| 16 |
|
|
| 17 |
%---------------------------------------------------------------------- |
%---------------------------------------------------------------------- |
| 18 |
\subsubsection{Introduction |
\subsubsection{Introduction |
| 19 |
\label{sec:pkg:exf:intro}} |
\label{sec:pkg:seaice:intro}} |
| 20 |
|
|
| 21 |
|
|
| 22 |
Package ``seaice'' provides a dynamic and thermodynamic interactive |
Package ``seaice'' provides a dynamic and thermodynamic interactive |
| 25 |
CPP options enable or disable different aspects of the package |
CPP options enable or disable different aspects of the package |
| 26 |
(Section \ref{sec:pkg:seaice:config}). |
(Section \ref{sec:pkg:seaice:config}). |
| 27 |
Run-Time options, flags, filenames and field-related dates/times are |
Run-Time options, flags, filenames and field-related dates/times are |
| 28 |
set in \texttt{data.seaice} |
set in \code{data.seaice} |
| 29 |
(Section \ref{sec:pkg:seaice:runtime}). |
(Section \ref{sec:pkg:seaice:runtime}). |
| 30 |
A description of key subroutines is given in Section |
A description of key subroutines is given in Section |
| 31 |
\ref{sec:pkg:seaice:subroutines}. |
\ref{sec:pkg:seaice:subroutines}. |
| 32 |
Input fields, units and sign conventions are summarized in |
Input fields, units and sign conventions are summarized in |
| 33 |
Section \ref{sec:pkg:seaice:fields_units}, and available diagnostics |
Section \ref{sec:pkg:seaice:fields_units}, and available diagnostics |
| 34 |
output is listed in Section \ref{sec:pkg:seaice:fields_diagnostics}. |
output is listed in Section \ref{sec:pkg:seaice:diagnostics}. |
| 35 |
|
|
| 36 |
%---------------------------------------------------------------------- |
%---------------------------------------------------------------------- |
| 37 |
|
|
| 46 |
\begin{itemize} |
\begin{itemize} |
| 47 |
% |
% |
| 48 |
\item |
\item |
| 49 |
using the \texttt{packages.conf} file by adding \texttt{seaice} to it, |
using the \code{packages.conf} file by adding \code{seaice} to it, |
| 50 |
% |
% |
| 51 |
\item |
\item |
| 52 |
or using \texttt{genmake2} adding |
or using \code{genmake2} adding |
| 53 |
\texttt{-enable=seaice} or \texttt{-disable=seaice} switches |
\code{-enable=seaice} or \code{-disable=seaice} switches |
| 54 |
% |
% |
| 55 |
\item |
\item |
| 56 |
\textit{required packages and CPP options}: \\ |
\textit{required packages and CPP options}: \\ |
| 57 |
SEAICE requires the external forcing package \texttt{exf} to be enabled; |
SEAICE requires the external forcing package \code{exf} to be enabled; |
| 58 |
no additional CPP options are required. |
no additional CPP options are required. |
| 59 |
% |
% |
| 60 |
\end{itemize} |
\end{itemize} |
| 61 |
(see Section \ref{sect:buildingCode}). |
(see Section \ref{sec:buildingCode}). |
| 62 |
|
|
| 63 |
Parts of the SEAICE code can be enabled or disabled at compile time |
Parts of the SEAICE code can be enabled or disabled at compile time |
| 64 |
via CPP preprocessor flags. These options are set in either |
via CPP preprocessor flags. These options are set in |
| 65 |
\texttt{SEAICE\_OPTIONS.h} or in \texttt{ECCO\_CPPOPTIONS.h}. |
\code{SEAICE\_OPTIONS.h}. |
| 66 |
Table \ref{tab:pkg:seaice:cpp} summarizes these options. |
Table \ref{tab:pkg:seaice:cpp} summarizes the most important ones. |
| 67 |
|
|
| 68 |
\begin{table}[h!] |
\begin{table}[!ht] |
| 69 |
\centering |
\centering |
| 70 |
\label{tab:pkg:seaice:cpp} |
\label{tab:pkg:seaice:cpp} |
| 71 |
{\footnotesize |
{\footnotesize |
| 73 |
\hline |
\hline |
| 74 |
\textbf{CPP option} & \textbf{Description} \\ |
\textbf{CPP option} & \textbf{Description} \\ |
| 75 |
\hline \hline |
\hline \hline |
| 76 |
\texttt{SEAICE\_DEBUG} & |
\code{SEAICE\_DEBUG} & |
| 77 |
Enhance STDOUT for debugging \\ |
Enhance STDOUT for debugging \\ |
| 78 |
\texttt{SEAICE\_ALLOW\_DYNAMICS} & |
\code{SEAICE\_ALLOW\_DYNAMICS} & |
| 79 |
sea-ice dynamics code \\ |
sea-ice dynamics code \\ |
| 80 |
\texttt{SEAICE\_CGRID} & |
\code{SEAICE\_CGRID} & |
| 81 |
LSR solver on C-grid (rather than original B-grid) \\ |
LSR solver on C-grid (rather than original B-grid) \\ |
| 82 |
\texttt{SEAICE\_ALLOW\_EVP} & |
\code{SEAICE\_ALLOW\_EVP} & |
| 83 |
use EVP rather than LSR rheology solver \\ |
enable use of EVP rheology solver \\ |
| 84 |
\texttt{SEAICE\_EXTERNAL\_FLUXES} & |
\code{SEAICE\_ALLOW\_JFNK} & |
| 85 |
|
enable use of JFNK rheology solver \\ |
| 86 |
|
\code{SEAICE\_EXTERNAL\_FLUXES} & |
| 87 |
use EXF-computed fluxes as starting point \\ |
use EXF-computed fluxes as starting point \\ |
| 88 |
\texttt{SEAICE\_MULTICATEGORY} & |
\code{SEAICE\_ZETA\_SMOOTHREG} & |
| 89 |
enable 8-category thermodynamics (by default undefined)\\ |
use differentialable regularization for viscosities \\ |
| 90 |
\texttt{SEAICE\_VARIABLE\_FREEZING\_POINT} & |
\code{SEAICE\_VARIABLE\_FREEZING\_POINT} & |
| 91 |
enable linear dependence of the freezing point on salinity |
enable linear dependence of the freezing point on salinity |
| 92 |
(by default undefined)\\ |
(by default undefined)\\ |
| 93 |
\texttt{ALLOW\_SEAICE\_FLOODING} & |
\code{ALLOW\_SEAICE\_FLOODING} & |
| 94 |
enable snow to ice conversion for submerged sea-ice \\ |
enable snow to ice conversion for submerged sea-ice \\ |
| 95 |
\texttt{SEAICE\_SALINITY} & |
\code{SEAICE\_VARIABLE\_SALINITY} & |
| 96 |
enable "salty" sea-ice (by default undefined) \\ |
enable sea-ice with variable salinity (by default undefined) \\ |
| 97 |
\texttt{SEAICE\_AGE} & |
\code{SEAICE\_SITRACER} & |
| 98 |
enable "age tracer" sea-ice (by default undefined) \\ |
enable sea-ice tracer package (by default undefined) \\ |
| 99 |
\texttt{SEAICE\_CAP\_HEFF} & |
\code{SEAICE\_BICE\_STRESS} & |
|
enable capping of sea-ice thickness to MAX\_HEFF \\ \hline |
|
|
\texttt{SEAICE\_BICE\_STRESS} & |
|
| 100 |
B-grid only for backward compatiblity: turn on ice-stress on |
B-grid only for backward compatiblity: turn on ice-stress on |
| 101 |
ocean\\ |
ocean\\ |
| 102 |
\texttt{EXPLICIT\_SSH\_SLOPE} & |
\code{EXPLICIT\_SSH\_SLOPE} & |
| 103 |
B-grid only for backward compatiblity: use ETAN for tilt |
B-grid only for backward compatiblity: use ETAN for tilt |
| 104 |
computations rather than geostrophic velocities \\ |
computations rather than geostrophic velocities \\ |
| 105 |
\hline |
\hline |
| 106 |
\end{tabular} |
\end{tabular} |
| 107 |
} |
} |
| 108 |
\caption{~} |
\caption{Some of the most relevant CPP preprocessor flags in the |
| 109 |
|
\code{seaice}-package.} |
| 110 |
\end{table} |
\end{table} |
| 111 |
|
|
| 112 |
%---------------------------------------------------------------------- |
%---------------------------------------------------------------------- |
| 114 |
\subsubsection{Run-time parameters |
\subsubsection{Run-time parameters |
| 115 |
\label{sec:pkg:seaice:runtime}} |
\label{sec:pkg:seaice:runtime}} |
| 116 |
|
|
| 117 |
Run-time parameters are set in files |
Run-time parameters (see Table~\ref{tab:pkg:seaice:runtimeparms}) are set |
| 118 |
\texttt{data.pkg} (read in \texttt{packages\_readparms.F}), |
in files \code{data.pkg} (read in \code{packages\_readparms.F}), and |
| 119 |
and \texttt{data.seaice} (read in \texttt{seaice\_readparms.F}). |
\code{data.seaice} (read in \code{seaice\_readparms.F}). |
| 120 |
|
|
| 121 |
\paragraph{Enabling the package} |
\paragraph{Enabling the package} |
| 122 |
~ \\ |
~ \\ |
| 123 |
% |
% |
| 124 |
A package is switched on/off at run-time by setting |
A package is switched on/off at run-time by setting |
| 125 |
(e.g. for SEAICE) \texttt{useSEAICE = .TRUE.} in \texttt{data.pkg}. |
(e.g. for SEAICE) \code{useSEAICE = .TRUE.} in \code{data.pkg}. |
| 126 |
|
|
| 127 |
