Considerations in the Calculation of Vertical Velocity in Three-Dimensional Circulation Models

The vertical velocity, w, in three-dimensional circulation models is typically computed from the three-dimensional continuity equation given the free-surface elevation and depth-varying horizontal velocity. This problem appears to be overdetermined, since the continuity equation is first order, yet w must satisfy boundary conditions at both the free surface and the bottom. At least three methods have been previously proposed to compute w: (i) a ‘‘traditional’’ method that solves the continuity equation using only the bottom boundary condition, (ii) a ‘‘vertical derivative’’ method that solves the vertical derivative of the continuity equation using both boundary conditions, and (iii) an ‘‘adjoint’’ method that solves the continuity equation and both boundary conditions in a least squares sense. The latter solution is equivalent to the traditional solution plus a correction that varies linearly over the depth. It is shown here that the vertical derivative method is mathematically and physically inconsistent if discretized as previously proposed. However, if properly discretized it is equivalent to the adjoint method if the boundary conditions are weighted so that they are satisfied exactly. Furthermore, if the surface elevation and horizontal velocity fields satisfy the depth-integrated continuity equation locally, one of the boundary conditions is redundant. In this case, the traditional, adjoint, and properly discretized vertical derivative approaches yield the same results for w. If the elevation and horizontal velocity are not locally mass conserving, the mass error is transferred into w. This is important for models that do not guarantee local mass conservation, such as some finite element models.