| 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 |
| 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 ($\phi_s$), |
| 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 |
| 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*} |
| 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 |
| 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 |
|
\begin{eqnarray} |
| 317 |
\eta^{n+1/2} \hspace{-2mm} & = & |
\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 \\ |
| 354 |
thickness $h^{n+1}$ can be different from $\eta^{n+1/2} + H$~: |
thickness $h^{n+1}$ can be different from $\eta^{n+1/2} + H$~: |
| 355 |
\begin{itemize} |
\begin{itemize} |
| 356 |
\item when Crank-Nickelson time-stepping is used (see section |
\item when Crank-Nickelson time-stepping is used (see section |
| 357 |
\ref{sect:freesurf-CrankNick}). |
\ref{sec:freesurf-CrankNick}). |
| 358 |
\item when filters are applied to the flow field, after |
\item when filters are applied to the flow field, after |
| 359 |
(\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. |
| 360 |
\item when the solver does not iterate until convergence~; |
\item when the solver does not iterate until convergence~; |
| 366 |
are computed using $dh^{n-1}$ geometry factors. |
are computed using $dh^{n-1}$ geometry factors. |
| 367 |
(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}, |
| 368 |
(eq.\ref{eq:vstar-nlfs}), similarly to tracer tendencies (see section |
(eq.\ref{eq:vstar-nlfs}), similarly to tracer tendencies (see section |
| 369 |
\ref{sect:freesurf-tracer-advection}). |
\ref{sec:freesurf-tracer-advection}). |
| 370 |
The tracers are stepped forward later, using the recently updated |
The tracers are stepped forward later, using the recently updated |
| 371 |
flow field ${\bf v}^{n+1/2}$ and the corresponding model geometry |
flow field ${\bf v}^{n+1/2}$ and the corresponding model geometry |
| 372 |
$dh^{n}$ to compute the tendencies (eq.\ref{eq:Gt-n-nlfs}); |
$dh^{n}$ to compute the tendencies (eq.\ref{eq:Gt-n-nlfs}); |
| 404 |
\end{minipage} } |
\end{minipage} } |
| 405 |
|
|
| 406 |
\subsubsection{Non-linear free-surface and vertical resolution} |
\subsubsection{Non-linear free-surface and vertical resolution} |
| 407 |
\label{sect:nonlin-freesurf-dz_surf} |
\label{sec:nonlin-freesurf-dz_surf} |
| 408 |
|
|
| 409 |
When the amplitude of the free-surface variations becomes |
When the amplitude of the free-surface variations becomes |
| 410 |
as large as the vertical resolution near the surface, |
as large as the vertical resolution near the surface, |
Parent Directory
| 
Revision Log
| 
Revision Graph
| 
Patch