samples/texts/1513018/page_10.md

with $\mathbf{U}^N = (u^N, v^N, w^N)^T$. Simple computations give

$$0={\int }_{\mathbb{R}}{\mathbf{1}}_{z\in L_{\alpha }(t,x,y)}\mathrm{\nabla }\cdot {\mathbf{U}}^{N}dz=\frac{\mathrm{\partial }{h}_{\alpha }}{\mathrm{\partial }t}+\frac{\mathrm{\partial }}{\mathrm{\partial }x}{\int }_{z_{\alpha -1\mathrm{/}2}}^{z_{\alpha +1\mathrm{/}2}}udz+\frac{\mathrm{\partial }}{\mathrm{\partial }y}{\int }_{z_{\alpha -1\mathrm{/}2}}^{z_{\alpha +1\mathrm{/}2}}vdz-G_{\alpha +1\mathrm{/}2}+G_{\alpha -1\mathrm{/}2},0\; =\; \backslash int\_\{\backslash mathbb\{R\}\}\; \backslash mathbf\{1\}\_\{z\; \backslash in\; L\_\backslash alpha(t,x,y)\}\; \backslash nabla\; \backslash cdot\; \backslash mathbf\{U\}^N\; dz\; =\; \backslash frac\{\backslash partial\; h\_\backslash alpha\}\{\backslash partial\; t\}\; +\; \backslash frac\{\backslash partial\}\{\backslash partial\; x\}\; \backslash int\_\{z\_\{\backslash alpha-1/2\}\}^\{z\_\{\backslash alpha+1/2\}\}\; u\; \backslash ,\; dz\; +\; \backslash frac\{\backslash partial\}\{\backslash partial\; y\}\; \backslash int\_\{z\_\{\backslash alpha-1/2\}\}^\{z\_\{\backslash alpha+1/2\}\}\; v\; \backslash ,\; dz\; -\; G\_\{\backslash alpha+1/2\}\; +\; G\_\{\backslash alpha-1/2\},$$

leading to

$$\frac{\mathrm{\partial }{h}_{\alpha }}{\mathrm{\partial }t}+{\mathrm{\nabla }}_{x,y}(h_{\alpha }{\mathbf{u}}_{\alpha })=G_{\alpha +1\mathrm{/}2}-G_{\alpha -1\mathrm{/}2},(28)\backslash frac\{\backslash partial\; h\_\{\backslash alpha\}\}\{\backslash partial\; t\}\; +\; \backslash nabla\_\{x,y\}\; (h\_\{\backslash alpha\}\; \backslash mathbf\{u\}\_\{\backslash alpha\})\; =\; G\_\{\backslash alpha+1/2\}\; -\; G\_\{\backslash alpha-1/2\},\; \backslash quad\; (28)$$

with $G_{\alpha \pm 1/2}$ defined by

$$G_{\alpha +1\mathrm{/}2}=\frac{\mathrm{\partial }{z}_{\alpha +1\mathrm{/}2}}{\mathrm{\partial }t}+{\mathbf{u}}_{\alpha +1\mathrm{/}2}\cdot {\mathrm{\nabla }}_{x,y}{z}_{\alpha +1\mathrm{/}2}-w_{\alpha +1\mathrm{/}2}\mathrm{.}G\_\{\backslash alpha+1/2\}\; =\; \backslash frac\{\backslash partial\; z\_\{\backslash alpha+1/2\}\}\{\backslash partial\; t\}\; +\; \backslash mathbf\{u\}\_\{\backslash alpha+1/2\}\; \backslash cdot\; \backslash nabla\_\{x,y\}\; z\_\{\backslash alpha+1/2\}\; -\; w\_\{\backslash alpha+1/2\}.$$

The sum for $\alpha = 1, \dots, N$ of the above relations gives Eq. (18) where the kinematic boundary conditions (4), (5) corresponding to

$$G_{1\mathrm{/}2}=G_{N+1\mathrm{/}2}=0,(29)G\_\{1/2\}\; =\; G\_\{N+1/2\}\; =\; 0,\; \backslash quad\; (29)$$

have been used. Similarly, the sum for $j = 1, \dots, \alpha$ of the relations (28) with (29) gives the expression (20) for $G_{\alpha+1/2}$.