\paragraph{General flags and parameters} |
\paragraph{General flags and parameters} |
| 128 |
~ \\ |
~ \\ |
| 129 |
% |
% |
| 130 |
Table~\ref{tab:pkg:seaice:runtimeparms} lists most run-time parameters. |
Table~\ref{tab:pkg:seaice:runtimeparms} lists most run-time parameters. |
| 131 |
\input{part6/seaice-parms.tex} |
\input{s_phys_pkgs/text/seaice-parms.tex} |
|
|
|
| 132 |
|
|
| 133 |
|
\paragraph{Input fields and units\label{sec:pkg:seaice:fields_units}} |
| 134 |
|
\begin{description} |
| 135 |
|
\item[\code{HeffFile}:] Initial sea ice thickness averaged over grid cell |
| 136 |
|
in meters; initializes variable \code{HEFF}; |
| 137 |
|
\item[\code{AreaFile}:] Initial fractional sea ice cover, range $[0,1]$; |
| 138 |
|
initializes variable \code{AREA}; |
| 139 |
|
\item[\code{HsnowFile}:] Initial snow thickness on sea ice averaged |
| 140 |
|
over grid cell in meters; initializes variable \code{HSNOW}; |
| 141 |
|
\item[\code{HsaltFile}:] Initial salinity of sea ice averaged over grid |
| 142 |
|
cell in g/m$^2$; initializes variable \code{HSALT}; |
| 143 |
|
\end{description} |
| 144 |
|
|
| 145 |
%---------------------------------------------------------------------- |
%---------------------------------------------------------------------- |
| 146 |
\subsubsection{Description |
\subsubsection{Description |
| 181 |
first introduced by \citet{hib79, hib80}. In order to adapt this model |
first introduced by \citet{hib79, hib80}. In order to adapt this model |
| 182 |
to the requirements of coupled ice-ocean state estimation, many |
to the requirements of coupled ice-ocean state estimation, many |
| 183 |
important aspects of the original code have been modified and |
important aspects of the original code have been modified and |
| 184 |
improved: |
improved \citep{losch10:_mitsim}: |
| 185 |
\begin{itemize} |
\begin{itemize} |
| 186 |
\item the code has been rewritten for an Arakawa C-grid, both B- and |
\item the code has been rewritten for an Arakawa C-grid, both B- and |
| 187 |
C-grid variants are available; the C-grid code allows for no-slip |
C-grid variants are available; the C-grid code allows for no-slip |
| 188 |
and free-slip lateral boundary conditions; |
and free-slip lateral boundary conditions; |
| 189 |
\item two different solution methods for solving the nonlinear |
\item three different solution methods for solving the nonlinear |
| 190 |
momentum equations have been adopted: LSOR \citep{zhang97}, and EVP |
momentum equations have been adopted: LSOR \citep{zhang97}, EVP |
| 191 |
\citep{hun97}; |
\citep{hun97}, JFNK \citep{lemieux10,losch14:_jfnk}; |
| 192 |
\item ice-ocean stress can be formulated as in \citet{hibler87} or as in |
\item ice-ocean stress can be formulated as in \citet{hibler87} or as in |
| 193 |
\citet{cam08}; |
\citet{cam08}; |
| 194 |
\item ice variables are advected by sophisticated, conservative |
\item ice variables are advected by sophisticated, conservative |
| 213 |
to use the VP model as the default dynamic component of our ice |
to use the VP model as the default dynamic component of our ice |
| 214 |
model. To do this we extended the line successive over relaxation |
model. To do this we extended the line successive over relaxation |
| 215 |
(LSOR) method of \citet{zhang97} for use in a parallel |
(LSOR) method of \citet{zhang97} for use in a parallel |
| 216 |
configuration. |
configuration. An EVP model and a free-drift implemtation can be |
| 217 |
|
selected with runtime flags. |
| 218 |
|
|
| 219 |
Note, that by default the seaice-package includes the orginial |
\paragraph{Compatibility with ice-thermodynamics package \code{thsice}\label{sec:pkg:seaice:thsice}}~\\ |
| 220 |
|
% |
| 221 |
|
Note, that by default the \code{seaice}-package includes the orginial |
| 222 |
so-called zero-layer thermodynamics following \citet{hib80} with a |
so-called zero-layer thermodynamics following \citet{hib80} with a |
| 223 |
snow cover as in \citet{zha98a}. The zero-layer thermodynamic model |
snow cover as in \citet{zha98a}. The zero-layer thermodynamic model |
| 224 |
assumes that ice does not store heat and, therefore, tends to |
assumes that ice does not store heat and, therefore, tends to |
| 225 |
exaggerate the seasonal variability in ice thickness. This |
exaggerate the seasonal variability in ice thickness. This |
| 226 |
exaggeration can be significantly reduced by using |
exaggeration can be significantly reduced by using |
| 227 |
\citeauthor{sem76}'s~[\citeyear{sem76}] three-layer thermodynamic model |
\citeauthor{sem76}'s~[\citeyear{sem76}] three-layer thermodynamic |
| 228 |
that permits heat storage in ice. Recently, the three-layer |
model that permits heat storage in ice. Recently, the three-layer thermodynamic model has been reformulated by |
| 229 |
thermodynamic model has been reformulated by \citet{win00}. The |
\citet{win00}. The reformulation improves model physics by |
| 230 |
reformulation improves model physics by representing the brine content |
representing the brine content of the upper ice with a variable heat |
| 231 |
of the upper ice with a variable heat capacity. It also improves |
capacity. It also improves model numerics and consumes less computer |
| 232 |
model numerics and consumes less computer time and memory. The Winton |
time and memory. |
| 233 |
sea-ice thermodynamics have been ported to the MIT GCM; they currently |
|
| 234 |
reside under pkg/thsice. The package pkg/thsice is fully compatible |
The Winton sea-ice thermodynamics have been ported to the MIT GCM; |
| 235 |
with pkg/seaice and with pkg/exf. When turned on together with |
they currently reside under \code{pkg/thsice}. The package |
| 236 |
pkg/seaice, the zero-layer thermodynamics are replaced by the Winton |
\code{thsice} is described in section~\ref{sec:pkg:thsice}; it is |
| 237 |
thermodynamics. |
fully compatible with the packages \code{seaice} and \code{exf}. When |
| 238 |
|
turned on together with \code{seaice}, the zero-layer thermodynamics |
| 239 |
|
are replaced by the Winton thermodynamics. In order to use the |
| 240 |
|
\code{seaice}-package with the thermodynamics of \code{thsice}, |
| 241 |
|
compile both packages and turn both package on in \code{data.pkg}; see |
| 242 |
|
an example in \code{global\_ocean.cs32x15/input.icedyn}. Note, that |
| 243 |
|
once \code{thsice} is turned on, the variables and diagnostics |
| 244 |
|
associated to the default thermodynamics are meaningless, and the |
| 245 |
|
diagnostics of \code{thsice} have to be used instead. |
| 246 |
|
|
| 247 |
|
\paragraph{Surface forcing\label{sec:pkg:seaice:surfaceforcing}}~\\ |
| 248 |
|
% |
| 249 |
The sea ice model requires the following input fields: 10-m winds, 2-m |
The sea ice model requires the following input fields: 10-m winds, 2-m |
| 250 |
air temperature and specific humidity, downward longwave and shortwave |
air temperature and specific humidity, downward longwave and shortwave |
| 251 |
radiations, precipitation, evaporation, and river and glacier runoff. |
radiations, precipitation, evaporation, and river and glacier runoff. |
| 256 |
global: in ice-free regions bulk formulae are used to estimate oceanic |
global: in ice-free regions bulk formulae are used to estimate oceanic |
| 257 |
forcing from the atmospheric fields. |
forcing from the atmospheric fields. |
| 258 |
|
|
| 259 |
\subsubsection{Dynamics} |
\paragraph{Dynamics\label{sec:pkg:seaice:dynamics}}~\\ |
| 260 |
\label{sec:pkg:seaice:dynamics} |
% |
|
|
|
| 261 |
\newcommand{\vek}[1]{\ensuremath{\vec{\mathbf{#1}}}} |
\newcommand{\vek}[1]{\ensuremath{\vec{\mathbf{#1}}}} |
| 262 |
\newcommand{\vtau}{\vek{\mathbf{\tau}}} |
