| 4 |
|
|
| 5 |
|
|
| 6 |
\subsection{Non-linear free-surface} |
\subsection{Non-linear free-surface} |
| 7 |
\label{sect:nonlinear-freesurface} |
\label{sec:nonlinear-freesurface} |
| 8 |
|
|
| 9 |
Recently, new options have been added to the model |
Recently, new options have been added to the model |
| 10 |
that concern the free surface formulation. |
that concern the free surface formulation. |
| 11 |
|
|
| 12 |
|
|
| 13 |
\subsubsection{pressure/geo-potential and free surface} |
\subsubsection{pressure/geo-potential and free surface} |
| 14 |
|
\label{sec:phi-freesurface} |
| 15 |
|
|
| 16 |
For the atmosphere, since $\phi = \phi_{topo} - \int^p_{p_s} \alpha dp$, |
For the atmosphere, since $\phi = \phi_{topo} - \int^p_{p_s} \alpha dp$, |
| 17 |
subtracting the reference state defined in section |
subtracting the reference state defined in section |
| 18 |
\ref{sec:hpe-p-geo-potential-split}~:\\ |
\ref{sec:hpe-p-geo-potential-split}~:\\ |
| 19 |
$$ |
$$ |
| 20 |
\phi_o = \phi_{topo} - \int^p_{p_o} \alpha_o dp |
\phi_o = \phi_{topo} - \int^p_{p_o} \alpha_o dp |
| 21 |
\hspace{5mm}\mathrm{with}\hspace{3mm} \phi_o(p_o)=\phi_{topo} |
\hspace{5mm}\mathrm{with}\hspace{3mm} \phi_o(p_o)=\phi_{topo} |
| 22 |
$$ |
$$ |
| 23 |
we get: |
we get: |
| 28 |
on $z$ ($b_o=g$) and the surface reference position is uniformly $z=0$ ($R_o=0$), |
on $z$ ($b_o=g$) and the surface reference position is uniformly $z=0$ ($R_o=0$), |
| 29 |
and the same subtraction leads to a similar relation. |
and the same subtraction leads to a similar relation. |
| 30 |
For both fluid, using the isomorphic notations, we can write: |
For both fluid, using the isomorphic notations, we can write: |
| 31 |
$$ |
$$ |
| 32 |
\phi' = \int^{r_{surf}}_r b~ dr - \int^{R_o}_r b_o dr |
\phi' = \int^{r_{surf}}_r b~ dr - \int^{R_o}_r b_o dr |
| 33 |
$$ |
$$ |
| 34 |
\begin{eqnarray} |
and re-write as: |
| 35 |
\mathrm{and~re~write:}\hspace{10mm} |
\begin{equation} |
| 36 |
\phi' = \int^{r_{surf}}_{R_o} b~ dr & + & \int^{R_o}_r (b - b_o) dr |
\phi' = \int^{r_{surf}}_{R_o} b~ dr + \int^{R_o}_r (b - b_o) dr |
| 37 |
\label{eq:split-phi-Ro} \\ |
\label{eq:split-phi-Ro} |
| 38 |
\mathrm{or:}\hspace{10mm} |
\end{equation} |
| 39 |
\phi' = \int^{r_{surf}}_{R_o} b_o dr & + & \int^{r_{surf}}_r (b - b_o) dr |
or: |
| 40 |
|
\begin{equation} |
| 41 |
|
\phi' = \int^{r_{surf}}_{R_o} b_o dr + \int^{r_{surf}}_r (b - b_o) dr |
| 42 |
\label{eq:split-phi-bo} |
\label{eq:split-phi-bo} |
| 43 |
\end{eqnarray} |
\end{equation} |
| 44 |
|
|
| 45 |
In section \ref{sec:finding_the_pressure_field}, following eq.\ref{eq:split-phi-Ro}, |
In section \ref{sec:finding_the_pressure_field}, following eq.\ref{eq:split-phi-Ro}, |
| 46 |
the pressure/geo-potential $\phi'$ has been separated into surface ($\phi_s$), |
the pressure/geo-potential $\phi'$ has been separated into surface ($\phi_s$), |
| 47 |
and hydrostatic anomaly ($\phi'_{hyd}$). |
and hydrostatic anomaly ($\phi'_{hyd}$). |
| 48 |
In this section, the split between $\phi_s$ and $\phi'_{hyd}$ is |
In this section, the split between $\phi_s$ and $\phi'_{hyd}$ is |
| 49 |
made according to equation \ref{eq:split-phi-bo}. This slightly |
made according to equation \ref{eq:split-phi-bo}. This slightly |
| 50 |
different definition reflects the actual implementation in the code |
different definition reflects the actual implementation in the code |
| 51 |
and is valid for both linear and non-linear |
and is valid for both linear and non-linear |
| 52 |
free-surface formulation, in both r-coordinate and r*-coordinate. |
free-surface formulation, in both r-coordinate and r*-coordinate. |
| 53 |
|
|
| 54 |
Because the linear free-surface approximation ignore the tracer content |
Because the linear free-surface approximation ignore the tracer content |
| 55 |
of the fluid parcel between $R_o$ and $r_{surf}=R_o+\eta$, |
of the fluid parcel between $R_o$ and $r_{surf}=R_o+\eta$, |
