| 2 |
% $Name$ |
% $Name$ |
| 3 |
|
|
| 4 |
The convention used in this section is as follows: |
The convention used in this section is as follows: |
| 5 |
Time is "discretize" using a time step $\Delta t$ |
Time is ``discretized'' using a time step $\Delta t$ |
| 6 |
and $\Phi^n$ refers to the variable $\Phi$ |
and $\Phi^n$ refers to the variable $\Phi$ |
| 7 |
at time $t = n \Delta t$ . We used the notation $\Phi^{(n)}$ |
at time $t = n \Delta t$ . We use the notation $\Phi^{(n)}$ |
| 8 |
when time interpolation is required to estimate the value of $\phi$ |
when time interpolation is required to estimate the value of $\phi$ |
| 9 |
at the time $n \Delta t$. |
at the time $n \Delta t$. |
| 10 |
|
|
| 12 |
|
|
| 13 |
The discretization in time of the model equations (cf section I ) |
The discretization in time of the model equations (cf section I ) |
| 14 |
does not depend of the discretization in space of each |
does not depend of the discretization in space of each |
| 15 |
term, so that this section can be read independently. |
term and so this section can be read independently. |
|
Also for this purpose, we will refers to the continuous |
|
|
space-derivative form of model equations, and focus on |
|
|
the time discretization. |
|
| 16 |
|
|
| 17 |
The continuous form of the model equations is: |
The continuous form of the model equations is: |
| 18 |
|
|
| 19 |
\begin{eqnarray} |
\begin{eqnarray} |
| 20 |
\partial_t \theta & = & G_\theta |
\partial_t \theta & = & G_\theta |
| 21 |
|
\label{eq-tCsC-theta} |
| 22 |
\\ |
\\ |
| 23 |
\partial_t S & = & G_s |
\partial_t S & = & G_s |
| 24 |
|
\label{eq-tCsC-salt} |
| 25 |
\\ |
\\ |
| 26 |
b' & = & b'(\theta,S,r) |
b' & = & b'(\theta,S,r) |
| 27 |
\\ |
\\ |
| 28 |
\partial_r \phi'_{hyd} & = & -b' |
\partial_r \phi'_{hyd} & = & -b' |
| 29 |
\label{eq-r-split-hyd} |
\label{eq-tCsC-hyd} |
| 30 |
\\ |
\\ |
| 31 |
\partial_t \vec{\bf v} |
\partial_t \vec{\bf v} |
| 32 |
+ {\bf \nabla}_h b_s \eta |
+ {\bf \nabla}_h b_s \eta |
| 33 |
+ \epsilon_{nh} {\bf \nabla}_h \phi'_{nh} |
+ \epsilon_{nh} {\bf \nabla}_h \phi'_{nh} |
| 34 |
& = & \vec{\bf G}_{\vec{\bf v}} |
& = & \vec{\bf G}_{\vec{\bf v}} |
| 35 |
- {\bf \nabla}_h \phi'_{hyd} |
- {\bf \nabla}_h \phi'_{hyd} |
| 36 |
\label{eq-r-split-hmom} |
\label{eq-tCsC-Hmom} |
| 37 |
\\ |
\\ |
| 38 |
\epsilon_{nh} \frac {\partial{\dot{r}}}{\partial{t}} |
\epsilon_{nh} \frac {\partial{\dot{r}}}{\partial{t}} |
| 39 |
+ \epsilon_{nh} \partial_r \phi'_{nh} |
+ \epsilon_{nh} \partial_r \phi'_{nh} |
| 40 |
& = & \epsilon_{nh} G_{\dot{r}} |
& = & \epsilon_{nh} G_{\dot{r}} |
| 41 |
\label{eq-r-split-rdot} |
\label{eq-tCsC-Vmom} |
| 42 |
\\ |
\\ |
| 43 |
{\bf \nabla}_h \cdot \vec{\bf v} + \partial_r \dot{r} |
{\bf \nabla}_h \cdot \vec{\bf v} + \partial_r \dot{r} |
| 44 |
& = & 0 |
& = & 0 |
| 45 |
\label{eq-r-cont} |
\label{eq-tCsC-cont} |
| 46 |
\end{eqnarray} |
\end{eqnarray} |
| 47 |
where |
where |
| 48 |
\begin{eqnarray*} |
\begin{eqnarray*} |
| 63 |
- \vec{\bf v} \cdot {\bf \nabla} \dot{r} |
- \vec{\bf v} \cdot {\bf \nabla} \dot{r} |
| 64 |
+ {\cal F}_{\dot{r}} |
+ {\cal F}_{\dot{r}} |
| 65 |
\end{eqnarray*} |
\end{eqnarray*} |
| 66 |
The exact form of all the "{\it G}"s terms is described in the next |
The exact form of all the ``{\it G}''s terms is described in the next |
| 67 |
section (). Here its sufficient to mention that they contains |
section \ref{sect:discrete}. Here its sufficient to mention that they contains |
| 68 |
all the RHS terms except the pressure / geo- potential terms. |
all the RHS terms except the pressure/geo-potential terms. |
| 69 |
|
|
| 70 |
The switch $\epsilon_{nh}$ allows to activate the non hydrostatic |
The switch $\epsilon_{nh}$ allows one to activate the non-hydrostatic |
| 71 |
mode ($\epsilon_{nh}=1$) for the ocean model. Otherwise, |
mode ($\epsilon_{nh}=1$) for the ocean model. Otherwise, in the |
| 72 |
in the hydrostatic limit $\epsilon_{nh} = 0$ |
hydrostatic limit $\epsilon_{nh} = 0$ and equation \ref{eq-tCsC-Vmom} |
| 73 |
and equation \ref{eq-r-split-rdot} vanishes. |
is not used. |
| 74 |
|
|
| 75 |
The equation for $\eta$ is obtained by integrating the |
As discussed in section \ref{sect:1.3.6.2}, the equation for $\eta$ is |
| 76 |
continuity equation over the entire depth of the fluid, |
