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Replacing formally in the above equation $\mathbf{u}$ (resp. $w$) by $\mathbf{u}^N$ (resp. $w^N$) defined by (17) and performing an integration over the layer $L_1$ of the obtained relation yields

$h_1 w_1 = - \int_{z_b}^{z_{3/2}} \nabla_{x,y} \cdot \int_{z_b}^{z} \mathbf{u}_1 \, dz \, dz_1 = h_1 \nabla_{x,y} \cdot (z_b \mathbf{u}_1) - \frac{z_{3/2}^2 - z_b^2}{2} \nabla_{x,y} \cdot \mathbf{u}_1,$

i.e. $w_1 = \nabla_{x,y} \cdot (z_b \mathbf{u}1) - z_1 \nabla{x,y} \cdot \mathbf{u}_1$, corresponding to (27) for $\alpha = 1$. A similar computation for the layers $L_2, \dots, L_N$ proves the result (27) for $\alpha = 2, \dots, N$. A more detailed version of this proof is given in [24]. ■

3.2 The layer-averaged Navier-Stokes system

In paragraph 3.1, we have applied the layer-averaging to the Euler system, we now use the same process for the hydrostatic Navier-Stokes system. First, we consider the Navier-Stokes system (1)-(7) for a Newtonian fluid and then with a simplified rheology.

3.2.1 Complete model

The layer-averaging process applied to the Navier-Stokes system (1)-(7) leads to the following proposition.

Proposition 3.4 The layer-averaged hydrostatic Navier-Stokes system (1)-(7) is given by

$\begin{align} & \sum_{\alpha=1}^{N} \frac{\partial h_{\alpha}}{\partial t} + \sum_{\alpha=1}^{N} \nabla_{x,y} \cdot (h_{\alpha} \mathbf{u}_{\alpha}) = 0, \tag{30} \\ & \frac{\partial h_{\alpha} \mathbf{u}_{\alpha}}{\partial t} + \nabla_{x,y} \cdot (h_{\alpha} \mathbf{u}_{\alpha} \otimes \mathbf{u}_{\alpha}) + \nabla_{x,y} \cdot (\frac{g}{2} h h_{\alpha}) = -gh_{\alpha} \nabla_{x,y} z_{b} \nonumber \\ & + \mathbf{u}_{\alpha+1/2} G_{\alpha+1/2} - \mathbf{u}_{\alpha-1/2} G_{\alpha-1/2} + \nabla_{x,y} \cdot (h_{\alpha} \boldsymbol{\Sigma}_{\alpha}) \nonumber \\ & - \boldsymbol{\Sigma}_{\alpha+1/2} \nabla_{x,y} z_{b} z_{\alpha+1/2} + \boldsymbol{\Sigma}_{\alpha-1/2} \nabla_{x,y} z_{b} z_{\alpha-1/2} \nonumber \\ & + 2\nu_{\alpha+1/2} \frac{\mathbf{u}_{\alpha+1} - \mathbf{u}_{\alpha}}{h_{\alpha+1} + h_{\alpha}} - 2\nu_{\alpha-1/2} \frac{\mathbf{u}_{\alpha} - \mathbf{u}_{\alpha-1}}{h_{\alpha} + h_{\alpha-1}} - \kappa_{\alpha} \mathbf{u}_{\alpha} + W_{\alpha} \mathbf{t}_{s}, \quad \alpha = 1, \dots, N, \tag{31} \end{align}$

with

$\Sigma_{\alpha+1/2} = \begin{pmatrix} \Sigma_{xx,\alpha+1/2} & \Sigma_{xy,\alpha+1/2} \\ \Sigma_{yx,\alpha+1/2} & \Sigma_{yy,\alpha+1/2} \end{pmatrix}, \qquad (32)$

$\Sigma_{xx,\alpha+1/2} = \frac{\nu_{\alpha+1/2}}{h_{\alpha+1} + h_\alpha} \left(h_\alpha \frac{\partial u_\alpha}{\partial x} + h_{\alpha+1} \frac{\partial u_{\alpha+1}}{\partial x}\right) - 2\nu_{\alpha+1/2} \frac{\partial z_{\alpha+1/2}}{\partial x} \frac{u_{\alpha+1}-u_\alpha}{h_{\alpha+1}+h_\alpha}, \qquad (33)$

$\Sigma_{xy,\alpha+1/2} = \frac{\nu_{\alpha+1/2}}{h_{\alpha+1} + h_\alpha} \left( h_\alpha \frac{\partial u_\alpha}{\partial y} + h_{\alpha+1} \frac{\partial u_{\alpha+1}}{\partial y} \right) - 2\nu_{\alpha+1/2} \frac{\partial z_{\alpha+1/2}}{\partial y} \frac{u_{\alpha+1}-u_\alpha}{h_{\alpha+1}+h_\alpha}, \qquad (34)$

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

$\Sigma_\alpha = \begin{pmatrix} \Sigma_{xx,\alpha} & \Sigma_{xy,\alpha} \\ \Sigma_{yx,\alpha} & \Sigma_{yy,\alpha} \end{pmatrix} = \frac{\Sigma_{\alpha+1/2} + \Sigma_{\alpha-1/2}}{2}, \qquad (35)$