| 56 |
for consistency reasons, this part is also neglected in $\phi'_{hyd}$~: |
for consistency reasons, this part is also neglected in $\phi'_{hyd}$~: |
| 57 |
$$ |
$$ |
| 58 |
\phi'_{hyd} = \int^{r_{surf}}_r (b - b_o) dr \simeq \int^{R_o}_r (b - b_o) dr |
\phi'_{hyd} = \int^{r_{surf}}_r (b - b_o) dr \simeq \int^{R_o}_r (b - b_o) dr |
| 59 |
$$ |
$$ |
| 60 |
Note that in this case, the two definitions of $\phi_s$ and $\phi'_{hyd}$ |
Note that in this case, the two definitions of $\phi_s$ and $\phi'_{hyd}$ |
| 61 |
from equation \ref{eq:split-phi-Ro} and \ref{eq:split-phi-bo} converge toward |
from equation \ref{eq:split-phi-Ro} and \ref{eq:split-phi-bo} converge toward |
| 62 |
the same (approximated) expressions: $\phi_s = \int^{r_{surf}}_{R_o} b_o dr$ |
the same (approximated) expressions: $\phi_s = \int^{r_{surf}}_{R_o} b_o dr$ |
| 63 |
and $\phi'_{hyd}=\int^{R_o}_r b' dr$.\\ |
and $\phi'_{hyd}=\int^{R_o}_r b' dr$.\\ |
| 64 |
On the contrary, the unapproximated formulation ("non-linear free-surface", |
On the contrary, the unapproximated formulation ("non-linear free-surface", |
| 65 |
see the next section) retains the full expression: |
see the next section) retains the full expression: |
| 67 |
This is obtained by selecting {\bf nonlinFreeSurf}=4 in parameter |
This is obtained by selecting {\bf nonlinFreeSurf}=4 in parameter |
| 68 |
file {\em data}.\\ |
file {\em data}.\\ |
| 69 |
|
|
| 70 |
Regarding the surface potential: |
Regarding the surface potential: |
| 71 |
$$\phi_s = \int_{R_o}^{R_o+\eta} b_o dr = b_s \eta |
$$\phi_s = \int_{R_o}^{R_o+\eta} b_o dr = b_s \eta |
| 72 |
\hspace{5mm}\mathrm{with}\hspace{5mm} |
\hspace{5mm}\mathrm{with}\hspace{5mm} |
| 73 |
b_s = \frac{1}{\eta} \int_{R_o}^{R_o+\eta} b_o dr $$ |
b_s = \frac{1}{\eta} \int_{R_o}^{R_o+\eta} b_o dr $$ |
| 74 |
$b_s \simeq b_o(R_o)$ is an excellent approximation (better than |
$b_s \simeq b_o(R_o)$ is an excellent approximation (better than |
| 75 |
the usual numerical truncation, since generally $|\eta|$ is smaller |
the usual numerical truncation, since generally $|\eta|$ is smaller |
| 76 |
than the vertical grid increment). |
than the vertical grid increment). |
| 77 |
|
|
| 78 |
For the ocean, $\phi_s = g \eta$ and $b_s = g$ is uniform. |
For the ocean, $\phi_s = g \eta$ and $b_s = g$ is uniform. |
| 79 |
For the atmosphere, however, because of topographic effects, the |
For the atmosphere, however, because of topographic effects, the |
| 80 |
reference surface pressure $R_o=p_o$ has large spatial variations that |
reference surface pressure $R_o=p_o$ has large spatial variations that |
| 81 |
are responsible for significant $b_s$ variations (from 0.8 to 1.2 |
are responsible for significant $b_s$ variations (from 0.8 to 1.2 |
| 82 |
$[m^3/kg]$). For this reason, when {\bf uniformLin\_PhiSurf} {\em=.FALSE.} |
$[m^3/kg]$). For this reason, when {\bf uniformLin\_PhiSurf} {\em=.FALSE.} |
| 83 |
(parameter file {\em data}, namelist {\em PARAM01}) |
(parameter file {\em data}, namelist {\em PARAM01}) |
| 84 |
a non-uniform linear coefficient $b_s$ is used and computed |
a non-uniform linear coefficient $b_s$ is used and computed |
| 85 |
({\it S/R INI\_LINEAR\_PHISURF}) according to the reference surface |
({\it S/R INI\_LINEAR\_PHISURF}) according to the reference surface |
| 86 |
pressure $p_o$: |
pressure $p_o$: |
| 87 |
$b_s = b_o(R_o) = c_p \kappa (p_o / P^o_{SL})^{(\kappa - 1)} \theta_{ref}(p_o)$. |
$b_s = b_o(R_o) = c_p \kappa (p_o / P^o_{SL})^{(\kappa - 1)} \theta_{ref}(p_o)$. |
| 88 |
with $P^o_{SL}$ the mean sea-level pressure. |
with $P^o_{SL}$ the mean sea-level pressure. |
| 92 |
(Non-linear free-surface)} |
(Non-linear free-surface)} |
| 93 |
|
|
| 94 |
The total thickness of the fluid column is $r_{surf} - R_{fixed} = |
The total thickness of the fluid column is $r_{surf} - R_{fixed} = |
| 95 |