obtained by integrating the continuity equation over the entire depth |
| 77 |
from $R_{min}(x,y)$ up to $R_o(x,y)$ |
of the fluid, from $R_{fixed}(x,y)$ up to $R_o(x,y)$. The linear free |
| 78 |
(Linear free surface): |
surface evolves according to: |
| 79 |
\begin{eqnarray} |
\begin{eqnarray} |
| 80 |
\epsilon_{fs} \partial_t \eta = |
\epsilon_{fs} \partial_t \eta = |
| 81 |
\left. \dot{r} \right|_{r=r_{surf}} + \epsilon_{fw} (P-E) = |
\left. \dot{r} \right|_{r=r_{surf}} + \epsilon_{fw} (P-E) = |
| 82 |
- {\bf \nabla} \cdot \int_{R_{min}}^{R_o} \vec{\bf v} dr |
- {\bf \nabla} \cdot \int_{R_{fixed}}^{R_o} \vec{\bf v} dr |
| 83 |
+ \epsilon_{fw} (P-E) |
+ \epsilon_{fw} (P-E) |
| 84 |
\label{eq-cont-2D} |
\label{eq-tCsC-eta} |
| 85 |
\end{eqnarray} |
\end{eqnarray} |
| 86 |
|
|
| 87 |
Where $\epsilon_{fs}$,$\epsilon_{fw}$ are two flags to |
Here, $\epsilon_{fs}$ is a flag to switch between the free-surface, |
| 88 |
distinguish between a free-surface equation ($\epsilon_{fs}=1$) |
$\epsilon_{fs}=1$, and a rigid-lid, $\epsilon_{fs}=0$. The flag |
| 89 |
or the rigid-lid approximation ($\epsilon_{fs}=0$); |
$\epsilon_{fw}$ determines whether an exchange of fresh water is |
| 90 |
and to distinguish when exchange of Fresh-Water is included |
included at the ocean surface (natural BC) ($\epsilon_{fw} = 1$) or |
| 91 |
at the ocean surface (natural BC) ($\epsilon_{fw} = 1$) |
not ($\epsilon_{fw} = 0$). |
|
or not ($\epsilon_{fw} = 0$). |
|
| 92 |
|
|
| 93 |
The hydrostatic potential is found by |
The hydrostatic potential is found by integrating (equation |
| 94 |
integrating \ref{eq-r-split-hyd} with the boundary condition that |
\ref{eq-tCsC-hyd}) with the boundary condition that |
| 95 |
$\phi'_{hyd}(r=R_o) = 0$: |
$\phi'_{hyd}(r=R_o) = 0$: |
| 96 |
\begin{eqnarray*} |
\begin{eqnarray*} |
| 97 |
& & |
& & |
| 105 |
|
|
| 106 |
\subsection{General method} |
\subsection{General method} |
| 107 |
|
|
| 108 |
The general algorithm consist in a "predictor step" that computes |
An overview of the general method is now presented with explicit |
| 109 |
the forward tendencies ("G" terms") and all |
references to the Fortran code. We often refer to the discretized |
| 110 |
the "first guess" values star notation): |
equations of the model that are detailed in the following sections. |
| 111 |
$\vec{\bf v}^*, \theta^*, S^*$ (and $\dot{r}^*$ |
|
| 112 |
in non-hydrostatic mode). This is done in the routine {\it DYNAMICS}. |
The general algorithm consist of a ``predictor step'' that computes |
| 113 |
|
the forward tendencies ("G" terms") comprising of explicit-in-time |
| 114 |
Then the implicit terms that appear here on the left hand side (LHS), |
terms and the ``first guess'' values (star notation): $\theta^*, S^*, |
| 115 |
are solved as follows: |
\vec{\bf v}^*$ (and $\dot{r}^*$ in non-hydrostatic mode). This is done |
| 116 |
Since implicit vertical diffusion and viscosity terms |
in the two routines {\it THERMODYNAMICS} and {\it DYNAMICS}. |
| 117 |
are independent from the barotropic flow adjustment, |
|
| 118 |
they are computed first, solving a 3 diagonal Nr x Nr linear system, |
Terms that are integrated implicitly in time are handled at the end of |
| 119 |
and incorporated at the end of the {\it DYNAMICS} routines. |
the {\it THERMODYNAMICS} and {\it DYNAMICS} routines. Then the |
| 120 |
Then the surface pressure and non hydrostatic pressure |
surface pressure and non hydrostatic pressure are solved for in ({\it |
| 121 |
are evaluated ({\it SOLVE\_FOR\_PRESSURE}); |
SOLVE\_FOR\_PRESSURE}). |
| 122 |
|
|
| 123 |
Finally, within a "corrector step', |
Finally, in the ``corrector step'', (routine {\it |
| 124 |
(routine {\it THE\_CORRECTION\_STEP}) |
THE\_CORRECTION\_STEP}) the new values of $u,v,w,\theta,S$ are |
| 125 |
the new values of $u,v,w,\theta,S$ |
determined (see details in \ref{sect:II.1.3}). |
| 126 |
are derived according to the above equations |
|
| 127 |
(see details in II.1.3). |
At this point, the regular time stepping process is complete. However, |
| 128 |
|
after the correction step there are optional adjustments such as |
| 129 |
At this point, the regular time step is over, but |
convective adjustment or filters (zonal FFT filter, shapiro filter) |
| 130 |
