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 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250  /** # Multilayer Saint-Venant system with mass exchanges The [Saint-Venant system](saint-venant.h) is extended to multiple layers following [Audusse et al, 2011](references.bib#audusse2011) as $$\partial_th + \partial_x\sum_{l=0}^{nl-1}h_lu_l = 0$$ with $$h_l = \mathrm{layer}_lh$$ with $\mathrm{layer}_l$ the relative thickness of the layers satisfying $$\mathrm{layer}_l >= 0,\;\sum_{l=0}^{nl - 1}\mathrm{layer}_l = 1.$$ The momentum equation in each layer is thus $$\partial_t(h\mathbf{u}_l) + \nabla\cdot\left(h\mathbf{u}_l\otimes\mathbf{u}_l + \frac{gh^2}{2}\mathbf{I}\right) = - gh\nabla z_b + \frac{1}{\mathrm{layer}_l}\left[\mathbf{u}_{l+1/2}G_{l+1/2} - \mathbf{u}_{l-1/2}G_{l-1/2} + \nu\left(\frac{u_{l+1} - u_l}{h_{l+1/2}} - \frac{u_{l} - u_{l-1}}{h_{l-1/2}}\right)\right]$$ 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](saint-venant.h). 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. $$\partial_z u |_t = \dot{u}_t$$ and a Navier slip condition on the bottom i.e. $$u|_b = u_b + \lambda_b \partial_z u|_b$$ By default the viscosity is zero and we impose free-slip on the top boundary and no-slip on the bottom boundary i.e. $\dot{u}_t = 0$, $\lambda_b = 0$, $u_b = 0$. */ double nu = 0.; (const) scalar lambda_b = zeroc, dut = zeroc, u_b = zeroc; /** For stability, we discretise the viscous friction term implicitly as $$\frac{(hu_l)_{n + 1} - (hu_l)_{\star}}{\Delta t} = \frac{\nu}{\mathrm{layer}_l} \left( \frac{u_{l + 1} - u_l}{h_{l + 1 / 2}} - \frac{u_l - u_{l - 1}}{h_{l - 1 / 2}} \right)_{n + 1}$$ which can be expressed as the linear system $$\mathbf{Mu}_{n + 1} = \mathrm{rhs}$$ where $\mathbf{M}$ is a [tridiagonal matrix](https://en.wikipedia.org/wiki/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 (nu == 0.) return; double a[nl], b[nl], c[nl], rhs[nl]; foreach_dimension() { /** The *rhs* of the tridiagonal system is $h_lu_l = h\mathrm{layer}_lu_l$. */ 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 $$a_{l > 0} = - \left( \frac{\nu \Delta t}{h_{l - 1 / 2}} \right)_{n + 1}$$ $$c_{l < \mathrm{nl} - 1} = - \left( \frac{\nu \Delta t}{h_{l + 1 / 2}} \right)_{n + 1}$$ $$b_{0 < l < \mathrm{nl} - 1} = \mathrm{layer}_l h_{n + 1} - a_l - c_l$$ */ for (l = 1; l < nl - 1; l++) { a[l] = - 2.*nu*dt/(h*(layer[l-1] + layer[l])); c[l] = - 2.*nu*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 $$u_{\mathrm{nl}} = u_{\mathrm{nl} - 1} + \dot{u}_t h_{\mathrm{nl} - 1},$$ which gives the diagonal coefficient and right-hand-side $$b_{\mathrm{nl} - 1} = \mathrm{layer}_{\mathrm{nl} - 1} h_{n + 1} - a_{\mathrm{nl} - 1}$$ $$\mathrm{rhs}_{\mathrm{nl} - 1} = \mathrm{layer}_{\mathrm{nl} - 1} (hu_{\mathrm{nl} - 1})_{\star} + \nu \Delta t \dot{u}_t$$ */ a[nl-1] = - 2.*nu*dt/(h*(layer[nl-2] + layer[nl-1])); b[nl-1] = layer[nl-1]*h - a[nl-1]; rhs[nl-1] += nu*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 $$b_0 = \mathrm{layer}_0 h_{n + 1} - c_0 + \frac{2 \nu \Delta t}{2 \lambda_b + h_0}$$ $$\mathrm{rhs}_0 = \mathrm{layer}_0 (hu_0)_{\star} + \frac{2 \nu \Delta t}{2 \lambda_b + h_0} u_b$$ */ c[0] = - 2.*dt*nu/(h*(layer[0] + layer[1])); b[0] = layer[0]*h - c[0] + 2.*nu*dt/(2.*lambda_b[] + h*layer[0]); rhs[0] += 2.*nu*dt/(2.*lambda_b[] + h*layer[0])*u_b[]; /** We can now solve the tridiagonal system using the [Thomas algorithm](https://en.wikipedia.org/wiki/Tridiagonal_matrix_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](references.bib#audusse2011)) $$G_{l+1/2} = \sum_{j=0}^{l}(\mathrm{div}_j + \mathrm{layer}_j\mathrm{dh})$$ with $$\mathrm{div}_l = \nabla\cdot(h_l\mathbf{u}_l)$$ $$\mathrm{dh} = - \sum_{l=0}^{nl-1} \mathrm{div}_l$$ */ 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 $\mathrm{Gi} = G_{l+1/2}$ 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](references.bib#audusse2011) and (77) of [Audusse et al, 2011b](references.bib#audusse2011b) but not with eq. (2.23) of [Audusse et al, 2011](references.bib#audusse2011). */ 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; scalar du_b = updates[l].x, du_t = updates[l + 1].x; du_b[] += flux/layer[l]; du_t[] -= flux/layer[l + 1]; /** To compute the vertical velocity we use the definition of the mass flux term (eq. 2.13 of [Audusse et al, 2011](references.bib#audusse2011)): $$\mathrm{w}(\mathbf{x},z_{l+1/2}) = \partial_t z_{l+1/2} - G_{l+1/2} + \mathbf{u}_{l+1/2} \cdot \nabla z_{l+1/2}$$ 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: $$\mathrm{w}(\mathbf{x},z_{l+1/2}) = \mathrm{dh}\sum_{j=0}^{l}\mathrm{layer}_{j} - G_{l+1/2} + \mathbf{u}_{l+1/2} \cdot \left[\nabla z_{b} + \nabla h \sum_{j=0}^{l}\mathrm{layer}_{j}\right]$$ */ w[] += ui*((zb[1] - zb[-1]) + (h[1] - h[-1])*sumjl)/(2.*Delta); } } } }