\eta + R_o - R_{fixed}$. In most applications, the free surface |
\eta + R_o - R_{fixed}$. In most applications, the free surface |
| 96 |
displacements are small compared to the total thickness |
displacements are small compared to the total thickness |
| 97 |
$\eta \ll H_o = R_o - R_{fixed}$. |
$\eta \ll H_o = R_o - R_{fixed}$. |
| 98 |
In the previous sections and in older version of the model, |
In the previous sections and in older version of the model, |
| 99 |
the linearized free-surface approximation was made, assuming |
the linearized free-surface approximation was made, assuming |
| 100 |
$r_{surf} - R_{fixed} \simeq H_o$ when computing horizontal transports, |
$r_{surf} - R_{fixed} \simeq H_o$ when computing horizontal transports, |
| 101 |
either in the continuity equation or in tracer and momentum |
either in the continuity equation or in tracer and momentum |
| 102 |
advection terms. |
advection terms. |
| 103 |
This approximation is dropped when using the non-linear free-surface |
This approximation is dropped when using the non-linear free-surface |
| 104 |
formulation and the total thickness, including the time varying part |
formulation and the total thickness, including the time varying part |
| 105 |
$\eta$, is considered when computing horizontal transports. |
$\eta$, is considered when computing horizontal transports. |
| 106 |
Implications for the barotropic part are presented hereafter. |
Implications for the barotropic part are presented hereafter. |
| 107 |
In section \ref{sect:freesurf-tracer-advection} consequences for |
In section \ref{sec:freesurf-tracer-advection} consequences for |
| 108 |
tracer conservation is briefly discussed (more details can be |
tracer conservation is briefly discussed (more details can be |
| 109 |
found in \cite{campin:02})~; the general time-stepping is presented |
found in \cite{campin:02})~; the general time-stepping is presented |
| 110 |
in section \ref{sect:nonlin-freesurf-timestepping} with some |
in section \ref{sec:nonlin-freesurf-timestepping} with some |
| 111 |
limitations regarding the vertical resolution in section |
limitations regarding the vertical resolution in section |
| 112 |
\ref{sect:nonlin-freesurf-dz_surf}. |
\ref{sec:nonlin-freesurf-dz_surf}. |
| 113 |
|
|
| 114 |
In the non-linear formulation, the continuous form of the model |
In the non-linear formulation, the continuous form of the model |
| 115 |
equations remains unchanged, except for the 2D continuity equation |
equations remains unchanged, except for the 2D continuity equation |
| 116 |
(\ref{eq:discrete-time-backward-free-surface}) which is now |
(\ref{eq:discrete-time-backward-free-surface}) which is now |
| 117 |
integrated from $R_{fixed}(x,y)$ up to $r_{surf}=R_o+\eta$ : |
integrated from $R_{fixed}(x,y)$ up to $r_{surf}=R_o+\eta$ : |
| 118 |
|
|
| 142 |
{\eta}^* = \epsilon_{fs} \: {\eta}^{n} - |
{\eta}^* = \epsilon_{fs} \: {\eta}^{n} - |
| 143 |
\Delta t {\bf \nabla}_h \cdot \int_{R_{fixed}}^{R_o+\eta^n} \vec{\bf v}^* dr |
\Delta t {\bf \nabla}_h \cdot \int_{R_{fixed}}^{R_o+\eta^n} \vec{\bf v}^* dr |
| 144 |
\: + \: \epsilon_{fw} \Delta_t (P-E)^{n} |
\: + \: \epsilon_{fw} \Delta_t (P-E)^{n} |
| 145 |
\end{eqnarray*} |
\end{eqnarray*} |
| 146 |
This method requires us to update the solver matrix at each time step. |
This method requires us to update the solver matrix at each time step. |
| 147 |
|
|
| 148 |
Alternatively, the non-linear contribution can be evaluated fully |
Alternatively, the non-linear contribution can be evaluated fully |
| 154 |
= {\eta}^* |
= {\eta}^* |
| 155 |
+{\bf \nabla}_h \cdot \Delta t^2 (\eta^{n}) |
+{\bf \nabla}_h \cdot \Delta t^2 (\eta^{n}) |
| 156 |
{\bf \nabla}_h b_s {\eta}^{n} |
{\bf \nabla}_h b_s {\eta}^{n} |
| 157 |
\end{eqnarray*} |
\end{eqnarray*} |
| 158 |
This formulation allows one to keep the initial solver matrix |
This formulation allows one to keep the initial solver matrix |