the correction step contains also other optional |
that can be applied on both momentum and tracer fields, just prior to |
| 131 |
adjustments such as convective adjustment algorithm, or filters |
incrementing the time level to $n+1$. |
| 132 |
(zonal FFT filter, shapiro filter) |
|
| 133 |
that applied on both momentum and tracer fields. |
Since the pressure solver precision is of the order of the ``target |
| 134 |
just before setting the $n+1$ new time step value. |
residual'' and can be lower than the the computer truncation error, |
| 135 |
|
and also because some filters might alter the divergence part of the |
| 136 |
Since the pressure solver precision is of the order of |
flow field, a final evaluation of the surface r anomaly $\eta^{n+1}$ |
| 137 |
the "target residual" that could be lower than the |
is performed in {\it CALC\_EXACT\_ETA}. This ensures exact volume |
| 138 |
the computer truncation error, and also because some filters |
conservation. Note that there is no need for an equivalent |
| 139 |
might alter the divergence part of the flow field, |
non-hydrostatic ``exact conservation'' step, since by default $w$ is |
| 140 |
a final evaluation of the surface r anomaly $\eta^{n+1}$ |
already computed after the filters are applied. |
| 141 |
is performed, according to \ref{eq-rtd-eta} ({\it CALC\_EXACT\_ETA}). |
|
| 142 |
This ensures a perfect volume conservation. |
Optional forcing terms (usually part of a physics ``package''), that |
| 143 |
Note that there is no need for an equivalent Non-hydrostatic |
account for a specific source or sink process (e.g. condensation as a |
| 144 |
"exact conservation" step, since W is already computed after |
sink of water vapor Q) are generally incorporated in the main |
| 145 |
the filters applied. |
algorithm as follows: at the the beginning of the time step, the |
| 146 |
|
additional tendencies are computed as a function of the present state |
| 147 |
optionnal forcing terms (package):\\ |
(time step $n$) and external forcing; then within the main part of |
| 148 |
Regarding optional forcing terms (usually part of a "package"), |
model, only those new tendencies are added to the model variables. |
|
that a account for a specific source or sink term (e.g.: condensation |
|
|
as a sink of water vapor Q), they are generally incorporated |
|
|
in the main algorithm as follows; |
|
|
At the the beginning of the time step, |
|
|
the additionnal tendencies are computed |
|
|
as function of the present state (time step $n$) and external forcing ; |
|
|
Then within the main part of model, |
|
|
only those new tendencies are added to the model variables. |
|
| 149 |
|
|
| 150 |
[more details needed]\\ |
[more details needed]\\ |
| 151 |
|
|
| 152 |
The atmospheric physics follows this general scheme. |
The atmospheric physics follows this general scheme. |
| 153 |
|
|
| 154 |
|
[more about C\_grid, A\_grid conversion \& drag term]\\ |
| 155 |
|
|
| 156 |
|
|
| 157 |
|
|
| 158 |
\subsection{Standard synchronous time stepping} |
\subsection{Standard synchronous time stepping} |
| 159 |
|
|
| 160 |
In the standard formulation, the surface pressure is |
In the standard formulation, the surface pressure is evaluated at time |
| 161 |
evaluated at time step n+1 (implicit method). |
step n+1 (an implicit method). Equations \ref{eq-tCsC-theta} to |
| 162 |
The above set of equations is then discretized in time |
\ref{eq-tCsC-cont} are then discretized in time as follows: |
|
as follows: |
|
| 163 |
\begin{eqnarray} |
\begin{eqnarray} |
| 164 |
\left[ 1 - \partial_r \kappa_v^\theta \partial_r \right] |
\left[ 1 - \partial_r \kappa_v^\theta \partial_r \right] |
| 165 |
\theta^{n+1} & = & \theta^* |
\theta^{n+1} & = & \theta^* |
| 166 |
|
\label{eq-tDsC-theta} |
| 167 |
\\ |
\\ |
| 168 |
\left[ 1 - \partial_r \kappa_v^S \partial_r \right] |
\left[ 1 - \partial_r \kappa_v^S \partial_r \right] |
| 169 |
S^{n+1} & = & S^* |
S^{n+1} & = & S^* |
| 170 |
|
\label{eq-tDsC-salt} |
| 171 |
\\ |
\\ |
| 172 |
%{b'}^{n} & = & b'(\theta^{n},S^{n},r) |
%{b'}^{n} & = & b'(\theta^{n},S^{n},r) |
| 173 |
%\partial_r {\phi'_{hyd}}^{n} & = & {-b'}^{n} |
%\partial_r {\phi'_{hyd}}^{n} & = & {-b'}^{n} |
| 174 |
%\\ |
%\\ |
| 175 |
{\phi'_{hyd}}^{n} & = & \int_{r'}^{R_o} b'(\theta^{n},S^{n},r) dr |
