samples/texts/1513018/page_12.md

and

The vertical velocities {$w_\alpha$}$_{\alpha=1}^N$ are defined by (27). For smooth solutions, the system (30)-(31) admits the energy balance

$$\frac{\mathrm{\partial }}{\mathrm{\partial }t}\sum _{\alpha =1}^N{E}_{\alpha }+{\mathrm{\nabla }}_{x,y}\cdot \sum _{\alpha =1}^N{\mathbf{u}}_{\alpha }(E_{\alpha }+\frac{g}{2}{h}_{\alpha }h-h_{\alpha }{\mathbf{\Sigma }}_{\alpha })=-\sum _{\alpha =1}^{N-1}\frac{\mathrm{\mid }{\mathbf{u}}_{\alpha +1}-{\mathbf{u}}_{\alpha }{\mathrm{\mid }}^2}{2}\mathrm{\mid }{G}_{\alpha +1\mathrm{/}2}\mathrm{\mid }-\sum _{\alpha =1}^{N-1}\frac{h_{\alpha +1}+h_{\alpha }}{2\nu }{\mathbf{\Sigma }}_{\alpha +1\mathrm{/}2}^2-\sum _{\alpha =1}^{N-1}2\nu \frac{\mathrm{\mid }{\mathbf{u}}_{\alpha +1}-{\mathbf{u}}_{\alpha }{\mathrm{\mid }}^2}{h_{\alpha +1}+h_{\alpha }}-\kappa \mathrm{\mid }{\mathbf{u}}_1{\mathrm{\mid }}^2+W{\mathbf{u}}_N\cdot {\mathbf{t}}_s,(37)\backslash frac\{\backslash partial\}\{\backslash partial\; t\}\; \backslash sum\_\{\backslash alpha=1\}^\{N\}\; E\_\{\backslash alpha\}\; +\; \backslash nabla\_\{x,y\}\; \backslash cdot\; \backslash sum\_\{\backslash alpha=1\}^\{N\}\; \backslash mathbf\{u\}\_\{\backslash alpha\}\; \backslash left(\; E\_\{\backslash alpha\}\; +\; \backslash frac\{g\}\{2\}\; h\_\{\backslash alpha\}\; h\; -\; h\_\{\backslash alpha\}\; \backslash boldsymbol\{\backslash Sigma\}\_\{\backslash alpha\}\; \backslash right)\; =\; -\; \backslash sum\_\{\backslash alpha=1\}^\{N-1\}\; \backslash frac\{|\backslash mathbf\{u\}\_\{\backslash alpha+1\}\; -\; \backslash mathbf\{u\}\_\{\backslash alpha\}|^2\}\{2\}\; |G\_\{\backslash alpha+1/2\}|\; \backslash \backslash \; -\; \backslash sum\_\{\backslash alpha=1\}^\{N-1\}\; \backslash frac\{h\_\{\backslash alpha+1\}\; +\; h\_\{\backslash alpha\}\}\{2\backslash nu\}\; \backslash boldsymbol\{\backslash Sigma\}\_\{\backslash alpha+1/2\}^2\; -\; \backslash sum\_\{\backslash alpha=1\}^\{N-1\}\; 2\backslash nu\; \backslash frac\{|\backslash mathbf\{u\}\_\{\backslash alpha+1\}\; -\; \backslash mathbf\{u\}\_\{\backslash alpha\}|^2\}\{h\_\{\backslash alpha+1\}\; +\; h\_\{\backslash alpha\}\}\; -\; \backslash kappa\; |\backslash mathbf\{u\}\_1|^2\; +\; W\; \backslash mathbf\{u\}\_N\; \backslash cdot\; \backslash mathbf\{t\}\_s,\; \backslash quad\; (37)$$

with $E_\alpha$ defined by (23) and $\Sigma_{\alpha+1/2}^2 = \sum_{i,j} \Sigma_{i,j,\alpha+1/2}^2$. Relation (37) is consistent with a layer-averaged discretization of the equation (10).

Notice that in (37), we use the notation

$${\mathbf{u}}_{\alpha }{\mathbf{\Sigma }}_{\alpha }=\left(\begin{array}{c}{u}_{\alpha }{\mathbf{\Sigma }}_{xx,\alpha }+v_{\alpha }{\mathbf{\Sigma }}_{yx,\alpha }\\ u_{\alpha }{\mathbf{\Sigma }}_{xy,\alpha }+v_{\alpha }{\mathbf{\Sigma }}_{yy,\alpha }\end{array}\right)\mathrm{.}\backslash mathbf\{u\}\_\{\backslash alpha\}\backslash boldsymbol\{\backslash Sigma\}\_\{\backslash alpha\}\; =\; \backslash begin\{pmatrix\}\; u\_\{\backslash alpha\}\backslash boldsymbol\{\backslash Sigma\}\_\{xx,\backslash alpha\}\; +\; v\_\{\backslash alpha\}\backslash boldsymbol\{\backslash Sigma\}\_\{yx,\backslash alpha\}\; \backslash \backslash \; u\_\{\backslash alpha\}\backslash boldsymbol\{\backslash Sigma\}\_\{xy,\backslash alpha\}\; +\; v\_\{\backslash alpha\}\backslash boldsymbol\{\backslash Sigma\}\_\{yy,\backslash alpha\}\; \backslash end\{pmatrix\}.$$

Proof of prop. 3.4 The proof is given in appendix A. ■

Remark 3.2 Notice that in the definition (35), since we consider viscous terms we use a centered approximation.