| 159 |
unchanged though throughout the integration, since the non-linear free |
unchanged though throughout the integration, since the non-linear free |
| 160 |
surface only affects the RHS. |
surface only affects the RHS. |
| 166 |
|
|
| 167 |
Those different options (see Table \ref{tab:nonLinFreeSurf_flags}) |
Those different options (see Table \ref{tab:nonLinFreeSurf_flags}) |
| 168 |
have been tested and show little differences. However, we recommend |
have been tested and show little differences. However, we recommend |
| 169 |
the use of the most precise method (the 1rst one) since the |
the use of the most precise method (the 1rst one) since the |
| 170 |
computation cost involved in the solver matrix update is negligible. |
computation cost involved in the solver matrix update is negligible. |
| 171 |
|
|
| 172 |
\begin{table}[htb] |
\begin{table}[htb] |
| 174 |
\centering |
\centering |
| 175 |
\begin{tabular}[htb]{|l|c|l|} |
\begin{tabular}[htb]{|l|c|l|} |
| 176 |
\hline |
\hline |
| 177 |
parameter & value & description \\ |
parameter & value & description \\ |
| 178 |
\hline |
\hline |
| 179 |
& -1 & linear free-surface, restart from a pickup file \\ |
& -1 & linear free-surface, restart from a pickup file \\ |
| 180 |
& & produced with \#undef EXACT\_CONSERV code\\ |
& & produced with \#undef EXACT\_CONSERV code\\ |
| 181 |
\cline{2-3} |
\cline{2-3} |
| 182 |
& 0 & Linear free-surface \\ |
& 0 & Linear free-surface \\ |
| 183 |
\cline{2-3} |
\cline{2-3} |
| 184 |
nonlinFreeSurf & 4 & Non-linear free-surface \\ |
nonlinFreeSurf & 4 & Non-linear free-surface \\ |
| 185 |
\cline{2-3} |
\cline{2-3} |
| 205 |
|
|
| 206 |
|
|
| 207 |
\subsubsection{Tracer conservation with non-linear free-surface} |
\subsubsection{Tracer conservation with non-linear free-surface} |
| 208 |
\label{sect:freesurf-tracer-advection} |
\label{sec:freesurf-tracer-advection} |
| 209 |
|
|
| 210 |
To ensure global tracer conservation (i.e., the total amount) as well |
To ensure global tracer conservation (i.e., the total amount) as well |
| 211 |
as local conservation, the change in the surface level thickness must |
as local conservation, the change in the surface level thickness must |
| 225 |
= P \theta_{\mathrm{rain}} |
= P \theta_{\mathrm{rain}} |
| 226 |
$$ |
$$ |
| 227 |
Using the implicit (non-linear) free surface described above (section |
Using the implicit (non-linear) free surface described above (section |
| 228 |
\ref{sect:pressure-method-linear-backward}) we have: |
\ref{sec:pressure-method-linear-backward}) we have: |
| 229 |
\begin{eqnarray*} |
\begin{eqnarray*} |
| 230 |
h^{n+1} = h^{n} - \Delta t \nabla \cdot (h^n \, \vec{\bf v}^{n+1} ) + \Delta t P \\ |
h^{n+1} = h^{n} - \Delta t \nabla \cdot (h^n \, \vec{\bf v}^{n+1} ) + \Delta t P \\ |
| 231 |
\end{eqnarray*} |
\end{eqnarray*} |
| 233 |
``form'' in the computation of tracer fluxes, that is, the same value |
``form'' in the computation of tracer fluxes, that is, the same value |
| 234 |
of $h$, as used in the continuity equation: |
of $h$, as used in the continuity equation: |
| 235 |
\begin{eqnarray*} |
\begin{eqnarray*} |
| 236 |
h^{n+1} \, \theta^{n+1} = h^n \, \theta^n |
h^{n+1} \, \theta^{n+1} = h^n \, \theta^n |
| 237 |
- \Delta t \nabla \cdot (h^n \, \theta^n \, \vec{\bf v}^{n+1}) |
- \Delta t \nabla \cdot (h^n \, \theta^n \, \vec{\bf v}^{n+1}) |
| 238 |
+ \Delta t P \theta_{rain} |
+ \Delta t P \theta_{rain} |
| 239 |
\end{eqnarray*} |
\end{eqnarray*} |
| 250 |
surface formulation (with the "{\it surface correction}" turned "on", |
surface formulation (with the "{\it surface correction}" turned "on", |
| 251 |
see tracer section): |
see tracer section): |
| 252 |
$$ |
$$ |
| 253 |
G_\theta^n = \left(- \nabla \cdot (h^n \, \theta^n \, \vec{\bf v}^{n+1}) |
G_\theta^n = \left(- \nabla \cdot (h^n \, \theta^n \, \vec{\bf v}^{n+1}) |
| 254 |