{\phi'_{hyd}}^{n} & = & \int_{r'}^{R_o} b'(\theta^{n},S^{n},r) dr |
| 176 |
\label{eq-rtd-hyd} |
\label{eq-tDsC-hyd} |
| 177 |
\\ |
\\ |
| 178 |
\vec{\bf v} ^{n+1} |
\vec{\bf v} ^{n+1} |
| 179 |
+ \Delta t {\bf \nabla}_h b_s {\eta}^{n+1} |
+ \Delta t {\bf \nabla}_h b_s {\eta}^{n+1} |
| 181 |
- \partial_r A_v \partial_r \vec{\bf v}^{n+1} |
- \partial_r A_v \partial_r \vec{\bf v}^{n+1} |
| 182 |
& = & |
& = & |
| 183 |
\vec{\bf v}^* |
\vec{\bf v}^* |
| 184 |
\label{eq-rtd-hmom} |
\label{eq-tDsC-Hmom} |
| 185 |
\\ |
\\ |
| 186 |
\epsilon_{fs} {\eta}^{n+1} + \Delta t |
\epsilon_{fs} {\eta}^{n+1} + \Delta t |
| 187 |
{\bf \nabla}_h \cdot \int_{R_{min}}^{R_o} \vec{\bf v}^{n+1} dr |
{\bf \nabla}_h \cdot \int_{R_{fixed}}^{R_o} \vec{\bf v}^{n+1} dr |
| 188 |
& = & |
& = & |
| 189 |
\epsilon_{fs} {\eta}^{n} + \epsilon_{fw} \Delta_t (P-E)^{n} |
\epsilon_{fs} {\eta}^{n} + \epsilon_{fw} \Delta_t (P-E)^{n} |
| 190 |
\nonumber |
\nonumber |
| 191 |
\\ |
\\ |
| 192 |
% = \epsilon_{fs} {\eta}^{n} & + & \epsilon_{fw} \Delta_t (P-E)^{n} |
% = \epsilon_{fs} {\eta}^{n} & + & \epsilon_{fw} \Delta_t (P-E)^{n} |
| 193 |
\label{eq-rtd-eta} |
\label{eq-tDsC-eta} |
| 194 |
\\ |
\\ |
| 195 |
\epsilon_{nh} \left( \dot{r} ^{n+1} |
\epsilon_{nh} \left( \dot{r} ^{n+1} |
| 196 |
+ \Delta t \partial_r {\phi'_{nh}} ^{n+1} |
+ \Delta t \partial_r {\phi'_{nh}} ^{n+1} |
| 197 |
\right) |
\right) |
| 198 |
& = & \epsilon_{nh} \dot{r}^* |
& = & \epsilon_{nh} \dot{r}^* |
| 199 |
\label{eq-rtd-rdot} |
\label{eq-tDsC-Vmom} |
| 200 |
\\ |
\\ |
| 201 |
{\bf \nabla}_h \cdot \vec{\bf v}^{n+1} + \partial_r \dot{r}^{n+1} |
{\bf \nabla}_h \cdot \vec{\bf v}^{n+1} + \partial_r \dot{r}^{n+1} |
| 202 |
& = & 0 |
& = & 0 |
| 203 |
\label{eq-rtd-cont} |
\label{eq-tDsC-cont} |
| 204 |
\end{eqnarray} |
\end{eqnarray} |
| 205 |
where |
where |
| 206 |
\begin{eqnarray} |
\begin{eqnarray} |
| 218 |
\dot{r} ^{n} + \Delta t G_{\dot{r}} ^{(n+1/2)} |
\dot{r} ^{n} + \Delta t G_{\dot{r}} ^{(n+1/2)} |
| 219 |
\end{eqnarray} |
\end{eqnarray} |
| 220 |
|
|
| 221 |
Note that implicit vertical terms (viscosity and diffusivity) are |
Note that implicit vertical viscosity and diffusivity terms are not |
| 222 |
not considered as part of the "{\it G}" terms, but are |
considered as part of the ``{\it G}'' terms, but are written |
| 223 |
written separately here. |
separately here. |
| 224 |
|
|
| 225 |
To ensure a second order time discretization for both |
The default time-stepping method is the Adams-Bashforth quasi-second |
| 226 |
momentum and tracer, |
order scheme in which the ``G'' terms are extrapolated forward in time |
| 227 |
The "G" terms are "extrapolated" forward in time |
from time-levels $n-1$ and $n$ to time-level $n+1/2$ and provides a |
| 228 |
(Adams Bashforth time stepping) |
simple but stable algorithm: |
| 229 |
from the values computed at time step $n$ and $n-1$ |
\begin{equation} |
| 230 |
to the time $n+1/2$: |
G^{(n+1/2)} = G^n + (1/2+\epsilon_{AB}) (G^n - G^{n-1}) |
| 231 |
$$G^{(n+1/2)} = G^n + (1/2+\epsilon_{AB}) (G^n - G^{n-1})$$ |
\end{equation} |
| 232 |
A small number for the parameter $\epsilon_{AB}$ is generally used |
where $\epsilon_{AB}$ is a small number used to stabilize the time |
| 233 |
to stabilize this time stepping. |
stepping. |
| 234 |
|
|
| 235 |
In the standard non-stagger formulation, |
In the standard non-staggered formulation, the Adams-Bashforth time |
| 236 |
the Adams-Bashforth time stepping is also applied to the |
stepping is also applied to the hydrostatic pressure/geo-potential |
| 237 |
hydrostatic (pressure / geo-) potential term $\nabla_h \Phi'_{hyd}$. |
gradient, $\nabla_h \Phi'_{hyd}$. Note that presently, this term is in |
| 238 |
Note that presently, this term is in fact incorporated to the |
fact incorporated to the $\vec{\bf G}_{\vec{\bf v}}$ arrays ({\bf |
| 239 |
$\vec{\bf G}_{\vec{\bf v}}$ arrays ({\bf gU,gV}). |
gU,gV}). |
| 240 |
|
\marginpar{JMC: Clarify ``this term''?} |
| 241 |
|
|
| 242 |
|
\fbox{ \begin{minipage}{4.75in} |
| 243 |
|
{\em S/R TIMESTEP} ({\em timestep.F}) |
| 244 |
|
|
| 245 |
|
$G_u^n$: {\bf Gu} ({\em DYNVARS.h}) |
| 246 |
|
|
| 247 |
|
$G_u^{n-1}, u^*$: {\bf GuNm1} ({\em DYNVARS.h}) |
| 248 |
|
|
| 249 |
|
$G_v^n$: {\bf Gv} ({\em DYNVARS.h}) |
| 250 |
|
|
| 251 |
