary conditions alone. Internal torques should balance in the angular momentum budget. This requirement is fulfilled when the parameterization in the vertically integrated momentum equations is the divergence of a symmetric stress tensor. These equations solve for the mean transport, which includes implicitly the eddy-induced contribution. Another requirement on the parameterization of lateral stress follows from considering that it should imply kinetic energy dissipation. Both requirements fail with a commonly used parameterization and are fulfilled with the one proposed by C. Schär and R. B. Smith, which also is in near agreement with derivations of the shallow-water equations via vertical integrations of the Navier–Stokes equations. Here, the authors show two other parameterizations that are consistent with the angular momentum and energy requirements. One of the parameterizations follows from the symmetric component of a stress tensor in agreement with the parameterization shown by P. R. Gent to be energetically consistent. The other parameterization is related to the so-called biharmonic dissipation. In general, the difficulty for friction parameterizations is on the energy dissipation requirement, because the one on angular momentum is easily fulfilled.