Now we consider the Galerkin approximation of Eq. (13) i.e. the quantity

$${\int }_{\mathbb{R}}{\mathbf{1}}_{z\in L_{\alpha }(t,x,y)}(\frac{\mathrm{\partial }{\mathbf{u}}^N}{\mathrm{\partial }t}+{\mathrm{\nabla }}_{x,y}({\mathbf{u}}^N\otimes {\mathbf{u}}^N)+\frac{\mathrm{\partial }{\mathbf{u}}^N{w}^N}{\mathrm{\partial }z}+g{\mathrm{\nabla }}_{x,y}\eta )dz=0,\backslash int\_\{\backslash mathbb\{R\}\}\; \backslash mathbf\{1\}\_\{z\; \backslash in\; L\_\backslash alpha(t,x,y)\}\; \backslash left(\; \backslash frac\{\backslash partial\; \backslash mathbf\{u\}^N\}\{\backslash partial\; t\}\; +\; \backslash nabla\_\{x,y\}\; (\backslash mathbf\{u\}^N\; \backslash otimes\; \backslash mathbf\{u\}^N)\; +\; \backslash frac\{\backslash partial\; \backslash mathbf\{u\}^N\; w^N\}\{\backslash partial\; z\}\; +\; g\; \backslash nabla\_\{x,y\}\; \backslash eta\; \backslash right)\; dz\; =\; 0,$$

leading, after simple computations, to Eq. (19). $\blacksquare$

Proof of prop. 3.2 In order to obtain (22) we multiply (28) by $g(h+z_b)$ ($|\mathbf{u}_\alpha|^2/2$ and (19) by $\mathbf{u}_\alpha$ then we perform simple manipulations. More precisely, the momentum equation along the x axis multiplied by $u_\alpha$ gives

$$(\frac{\mathrm{\partial }}{\mathrm{\partial }t}(h_{\alpha }{u}_{\alpha })+\frac{\mathrm{\partial }}{\mathrm{\partial }x}(h_{\alpha }{u}_{\alpha }^2+\frac{g}{2}h{h}_{\alpha })+\frac{\mathrm{\partial }}{\mathrm{\partial }y}(h_{\alpha }{u}_{\alpha }{v}_{\alpha }))u_{\alpha }=(-g{h}_{\alpha }\frac{\mathrm{\partial }{z}_b}{\mathrm{\partial }x}+u_{\alpha +1\mathrm{/}2}{G}_{\alpha +1\mathrm{/}2}-u_{\alpha -1\mathrm{/}2}{G}_{\alpha -1\mathrm{/}2})u_{\alpha }\mathrm{.}\backslash left(\backslash frac\{\backslash partial\}\{\backslash partial\; t\}(h\_\backslash alpha\; u\_\backslash alpha)\; +\; \backslash frac\{\backslash partial\}\{\backslash partial\; x\}\backslash left(h\_\backslash alpha\; u\_\backslash alpha^2\; +\; \backslash frac\{g\}\{2\}hh\_\backslash alpha\backslash right)\; +\; \backslash frac\{\backslash partial\}\{\backslash partial\; y\}(h\_\backslash alpha\; u\_\backslash alpha\; v\_\backslash alpha)\backslash right)u\_\backslash alpha\; =\; \backslash \backslash \; \backslash left(-gh\_\backslash alpha\; \backslash frac\{\backslash partial\; z\_b\}\{\backslash partial\; x\}\; +\; u\_\{\backslash alpha+1/2\}G\_\{\backslash alpha+1/2\}\; -\; u\_\{\backslash alpha-1/2\}G\_\{\backslash alpha-1/2\}\backslash right)u\_\backslash alpha.$$

