# Multilayer Saint-Venant system with mass exchanges

The Saint-Venant system is extended to multiple layers following Audusse et al, 2011 as ${\partial }_{t}h+{\partial }_{x}\sum _{l=0}^{nl-1}{h}_{l}{u}_{l}=0$ with ${h}_{l}={\mathrm{\text{layer}}}_{l}h$ with ${\mathrm{\text{layer}}}_{l}$ the relative thickness of the layers satisfying ${\mathrm{\text{layer}}}_{l}>=0,\phantom{\rule{0.278em}{0ex}}\sum _{l=0}^{nl-1}{\mathrm{\text{layer}}}_{l}=1.$ The momentum equation in each layer is thus ${\partial }_{t}\left(h{\mathbf{\text{u}}}_{l}\right)+\nabla \cdot \left(h{\mathbf{\text{u}}}_{l}\otimes {\mathbf{\text{u}}}_{l}+\frac{g{h}^{2}}{2}\mathbf{\text{I}}\right)=-gh\nabla {z}_{b}+\frac{1}{{\mathrm{\text{layer}}}_{l}}\left[{\mathbf{\text{u}}}_{l+1/2}{G}_{l+1/2}-{\mathbf{\text{u}}}_{l-1/2}{G}_{l-1/2}+\nu \left(\frac{{u}_{l+1}-{u}_{l}}{{h}_{l+1/2}}-\frac{{u}_{l}-{u}_{l-1}}{{h}_{l-1/2}}\right)\right]$ where ${G}_{l+1/2}$ is the relative vertical transport velocity between layers and the second term corresponds to viscous friction between layers.

These last two terms are the only difference with the one layer system.

The horizontal velocity in each layer is stored in ul and the vertical velocity between layers in wl.

vector * ul = NULL;
scalar * wl = NULL;
double * layer;

## Viscous friction between layers

Boundary conditions on the top and bottom layers need to be added to close the system for the viscous stresses. We chose to impose a Neumann condition on the top boundary i.e. ${\partial }_{z}u{\mid }_{t}={\stackrel{̇}{u}}_{t}$ and a Navier slip condition on the bottom i.e. $u{\mid }_{b}={u}_{b}+{\lambda }_{b}{\partial }_{z}u{\mid }_{b}$ By default the viscosity is zero and we impose free-slip on the top boundary and no-slip on the bottom boundary i.e. ${\stackrel{̇}{u}}_{t}=0$, ${\lambda }_{b}=0$, ${u}_{b}=0$.

double ν = 0.;
(const) scalar lambda_b = zeroc, dut = zeroc, u_b = zeroc;

For stability, we discretise the viscous friction term implicitly as $\frac{\left(h{u}_{l}{\right)}_{n+1}-\left(h{u}_{l}{\right)}_{\star }}{\Delta t}=\frac{\nu }{{\mathrm{\text{layer}}}_{l}}{\left(\frac{{u}_{l+1}-{u}_{l}}{{h}_{l+1/2}}-\frac{{u}_{l}-{u}_{l-1}}{{h}_{l-1/2}}\right)}_{n+1}$ which can be expressed as the linear system ${\mathbf{\text{Mu}}}_{n+1}=\mathrm{\text{rhs}}$ where $\mathbf{\text{M}}$ is a tridiagonal matrix. The lower, principal and upper diagonals are a, b and c respectively.

void vertical_viscosity (Point point, double h, vector * ul, double dt)
{
if (ν == 0.)
return;

double a[nl], b[nl], c[nl], rhs[nl];

foreach_dimension() {

The rhs of the tridiagonal system is ${h}_{l}{u}_{l}=h{\mathrm{\text{layer}}}_{l}{u}_{l}$.

int l = 0;
for (vector u in ul)
rhs[l] = h*layer[l]*u.x[], l++;

The lower, principal and upper diagonals $a$, $b$ and $c$ are given by ${a}_{l>0}=-{\left(\frac{\nu \Delta t}{{h}_{l-1/2}}\right)}_{n+1}$ ${c}_{l<\mathrm{\text{nl}}-1}=-{\left(\frac{\nu \Delta t}{{h}_{l+1/2}}\right)}_{n+1}$ ${b}_{0

for (l = 1; l < nl - 1; l++) {
a[l] = - 2.*ν*dt/(h*(layer[l-1] + layer[l]));
c[l] = - 2.*ν*dt/(h*(layer[l] + layer[l+1]));
b[l] = layer[l]*h - a[l] - c[l];
}

For the top layer the boundary conditions give the (ghost) boundary value ${u}_{\mathrm{\text{nl}}}={u}_{\mathrm{\text{nl}}-1}+{\stackrel{̇}{u}}_{t}{h}_{\mathrm{\text{nl}}-1},$ which gives the diagonal coefficient and right-hand-side ${b}_{\mathrm{\text{nl}}-1}={\mathrm{\text{layer}}}_{\mathrm{\text{nl}}-1}{h}_{n+1}-{a}_{\mathrm{\text{nl}}-1}$ ${\mathrm{\text{rhs}}}_{\mathrm{\text{nl}}-1}={\mathrm{\text{layer}}}_{\mathrm{\text{nl}}-1}\left(h{u}_{\mathrm{\text{nl}}-1}{\right)}_{\star }+\nu \Delta t{\stackrel{̇}{u}}_{t}$

a[nl-1] = - 2.*ν*dt/(h*(layer[nl-2] + layer[nl-1]));
b[nl-1] = layer[nl-1]*h - a[nl-1];
rhs[nl-1] += ν*dt*dut[];