|
$G_v^{n-1}, v^*$: {\bf GvNm1} ({\em DYNVARS.h}) |
| 252 |
|
|
| 253 |
|
$G_u^{(n+1/2)}$: {\bf GuTmp} (local) |
| 254 |
|
|
| 255 |
|
$G_v^{(n+1/2)}$: {\bf GvTmp} (local) |
| 256 |
|
|
| 257 |
|
\end{minipage} } |
| 258 |
|
|
| 259 |
|
|
| 260 |
|
|
| 261 |
|
|
| 262 |
|
|
| 263 |
\subsection{Stagger baroclinic time stepping} |
\subsection{Stagger baroclinic time stepping} |
| 264 |
|
|
| 265 |
An alternative is to evaluate $\phi'_{hyd}$ with the |
An alternative to synchronous time-stepping is to stagger the momentum |
| 266 |
new tracer fields, and step forward the momentum after. |
and tracer fields in time. This allows the evaluation and gradient of |
| 267 |
This option is known as stagger baroclinic time stepping, |
$\phi'_{hyd}$ to be centered in time with out needing to use the |
| 268 |
since tracer and momentum are step forward in time one after the other. |
Adams-Bashforth extrapoltion. This option is known as staggered |
| 269 |
It can be activated turning on a running flag parameter |
baroclinic time stepping because tracer and momentum are stepped |
| 270 |
{\bf staggerTimeStep} in file "{\it data}"). |
forward-in-time one after the other. It can be activated by turning |
| 271 |
|
on a run-time parameter {\bf staggerTimeStep} in namelist ``{\it |
| 272 |
The main advantage of this time stepping compared to a synchronous one, |
PARM01}''. |
| 273 |
is a better stability, specially regarding internal gravity waves, |
|
| 274 |
and a very natural implementation of a 2nd order in time |
The main advantage of staggered time-stepping compared to synchronous, |
| 275 |
hydrostatic pressure / geo- potential term. |
is improved stability to internal gravity wave motions and a very |
| 276 |
In the other hand, a synchronous time step might be better |
natural implementation of a 2nd order in time hydrostatic |
| 277 |
for convection problems; Its also make simpler time dependent forcing |
pressure/geo-potential gradient term. However, synchronous |
| 278 |
and diagnostic implementation ; and allows a more efficient threading. |
time-stepping might be better for convection problems, time dependent |
| 279 |
|
forcing and diagnosing the model are also easier and it allows a more |
| 280 |
Although the stagger time step does not affect deeply the |
efficient parallel decomposition. |
|
structure of the code --- a switch allows to evaluate the |
|
|
hydrostatic pressure / geo- potential from new $\theta,S$ |
|
|
instead of the Adams-Bashforth estimation --- |
|
|
this affect the way the time discretization is presented : |
|
| 281 |
|
|
| 282 |
\begin{eqnarray*} |
The staggered baroclinic time-stepping scheme is equations |
| 283 |
\left[ 1 - \partial_r \kappa_v^\theta \partial_r \right] |
\ref{eq-tDsC-theta} to \ref{eq-tDsC-cont} except that \ref{eq-tDsC-hyd} is replaced with: |
| 284 |
\theta^{n+1/2} & = & \theta^* |
\begin{equation} |
| 285 |
\\ |
{\phi'_{hyd}}^{n+1/2} = \int_{r'}^{R_o} b'(\theta^{n+1/2},S^{n+1/2},r) |
| 286 |
\left[ 1 - \partial_r \kappa_v^S \partial_r \right] |
dr |
| 287 |
S^{n+1/2} & = & S^* |
\end{equation} |
| 288 |
\end{eqnarray*} |
and the time-level for tracers has been shifted back by half: |
|
with |
|
| 289 |
\begin{eqnarray*} |
\begin{eqnarray*} |
| 290 |
\theta^* & = & |
\theta^* & = & |
| 291 |
\theta ^{(n-1/2)} + \Delta t G_{\theta} ^{(n)} |
\theta ^{(n-1/2)} + \Delta t G_{\theta} ^{(n)} |
| 292 |
\\ |
\\ |
| 293 |
S^* & = & |
S^* & = & |
| 294 |
S ^{(n-1/2)} + \Delta t G_{S} ^{(n)} |
S ^{(n-1/2)} + \Delta t G_{S} ^{(n)} |
|
\end{eqnarray*} |
|
|
And |
|
|
\begin{eqnarray*} |
|
|
%{b'}^{n+1/2} & = & b'(\theta^{n+1/2},S^{n+1/2},r) |
|
|
%\\ |
|
|
%\partial_r {\phi'_{hyd}}^{n+1/2} & = & {-b'}^{n+1/2} |
|
|
{\phi'_{hyd}}^{n+1/2} & = & \int_{r'}^{R_o} b'(\theta^{n+1/2},S^{n+1/2},r) dr |
|
|
%\label{eq-rtd-hyd} |
|
|
\\ |
|
|
\vec{\bf v} ^{n+1} |
|
|
+ \Delta t {\bf \nabla}_h b_s {\eta}^{n+1} |
|
|
+ \epsilon_{nh} \Delta t {\bf \nabla}_h {\phi'_{nh}}^{n+1} |
|
|
- \partial_r A_v \partial_r \vec{\bf v}^{n+1} |
|
|
& = & |
|
|
\vec{\bf v}^* |
|
|
%\label{eq-rtd-hmom} |
|
|
\\ |
|
|
\epsilon_{fs} {\eta}^{n+1} + \Delta t |
|
|
{\bf \nabla}_h \cdot \int_{R_{min}}^{R_o} \vec{\bf v}^{n+1} dr |
|
|
& = & |
|
|
\epsilon_{fs} {\eta}^{n} + \epsilon_{fw} \Delta_t (P-E)^{n} |
|