- \dot{r}_{surf}^{n+1} \theta^n \right) / h^n |
- \dot{r}_{surf}^{n+1} \theta^n \right) / h^n |
| 255 |
$$ |
$$ |
| 256 |
Then, in a second step, the thickness variation (expansion/reduction) |
Then, in a second step, the thickness variation (expansion/reduction) |
| 257 |
is taken into account: |
is taken into account: |
| 258 |
$$ |
$$ |
| 259 |
\theta^{n+1} = \theta^n + \Delta t \frac{h^n}{h^{n+1}} |
\theta^{n+1} = \theta^n + \Delta t \frac{h^n}{h^{n+1}} |
| 260 |
\left( G_\theta^{(n+1/2)} + P (\theta_{\mathrm{rain}} - \theta^n )/h^n \right) |
\left( G_\theta^{(n+1/2)} + P (\theta_{\mathrm{rain}} - \theta^n )/h^n \right) |
| 261 |
%\theta^{n+1} = \theta^n + \frac{\Delta t}{h^{n+1}} |
%\theta^{n+1} = \theta^n + \frac{\Delta t}{h^{n+1}} |
| 262 |
% \left( h^n G_\theta^{(n+1/2)} + P (\theta_{\mathrm{rain}} - \theta^n ) \right) |
% \left( h^n G_\theta^{(n+1/2)} + P (\theta_{\mathrm{rain}} - \theta^n ) \right) |
| 263 |
$$ |
$$ |
| 264 |
Note that with a simple forward time step (no Adams-Bashforth), |
Note that with a simple forward time step (no Adams-Bashforth), |
| 265 |
these two formulations are equivalent, |
these two formulations are equivalent, |
| 266 |
since |
since |
| 267 |
$ |
$ |
| 268 |
(h^{n+1} - h^{n})/ \Delta t = |
(h^{n+1} - h^{n})/ \Delta t = |
| 269 |
P - \nabla \cdot (h^n \, \vec{\bf v}^{n+1} ) = P + \dot{r}_{surf}^{n+1} |
P - \nabla \cdot (h^n \, \vec{\bf v}^{n+1} ) = P + \dot{r}_{surf}^{n+1} |
| 270 |
$ |
$ |
| 271 |
|
|
| 272 |
\subsubsection{Time stepping implementation of the |
\subsubsection{Time stepping implementation of the |
| 273 |
non-linear free-surface} |
non-linear free-surface} |
| 274 |
\label{sect:nonlin-freesurf-timestepping} |
\label{sec:nonlin-freesurf-timestepping} |
| 275 |
|
|
| 276 |
The grid cell thickness was hold constant with the linear |
The grid cell thickness was hold constant with the linear |
| 277 |
free-surface~; with the non-linear free-surface, it is now varying |
free-surface~; with the non-linear free-surface, it is now varying |
| 278 |
in time, at least at the surface level. |
in time, at least at the surface level. |
| 279 |
This implies some modifications of the general algorithm described |
This implies some modifications of the general algorithm described |
| 280 |
earlier in sections \ref{sect:adams-bashforth-sync} and |
earlier in sections \ref{sec:adams-bashforth-sync} and |
| 281 |
\ref{sect:adams-bashforth-staggered}. |
\ref{sec:adams-bashforth-staggered}. |
| 282 |
|
|
| 283 |
A simplified version of the staggered in time, non-linear |
A simplified version of the staggered in time, non-linear |
| 284 |
free-surface algorithm is detailed hereafter, and can be compared |
free-surface algorithm is detailed hereafter, and can be compared |
| 285 |
to the equivalent linear free-surface case (eq. \ref{eq:Gv-n-staggered} |
to the equivalent linear free-surface case (eq. \ref{eq:Gv-n-staggered} |
| 286 |
to \ref{eq:t-n+1-staggered}) and can also be easily transposed |
to \ref{eq:t-n+1-staggered}) and can also be easily transposed |
| 287 |
to the synchronous time-stepping case. |
to the synchronous time-stepping case. |
| 294 |
\begin{eqnarray} |
\begin{eqnarray} |
| 295 |
\phi^{n}_{hyd} & = & \int b(\theta^{n},S^{n},r) dr |
\phi^{n}_{hyd} & = & \int b(\theta^{n},S^{n},r) dr |
| 296 |
\label{eq:phi-hyd-nlfs} \\ |
\label{eq:phi-hyd-nlfs} \\ |
| 297 |
\vec{\bf G}_{\vec{\bf v}}^{n-1/2}\hspace{-2mm} & = & |
\vec{\bf G}_{\vec{\bf v}}^{n-1/2}\hspace{-2mm} & = & |
| 298 |
\vec{\bf G}_{\vec{\bf v}} (dh^{n-1},\vec{\bf v}^{n-1/2}) |
\vec{\bf G}_{\vec{\bf v}} (dh^{n-1},\vec{\bf v}^{n-1/2}) |
| 299 |
\hspace{+2mm};\hspace{+2mm} |
\hspace{+2mm};\hspace{+2mm} |
| 300 |
\vec{\bf G}_{\vec{\bf v}}^{(n)} = |
\vec{\bf G}_{\vec{\bf v}}^{(n)} = |
| 301 |
\frac{3}{2} \vec{\bf G}_{\vec{\bf v}}^{n-1/2} |
\frac{3}{2} \vec{\bf G}_{\vec{\bf v}}^{n-1/2} |
| 302 |
- \frac{1}{2} \vec{\bf G}_{\vec{\bf v}}^{n-3/2} |