Considering first the left hand side of the preceding equation excluding the pressure terms, we denote

$$I_{u,\alpha }=(\frac{\mathrm{\partial }}{\mathrm{\partial }t}(h_{\alpha }{u}_{\alpha })+\frac{\mathrm{\partial }}{\mathrm{\partial }x}(h_{\alpha }{u}_{\alpha }^2)+\frac{\mathrm{\partial }}{\mathrm{\partial }y}(h_{\alpha }{u}_{\alpha }{v}_{\alpha }))u_{\alpha },I\_\{u,\backslash alpha\}\; =\; \backslash left(\; \backslash frac\{\backslash partial\}\{\backslash partial\; t\}(h\_\backslash alpha\; u\_\backslash alpha)\; +\; \backslash frac\{\backslash partial\}\{\backslash partial\; x\}(h\_\backslash alpha\; u\_\backslash alpha^2)\; +\; \backslash frac\{\backslash partial\}\{\backslash partial\; y\}(h\_\backslash alpha\; u\_\backslash alpha\; v\_\backslash alpha)\; \backslash right)\; u\_\backslash alpha,$$

and using (28) we have

$$I_{u,\alpha }=\frac{\mathrm{\partial }}{\mathrm{\partial }t}\left(\frac{h_{\alpha }{u}_{\alpha }^2}{2}\right)+\frac{\mathrm{\partial }}{\mathrm{\partial }x}\left(u_{\alpha }\frac{h_{\alpha }{u}_{\alpha }^2}{2}\right)+\frac{\mathrm{\partial }}{\mathrm{\partial }y}\left(v_{\alpha }\frac{h_{\alpha }{u}_{\alpha }^2}{2}\right)+\frac{u_{\alpha }^2}{2}(\frac{\mathrm{\partial }{h}_{\alpha }}{\mathrm{\partial }t}+{\mathrm{\nabla }}_{x,y}(h_{\alpha }{\mathbf{u}}_{\alpha }))\mathrm{.}I\_\{u,\backslash alpha\}\; =\; \backslash frac\{\backslash partial\}\{\backslash partial\; t\}\; \backslash left(\; \backslash frac\{h\_\backslash alpha\; u\_\backslash alpha^2\}\{2\}\; \backslash right)\; +\; \backslash frac\{\backslash partial\}\{\backslash partial\; x\}\; \backslash left(\; u\_\backslash alpha\; \backslash frac\{h\_\backslash alpha\; u\_\backslash alpha^2\}\{2\}\; \backslash right)\; +\; \backslash frac\{\backslash partial\}\{\backslash partial\; y\}\; \backslash left(\; v\_\backslash alpha\; \backslash frac\{h\_\backslash alpha\; u\_\backslash alpha^2\}\{2\}\; \backslash right)\; +\; \backslash frac\{u\_\backslash alpha^2\}\{2\}\; \backslash left(\; \backslash frac\{\backslash partial\; h\_\backslash alpha\}\{\backslash partial\; t\}\; +\; \backslash nabla\_\{x,y\}\; (h\_\backslash alpha\; \backslash mathbf\{u\}\_\backslash alpha)\; \backslash right).$$

Now we consider the contribution of the pressure terms over the energy balance i.e.

$$I_{p,u,\alpha }=(\frac{\mathrm{\partial }}{\mathrm{\partial }x}\left(\frac{g}{2}h{h}_{\alpha }\right)+g{h}_{\alpha }\frac{\mathrm{\partial }{z}_b}{\mathrm{\partial }x})u_{\alpha },I\_\{p,u,\backslash alpha\}\; =\; \backslash left(\; \backslash frac\{\backslash partial\}\{\backslash partial\; x\}\; \backslash left(\; \backslash frac\{g\}\{2\}\; hh\_\{\backslash alpha\}\; \backslash right)\; +\; gh\_\{\backslash alpha\}\; \backslash frac\{\backslash partial\; z\_\{b\}\}\{\backslash partial\; x\}\; \backslash right)\; u\_\{\backslash alpha\},$$

and it comes

Performing the same manipulations over the momentum equation along $y$ and adding the terms $I_{u,\alpha}, I_{v,\alpha}, I_{p,u,\alpha}, I_{p,v,\alpha}$ and (28) multiplied by $g(h+z_b)-|\mathbf{u}_\alpha|^2/2$ gives the result. $\blacksquare$