|
\\ |
|
|
\epsilon_{nh} \left( \dot{r} ^{n+1} |
|
|
+ \Delta t \partial_r {\phi'_{nh}} ^{n+1} |
|
|
\right) |
|
|
& = & \epsilon_{nh} \dot{r}^* |
|
|
%\label{eq-rtd-rdot} |
|
| 295 |
\\ |
\\ |
| 296 |
{\bf \nabla}_h \cdot \vec{\bf v}^{n+1} + \partial_r \dot{r}^{n+1} |
\left[ 1 - \partial_r \kappa_v^\theta \partial_r \right] |
| 297 |
& = & 0 |
\theta^{n+1/2} & = & \theta^* |
|
%\label{eq-rtd-cont} |
|
|
\end{eqnarray*} |
|
|
with |
|
|
\begin{eqnarray*} |
|
|
\vec{\bf v}^* & = & |
|
|
\vec{\bf v} ^{n} + \Delta t \vec{\bf G}_{\vec{\bf v}} ^{(n+1/2)} |
|
|
+ \Delta t {\bf \nabla}_h {\phi'_{hyd}}^{n+1/2} |
|
| 298 |
\\ |
\\ |
| 299 |
\dot{r}^* & = & |
\left[ 1 - \partial_r \kappa_v^S \partial_r \right] |
| 300 |
\dot{r} ^{n} + \Delta t G_{\dot{r}} ^{(n+1/2)} |
S^{n+1/2} & = & S^* |
| 301 |
\end{eqnarray*} |
\end{eqnarray*} |
| 302 |
|
|
|
%--------------------------------------------------------------------- |
|
| 303 |
|
|
| 304 |
\subsection{Surface pressure} |
\subsection{Surface pressure} |
| 305 |
|
|
| 306 |
Substituting \ref{eq-rtd-hmom} into \ref{eq-rtd-cont}, assuming |
Substituting \ref{eq-tDsC-Hmom} into \ref{eq-tDsC-cont}, assuming |
| 307 |
$\epsilon_{nh} = 0$ yields a Helmholtz equation for ${\eta}^{n+1}$: |
$\epsilon_{nh} = 0$ yields a Helmholtz equation for ${\eta}^{n+1}$: |
| 308 |
\begin{eqnarray} |
\begin{eqnarray} |
| 309 |
\epsilon_{fs} {\eta}^{n+1} - |
\epsilon_{fs} {\eta}^{n+1} - |
| 310 |
{\bf \nabla}_h \cdot \Delta t^2 (R_o-R_{min}) |
{\bf \nabla}_h \cdot \Delta t^2 (R_o-R_{fixed}) |
| 311 |
{\bf \nabla}_h b_s {\eta}^{n+1} |
{\bf \nabla}_h b_s {\eta}^{n+1} |
| 312 |
= {\eta}^* |
= {\eta}^* |
| 313 |
\label{solve_2d} |
\label{eq-solve2D} |
| 314 |
\end{eqnarray} |
\end{eqnarray} |
| 315 |
where |
where |
| 316 |
\begin{eqnarray} |
\begin{eqnarray} |
| 317 |
{\eta}^* = \epsilon_{fs} \: {\eta}^{n} - |
{\eta}^* = \epsilon_{fs} \: {\eta}^{n} - |
| 318 |
\Delta t {\bf \nabla}_h \cdot \int_{R_{min}}^{R_o} \vec{\bf v}^* dr |
\Delta t {\bf \nabla}_h \cdot \int_{R_{fixed}}^{R_o} \vec{\bf v}^* dr |
| 319 |
\: + \: \epsilon_{fw} \Delta_t (P-E)^{n} |
\: + \: \epsilon_{fw} \Delta_t (P-E)^{n} |
| 320 |
\label{solve_2d_rhs} |
\label{eq-solve2D_rhs} |
| 321 |
\end{eqnarray} |
\end{eqnarray} |
| 322 |
|
|
| 323 |
Once ${\eta}^{n+1}$ has been found substituting into \ref{eq-rtd-hmom} |
\fbox{ \begin{minipage}{4.75in} |
| 324 |
would yield $\vec{\bf v}^{n+1}$ if the model is hydrostatic |
{\em S/R SOLVE\_FOR\_PRESSURE} ({\em solve\_for\_pressure.F}) |
| 325 |
($\epsilon_{nh}=0$): |
|
| 326 |
|
$u^*$: {\bf GuNm1} ({\em DYNVARS.h}) |
| 327 |
|
|
| 328 |
|
$v^*$: {\bf GvNm1} ({\em DYNVARS.h}) |
| 329 |
|
|
| 330 |
|
$\eta^*$: {\bf cg2d\_b} (\em SOLVE\_FOR\_PRESSURE.h) |
| 331 |
|
|
| 332 |
|
$\eta^{n+1}$: {\bf etaN} (\em DYNVARS.h) |
| 333 |
|
|
| 334 |
|
\end{minipage} } |
| 335 |
|
|
| 336 |
|
|
| 337 |
|
Once ${\eta}^{n+1}$ has been found, substituting into |
| 338 |
|
\ref{eq-tDsC-Hmom} yields $\vec{\bf v}^{n+1}$ if the model is |
| 339 |
|
hydrostatic ($\epsilon_{nh}=0$): |
| 340 |
$$ |
$$ |
| 341 |
\vec{\bf v}^{n+1} = \vec{\bf v}^{*} |
\vec{\bf v}^{n+1} = \vec{\bf v}^{*} |
| 342 |
- \Delta t {\bf \nabla}_h b_s {\eta}^{n+1} |
- \Delta t {\bf \nabla}_h b_s {\eta}^{n+1} |
| 344 |
|
|
| 345 |
This is known as the correction step. However, when the model is |
This is known as the correction step. However, when the model is |
| 346 |
non-hydrostatic ($\epsilon_{nh}=1$) we need an additional step and an |
non-hydrostatic ($\epsilon_{nh}=1$) we need an additional step and an |
| 347 |
additional equation for $\phi'_{nh}$. This is obtained by |
additional equation for $\phi'_{nh}$. This is obtained by substituting |
| 348 |
substituting \ref{eq-rtd-hmom} and \ref{eq-rtd-rdot} into |
\ref{eq-tDsC-Hmom} and \ref{eq-tDsC-Vmom} into |
| 349 |
\ref{eq-rtd-cont}: |
\ref{eq-tDsC-cont}: |
| 350 |
\begin{equation} |
\begin{equation} |
| 351 |
\left[ {\bf \nabla}_h^2 + \partial_{rr} \right] {\phi'_{nh}}^{n+1} |
\left[ {\bf \nabla}_h^2 + \partial_{rr} \right] {\phi'_{nh}}^{n+1} |
| 352 |
= \frac{1}{\Delta t} \left( |
= \frac{1}{\Delta t} \left( |
| 367 |
- \epsilon_{nh} \Delta t {\bf \nabla}_h {\phi'_{nh}}^{n+1} |
- \epsilon_{nh} \Delta t {\bf \nabla}_h {\phi'_{nh}}^{n+1} |
| 368 |
\end{equation} |
\end{equation} |
