A study of hydrodynamic mass-consistent models

Abstract

Variational mass consistent models (MCM’s) provide a solenoidal field V in a region Ω by minimizing a metric $||\mathbf{V}-{\mathbf{V}}^0|{|}_{\mathrm{\Omega }\mathit{S}}^2$ subject to $\nabla ·\mathbf{V}=0$ , where ${\mathit{S}}_{\mathit{i}\mathit{j}}={\mathit{\alpha }}_{\mathit{i}}^2{\mathit{\delta }}_{\mathit{i}\mathit{j}}$ is a symmetric and positive definite matrix. A least-squares approach suggests that ${\mathit{S}}^{-1}$ 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 ${\mathit{\alpha }}_{\mathit{i}}$ ’s are distributed parameters that can be estimated by means regularization methods. The range of values of ${\mathit{\alpha }}^2={\mathit{\alpha }}_1^2\mathrm{/}{\mathit{\alpha }}_3^2$ considered in the literature is $[{10}^{-12}\mathrm{,}\infty \mathrm{)}$ . It is shown that V becomes singular as ${\mathit{\alpha }}^2\to 0$ 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 ${\mathit{\alpha }}^2$ . Several simulations with MCM’s use the boundary condition $\partial \mathit{\lambda }\mathrm{/}\partial \mathit{n}=0$ 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 $\partial \mathit{\lambda }\mathrm{/}\partial \mathit{n}=0$ is used.