\newcommand{\vtau}{\vek{\mathbf{\tau}}} |
| 263 |
The momentum equation of the sea-ice model is |
The momentum equation of the sea-ice model is |
| 287 |
\vtau_{air} = & \rho_{air} C_{air} |\vek{U}_{air} -\vek{u}| |
\vtau_{air} = & \rho_{air} C_{air} |\vek{U}_{air} -\vek{u}| |
| 288 |
R_{air} (\vek{U}_{air} -\vek{u}), \\ |
R_{air} (\vek{U}_{air} -\vek{u}), \\ |
| 289 |
\vtau_{ocean} = & \rho_{ocean}C_{ocean} |\vek{U}_{ocean}-\vek{u}| |
\vtau_{ocean} = & \rho_{ocean}C_{ocean} |\vek{U}_{ocean}-\vek{u}| |
| 290 |
R_{ocean}(\vek{U}_{ocean}-\vek{u}), \\ |
R_{ocean}(\vek{U}_{ocean}-\vek{u}), |
| 291 |
\end{align*} |
\end{align*} |
| 292 |
where $\vek{U}_{air/ocean}$ are the surface winds of the atmosphere |
where $\vek{U}_{air/ocean}$ are the surface winds of the atmosphere |
| 293 |
and surface currents of the ocean, respectively; $C_{air/ocean}$ are |
and surface currents of the ocean, respectively; $C_{air/ocean}$ are |
| 295 |
densities; and $R_{air/ocean}$ are rotation matrices that act on the |
densities; and $R_{air/ocean}$ are rotation matrices that act on the |
| 296 |
wind/current vectors. |
wind/current vectors. |
| 297 |
|
|
| 298 |
|
\paragraph{Viscous-Plastic (VP) Rheology\label{sec:pkg:seaice:VPrheology}}~\\ |
| 299 |
|
% |
| 300 |
For an isotropic system the stress tensor $\sigma_{ij}$ ($i,j=1,2$) can |
For an isotropic system the stress tensor $\sigma_{ij}$ ($i,j=1,2$) can |
| 301 |
be related to the ice strain rate and strength by a nonlinear |
be related to the ice strain rate and strength by a nonlinear |
| 302 |
viscous-plastic (VP) constitutive law \citep{hib79, zhang97}: |
viscous-plastic (VP) constitutive law \citep{hib79, zhang97}: |
| 316 |
The maximum ice pressure $P_{\max}$, a measure of ice strength, depends on |
The maximum ice pressure $P_{\max}$, a measure of ice strength, depends on |
| 317 |
both thickness $h$ and compactness (concentration) $c$: |
both thickness $h$ and compactness (concentration) $c$: |
| 318 |
\begin{equation} |
\begin{equation} |
| 319 |
P_{\max} = P^{*}c\,h\,e^{[C^{*}\cdot(1-c)]}, |
P_{\max} = P^{*}c\,h\,\exp\{-C^{*}\cdot(1-c)\}, |
| 320 |
\label{eq:icestrength} |
\label{eq:icestrength} |
| 321 |
\end{equation} |
\end{equation} |
| 322 |
with the constants $P^{*}$ (run-time parameter \texttt{SEAICE\_strength}) and |
with the constants $P^{*}$ (run-time parameter \code{SEAICE\_strength}) and |
| 323 |
$C^{*}=20$. The nonlinear bulk and shear |
$C^{*}=20$. The nonlinear bulk and shear |
| 324 |
viscosities $\eta$ and $\zeta$ are functions of ice strain rate |
viscosities $\eta$ and $\zeta$ are functions of ice strain rate |
| 325 |
invariants and ice strength such that the principal components of the |
invariants and ice strength such that the principal components of the |
| 338 |
\end{align*} |
\end{align*} |
| 339 |
The bulk viscosities are bounded above by imposing both a minimum |
The bulk viscosities are bounded above by imposing both a minimum |
| 340 |
$\Delta_{\min}$ (for numerical reasons, run-time parameter |
$\Delta_{\min}$ (for numerical reasons, run-time parameter |
| 341 |
\texttt{SEAICE\_EPS} with a default value of |
\code{SEAICE\_EPS} with a default value of |
| 342 |
$10^{-10}\text{\,s}^{-1}$) and a maximum $\zeta_{\max} = |
$10^{-10}\text{\,s}^{-1}$) and a maximum $\zeta_{\max} = |
| 343 |
P_{\max}/\Delta^*$, where |
P_{\max}/\Delta^*$, where |
| 344 |
$\Delta^*=(5\times10^{12}/2\times10^4)\text{\,s}^{-1}$. (There is also |
$\Delta^*=(5\times10^{12}/2\times10^4)\text{\,s}^{-1}$. (There is also |
| 345 |
the option of bounding $\zeta$ from below by setting run-time |
the option of bounding $\zeta$ from below by setting run-time |
| 346 |
parameter \texttt{SEAICE\_zetaMin} $>0$, but this is generally not |
parameter \code{SEAICE\_zetaMin} $>0$, but this is generally not |
| 347 |
recommended). For stress tensor computation the replacement pressure $P |
recommended). For stress tensor computation the replacement pressure $P |
| 348 |
= 2\,\Delta\zeta$ \citep{hibler95} is used so that the stress state |
= 2\,\Delta\zeta$ \citep{hibler95} is used so that the stress state |
| 349 |
always lies on the elliptic yield curve by definition. |
always lies on the elliptic yield curve by definition. |
| 350 |
|
|
| 351 |
In the so-called truncated ellipse method the shear viscosity $\eta$ |
Defining the CPP-flag \code{SEAICE\_ZETA\_SMOOTHREG} in |
| 352 |
is capped to suppress any tensile stress \citep{hibler97, geiger98}: |
\code{SEAICE\_OPTIONS.h} before compiling replaces the method for |
| 353 |
|
bounding $\zeta$ by a smooth (differentiable) expression: |
| 354 |
\begin{equation} |
\begin{equation} |
| 355 |
\label{eq:etatem} |
\label{eq:zetaregsmooth} |
| 356 |
\eta = \min\left(\frac{\zeta}{e^2}, |
\begin{split} |
| 357 |
\frac{\frac{P}{2}-\zeta(\dot{\epsilon}_{11}+\dot{\epsilon}_{22})} |
\zeta &= \zeta_{\max}\tanh\left(\frac{P}{2\,\min(\Delta,\Delta_{\min}) |
| 358 |
{\sqrt{(\dot{\epsilon}_{11}+\dot{\epsilon}_{22})^2 |
\,\zeta_{\max}}\right)\\ |
| 359 |
+4\dot{\epsilon}_{12}^2}}\right). |
&= \frac{P}{2\Delta^*} |
| 360 |
|
\tanh\left(\frac{\Delta^*}{\min(\Delta,\Delta_{\min})}\right) |
| 361 |
|
\end{split} |
| 362 |
\end{equation} |
\end{equation} |
| 363 |
To enable this method, set \texttt{\#define SEAICE\_ALLOW\_TEM} in |
where $\Delta_{\min}=10^{-20}\text{\,s}^{-1}$ is chosen to avoid divisions |
| 364 |
\texttt{SEAICE\_OPTIONS.h} and turn it on with |
by zero. |
|
\texttt{SEAICEuseTEM=.TRUE.} in \texttt{data.seaice}. |
|
|
|
|
|
In the current implementation, the VP-model is integrated with the |
|
|
semi-implicit line successive over relaxation (LSOR)-solver of |
|
|
\citet{zhang97}, which allows for long time steps that, in our case, |
|
|
are limited by the explicit treatment of the Coriolis term. The |
|
|
explicit treatment of the Coriolis term does not represent a severe |
|
|
limitation because it restricts the time step to approximately the |
|
|
same length as in the ocean model where the Coriolis term is also |
|
|
treated explicitly. |
|
| 365 |
|
|
| 366 |
|
\paragraph{LSR and JFNK solver \label{sec:pkg:seaice:LSRJFNK}}~\\ |
| 367 |
|
% |
| 368 |
|
% By default, the VP-model is integrated by a Pickwith the |
| 369 |
|
% semi-implicit line successive over relaxation (LSOR)-solver of |
| 370 |
|
% \citet{zhang97}, which allows for long time steps that, in our case, |
| 371 |
|
% are limited by the explicit treatment of the Coriolis term. The |
| 372 |
|
% explicit treatment of the Coriolis term does not represent a severe |
| 373 |
|
% limitation because it restricts the time step to approximately the |
| 374 |
|
% same length as in the ocean model where the Coriolis term is also |
| 375 |
|
% treated explicitly. |
| 376 |
|
|
| 377 |
|
\newcommand{\mat}[1]{\ensuremath{\mathbf{#1}}} |
| 378 |
|
% |
| 379 |
|
In the matrix notation, the discretized momentum equations can be |
| 380 |
|
written as |
| 381 |
|
\begin{equation} |
| 382 |
|
\label{eq:matrixmom} |
| 383 |
|
\mat{A}(\vek{x})\,\vek{x} = \vek{b}(\vek{x}). |
| 384 |
|
\end{equation} |
| 385 |
|
The solution vector $\vek{x}$ consists of the two velocity components |
| 386 |
|
$u$ and $v$ that contain the velocity variables at all grid points and |
| 387 |
|
at one time level. The standard (and default) method for solving |
| 388 |
|
Eq.\,(\ref{eq:matrixmom}) in the sea ice component of the |
| 389 |
|
\mbox{MITgcm}, as in many sea ice models, is an iterative Picard |
| 390 |
|
solver: in the $k$-th iteration a linearized form |
| 391 |
|
$\mat{A}(\vek{x}^{k-1})\,\vek{x}^{k} = \vek{b}(\vek{x}^{k-1})$ is |
| 392 |
|
solved (in the case of the MITgcm it is a Line Successive (over) |
| 393 |
|