| 369 |
and the vertical velocity is found by integrating the continuity |
and the vertical velocity is found by integrating the continuity |
| 370 |
equation vertically. |
equation vertically. Note that, for the convenience of the restart |
| 371 |
Note that, for convenience regarding the restart procedure, |
procedure, the vertical integration of the continuity equation has |
| 372 |
the integration of the continuity equation has been |
been moved to the beginning of the time step (instead of at the end), |
|
moved at the beginning of the time step (instead of at the end), |
|
| 373 |
without any consequence on the solution. |
without any consequence on the solution. |
| 374 |
|
|
| 375 |
Regarding the implementation, all those computation are done |
\fbox{ \begin{minipage}{4.75in} |
| 376 |
within the routine {\it SOLVE\_FOR\_PRESSURE} and its dependent |
{\em S/R CORRECTION\_STEP} ({\em correction\_step.F}) |
| 377 |
{\it CALL}s. |
|
| 378 |
The standard method to solve the 2D elliptic problem (\ref{solve_2d}) |
$\eta^{n+1}$: {\bf etaN} (\em DYNVARS.h) |
| 379 |
uses the conjugate gradient method (routine {\it CG2D}); The |
|
| 380 |
the solver matrix and conjugate gradient operator are only function |
$\phi_{nh}^{n+1}$: {\bf phi\_nh} (\em DYNVARS.h) |
| 381 |
of the discretized domain and are therefore evaluated separately, |
|
| 382 |
before the time iteration loop, within {\it INI\_CG2D}. |
$u^*$: {\bf GuNm1} ({\em DYNVARS.h}) |
| 383 |
The computation of the RHS $\eta^*$ is partly |
|
| 384 |
done in {\it CALC\_DIV\_GHAT} and in {\it SOLVE\_FOR\_PRESSURE}. |
$v^*$: {\bf GvNm1} ({\em DYNVARS.h}) |
| 385 |
|
|
| 386 |
The same method is applied for the non hydrostatic part, using |
$u^{n+1}$: {\bf uVel} ({\em DYNVARS.h}) |
| 387 |
a conjugate gradient 3D solver ({\it CG3D}) that is initialized |
|
| 388 |
in {\it INI\_CG3D}. The RHS terms of 2D and 3D problems |
$v^{n+1}$: {\bf vVel} ({\em DYNVARS.h}) |
| 389 |
are computed together, within the same part of the code. |
|
| 390 |
|
\end{minipage} } |
| 391 |
|
|
| 392 |
|
|
| 393 |
|
|
| 394 |
|
Regarding the implementation of the surface pressure solver, all |
| 395 |
|
computation are done within the routine {\it SOLVE\_FOR\_PRESSURE} and |
| 396 |
|
its dependent calls. The standard method to solve the 2D elliptic |
| 397 |
|
problem (\ref{eq-solve2D}) uses the conjugate gradient method (routine |
| 398 |
|
{\it CG2D}); the solver matrix and conjugate gradient operator are |
| 399 |
|
only function of the discretized domain and are therefore evaluated |
| 400 |
|
separately, before the time iteration loop, within {\it INI\_CG2D}. |
| 401 |
|
The computation of the RHS $\eta^*$ is partly done in {\it |
| 402 |
|
CALC\_DIV\_GHAT} and in {\it SOLVE\_FOR\_PRESSURE}. |
| 403 |
|
|
| 404 |
|
The same method is applied for the non hydrostatic part, using a |
| 405 |
|
conjugate gradient 3D solver ({\it CG3D}) that is initialized in {\it |
| 406 |
|
INI\_CG3D}. The RHS terms of 2D and 3D problems are computed together |
| 407 |
|
at the same point in the code. |
| 408 |
|
|
| 409 |
|
|
| 410 |
|
|
|
\newpage |
|
|
%----------------------------------------------------------------------------------- |
|
| 411 |
\subsection{Crank-Nickelson barotropic time stepping} |
\subsection{Crank-Nickelson barotropic time stepping} |
| 412 |
|
|
| 413 |
The full implicit time stepping described previously is unconditionally stable |
The full implicit time stepping described previously is |
| 414 |
but damps the fast gravity waves, resulting in a loss of |
unconditionally stable but damps the fast gravity waves, resulting in |
| 415 |
gravity potential energy. |
a loss of potential energy. The modification presented now allows one |
| 416 |
The modification presented hereafter allows to combine an implicit part |
to combine an implicit part ($\beta,\gamma$) and an explicit part |
| 417 |
($\beta,\gamma$) and an explicit part ($1-\beta,1-\gamma$) for the surface |
($1-\beta,1-\gamma$) for the surface pressure gradient ($\beta$) and |
| 418 |
pressure gradient ($\beta$) and for the barotropic flow divergence ($\gamma$). |
for the barotropic flow divergence ($\gamma$). |
| 419 |
\\ |
\\ |
| 420 |
For instance, $\beta=\gamma=1$ is the previous fully implicit scheme; |