- \frac{1}{2} \vec{\bf G}_{\vec{\bf v}}^{n-3/2} |
| 303 |
\label{eq:Gv-n-nlfs} \\ |
\label{eq:Gv-n-nlfs} \\ |
| 304 |
%\vec{\bf G}_{\vec{\bf v}}^{(n)} & = & |
%\vec{\bf G}_{\vec{\bf v}}^{(n)} & = & |
| 305 |
% \frac{3}{2} \vec{\bf G}_{\vec{\bf v}}^{n-1/2} |
% \frac{3}{2} \vec{\bf G}_{\vec{\bf v}}^{n-1/2} |
| 306 |
%- \frac{1}{2} \vec{\bf G}_{\vec{\bf v}}^{n-3/2} |
%- \frac{1}{2} \vec{\bf G}_{\vec{\bf v}}^{n-3/2} |
| 307 |
%\label{eq:Gv-n+5-nlfs} \\ |
%\label{eq:Gv-n+5-nlfs} \\ |
| 308 |
%\vec{\bf v}^{*} & = & \vec{\bf v}^{n-1/2} + \frac{\Delta t}{dh^{n}} \left( |
%\vec{\bf v}^{*} & = & \vec{\bf v}^{n-1/2} + \frac{\Delta t}{dh^{n}} \left( |
| 310 |
\vec{\bf v}^{*} & = & \vec{\bf v}^{n-1/2} + \Delta t \frac{dh^{n-1}}{dh^{n}} \left( |
\vec{\bf v}^{*} & = & \vec{\bf v}^{n-1/2} + \Delta t \frac{dh^{n-1}}{dh^{n}} \left( |
| 311 |
\vec{\bf G}_{\vec{\bf v}}^{(n)} + F_{\vec{\bf v}}^{n}/dh^{n-1} \right) |
\vec{\bf G}_{\vec{\bf v}}^{(n)} + F_{\vec{\bf v}}^{n}/dh^{n-1} \right) |
| 312 |
- \Delta t \nabla \phi_{hyd}^{n} |
- \Delta t \nabla \phi_{hyd}^{n} |
| 313 |
\label{eq:vstar-nlfs} \\ |
\label{eq:vstar-nlfs} |
| 314 |
\mathrm{update}\hspace{-4mm} & & \hspace{-4mm}\mathrm{ |
\end{eqnarray} |
| 315 |
model~geometry~:~{\bf hFac}}(dh^n)\nonumber \\ |
\hspace{3cm}$\longrightarrow$~~{\it update model~geometry~:~}${\bf hFac}(dh^n)$\\ |
| 316 |
\eta^{n+1/2} \hspace{-2mm} & = & |
\begin{eqnarray} |
| 317 |
|
\eta^{n+1/2} \hspace{-2mm} & = & |
| 318 |
\eta^{n-1/2} + \Delta t P^{n+1/2} - \Delta t |
\eta^{n-1/2} + \Delta t P^{n+1/2} - \Delta t |
| 319 |
\nabla \cdot \int \vec{\bf v}^{n+1/2} dh^{n} \nonumber \\ |
\nabla \cdot \int \vec{\bf v}^{n+1/2} dh^{n} \nonumber \\ |
| 320 |
& = & \eta^{n-1/2} + \Delta t P^{n+1/2} - \Delta t |
& = & \eta^{n-1/2} + \Delta t P^{n+1/2} - \Delta t |
| 321 |
\nabla \cdot \int \!\!\! \left( \vec{\bf v}^* - g \Delta t \nabla \eta^{n+1/2} \right) dh^{n} |
\nabla \cdot \int \!\!\! \left( \vec{\bf v}^* - g \Delta t \nabla \eta^{n+1/2} \right) dh^{n} |
| 322 |
\label{eq:nstar-nlfs} \\ |
\label{eq:nstar-nlfs} \\ |
| 323 |
\vec{\bf v}^{n+1/2}\hspace{-2mm} & = & |
\vec{\bf v}^{n+1/2}\hspace{-2mm} & = & |
| 324 |
\vec{\bf v}^{*} - g \Delta t \nabla \eta^{n+1/2} |
\vec{\bf v}^{*} - g \Delta t \nabla \eta^{n+1/2} |
| 325 |
\label{eq:v-n+1-nlfs} \\ |
\label{eq:v-n+1-nlfs} \\ |
| 326 |
h^{n+1} & = & h^{n} + \Delta t P^{n+1/2} - \Delta t |
h^{n+1} & = & h^{n} + \Delta t P^{n+1/2} - \Delta t |
| 330 |
\hspace{+2mm};\hspace{+2mm} |
\hspace{+2mm};\hspace{+2mm} |
| 331 |
G_{\theta}^{(n+1/2)} = \frac{3}{2} G_{\theta}^{n} - \frac{1}{2} G_{\theta}^{n-1} |
G_{\theta}^{(n+1/2)} = \frac{3}{2} G_{\theta}^{n} - \frac{1}{2} G_{\theta}^{n-1} |
| 332 |
\label{eq:Gt-n-nlfs} \\ |
\label{eq:Gt-n-nlfs} \\ |
| 333 |
%\theta^{n+1} & = &\theta^{n} + \frac{\Delta t}{dh^{n+1}} \left( dh^n |
%\theta^{n+1} & = &\theta^{n} + \frac{\Delta t}{dh^{n+1}} \left( dh^n |
| 334 |
%G_{\theta}^{(n+1/2)} + Q^{n+1/2} + P^{n+1/2} (\theta_{\mathrm{rain}}-\theta^n) \right) |
%G_{\theta}^{(n+1/2)} + Q^{n+1/2} + P^{n+1/2} (\theta_{\mathrm{rain}}-\theta^n) \right) |
| 335 |
\theta^{n+1} & = &\theta^{n} + \Delta t \frac{dh^n}{dh^{n+1}} \left( |
\theta^{n+1} & = &\theta^{n} + \Delta t \frac{dh^n}{dh^{n+1}} \left( |
| 336 |
G_{\theta}^{(n+1/2)} |
G_{\theta}^{(n+1/2)} |
| 337 |
+( P^{n+1/2} (\theta_{\mathrm{rain}}-\theta^n) + Q^{n+1/2})/dh^n \right) |
+( P^{n+1/2} (\theta_{\mathrm{rain}}-\theta^n) + Q^{n+1/2})/dh^n \right) |
| 338 |
\nonumber \\ |
\nonumber \\ |
| 339 |
& & \label{eq:t-n+1-nlfs} |
& & \label{eq:t-n+1-nlfs} |
| 340 |
\end{eqnarray} |
\end{eqnarray} |
| 341 |
%------------------------------------------------------------- |
%------------------------------------------------------------- |
| 342 |
Two steps have been added to linear free-surface algorithm |
Two steps have been added to linear free-surface algorithm |
| 343 |