Relaxation (LSR) algorithm \citep{zhang97}). Picard solvers converge |
| 394 |
|
slowly, but generally the iteration is terminated after only a few |
| 395 |
|
non-linear steps \citep{zhang97, lemieux09} and the calculation |
| 396 |
|
continues with the next time level. This method is the default method |
| 397 |
|
in the MITgcm. The number of non-linear iteration steps or pseudo-time |
| 398 |
|
steps can be controlled by the runtime parameter |
| 399 |
|
\code{NPSEUDOTIMESTEPS} (default is 2). |
| 400 |
|
|
| 401 |
|
In order to overcome the poor convergence of the Picard-solver, |
| 402 |
|
\citet{lemieux10} introduced a Jacobian-free Newton-Krylov solver for |
| 403 |
|
the sea ice momentum equations. This solver is also implemented in the |
| 404 |
|
MITgcm \citep{losch14:_jfnk}. The Newton method transforms minimizing |
| 405 |
|
the residual $\vek{F}(\vek{x}) = \mat{A}(\vek{x})\,\vek{x} - |
| 406 |
|
\vek{b}(\vek{x})$ to finding the roots of a multivariate Taylor |
| 407 |
|
expansion of the residual \vek{F} around the previous ($k-1$) estimate |
| 408 |
|
$\vek{x}^{k-1}$: |
| 409 |
|
\begin{equation} |
| 410 |
|
\label{eq:jfnktaylor} |
| 411 |
|
\vek{F}(\vek{x}^{k-1}+\delta\vek{x}^{k}) = |
| 412 |
|
\vek{F}(\vek{x}^{k-1}) + \vek{F}'(\vek{x}^{k-1})\,\delta\vek{x}^{k} |
| 413 |
|
\end{equation} |
| 414 |
|
with the Jacobian $\mat{J}\equiv\vek{F}'$. The root |
| 415 |
|
$\vek{F}(\vek{x}^{k-1}+\delta\vek{x}^{k})=0$ is found by solving |
| 416 |
|
\begin{equation} |
| 417 |
|
\label{eq:jfnklin} |
| 418 |
|
\mat{J}(\vek{x}^{k-1})\,\delta\vek{x}^{k} = -\vek{F}(\vek{x}^{k-1}) |
| 419 |
|
\end{equation} |
| 420 |
|
for $\delta\vek{x}^{k}$. The next ($k$-th) estimate is given by |
| 421 |
|
$\vek{x}^{k}=\vek{x}^{k-1}+a\,\delta\vek{x}^{k}$. In order to avoid |
| 422 |
|
overshoots the factor $a$ is iteratively reduced in a line search |
| 423 |
|
($a=1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \ldots$) until |
| 424 |
|
$\|\vek{F}(\vek{x}^k)\| < \|\vek{F}(\vek{x}^{k-1})\|$, where |
| 425 |
|
$\|\cdot\|=\int\cdot\,dx^2$ is the $L_2$-norm. In practice, the line |
| 426 |
|
search is stopped at $a=\frac{1}{8}$. The line search starts after |
| 427 |
|
$\code{SEAICE\_JFNK\_lsIter}$ non-linear Newton iterations (off by |
| 428 |
|
default). |
| 429 |
|
|
| 430 |
|
|
| 431 |
|
Forming the Jacobian $\mat{J}$ explicitly is often avoided as ``too |
| 432 |
|
error prone and time consuming'' \citep{knoll04:_jfnk}. Instead, |
| 433 |
|
Krylov methods only require the action of \mat{J} on an arbitrary |
| 434 |
|
vector \vek{w} and hence allow a matrix free algorithm for solving |
| 435 |
|
Eq.\,(\ref{eq:jfnklin}) \citep{knoll04:_jfnk}. The action of \mat{J} |
| 436 |
|
can be approximated by a first-order Taylor series expansion: |
| 437 |
|
\begin{equation} |
| 438 |
|
\label{eq:jfnkjacvecfd} |
| 439 |
|
\mat{J}(\vek{x}^{k-1})\,\vek{w} \approx |
| 440 |
|
\frac{\vek{F}(\vek{x}^{k-1}+\epsilon\vek{w}) - \vek{F}(\vek{x}^{k-1})} |
| 441 |
|
{\epsilon} |
| 442 |
|
\end{equation} |
| 443 |
|
or computed exactly with the help of automatic differentiation (AD) |
| 444 |
|
tools. \code{SEAICE\_JFNKepsilon} sets the step size |
| 445 |
|
$\epsilon$. |
| 446 |
|
|
| 447 |
|
We use the Flexible Generalized Minimum RESidual method |
| 448 |
|
\citep[FGMRES,][]{saad93:_fgmres} with right-hand side preconditioning |
| 449 |
|
to solve Eq.\,(\ref{eq:jfnklin}) iteratively starting from a first |
| 450 |
|
guess of $\delta\vek{x}^{k}_{0} = 0$. For the preconditioning matrix |
| 451 |
|
\mat{P} we choose a simplified form of the system matrix |
| 452 |
|
$\mat{A}(\vek{x}^{k-1})$ \citep{lemieux10} where $\vek{x}^{k-1}$ is |
| 453 |
|
the estimate of the previous Newton step $k-1$. The transformed |
| 454 |
|
equation\,(\ref{eq:jfnklin}) becomes |
| 455 |
|
\begin{equation} |
| 456 |
|
\label{eq:jfnklinpc} |
| 457 |
|
\mat{J}(\vek{x}^{k-1})\,\mat{P}^{-1}\delta\vek{z} = |
| 458 |
|
-\vek{F}(\vek{x}^{k-1}), |
| 459 |
|
\quad\text{with}\quad \delta\vek{z}=\mat{P}\delta\vek{x}^{k}. |
| 460 |
|
\end{equation} |
| 461 |
|
The Krylov method iteratively improves the approximate solution |
| 462 |
|
to~(\ref{eq:jfnklinpc}) in subspace ($\vek{r}_0$, |
| 463 |
|
$\mat{J}\mat{P}^{-1}\vek{r}_0$, $(\mat{J}\mat{P}^{-1})^2\vek{r}_0$, |
| 464 |
|
\ldots, $(\mat{J}\mat{P}^{-1})^m\vek{r}_0$) with increasing $m$; |
| 465 |
|
$\vek{r}_0 = -\vek{F}(\vek{x}^{k-1}) |
| 466 |
|
-\mat{J}(\vek{x}^{k-1})\,\delta\vek{x}^{k}_{0}$ |
| 467 |
|
%-\vek{F}(\vek{x}^{k-1}) |
| 468 |
|
%-\mat{J}(\vek{x}^{k-1})\,\mat{P}^{-1}\delta\vek{z}$ |
| 469 |
|
is the initial residual of |
| 470 |
|
(\ref{eq:jfnklin}); $\vek{r}_0=-\vek{F}(\vek{x}^{k-1})$ with the first |
| 471 |
|
guess $\delta\vek{x}^{k}_{0}=0$. We allow a Krylov-subspace of |
| 472 |
|
dimension~$m=50$ and we do not use restarts. The preconditioning operation |
| 473 |
|
involves applying $\mat{P}^{-1}$ to the basis vectors $\vek{v}_0, |
| 474 |
|
\vek{v}_1, \vek{v}_2, \ldots, \vek{v}_m$ of the Krylov subspace. This |
| 475 |
|
operation is approximated by solving the linear system |
| 476 |
|
$\mat{P}\,\vek{w}=\vek{v}_i$. Because $\mat{P} \approx |
| 477 |
|
\mat{A}(\vek{x}^{k-1})$, we can use the LSR-algorithm \citep{zhang97} |
| 478 |
|
already implemented in the Picard solver. Each preconditioning |
| 479 |
|
operation uses a fixed number of 10~LSR-iterations avoiding any |
| 480 |
|
termination criterion. More details and results can be found in |
| 481 |
|
\citet{lemieux10, losch14:_jfnk}. |
| 482 |
|
|
| 483 |
|
To use the JFNK-solver set \code{SEAICEuseJFNK = .TRUE.} in the |
| 484 |
|
namelist file \code{data.seaice}; \code{SEAICE\_ALLOW\_JFNK} needs to |
| 485 |
|
be defined in \code{SEAICE\_OPTIONS.h} and we recommend using a smooth |
| 486 |
|
regularization of $\zeta$ by defining \code{SEAICE\_ZETA\_SMOOTHREG} |
| 487 |
|
(see above) for better convergence. The non-linear Newton iteration |
| 488 |
|
is terminated when the $L_2$-norm of the residual is reduced by |
| 489 |
|
$\gamma_{\mathrm{nl}}$ (runtime parameter \code{JFNKgamma\_nonlin = |
| 490 |
|
1.e-4} will already lead to expensive simulations) with respect to |
| 491 |
|
the initial norm: $\|\vek{F}(\vek{x}^k)\| < |
| 492 |
|
\gamma_{\mathrm{nl}}\|\vek{F}(\vek{x}^0)\|$. Within a non-linear |
| 493 |
|
iteration, the linear FGMRES solver is terminated when the residual is |
| 494 |
|
smaller than $\gamma_k\|\vek{F}(\vek{x}^{k-1})\|$ where $\gamma_k$ is |
| 495 |
|
determined by |
| 496 |
|
\begin{equation} |
| 497 |
|
\label{eq:jfnkgammalin} |
| 498 |
|
\gamma_k = |
| 499 |
|
\begin{cases} |
| 500 |
|
\gamma_0 &\text{for $\|\vek{F}(\vek{x}^{k-1})\| \geq r$}, \\ |
| 501 |
|
\max\left(\gamma_{\min}, |
| 502 |
|
\frac{\|\vek{F}(\vek{x}^{k-1})\|}{\|\vek{F}(\vek{x}^{k-2})\|}\right) |
| 503 |
|
% \phi\left(\frac{\|\vek{F}(\vek{x}^{k-1})\|}{\|\vek{F}(\vek{x}^{k-2})\|}\right)^\alpha\right) |
| 504 |
|
&\text{for $\|\vek{F}(\vek{x}^{k-1})\| < r$,} |
| 505 |
|
\end{cases} |
| 506 |
|
\end{equation} |
| 507 |
|
so that the linear tolerance parameter $\gamma_k$ decreases with the |
| 508 |
|
non-linear Newton step as the non-linear solution is approached. This |
| 509 |
|
inexact Newton method is generally more robust and computationally |
| 510 |
|
more efficient than exact methods \citep[e.g.,][]{knoll04:_jfnk}. |
| 511 |
|
% \footnote{The general idea behind |
| 512 |
|
% inexact Newton methods is this: The Krylov solver is ``only'' |
| 513 |
|
% used to find an intermediate solution of the linear |
| 514 |
|
% equation\,(\ref{eq:jfnklin}) that is used to improve the approximation of |
| 515 |
|
% the actual equation\,(\ref{eq:matrixmom}). With the choice of a |
| 516 |
|
% relatively weak lower limit for FGMRES convergence |