3.2.2 Simplified rheology

The viscous terms in the layer-averaged Navier-Stokes system are difficult to discretize especially when a discrete version of the energy balance has to be preserved. Hence, we propose a modified version of the model given in prop. 3.4.

Proposition 3.5 The layer-averaged Navier-Stokes can be written under the form

$$\begin{array}{cccc}& \sum _{\alpha =1}^N\frac{\mathrm{\partial }{h}_{\alpha }}{\mathrm{\partial }t}+\sum _{\alpha =1}^N{\mathrm{\nabla }}_{x,y}\mathrm{.}(h_{\alpha }{\mathbf{u}}_{\alpha })=0,& & \text{(38)}\end{array}\backslash sum\_\{\backslash alpha=1\}^\{N\}\; \backslash frac\{\backslash partial\; h\_\{\backslash alpha\}\}\{\backslash partial\; t\}\; +\; \backslash sum\_\{\backslash alpha=1\}^\{N\}\; \backslash nabla\_\{x,y\}.(h\_\{\backslash alpha\}\; \backslash mathbf\{u\}\_\{\backslash alpha\})\; =\; 0,\; \backslash tag\{38\}$$

and

$${\mathrm{\Sigma }}_{xx,\alpha +1\mathrm{/}2}^0=\frac{{\nu }_{\alpha +1\mathrm{/}2}}{h_{\alpha +1}+h_{\alpha }}(h_{\alpha }\frac{\mathrm{\partial }{u}_{\alpha }}{\mathrm{\partial }x}+h_{\alpha +1}\frac{\mathrm{\partial }{u}_{\alpha +1}}{\mathrm{\partial }x}),(40)\backslash Sigma\_\{xx,\backslash alpha+1/2\}^0\; =\; \backslash frac\{\backslash nu\_\{\backslash alpha+1/2\}\}\{h\_\{\backslash alpha+1\}\; +\; h\_\{\backslash alpha\}\}\; \backslash left(h\_\{\backslash alpha\}\; \backslash frac\{\backslash partial\; u\_\{\backslash alpha\}\}\{\backslash partial\; x\}\; +\; h\_\{\backslash alpha+1\}\; \backslash frac\{\backslash partial\; u\_\{\backslash alpha+1\}\}\{\backslash partial\; x\}\backslash right),\; \backslash quad\; (40)$$

$${\mathrm{\Sigma }}_{xy,\alpha +1\mathrm{/}2}^0=\frac{{\nu }_{\alpha +1\mathrm{/}2}}{h_{\alpha +1}+h_{\alpha }}(h_{\alpha }\frac{\mathrm{\partial }{u}_{\alpha }}{\mathrm{\partial }y}+h_{\alpha +1}\frac{\mathrm{\partial }{u}_{\alpha +1}}{\mathrm{\partial }y}),(41)\backslash Sigma^0\_\{xy,\backslash alpha+1/2\}\; =\; \backslash frac\{\backslash nu\_\{\backslash alpha+1/2\}\}\{h\_\{\backslash alpha+1\}\; +\; h\_\{\backslash alpha\}\}\; \backslash left(\; h\_\{\backslash alpha\}\; \backslash frac\{\backslash partial\; u\_\{\backslash alpha\}\}\{\backslash partial\; y\}\; +\; h\_\{\backslash alpha+1\}\; \backslash frac\{\backslash partial\; u\_\{\backslash alpha+1\}\}\{\backslash partial\; y\}\; \backslash right),\; \backslash quad\; (41)$$

and similar definitions for $\Sigma^0_{yx,\alpha+1/2}$, $\Sigma^0_{yy,\alpha+1/2}$. We also denote

$$T_{x,\alpha +1\mathrm{/}2}=\frac{2{\nu }_{\alpha +1\mathrm{/}2}}{h_{\alpha +1}+h_{\alpha }}{\mathrm{\nabla }}_{x,y}{z}_{\alpha +1\mathrm{/}2}(h_{\alpha }{\mathrm{\nabla }}_{x,y}{u}_{\alpha }+h_{\alpha +1}{\mathrm{\nabla }}_{x,y}{u}_{\alpha +1}),(42)T\_\{x,\backslash alpha+1/2\}\; =\; \backslash frac\{2\backslash nu\_\{\backslash alpha+1/2\}\}\{h\_\{\backslash alpha+1\}\; +\; h\_\{\backslash alpha\}\}\; \backslash nabla\_\{x,y\}\; z\_\{\backslash alpha+1/2\}\; (h\_\{\backslash alpha\}\; \backslash nabla\_\{x,y\}\; u\_\{\backslash alpha\}\; +\; h\_\{\backslash alpha+1\}\; \backslash nabla\_\{x,y\}\; u\_\{\backslash alpha+1\}),\; \backslash quad\; (42)$$