(eq. \ref{eq:Gv-n-staggered} to \ref{eq:t-n+1-staggered}): |
(eq. \ref{eq:Gv-n-staggered} to \ref{eq:t-n+1-staggered}): |
| 344 |
Firstly, the model ``geometry'' |
Firstly, the model ``geometry'' |
| 345 |
(here the {\bf hFacC,W,S}) is updated just before entering {\it |
(here the {\bf hFacC,W,S}) is updated just before entering {\it |
| 346 |
SOLVE\_FOR\_PRESSURE}, using the current $dh^{n}$ field. |
SOLVE\_FOR\_PRESSURE}, using the current $dh^{n}$ field. |
| 347 |
Secondly, the vertically integrated continuity equation |
Secondly, the vertically integrated continuity equation |
| 348 |
(eq.\ref{eq:h-n+1-nlfs}) has been added ({\bf exactConserv}{\em =TRUE}, |
(eq.\ref{eq:h-n+1-nlfs}) has been added ({\bf exactConserv}{\em =TRUE}, |
| 349 |
in parameter file {\em data}, namelist {\em PARM01}) |
in parameter file {\em data}, namelist {\em PARM01}) |
| 350 |
just before computing the vertical velocity, in subroutine |
just before computing the vertical velocity, in subroutine |
| 351 |
{\em INTEGR\_CONTINUITY}. This ensures that tracer and continuity equation |
{\em INTEGR\_CONTINUITY}. |
| 352 |
discretization a Although this equation might appear |
%This ensures that tracer and continuity equation discretization a |
| 353 |
redundant with eq.\ref{eq:nstar-nlfs}, the integrated column |
Although this equation might appear redundant with eq.\ref{eq:nstar-nlfs}, |
| 354 |
thickness $h^{n+1}$ can be different from $\eta^{n+1/2} + H$~: |
the integrated column thickness $h^{n+1}$ will be different from |
| 355 |
|
$\eta^{n+1/2} + H$~ in the following cases: |
| 356 |
\begin{itemize} |
\begin{itemize} |
| 357 |
\item when Crank-Nickelson time-stepping is used (see section |
\item when Crank-Nickelson time-stepping is used (see section |
| 358 |
\ref{sect:freesurf-CrankNick}). |
\ref{sec:freesurf-CrankNick}). |
| 359 |
\item when filters are applied to the flow field, after |
\item when filters are applied to the flow field, after |
| 360 |
(\ref{eq:v-n+1-nlfs}) and alter the divergence of the flow. |
(\ref{eq:v-n+1-nlfs}) and alter the divergence of the flow. |
| 361 |
\item when the solver does not iterate until convergence~; |
\item when the solver does not iterate until convergence~; |
| 367 |
are computed using $dh^{n-1}$ geometry factors. |
are computed using $dh^{n-1}$ geometry factors. |
| 368 |
(eq.\ref{eq:Gv-n-nlfs}) and then rescaled in subroutine {\it TIMESTEP}, |
(eq.\ref{eq:Gv-n-nlfs}) and then rescaled in subroutine {\it TIMESTEP}, |
| 369 |
(eq.\ref{eq:vstar-nlfs}), similarly to tracer tendencies (see section |
(eq.\ref{eq:vstar-nlfs}), similarly to tracer tendencies (see section |
| 370 |
\ref{sect:freesurf-tracer-advection}). |
\ref{sec:freesurf-tracer-advection}). |
| 371 |
The tracers are stepped forward later, using the recently updated |
The tracers are stepped forward later, using the recently updated |
| 372 |
flow field ${\bf v}^{n+1/2}$ and the corresponding model geometry |
flow field ${\bf v}^{n+1/2}$ and the corresponding model geometry |
| 373 |
$dh^{n}$ to compute the tendencies (eq.\ref{eq:Gt-n-nlfs}); |
$dh^{n}$ to compute the tendencies (eq.\ref{eq:Gt-n-nlfs}); |
| 375 |
the new tracers values $(\theta,S)^{n+1}$ (eq.\ref{eq:t-n+1-nlfs}, |
the new tracers values $(\theta,S)^{n+1}$ (eq.\ref{eq:t-n+1-nlfs}, |
| 376 |
in subroutine {\em CALC\_GT, CALC\_GS}). |
in subroutine {\em CALC\_GT, CALC\_GS}). |
| 377 |
|
|
| 378 |
Note that the fresh-water input is added in a consistent way in the |
Note that the fresh-water input is added in a consistent way in the |
| 379 |
continuity equation and in the tracer equation, taking into account |
continuity equation and in the tracer equation, taking into account |
| 380 |
the fresh-water temperature $\theta_{\mathrm{rain}}$. |
the fresh-water temperature $\theta_{\mathrm{rain}}$. |
| 381 |
|
|
| 382 |
Regarding the restart procedure, |
Regarding the restart procedure, |
| 383 |