| 517 |
|
% $\gamma_{\min}$ we make sure that the time spent in the FGMRES |
| 518 |
|
% solver is reduced at the cost of more Newton iterations. Newton |
| 519 |
|
% iterations are cheaper than Krylov iterations so that this choice |
| 520 |
|
% improves the overall efficiency.} |
| 521 |
|
Typical parameter choices are |
| 522 |
|
$\gamma_0=\code{JFNKgamma\_lin\_max}=0.99$, |
| 523 |
|
$\gamma_{\min}=\code{JFNKgamma\_lin\_min}=0.1$, and $r = |
| 524 |
|
\code{JFNKres\_tFac}\times\|\vek{F}(\vek{x}^{0})\|$ with |
| 525 |
|
$\code{JFNKres\_tFac} = \frac{1}{2}$. We recommend a maximum number of |
| 526 |
|
non-linear iterations $\code{SEAICEnewtonIterMax} = 100$ and a maximum |
| 527 |
|
number of Krylov iterations $\code{SEAICEkrylovIterMax} = 50$, because |
| 528 |
|
the Krylov subspace has a fixed dimension of 50. |
| 529 |
|
|
| 530 |
|
\paragraph{Elastic-Viscous-Plastic (EVP) Dynamics\label{sec:pkg:seaice:EVPdynamics}}~\\ |
| 531 |
|
% |
| 532 |
\citet{hun97}'s introduced an elastic contribution to the strain |
\citet{hun97}'s introduced an elastic contribution to the strain |
| 533 |
rate in order to regularize Eq.~\ref{eq:vpequation} in such a way that |
rate in order to regularize Eq.~\ref{eq:vpequation} in such a way that |
| 534 |
the resulting elastic-viscous-plastic (EVP) and VP models are |
the resulting elastic-viscous-plastic (EVP) and VP models are |
| 546 |
%used and compared the present sea-ice model study. |
%used and compared the present sea-ice model study. |
| 547 |
The EVP-model uses an explicit time stepping scheme with a short |
The EVP-model uses an explicit time stepping scheme with a short |
| 548 |
timestep. According to the recommendation of \citet{hun97}, the |
timestep. According to the recommendation of \citet{hun97}, the |
| 549 |
EVP-model is stepped forward in time 120 times within the physical |
EVP-model should be stepped forward in time 120 times |
| 550 |
ocean model time step (although this parameter is under debate), to |
($\code{SEAICE\_deltaTevp} = \code{SEAICIE\_deltaTdyn}/120$) within |
| 551 |
allow for elastic waves to disappear. Because the scheme does not |
the physical ocean model time step (although this parameter is under |
| 552 |
require a matrix inversion it is fast in spite of the small internal |
debate), to allow for elastic waves to disappear. Because the scheme |
| 553 |
timestep and simple to implement on parallel computers |
does not require a matrix inversion it is fast in spite of the small |
| 554 |
|
internal timestep and simple to implement on parallel computers |
| 555 |
\citep{hun97}. For completeness, we repeat the equations for the |
\citep{hun97}. For completeness, we repeat the equations for the |
| 556 |
components of the stress tensor $\sigma_{1} = |
components of the stress tensor $\sigma_{1} = |
| 557 |
\sigma_{11}+\sigma_{22}$, $\sigma_{2}= \sigma_{11}-\sigma_{22}$, and |
\sigma_{11}+\sigma_{22}$, $\sigma_{2}= \sigma_{11}-\sigma_{22}$, and |
| 572 |
\frac{\partial\sigma_{12}}{\partial{t}} + \frac{\sigma_{12} e^{2}}{2T} |
\frac{\partial\sigma_{12}}{\partial{t}} + \frac{\sigma_{12} e^{2}}{2T} |
| 573 |
&= \frac{P}{4T\Delta} D_S |
&= \frac{P}{4T\Delta} D_S |
| 574 |
\end{align} |
\end{align} |
| 575 |
Here, the elastic parameter $E$ is redefined in terms of a damping timescale |
Here, the elastic parameter $E$ is redefined in terms of a damping |
| 576 |
$T$ for elastic waves \[E=\frac{\zeta}{T}.\] |
timescale $T$ for elastic waves \[E=\frac{\zeta}{T}.\] |
| 577 |
$T=E_{0}\Delta{t}$ with the tunable parameter $E_0<1$ and |
$T=E_{0}\Delta{t}$ with the tunable parameter $E_0<1$ and the external |
| 578 |
the external (long) timestep $\Delta{t}$. \citet{hun97} recommend |
(long) timestep $\Delta{t}$. $E_{0} = \frac{1}{3}$ is the default |
| 579 |
$E_{0} = \frac{1}{3}$ (which is the default value in the code). |
value in the code and close to what \citet{hun97} and |
| 580 |
|
\citet{hun01} recommend. |
| 581 |
|
|
| 582 |
To use the EVP solver, make sure that both \texttt{SEAICE\_CGRID} and |
To use the EVP solver, make sure that both \code{SEAICE\_CGRID} and |
| 583 |
\texttt{SEAICE\_ALLOW\_EVP} are defined in \texttt{SEAICE\_OPTIONS.h} |
\code{SEAICE\_ALLOW\_EVP} are defined in \code{SEAICE\_OPTIONS.h} |
| 584 |
(default). The solver is turned on by setting the sub-cycling time |
(default). The solver is turned on by setting the sub-cycling time |
| 585 |
step \texttt{SEAICE\_deltaTevp} to a value larger than zero. The |
step \code{SEAICE\_deltaTevp} to a value larger than zero. The |
| 586 |
choice of this time step is under debate. \citet{hun97} recommend |
choice of this time step is under debate. \citet{hun97} recommend |
| 587 |
order(120) time steps for the EVP solver within one model time step |
order(120) time steps for the EVP solver within one model time step |
| 588 |
$\Delta{t}$ (\texttt{deltaTmom}). One can also choose order(120) time |
$\Delta{t}$ (\code{deltaTmom}). One can also choose order(120) time |
| 589 |
steps within the forcing time scale, but then we recommend adjusting |
steps within the forcing time scale, but then we recommend adjusting |
| 590 |
the damping time scale $T$ accordingly, by setting either |
the damping time scale $T$ accordingly, by setting either |
| 591 |
\texttt{SEAICE\_elasticParm} ($E_{0}$), so that |
\code{SEAICE\_elasticParm} ($E_{0}$), so that |
| 592 |
$E_{0}\Delta{t}=\mbox{forcing time scale}$, or directly |
$E_{0}\Delta{t}=\mbox{forcing time scale}$, or directly |
| 593 |
\texttt{SEAICE\_evpTauRelax} ($T$) to the forcing time scale. |
\code{SEAICE\_evpTauRelax} ($T$) to the forcing time scale. |
| 594 |
|
|
| 595 |
|
\paragraph{More stable variant of Elastic-Viscous-Plastic Dynamics: EVP*\label{sec:pkg:seaice:EVPstar}}~\\ |
| 596 |
|
% |
| 597 |
|
The genuine EVP schemes appears to give noisy solutions \citep{hun01, |
| 598 |
|
lemieux12, bouillon13}. This has lead to a modified EVP or EVP* |
| 599 |
|
\citep{lemieux12, bouillon13, kimmritz15}; here, refer to these |
| 600 |
|
variants by EVP*. The main idea is to modify the ``natural'' |
| 601 |
|
time-discretization of the momentum equations: |
| 602 |
|
\begin{equation} |
| 603 |
|
\label{eq:evpstar} |
| 604 |
|
m\frac{D\vec{u}}{Dt} \approx m\frac{u^{p+1}-u^{n}}{\Delta{t}} |
| 605 |
|
+ \beta^{*}\frac{u^{p+1}-u^{p}}{\Delta{t}_{\mathrm{EVP}}} |
| 606 |
|
\end{equation} |
| 607 |
|
where $n$ is the previous time step index, and $p$ is the previous |
| 608 |
|
sub-cycling index. The term allows the definition of a residual |
| 609 |
|
$|u^{p+1}-u^{p}|$ that, as $u^{p+1} \rightarrow u^{n+1}$, converges to |
| 610 |
|
$0$ and a re-interpretation of EVP as a pure iterative solver where |
| 611 |
|
the sub-cycling has lost all time-relation \citep{bouillon13, |
| 612 |
|
kimmritz15}. Using the terminology of \citet{kimmritz15}, the |
| 613 |
|
evolution equations of stress $\sigma_{ij}$ and momentum $\vec{u}$ can |
| 614 |
|
be written as: |
| 615 |
|
\begin{align} |
| 616 |
|
\label{eq:evpstarsigma} |
| 617 |
|
\sigma_{ij}^{p+1}&=\sigma_{ij}^p+\frac{1}{\alpha} |
| 618 |
|
\Big(\sigma_{ij}(\vec{u}^p)-\sigma_{ij}^p\Big), |
| 619 |
|
\phantom{\int}\\ |
| 620 |
|
\label{eq:evpstarmom} |
| 621 |
|
\vec{u}^{p+1}&=\vec{u}^p+\frac{1}{\beta} |
| 622 |
|
\Big(\frac{\Delta t}{m}\nabla \cdot{\bf \sigma}^{p+1}+ |
| 623 |
|
\frac{\Delta t}{m}\vec{R}^{p+1/2}+\vec{u}_n-\vec{u}^p\Big). |
| 624 |
|
\end{align} |
| 625 |
|
$\vec{R}$ contains all terms in the momentum equations except for the |
| 626 |
|
rheology terms and the time derivative, $\alpha$ and $\beta$ are free |
| 627 |
|
parameters (\code{SEAICE\_evpAlpha}, \code{SEAICE\_evpBeta}) that |
| 628 |
|
