| 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 |
| 24 |
|
|
| 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 |
Runtime 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 either |
| 65 |
\texttt{SEAICE\_OPTIONS.h} or in \texttt{ECCO\_CPPOPTIONS.h}. |
\code{SEAICE\_OPTIONS.h} or in \code{ECCO\_CPPOPTIONS.h}. |
| 66 |
Table \ref{tab:pkg:seaice:cpp} summarizes these options. |
Table \ref{tab:pkg:seaice:cpp} summarizes these options. |
| 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 |
| 72 |
\begin{tabular}{|l|l|} |
\begin{tabular}{|l|p{10cm}|} |
| 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 \\ |
use EVP rather than LSR rheology solver \\ |
| 84 |
\texttt{SEAICE\_EXTERNAL\_FLUXES} & |
\code{SEAICE\_EXTERNAL\_FLUXES} & |
| 85 |
use EXF-computed fluxes as starting point \\ |
use EXF-computed fluxes as starting point \\ |
| 86 |
\texttt{SEAICE\_MULTICATEGORY} & |
\code{SEAICE\_MULTICATEGORY} & |
| 87 |
enable 8-category thermodynamics \\ |
enable 8-category thermodynamics (by default undefined)\\ |
| 88 |
\texttt{SEAICE\_VARIABLE\_FREEZING\_POINT} & |
\code{SEAICE\_VARIABLE\_FREEZING\_POINT} & |
| 89 |
enable linear dependence of the freezing point on salinity \\ |
enable linear dependence of the freezing point on salinity |
| 90 |
\texttt{ALLOW\_SEAICE\_FLOODING} & |
(by default undefined)\\ |
| 91 |
|
\code{ALLOW\_SEAICE\_FLOODING} & |
| 92 |
enable snow to ice conversion for submerged sea-ice \\ |
enable snow to ice conversion for submerged sea-ice \\ |
| 93 |
\texttt{SEAICE\_SALINITY} & |
\code{SEAICE\_SALINITY} & |
| 94 |
enable "salty" sea-ice \\ |
enable "salty" sea-ice (by default undefined) \\ |
| 95 |
\texttt{SEAICE\_CAP\_HEFF} & |
\code{SEAICE\_AGE} & |
| 96 |
enable capping of sea-ice thickness to MAX\_HEFF \\ |
enable "age tracer" sea-ice (by default undefined) \\ |
| 97 |
|
\code{SEAICE\_CAP\_HEFF} & |
| 98 |
|
enable capping of sea-ice thickness to MAX\_HEFF \\ \hline |
| 99 |
|
\code{SEAICE\_BICE\_STRESS} & |
| 100 |
|
B-grid only for backward compatiblity: turn on ice-stress on |
| 101 |
|
ocean\\ |
| 102 |
|
\code{EXPLICIT\_SSH\_SLOPE} & |
| 103 |
|
B-grid only for backward compatiblity: use ETAN for tilt |
| 104 |
|
computations rather than geostrophic velocities \\ |
| 105 |
\hline |
\hline |
| 106 |
\end{tabular} |
\end{tabular} |
| 107 |
} |
} |
| 113 |
\subsubsection{Run-time parameters |
\subsubsection{Run-time parameters |
| 114 |
\label{sec:pkg:seaice:runtime}} |
\label{sec:pkg:seaice:runtime}} |
| 115 |
|
|
| 116 |
Run-time parameters are set in files |
Run-time parameters (see Table~\ref{tab:pkg:seaice:runtimeparms}) are set |
| 117 |
\texttt{data.pkg} (read in \texttt{packages\_readparms.F}), |
in files \code{data.pkg} (read in \code{packages\_readparms.F}), and |
| 118 |
and \texttt{data.seaice} (read in \texttt{seaice\_readparms.F}). |
\code{data.seaice} (read in \code{seaice\_readparms.F}). |
| 119 |
|
|
| 120 |
\paragraph{Enabling the package} |
\paragraph{Enabling the package} |
| 121 |
~ \\ |
~ \\ |
| 122 |
% |
% |
| 123 |
A package is switched on/off at runtime by setting |
A package is switched on/off at run-time by setting |
| 124 |
(e.g. for SEAICE) \texttt{useSEAICE = .TRUE.} in \texttt{data.pkg}. |
(e.g. for SEAICE) \code{useSEAICE = .TRUE.} in \code{data.pkg}. |
| 125 |
|
|
| 126 |
\paragraph{General flags and parameters} |
\paragraph{General flags and parameters} |
| 127 |
~ \\ |
~ \\ |
| 128 |
% |
% |
| 129 |
\input{part6/seaice-parms.tex} |
Table~\ref{tab:pkg:seaice:runtimeparms} lists most run-time parameters. |
| 130 |
|
\input{s_phys_pkgs/text/seaice-parms.tex} |
| 131 |
|
|
| 132 |