two 2.D fields $h^{n-1}$ and $(h^n-h^{n-1})/\Delta t$ |
two 2.D fields $h^{n-1}$ and $(h^n-h^{n-1})/\Delta t$ |
| 384 |
in addition to the standard |
in addition to the standard |
| 385 |
state variables and tendencies ($\eta^{n-1/2}$, ${\bf v}^{n-1/2}$, |
state variables and tendencies ($\eta^{n-1/2}$, ${\bf v}^{n-1/2}$, |
| 386 |
$\theta^n$, $S^n$, ${\bf G}_{\bf v}^{n-3/2}$, $G_{\theta,S}^{n-1}$) |
$\theta^n$, $S^n$, ${\bf G}_{\bf v}^{n-3/2}$, $G_{\theta,S}^{n-1}$) |
| 389 |
then update the model geometry according to $h^{n-1}$, |
then update the model geometry according to $h^{n-1}$, |
| 390 |
and compute $h^n$ and the vertical velocity |
and compute $h^n$ and the vertical velocity |
| 391 |
%$h^n=h^{n-1} + \Delta t [(h^n-h^{n-1})/\Delta t]$, |
%$h^n=h^{n-1} + \Delta t [(h^n-h^{n-1})/\Delta t]$, |
| 392 |
before starting the main calling sequence (eq.\ref{eq:phi-hyd-nlfs} |
before starting the main calling sequence (eq.\ref{eq:phi-hyd-nlfs} |
| 393 |
to \ref{eq:t-n+1-nlfs}, {\em S/R FORWARD\_STEP}). |
to \ref{eq:t-n+1-nlfs}, {\em S/R FORWARD\_STEP}). |
| 394 |
\\ |
\\ |
| 395 |
|
|
| 400 |
|
|
| 401 |
$h^{n} -H_o$: {\bf etaHnm1} ({\em SURFACE.h}) |
$h^{n} -H_o$: {\bf etaHnm1} ({\em SURFACE.h}) |
| 402 |
|
|
| 403 |
$h^{n+1}-h^{n}/\Delta t$: {\bf dEtaHdt} ({\em SURFACE.h}) |
$(h^{n+1}-h^{n})/\Delta t$: {\bf dEtaHdt} ({\em SURFACE.h}) |
| 404 |
|
|
| 405 |
\end{minipage} } |
\end{minipage} } |
| 406 |
|
|
| 407 |
\subsubsection{Non-linear free-surface and vertical resolution} |
\subsubsection{Non-linear free-surface and vertical resolution} |
| 408 |
\label{sect:nonlin-freesurf-dz_surf} |
\label{sec:nonlin-freesurf-dz_surf} |
| 409 |
|
|
| 410 |
When the amplitude of the free-surface variations becomes |
When the amplitude of the free-surface variations becomes |
| 411 |
as large as the vertical resolution near the surface, |
as large as the vertical resolution near the surface, |
| 412 |
the surface layer thickness can decrease to nearly zero or |
the surface layer thickness can decrease to nearly zero or |
| 413 |
can even vanish completely. |
can even vanish completely. |
| 414 |
This later possibility has not been implemented, and a |
This later possibility has not been implemented, and a |
| 415 |
minimum relative thickness is imposed ({\bf hFacInf}, |
minimum relative thickness is imposed ({\bf hFacInf}, |
| 416 |
parameter file {\em data}, namelist {\em PARM01}) to prevent |
parameter file {\em data}, namelist {\em PARM01}) to prevent |
| 417 |
numerical instabilities caused by very thin surface level. |
numerical instabilities caused by very thin surface level. |
| 418 |
|
|
| 419 |
A better alternative to the vanishing level problem has been |
A better alternative to the vanishing level problem has been |
| 420 |
found and implemented recently, and rely on a different |
found and implemented recently, and rely on a different |
| 421 |
vertical coordinate $r^*$~: |
vertical coordinate $r^*$~: |
| 422 |
The time variation ot the total column thickness becomes |
The time variation ot the total column thickness becomes |
| 423 |
part of the r* coordinate motion, as in a $\sigma_{z},\sigma_{p}$ |
part of the r* coordinate motion, as in a $\sigma_{z},\sigma_{p}$ |
| 424 |
model, but the fixed part related to topography is treated |
model, but the fixed part related to topography is treated |
| 425 |
as in a height or pressure coordinate model. |
as in a height or pressure coordinate model. |
| 426 |
A complete description is given in \cite{adcroft:04}. |
A complete description is given in \cite{adcroft:04a}. |
| 427 |
|
|
| 428 |
The time-stepping implementation of the $r^*$ coordinate is |
The time-stepping implementation of the $r^*$ coordinate is |
| 429 |
identical to the non-linear free-surface in $r$ coordinate, |
identical to the non-linear free-surface in $r$ coordinate, |
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