replace the time stepping parameters \code{SEAICE\_deltaTevp} |
| 629 |
|
($\Delta{T}_{\mathrm{EVP}}$), \code{SEAICE\_elasticParm} ($E_{0}$), or |
| 630 |
|
\code{SEAICE\_evpTauRelax} ($T$). $\alpha$ and $\beta$ determine the |
| 631 |
|
speed of convergence and the stability. Usually, it makes sense to use |
| 632 |
|
$\alpha = \beta$, and \code{SEAICEnEVPstarSteps} $>> \alpha = \beta$ |
| 633 |
|
\citep{kimmritz15}. |
| 634 |
|
|
| 635 |
|
In order to use EVP* in the MITgcm, set \code{SEAICEuseEVPstar = |
| 636 |
|
.TRUE.,} in \code{data.seaice}. \code{SEAICEuseEVPrev =.TRUE.,} uses |
| 637 |
|
the actual form of equations (\ref{eq:evpstarsigma}) and |
| 638 |
|
(\ref{eq:evpstarmom}) with fewer implicit terms and the factor of |
| 639 |
|
$e^{2}$ dropped in the stress equations (\ref{eq:evpstresstensor2}) |
| 640 |
|
and (\ref{eq:evpstresstensor12}). This turns out to improve |
| 641 |
|
convergence \citep{bouillon13}. |
| 642 |
|
|
| 643 |
|
Note, that for historical reasons, \code{SEAICE\_deltaTevp} needs to |
| 644 |
|
be set to some value in order to use also EVP*. Also note, that |
| 645 |
|
probably because of the C-grid staggering of velocities and stresses, |
| 646 |
|
EVP* does not converge as successfully as in \citet{kimmritz15}. |
| 647 |
|
|
| 648 |
|
|
| 649 |
|
\paragraph{Truncated ellipse method (TEM) for yield curve \label{sec:pkg:seaice:TEM}}~\\ |
| 650 |
|
% |
| 651 |
|
In the so-called truncated ellipse method the shear viscosity $\eta$ |
| 652 |
|
is capped to suppress any tensile stress \citep{hibler97, geiger98}: |
| 653 |
|
\begin{equation} |
| 654 |
|
\label{eq:etatem} |
| 655 |
|
\eta = \min\left(\frac{\zeta}{e^2}, |
| 656 |
|
\frac{\frac{P}{2}-\zeta(\dot{\epsilon}_{11}+\dot{\epsilon}_{22})} |
| 657 |
|
{\sqrt{(\dot{\epsilon}_{11}+\dot{\epsilon}_{22})^2 |
| 658 |
|
+4\dot{\epsilon}_{12}^2}}\right). |
| 659 |
|
\end{equation} |
| 660 |
|
To enable this method, set \code{\#define SEAICE\_ALLOW\_TEM} in |
| 661 |
|
\code{SEAICE\_OPTIONS.h} and turn it on with |
| 662 |
|
\code{SEAICEuseTEM} in \code{data.seaice}. |
| 663 |
|
|
| 664 |
|
\paragraph{Ice-Ocean stress \label{sec:pkg:seaice:iceoceanstress}}~\\ |
| 665 |
|
% |
| 666 |
Moving sea ice exerts a stress on the ocean which is the opposite of |
Moving sea ice exerts a stress on the ocean which is the opposite of |
| 667 |
the stress $\vtau_{ocean}$ in Eq.~\ref{eq:momseaice}. This stess is |
the stress $\vtau_{ocean}$ in Eq.~\ref{eq:momseaice}. This stess is |
| 668 |
applied directly to the surface layer of the ocean model. An |
applied directly to the surface layer of the ocean model. An |
| 679 |
velocity and the ice velocity leading to an inconsistency as the ice |
velocity and the ice velocity leading to an inconsistency as the ice |
| 680 |
temperature and salinity are different from the oceanic variables. |
temperature and salinity are different from the oceanic variables. |
| 681 |
To turn on the stress formulation of \citet{hibler87}, set |
To turn on the stress formulation of \citet{hibler87}, set |
| 682 |
\texttt{useHB87StressCoupling=.TRUE.} in \texttt{data.seaice}. |
\code{useHB87StressCoupling=.TRUE.} in \code{data.seaice}. |
| 683 |
|
|
| 684 |
|
|
| 685 |
% Our discretization differs from \citet{zhang97, zhang03} in the |
% Our discretization differs from \citet{zhang97, zhang03} in the |
| 691 |
% differences and averaging is only involved in computing $\Delta$ and |
% differences and averaging is only involved in computing $\Delta$ and |
| 692 |
% $P$ at vorticity points. |
% $P$ at vorticity points. |
| 693 |
|
|
| 694 |
\subsubsection{Finite-volume discretization of the stress tensor |
\paragraph{Finite-volume discretization of the stress tensor |
| 695 |
divergence} |
divergence\label{sec:pkg:seaice:discretization}}~\\ |
| 696 |
\label{sec:pkg:seaice:discretization} |
% |
| 697 |
On an Arakawa C~grid, ice thickness and concentration and thus ice |
On an Arakawa C~grid, ice thickness and concentration and thus ice |
| 698 |
strength $P$ and bulk and shear viscosities $\zeta$ and $\eta$ are |
strength $P$ and bulk and shear viscosities $\zeta$ and $\eta$ are |
| 699 |
naturally defined a C-points in the center of the grid |
naturally defined a C-points in the center of the grid |
| 755 |
[(\zeta-\eta)\dot{\epsilon}_{\gamma\gamma} - P/2 |
[(\zeta-\eta)\dot{\epsilon}_{\gamma\gamma} - P/2 |
| 756 |
]\delta_{\alpha\beta}$ \citep{hib79}. The stress tensor divergence |
]\delta_{\alpha\beta}$ \citep{hib79}. The stress tensor divergence |
| 757 |
$(\nabla\sigma)_{\alpha} = \partial_\beta\sigma_{\beta\alpha}$, is |
$(\nabla\sigma)_{\alpha} = \partial_\beta\sigma_{\beta\alpha}$, is |
| 758 |
discretized in finite volumes. This conveniently avoids dealing with |
discretized in finite volumes \citep[see |
| 759 |
|
also][]{losch10:_mitsim}. This conveniently avoids dealing with |
| 760 |
further metric terms, as these are ``hidden'' in the differential cell |
further metric terms, as these are ``hidden'' in the differential cell |
| 761 |
widths. For the $u$-equation ($\alpha=1$) we have: |
widths. For the $u$-equation ($\alpha=1$) we have: |
| 762 |
\begin{align} |
\begin{align} |
| 773 |
+ \Delta{x}_1\sigma_{21}\biggl|_{x_{2}}^{x_{2}+\Delta{x}_{2}} |
+ \Delta{x}_1\sigma_{21}\biggl|_{x_{2}}^{x_{2}+\Delta{x}_{2}} |
| 774 |
\biggr\} \\ \notag |
\biggr\} \\ \notag |
| 775 |
=& \frac{1}{A_{i,j}^w} \biggl\{ |
=& \frac{1}{A_{i,j}^w} \biggl\{ |
| 776 |
(\Delta{x}_2\sigma_{11})_{i,j}^C - (\Delta{x}_2\sigma_{11})_{i-1,j}^C \\\notag |
(\Delta{x}_2\sigma_{11})_{i,j}^C - |
| 777 |
|
(\Delta{x}_2\sigma_{11})_{i-1,j}^C |
| 778 |
|
\\\notag |
| 779 |
\phantom{=}& \phantom{\frac{1}{A_{i,j}^w} \biggl\{} |
\phantom{=}& \phantom{\frac{1}{A_{i,j}^w} \biggl\{} |
| 780 |
+ (\Delta{x}_1\sigma_{21})_{i,j+1}^Z - (\Delta{x}_1\sigma_{21})_{i,j}^Z |
+ (\Delta{x}_1\sigma_{21})_{i,j+1}^Z - (\Delta{x}_1\sigma_{21})_{i,j}^Z |
| 781 |
\biggr\} |
\biggr\} |
| 782 |
\intertext{with} |
\end{align} |
| 783 |
|
with |
| 784 |
|
\begin{align} |
| 785 |
(\Delta{x}_2\sigma_{11})_{i,j}^C =& \phantom{+} |
(\Delta{x}_2\sigma_{11})_{i,j}^C =& \phantom{+} |
| 786 |
\Delta{y}_{i,j}^{F}(\zeta + \eta)^{C}_{i,j} |
\Delta{y}_{i,j}^{F}(\zeta + \eta)^{C}_{i,j} |
| 787 |
\frac{u_{i+1,j}-u_{i,j}}{\Delta{x}_{i,j}^{F}} \\ \notag |
\frac{u_{i+1,j}-u_{i,j}}{\Delta{x}_{i,j}^{F}} \\ \notag |
| 792 |
\phantom{=}& + \Delta{y}_{i,j}^{F}(\zeta - \eta)^{C}_{i,j} |
\phantom{=}& + \Delta{y}_{i,j}^{F}(\zeta - \eta)^{C}_{i,j} |
| 793 |
k_{1,i,j}^{C}\frac{u_{i+1,j}+u_{i,j}}{2} \\ \notag |
k_{1,i,j}^{C}\frac{u_{i+1,j}+u_{i,j}}{2} \\ \notag |
| 794 |
\phantom{=}& - \Delta{y}_{i,j}^{F} \frac{P}{2} \\ |
\phantom{=}& - \Delta{y}_{i,j}^{F} \frac{P}{2} \\ |
|
% |
|
| 795 |
(\Delta{x}_1\sigma_{21})_{i,j}^Z =& \phantom{+} |
(\Delta{x}_1\sigma_{21})_{i,j}^Z =& \phantom{+} |
| 796 |
\Delta{x}_{i,j}^{V}\overline{\eta}^{Z}_{i,j} |
\Delta{x}_{i,j}^{V}\overline{\eta}^{Z}_{i,j} |
| 797 |
\frac{u_{i,j}-u_{i,j-1}}{\Delta{y}_{i,j}^{U}} \\ \notag |
\frac{u_{i,j}-u_{i,j-1}}{\Delta{y}_{i,j}^{U}} \\ \notag |
| 823 |
\phantom{=}& \phantom{\frac{1}{A_{i,j}^s} \biggl\{} |
\phantom{=}& \phantom{\frac{1}{A_{i,j}^s} \biggl\{} |
| 824 |
+ (\Delta{x}_1\sigma_{22})_{i,j}^C - (\Delta{x}_1\sigma_{22})_{i,j-1}^C |
+ (\Delta{x}_1\sigma_{22})_{i,j}^C - (\Delta{x}_1\sigma_{22})_{i,j-1}^C |
| 825 |
\biggr\} |
\biggr\} |
| 826 |
\intertext{with} |
\end{align} |
| 827 |
|
with |
| 828 |
|
\begin{align} |
| 829 |
(\Delta{x}_1\sigma_{12})_{i,j}^Z =& \phantom{+} |
(\Delta{x}_1\sigma_{12})_{i,j}^Z =& \phantom{+} |
| 830 |
\Delta{y}_{i,j}^{U}\overline{\eta}^{Z}_{i,j} |
\Delta{y}_{i,j}^{U}\overline{\eta}^{Z}_{i,j} |
| 831 |
\frac{u_{i,j}-u_{i,j-1}}{\Delta{y}_{i,j}^{U}} \\\notag |