For instance, $\beta=\gamma=1$ is the previous fully implicit scheme; |
| 421 |
$\beta=\gamma=1/2$ is the non damping (energy conserving), unconditionally |
$\beta=\gamma=1/2$ is the non damping (energy conserving), unconditionally |
| 426 |
{\it implicSurfPress}, {\it implicDiv2DFlow}. They are read from |
{\it implicSurfPress}, {\it implicDiv2DFlow}. They are read from |
| 427 |
the main data file "{\it data}" and are set by default to 1,1. |
the main data file "{\it data}" and are set by default to 1,1. |
| 428 |
|
|
| 429 |
Equations \ref{eq-rtd-hmom} and \ref{eq-rtd-eta} are modified as follows: |
Equations \ref{eq-tDsC-Hmom} and \ref{eq-tDsC-eta} are modified as follows: |
| 430 |
$$ |
$$ |
| 431 |
\frac{ \vec{\bf v}^{n+1} }{ \Delta t } |
\frac{ \vec{\bf v}^{n+1} }{ \Delta t } |
| 432 |
+ {\bf \nabla}_h b_s [ \beta {\eta}^{n+1} + (1-\beta) {\eta}^{n} ] |
+ {\bf \nabla}_h b_s [ \beta {\eta}^{n+1} + (1-\beta) {\eta}^{n} ] |
| 435 |
$$ |
$$ |
| 436 |
$$ |
$$ |
| 437 |
\epsilon_{fs} \frac{ {\eta}^{n+1} - {\eta}^{n} }{ \Delta t} |
\epsilon_{fs} \frac{ {\eta}^{n+1} - {\eta}^{n} }{ \Delta t} |
| 438 |
+ {\bf \nabla}_h \cdot \int_{R_{min}}^{R_o} |
+ {\bf \nabla}_h \cdot \int_{R_{fixed}}^{R_o} |
| 439 |
[ \gamma \vec{\bf v}^{n+1} + (1-\gamma) \vec{\bf v}^{n}] dr |
[ \gamma \vec{\bf v}^{n+1} + (1-\gamma) \vec{\bf v}^{n}] dr |
| 440 |
= \epsilon_{fw} (P-E) |
= \epsilon_{fw} (P-E) |
| 441 |
$$ |
$$ |
| 448 |
\\ |
\\ |
| 449 |
{\eta}^* & = & |
{\eta}^* & = & |
| 450 |
\epsilon_{fs} {\eta}^{n} + \epsilon_{fw} \Delta t (P-E) |
\epsilon_{fs} {\eta}^{n} + \epsilon_{fw} \Delta t (P-E) |
| 451 |
- \Delta t {\bf \nabla}_h \cdot \int_{R_{min}}^{R_o} |
- \Delta t {\bf \nabla}_h \cdot \int_{R_{fixed}}^{R_o} |
| 452 |
[ \gamma \vec{\bf v}^* + (1-\gamma) \vec{\bf v}^{n}] dr |
[ \gamma \vec{\bf v}^* + (1-\gamma) \vec{\bf v}^{n}] dr |
| 453 |
\end{eqnarray*} |
\end{eqnarray*} |
| 454 |
\\ |
\\ |
| 455 |
In the hydrostatic case ($\epsilon_{nh}=0$), |
In the hydrostatic case ($\epsilon_{nh}=0$), allowing us to find |
| 456 |
this allow to find ${\eta}^{n+1}$, according to: |
${\eta}^{n+1}$, thus: |
| 457 |
$$ |
$$ |
| 458 |
\epsilon_{fs} {\eta}^{n+1} - |
\epsilon_{fs} {\eta}^{n+1} - |
| 459 |
{\bf \nabla}_h \cdot \beta\gamma \Delta t^2 b_s (R_o - R_{min}) |
{\bf \nabla}_h \cdot \beta\gamma \Delta t^2 b_s (R_o - R_{fixed}) |
| 460 |
{\bf \nabla}_h {\eta}^{n+1} |
{\bf \nabla}_h {\eta}^{n+1} |
| 461 |
= {\eta}^* |
= {\eta}^* |
| 462 |
$$ |
$$ |
| 467 |
$$ |
$$ |
| 468 |
|
|
| 469 |
The non-hydrostatic part is solved as described previously. |
The non-hydrostatic part is solved as described previously. |
| 470 |
\\ \\ |
|
| 471 |
N.B: |
Note that: |
| 472 |
\\ |
\begin{enumerate} |
| 473 |
a) The non-hydrostatic part of the code has not yet been |
\item The non-hydrostatic part of the code has not yet been |
| 474 |
updated, %since it falls out of the purpose of this test, |
updated, so that this option cannot be used with $(\beta,\gamma) \neq (1,1)$. |
| 475 |
so that this option cannot be used with $(\beta,\gamma) \neq (1,1)$. |
\item The stability criteria with Crank-Nickelson time stepping |
| 476 |
\\ |
for the pure linear gravity wave problem in cartesian coordinates is: |
| 477 |
b) To remind, the stability criteria with the Crank-Nickelson time stepping |
\begin{itemize} |
| 478 |
for the pure linear gravity wave problem in cartesian coordinate is: |
\item $\beta + \gamma < 1$ : unstable |
| 479 |
\\ |
\item $\beta \geq 1/2$ and $ \gamma \geq 1/2$ : stable |
| 480 |
$\star$~ $\beta + \gamma < 1$ : unstable |
\item $\beta + \gamma \geq 1$ : stable if |
|
\\ |
|
|
$\star$~ $\beta \geq 1/2$ and $ \gamma \geq 1/2$ : stable |
|
|
\\ |
|
|
$\star$~ $\beta + \gamma \geq 1$ : stable if |
|
|
%, for all wave length $(k\Delta x,l\Delta y)$ |
|
| 481 |
$$ |
$$ |
| 482 |
c_{max}^2 (\beta - 1/2)(\gamma - 1/2) + 1 \geq 0 |
c_{max}^2 (\beta - 1/2)(\gamma - 1/2) + 1 \geq 0 |
| 483 |
$$ |
$$ |
| 492 |
c_{max} = 2 \Delta t \: \sqrt{g H} \: |
c_{max} = 2 \Delta t \: \sqrt{g H} \: |
| 493 |
\sqrt{ \frac{1}{\Delta x^2} + \frac{1}{\Delta y^2} } |
\sqrt{ \frac{1}{\Delta x^2} + \frac{1}{\Delta y^2} } |
| 494 |
$$ |
$$ |
| 495 |
|
\end{itemize} |
| 496 |
|
\end{enumerate} |
| 497 |
|
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