A study of hydrodynamic mass-consistent models

Variational mass consistent models (MCM's) provide a solenoidal field V in a region Ω by minimizing a metric $||V-V^0||_{ΩS}^2$ subject to ∇ · V=0, where $S_{ij}=α _{i}^{2}δ _{ij}$ is a symmetric and positive definite matrix. A least-squares approach suggests that $S^{-1}$ can be estimated with a Gaussian statistics. This approach is not valid in general, instead, it is shown that MCM's constitute a general scheme to get solenoidal fields V where the $α _i $'s are distributed parameters that can be estimated by means regularization methods. The range of values of $α ^{2}=α _1^2/α_3^2$ considered in the literature is $[10^{-12},∞)$. It is shown that V becomes singular as $α ^{2}→ 0$ or ∞ and this behavior together with the loss of regularity of λ on the boundary of Ω produce a spurious sensitivity of the residual divergence, which was proposed to estimate the optimal ratio $α^2$. Several simulations with MCM's use the boundary condition ∂ λ/∂ n=0 but it is not valid general. A deduction of MCM's in terrain-following coordinates is given and it is shown that such coordinates may hide the use of inconsistent boundary conditions. Numerical examples illustrate significative changes in the field V when the condition ∂ λ/∂ n=0 is used.