\frac{u_{i,j}-u_{i,j-1}}{\Delta{y}_{i,j}^{U}} |
| 832 |
&+ \Delta{y}_{i,j}^{U}\overline{\eta}^{Z}_{i,j} |
\\\notag & |
| 833 |
\frac{v_{i,j}-v_{i-1,j}}{\Delta{x}_{i,j}^{V}} \\ \notag |
+ \Delta{y}_{i,j}^{U}\overline{\eta}^{Z}_{i,j} |
| 834 |
&- \Delta{y}_{i,j}^{U}\overline{\eta}^{Z}_{i,j} |
\frac{v_{i,j}-v_{i-1,j}}{\Delta{x}_{i,j}^{V}} \\\notag |
|
k_{2,i,j}^{Z}\frac{u_{i,j}+u_{i,j-1}}{2} \\ \notag |
|
| 835 |
&- \Delta{y}_{i,j}^{U}\overline{\eta}^{Z}_{i,j} |
&- \Delta{y}_{i,j}^{U}\overline{\eta}^{Z}_{i,j} |
| 836 |
|
k_{2,i,j}^{Z}\frac{u_{i,j}+u_{i,j-1}}{2} |
| 837 |
|
\\\notag & |
| 838 |
|
- \Delta{y}_{i,j}^{U}\overline{\eta}^{Z}_{i,j} |
| 839 |
k_{1,i,j}^{Z}\frac{v_{i,j}+v_{i-1,j}}{2} \\ \notag |
k_{1,i,j}^{Z}\frac{v_{i,j}+v_{i-1,j}}{2} \\ \notag |
|
% |
|
| 840 |
(\Delta{x}_2\sigma_{22})_{i,j}^C =& \phantom{+} |
(\Delta{x}_2\sigma_{22})_{i,j}^C =& \phantom{+} |
| 841 |
\Delta{x}_{i,j}^{F}(\zeta - \eta)^{C}_{i,j} |
\Delta{x}_{i,j}^{F}(\zeta - \eta)^{C}_{i,j} |
| 842 |
\frac{u_{i+1,j}-u_{i,j}}{\Delta{x}_{i,j}^{F}} \\ \notag |
\frac{u_{i+1,j}-u_{i,j}}{\Delta{x}_{i,j}^{F}} \\ \notag |
| 855 |
analogy to $(\epsilon_{12})^Z=0$ on boundaries, we set |
analogy to $(\epsilon_{12})^Z=0$ on boundaries, we set |
| 856 |
$\sigma_{21}^{Z}=0$, or equivalently $\eta_{i,j}^{Z}=0$, on boundaries. |
$\sigma_{21}^{Z}=0$, or equivalently $\eta_{i,j}^{Z}=0$, on boundaries. |
| 857 |
|
|
| 858 |
\subsubsection{Thermodynamics} |
\paragraph{Thermodynamics\label{sec:pkg:seaice:thermodynamics}}~\\ |
| 859 |
\label{sec:pkg:seaice:thermodynamics} |
% |
|
|
|
| 860 |
In its original formulation the sea ice model \citep{menemenlis05} |
In its original formulation the sea ice model \citep{menemenlis05} |
| 861 |
uses simple thermodynamics following the appendix of |
uses simple thermodynamics following the appendix of |
| 862 |
\citet{sem76}. This formulation does not allow storage of heat, |
\citet{sem76}. This formulation does not allow storage of heat, |
| 879 |
\frac{2n-1}{7}\,h$ for $n\in[1,7]$. The heat fluxes computed for each |
\frac{2n-1}{7}\,h$ for $n\in[1,7]$. The heat fluxes computed for each |
| 880 |
thickness category is area-averaged to give the total heat flux |
thickness category is area-averaged to give the total heat flux |
| 881 |
\citep{hibler84}. To use this thickness category parameterization set |
\citep{hibler84}. To use this thickness category parameterization set |
| 882 |
\texttt{\#define SEAICE\_MULTICATEGORY}; note that this requires |
\code{\#define SEAICE\_MULTICATEGORY}; note that this requires |
| 883 |
different restart files and switching this flag on in the middle of an |
different restart files and switching this flag on in the middle of an |
| 884 |
integration is not possible. |
integration is not possible. |
| 885 |
|
|
| 889 |
the density and heat capacity of sea water and $T_{fr}$ is the local |
the density and heat capacity of sea water and $T_{fr}$ is the local |
| 890 |
freezing point temperature that is a function of salinity. This flux |
freezing point temperature that is a function of salinity. This flux |
| 891 |
is not assumed to instantaneously melt or create ice, but a time scale |
is not assumed to instantaneously melt or create ice, but a time scale |
| 892 |
of three days (run-time parameter \texttt{SEAICE\_gamma\_t}) is used |
of three days (run-time parameter \code{SEAICE\_gamma\_t}) is used |
| 893 |
to relax $T_{w}$ to the freezing point. |
to relax $T_{w}$ to the freezing point. |
| 894 |
% |
% |
| 895 |
The parameterization of lateral and vertical growth of sea ice follows |
The parameterization of lateral and vertical growth of sea ice follows |
| 896 |
that of \citet{hib79, hib80}; the so-called lead closing parameter |
that of \citet{hib79, hib80}; the so-called lead closing parameter |
| 897 |
$h_{0}$ (run-time parameter \texttt{HO}) has a default value of |
$h_{0}$ (run-time parameter \code{HO}) has a default value of |
| 898 |
0.5~meters. |
0.5~meters. |
| 899 |
|
|
| 900 |
On top of the ice there is a layer of snow that modifies the heat flux |
On top of the ice there is a layer of snow that modifies the heat flux |
| 907 |
snowice formation (a flood-freeze algorithm following Archimedes' |
snowice formation (a flood-freeze algorithm following Archimedes' |
| 908 |
principle) turns snow into ice until the ice surface is back at $z=0$ |
principle) turns snow into ice until the ice surface is back at $z=0$ |
| 909 |
\citep{leppaeranta83}. The flood-freeze algorithm is enabled with the CPP-flag |
\citep{leppaeranta83}. The flood-freeze algorithm is enabled with the CPP-flag |
| 910 |
\texttt{SEAICE\_ALLOW\_FLOODING} and turned on with run-time parameter |
\code{SEAICE\_ALLOW\_FLOODING} and turned on with run-time parameter |
| 911 |
\texttt{SEAICEuseFlooding=.true.}. |
\code{SEAICEuseFlooding=.true.}. |
| 912 |
|
|
| 913 |
|
\paragraph{Advection of thermodynamic variables\label{sec:pkg:seaice:advection}}~\\ |
| 914 |
|
% |
| 915 |
Effective ice thickness (ice volume per unit area, |
Effective ice thickness (ice volume per unit area, |
| 916 |
$c\cdot{h}$), concentration $c$ and effective snow thickness |
$c\cdot{h}$), concentration $c$ and effective snow thickness |
| 917 |
($c\cdot{h}_{s}$) are advected by ice velocities: |
($c\cdot{h}_{s}$) are advected by ice velocities: |
| 924 |
diffusive terms for quantities $X=(c\cdot{h}), c, (c\cdot{h}_{s})$. |
diffusive terms for quantities $X=(c\cdot{h}), c, (c\cdot{h}_{s})$. |
| 925 |
% |
% |
| 926 |
From the various advection scheme that are available in the MITgcm, we |
From the various advection scheme that are available in the MITgcm, we |
| 927 |
choose flux-limited schemes \citep[multidimensional 2nd and 3rd-order |
recommend flux-limited schemes \citep[multidimensional 2nd and |
| 928 |
advection scheme with flux limiter][]{roe:85, hundsdorfer94} to |
3rd-order advection scheme with flux limiter][]{roe:85, hundsdorfer94} |
| 929 |
preserve sharp gradients and edges that are typical of sea ice |
to preserve sharp gradients and edges that are typical of sea ice |
| 930 |
distributions and to rule out unphysical over- and undershoots |
distributions and to rule out unphysical over- and undershoots |
| 931 |
(negative thickness or concentration). These scheme conserve volume |
(negative thickness or concentration). These schemes conserve volume |
| 932 |
and horizontal area and are unconditionally stable, so that we can set |
and horizontal area and are unconditionally stable, so that we can set |
| 933 |
$D_{X}=0$. Run-timeflags: \texttt{SEAICEadvScheme} (default=2), |
$D_{X}=0$. Run-timeflags: \code{SEAICEadvScheme} (default=2, is the |
| 934 |
\texttt{DIFF1} (default=0.004). |
historic 2nd-order, centered difference scheme), \code{DIFF1} = |
| 935 |
|
$D_{X}/\Delta{x}$ |
| 936 |
There is considerable doubt about the reliability of a ``zero-layer'' |
(default=0.004). |
| 937 |
thermodynamic model --- \citet{semtner84} found significant errors in |
|
| 938 |
phase (one month lead) and amplitude ($\approx$50\%\,overestimate) in |
The MITgcm sea ice model provides the option to use |
| 939 |
such models --- so that today many sea ice models employ more complex |
the thermodynamics model of \citet{win00}, which in turn is based on |
| 940 |
thermodynamics. The MITgcm sea ice model provides the option to use |
the 3-layer model of \citet{sem76} and which treats brine content by |
| 941 |
the thermodynamics model of \citet{win00}, which in turn is based |
means of enthalpy conservation; the corresponding package |
| 942 |
on the 3-layer model of \citet{sem76} and which treats brine |
\code{thsice} is described in section~\ref{sec:pkg:thsice}. This |
