On the form of the viscous term for two dimensional Navier-Stokes flows

Abstract

The form of the viscous term is discussed for incompressible flow on a two-dimensional curved surface S and for the shallow water equations. In the case of flow on a surface three versions are considered. These correspond to taking curl twice, to applying the Laplacian defined in terms of a metric, and to taking the divergence of a symmetric stress tensor. These differ on a curved surface, for example a sphere. The three terms are related and their properties discussed, in particular energy and angular momentum conservation. In the case of the shallow water equations again three forms of dissipation are considered, the last of which involves the divergence of a stress tensor. Their properties are discussed, including energy conservation and whether the rotating bucket solution of the three-dimensional Navier–Stokes equation is reproduced. A derivation of the viscous term is also given based on shallow water equations as a truncation of the Navier–Stokes equation, with forces on a column determined by integration over the vertical. For both incompressible flow on a surface and for the shallow water equations, it is argued that a viscous term based on a symmetric stress tensor should be used as this leads to correct treatment of angular momentum.