For the bottom layer, the boundary conditions give the (ghost) boundary value ${u}_{-1}$ ${u}_{-1}=\frac{2{h}_{0}}{2{\lambda }_{b}+{h}_{0}}{u}_{b}+\frac{2{\lambda }_{b}-{h}_{0}}{2{\lambda }_{b}+{h}_{0}}{u}_{0},$ which gives the diagonal coefficient and right-hand-side ${b}_{0}={\mathrm{\text{layer}}}_{0}{h}_{n+1}-{c}_{0}+\frac{2\nu \Delta t}{2{\lambda }_{b}+{h}_{0}}$ ${\mathrm{\text{rhs}}}_{0}={\mathrm{\text{layer}}}_{0}\left(h{u}_{0}{\right)}_{\star }+\frac{2\nu \Delta t}{2{\lambda }_{b}+{h}_{0}}{u}_{b}$

c[0] = - 2.*dt*ν/(h*(layer[0] + layer[1]));
b[0] = layer[0]*h - c[0] + 2.*ν*dt/(2.*lambda_b[] + h*layer[0]);
rhs[0] += 2.*ν*dt/(2.*lambda_b[] + h*layer[0])*u_b[];

We can now solve the tridiagonal system using the Thomas algorithm.

for (l = 1; l < nl; l++) {
b[l] -= a[l]*c[l-1]/b[l-1];
rhs[l] -= a[l]*rhs[l-1]/b[l-1];
}
vector u = ul[nl-1];
u.x[] = a[nl-1] = rhs[nl-1]/b[nl-1];
for (l = nl - 2; l >= 0; l--) {
u = ul[l];
u.x[] = a[l] = (rhs[l] - c[l]*a[l+1])/b[l];
}
}
}

## Fluxes between layers

The relative vertical velocity between layers $l$ and $l+1$ is defined as (eq. (2.22) of Audusse et al, 2011) ${G}_{l+1/2}=\sum _{j=0}^{l}\left({\mathrm{\text{div}}}_{j}+{\mathrm{\text{layer}}}_{j}\mathrm{\text{dh}}\right)$ with ${\mathrm{\text{div}}}_{l}=\nabla \cdot \left({h}_{l}{\mathbf{\text{u}}}_{l}\right)$ $\mathrm{\text{dh}}=-\sum _{l=0}^{nl-1}{\mathrm{\text{div}}}_{l}$

void vertical_fluxes (vector * evolving, vector * updates,
scalar * divl, scalar dh)
{
foreach() {
double Gi = 0., sumjl = 0.;
for (int l = 0; l < nl - 1; l++) {
scalar div = divl[l];
Gi += div[] + layer[l]*dh[];
sumjl += layer[l];
scalar w = div;
w[] = dh[]*sumjl - Gi;
foreach_dimension() {

To compute the vertical advection term, we need an estimate of the velocity at $l+1/2$. This is obtained using simple upwinding according to the sign of the interface velocity $\mathrm{\text{Gi}}={G}_{l+1/2}$ and the values of the velocity in the $l$ and $l+1$ layers. Note that the inequality of upwinding is consistent with equs. (5.110) of Audusse et al, 2011 and (77) of Audusse et al, 2011b but not with eq. (2.23) of Audusse et al, 2011.

scalar ub = evolving[l].x, ut = evolving[l + 1].x;
double ui = Gi < 0. ? ub[] : ut[];

The flux at $l+1/2$ is then added to the updates of the bottom layer and substracted from the updates of the top layer.

double flux = Gi*ui;
To compute the vertical velocity we use the definition of the mass flux term (eq. 2.13 of Audusse et al, 2011): $\mathrm{\text{w}}\left(\mathbf{\text{x}},{z}_{l+1/2}\right)={\partial }_{t}{z}_{l+1/2}-{G}_{l+1/2}+{\mathbf{\text{u}}}_{l+1/2}\cdot \nabla {z}_{l+1/2}$ We can write the vertical position of the interface as: ${z}_{l+1/2}={z}_{b}+\sum _{j=0}^{l}{h}_{j}$ so that the vertical velocity is: $\mathrm{\text{w}}\left(\mathbf{\text{x}},{z}_{l+1/2}\right)=\mathrm{\text{dh}}\sum _{j=0}^{l}{\mathrm{\text{layer}}}_{j}-{G}_{l+1/2}+{\mathbf{\text{u}}}_{l+1/2}\cdot \left[\nabla {z}_{b}+\nabla h\sum _{j=0}^{l}{\mathrm{\text{layer}}}_{j}\right]$