| 943 |
content by means of enthalpy conservation. This scheme requires |
scheme requires additional state variables, namely the enthalpy of the |
| 944 |
additional state variables, namely the enthalpy of the two ice layers |
two ice layers (instead of effective ice salinity), to be advected by |
| 945 |
(instead of effective ice salinity), to be advected by ice velocities. |
ice velocities. |
| 946 |
% |
% |
| 947 |
The internal sea ice temperature is inferred from ice enthalpy. To |
The internal sea ice temperature is inferred from ice enthalpy. To |
| 948 |
avoid unphysical (negative) values for ice thickness and |
avoid unphysical (negative) values for ice thickness and |
| 949 |
concentration, a positive 2nd-order advection scheme with a SuperBee |
concentration, a positive 2nd-order advection scheme with a SuperBee |
| 950 |
flux limiter \citep{roe:85} is used in this study to advect all |
flux limiter \citep{roe:85} should be used to advect all |
| 951 |
sea-ice-related quantities of the \citet{win00} thermodynamic |
sea-ice-related quantities of the \citet{win00} thermodynamic model |
| 952 |
model. Because of the non-linearity of the advection scheme, care |
(runtime flag \code{thSIceAdvScheme=77} and |
| 953 |
must be taken in advecting these quantities: when simply using ice |
\code{thSIce\_diffK}=$D_{X}$=0 in \code{data.ice}, defaults are 0). Because of the |
| 954 |
velocity to advect enthalpy, the total energy (i.e., the volume |
non-linearity of the advection scheme, care must be taken in advecting |
| 955 |
integral of enthalpy) is not conserved. Alternatively, one can advect |
these quantities: when simply using ice velocity to advect enthalpy, |
| 956 |
the energy content (i.e., product of ice-volume and enthalpy) but then |
the total energy (i.e., the volume integral of enthalpy) is not |
| 957 |
false enthalpy extrema can occur, which then leads to unrealistic ice |
conserved. Alternatively, one can advect the energy content (i.e., |
| 958 |
temperature. In the currently implemented solution, the sea-ice mass |
product of ice-volume and enthalpy) but then false enthalpy extrema |
| 959 |
flux is used to advect the enthalpy in order to ensure conservation of |
can occur, which then leads to unrealistic ice temperature. In the |
| 960 |
enthalpy and to prevent false enthalpy extrema. |
currently implemented solution, the sea-ice mass flux is used to |
| 961 |
|
advect the enthalpy in order to ensure conservation of enthalpy and to |
| 962 |
|
prevent false enthalpy extrema. % |
| 963 |
|
|
| 964 |
%---------------------------------------------------------------------- |
%---------------------------------------------------------------------- |
| 965 |
|
|
| 966 |
\subsubsection{Key subroutines |
\subsubsection{Key subroutines |
| 967 |
\label{sec:pkg:seaice:subroutines}} |
\label{sec:pkg:seaice:subroutines}} |
| 968 |
|
|
| 969 |
Top-level routine: \texttt{seaice\_model.F} |
Top-level routine: \code{seaice\_model.F} |
| 970 |
|
|
| 971 |
{\footnotesize |
{\footnotesize |
| 972 |
\begin{verbatim} |
\begin{verbatim} |
| 1025 |
Available output fields are summarized in |
Available output fields are summarized in |
| 1026 |
Table \ref{tab:pkg:seaice:diagnostics}. |
Table \ref{tab:pkg:seaice:diagnostics}. |
| 1027 |
|
|
| 1028 |
\begin{table}[h!] |
\input{s_phys_pkgs/text/seaice_diags.tex} |
|
\centering |
|
|
\label{tab:pkg:seaice:diagnostics} |
|
|
{\footnotesize |
|
|
\begin{verbatim} |
|
|
---------+----+----+----------------+----------------- |
|
|
<-Name->|Levs|grid|<-- Units -->|<- Tile (max=80c) |
|
|
---------+----+----+----------------+----------------- |
|
|
SIarea | 1 |SM |m^2/m^2 |SEAICE fractional ice-covered area [0 to 1] |
|
|
SIheff | 1 |SM |m |SEAICE effective ice thickness |
|
|
SIuice | 1 |UU |m/s |SEAICE zonal ice velocity, >0 from West to East |
|
|
SIvice | 1 |VV |m/s |SEAICE merid. ice velocity, >0 from South to North |
|
|
SIhsnow | 1 |SM |m |SEAICE snow thickness |
|
|
SIhsalt | 1 |SM |g/m^2 |SEAICE effective salinity |
|
|
SIatmFW | 1 |SM |kg/m^2/s |Net freshwater flux from the atmosphere (+=down) |
|
|
SIuwind | 1 |SM |m/s |SEAICE zonal 10-m wind speed, >0 increases uVel |
|
|
SIvwind | 1 |SM |m/s |SEAICE meridional 10-m wind speed, >0 increases uVel |
|
|
SIfu | 1 |UU |N/m^2 |SEAICE zonal surface wind stress, >0 increases uVel |
|
|
SIfv | 1 |VV |N/m^2 |SEAICE merid. surface wind stress, >0 increases vVel |
|
|
SIempmr | 1 |SM |kg/m^2/s |SEAICE upward freshwater flux, > 0 increases salt |
|
|
SIqnet | 1 |SM |W/m^2 |SEAICE upward heatflux, turb+rad, >0 decreases theta |
|
|
SIqsw | 1 |SM |W/m^2 |SEAICE upward shortwave radiat., >0 decreases theta |
|
|
SIpress | 1 |SM |m^2/s^2 |SEAICE strength (with upper and lower limit) |
|
|
SIzeta | 1 |SM |m^2/s |SEAICE nonlinear bulk viscosity |
|
|
SIeta | 1 |SM |m^2/s |SEAICE nonlinear shear viscosity |
|
|
SIsigI | 1 |SM |no units |SEAICE normalized principle stress, component one |
|
|
SIsigII | 1 |SM |no units |SEAICE normalized principle stress, component two |
|
|
SIthdgrh| 1 |SM |m/s |SEAICE thermodynamic growth rate of effective ice thickness |
|
|
SIsnwice| 1 |SM |m/s |SEAICE ice formation rate due to flooding |
|
|
SIuheff | 1 |UU |m^2/s |Zonal Transport of effective ice thickness |
|
|
SIvheff | 1 |VV |m^2/s |Meridional Transport of effective ice thickness |
|
|
ADVxHEFF| 1 |UU |m.m^2/s |Zonal Advective Flux of eff ice thickn |
|
|
ADVyHEFF| 1 |VV |m.m^2/s |Meridional Advective Flux of eff ice thickn |
|
|
DFxEHEFF| 1 |UU |m.m^2/s |Zonal Diffusive Flux of eff ice thickn |
|
|
DFyEHEFF| 1 |VV |m.m^2/s |Meridional Diffusive Flux of eff ice thickn |
|
|
ADVxAREA| 1 |UU |m^2/m^2.m^2/s |Zonal Advective Flux of fract area |
|
|
ADVyAREA| 1 |VV |m^2/m^2.m^2/s |Meridional Advective Flux of fract area |
|
|
DFxEAREA| 1 |UU |m^2/m^2.m^2/s |Zonal Diffusive Flux of fract area |
|
|
DFyEAREA| 1 |VV |m^2/m^2.m^2/s |Meridional Diffusive Flux of fract area |
|
|
ADVxSNOW| 1 |UU |m.m^2/s |Zonal Advective Flux of eff snow thickn |
|
|
ADVySNOW| 1 |VV |m.m^2/s |Meridional Advective Flux of eff snow thickn |
|
|
DFxESNOW| 1 |UU |m.m^2/s |Zonal Diffusive Flux of eff snow thickn |
|
|
DFyESNOW| 1 |VV |m.m^2/s |Meridional Diffusive Flux of eff snow thickn |
|
|
ADVxSSLT| 1 |UU |psu.m^2/s |Zonal Advective Flux of seaice salinity |
|
|
ADVySSLT| 1 |VV |psu.m^2/s |Meridional Advective Flux of seaice salinity |
|
|
DFxESSLT| 1 |UU |psu.m^2/s |Zonal Diffusive Flux of seaice salinity |
|
|
DFyESSLT| 1 |VV |psu.m^2/s |Meridional Diffusive Flux of seaice salinity |
|
|
\end{verbatim} |
|
|
} |
|
|
\caption{Available diagnostics of the seaice-package} |
|
|
\end{table} |
|
|
|
|
| 1029 |
|
|
| 1030 |
%\subsubsection{Package Reference} |
%\subsubsection{Package Reference} |
| 1031 |
|
|
| 1033 |
\label{sec:pkg:seaice:experiments} |
\label{sec:pkg:seaice:experiments} |
| 1034 |
|
|
| 1035 |
\begin{itemize} |
\begin{itemize} |
| 1036 |
\item{Labrador Sea experiment in lab\_sea verification directory. } |
\item{Labrador Sea experiment in \code{lab\_sea} verification directory. } |
| 1037 |
|
\item \code{seaice\_obcs}, based on \code{lab\_sea} |
| 1038 |
|
\item \code{offline\_exf\_seaice/input.seaicetd}, based on \code{lab\_sea} |
| 1039 |
|
\item \code{global\_ocean.cs32x15/input.icedyn} and |
| 1040 |
|
\code{global\_ocean.cs32x15/input.seaice}, global |
| 1041 |
|
cubed-sphere-experiment with combinations of \code{seaice} and |
| 1042 |
|
\code{thsice} |
| 1043 |
\end{itemize} |
\end{itemize} |
| 1044 |
|
|
| 1045 |
|
|
| 1046 |
%%% Local Variables: |
%%% Local Variables: |
| 1047 |
%%% mode: latex |
%%% mode: latex |
| 1048 |
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