|
\paragraph{Input fields and units\label{sec:pkg:seaice:fields_units}} |
| 133 |
|
\begin{description} |
| 134 |
|
\item[\code{HeffFile}:] Initial sea ice thickness averaged over grid cell |
| 135 |
|
in meters; initializes variable \code{HEFF}; |
| 136 |
|
\item[\code{AreaFile}:] Initial fractional sea ice cover, range $[0,1]$; |
| 137 |
|
initializes variable \code{AREA}; |
| 138 |
|
\item[\code{HsnowFile}:] Initial snow thickness on sea ice averaged |
| 139 |
|
over grid cell in meters; initializes variable \code{HSNOW}; |
| 140 |
|
\item[\code{HsaltFile}:] Initial salinity of sea ice averaged over grid |
| 141 |
|
cell in g/m$^2$; initializes variable \code{HSALT}; |
| 142 |
|
\item[\code{IceAgeFile}:] Initial ice age of sea ice averaged over grid |
| 143 |
|
cell in seconds; initializes variable \code{ICEAGE}; |
| 144 |
|
\end{description} |
| 145 |
|
|
| 146 |
%---------------------------------------------------------------------- |
%---------------------------------------------------------------------- |
| 147 |
\subsubsection{Description |
\subsubsection{Description |
| 149 |
|
|
| 150 |
[TO BE CONTINUED/MODIFIED] |
[TO BE CONTINUED/MODIFIED] |
| 151 |
|
|
| 152 |
Sea-ice model thermodynamics are based on Hibler |
% Sea-ice model thermodynamics are based on Hibler |
| 153 |
\cite{hib80}, that is, a 2-category model that simulates ice thickness |
% \cite{hib80}, that is, a 2-category model that simulates ice thickness |
| 154 |
and concentration. Snow is simulated as per Zhang et al. |
% and concentration. Snow is simulated as per Zhang et al. |
| 155 |
\cite{zha98a}. Although recent years have seen an increased use of |
% \cite{zha98a}. Although recent years have seen an increased use of |
| 156 |
multi-category thickness distribution sea-ice models for climate |
% multi-category thickness distribution sea-ice models for climate |
| 157 |
studies, the Hibler 2-category ice model is still the most widely used |
% studies, the Hibler 2-category ice model is still the most widely used |
| 158 |
model and has resulted in realistic simulation of sea-ice variability |
% model and has resulted in realistic simulation of sea-ice variability |
| 159 |
on regional and global scales. Being less complicated, compared to |
% on regional and global scales. Being less complicated, compared to |
| 160 |
multi-category models, the 2-category model permits easier application |
% multi-category models, the 2-category model permits easier application |
| 161 |
of adjoint model optimization methods. |
% of adjoint model optimization methods. |
| 162 |
|
|
| 163 |
Note, however, that the Hibler 2-category model and its variants use a |
% Note, however, that the Hibler 2-category model and its variants use a |
| 164 |
so-called zero-layer thermodynamic model to estimate ice growth and |
% so-called zero-layer thermodynamic model to estimate ice growth and |
| 165 |
decay. The zero-layer thermodynamic model assumes that ice does not |
% decay. The zero-layer thermodynamic model assumes that ice does not |
| 166 |
store heat and, therefore, tends to exaggerate the seasonal |
% store heat and, therefore, tends to exaggerate the seasonal |
| 167 |
variability in ice thickness. This exaggeration can be significantly |
% variability in ice thickness. This exaggeration can be significantly |
| 168 |
reduced by using Semtner's \cite{sem76} three-layer thermodynamic |
% reduced by using Semtner's \cite{sem76} three-layer thermodynamic |
| 169 |
model that permits heat storage in ice. Recently, the three-layer |
% model that permits heat storage in ice. Recently, the three-layer |
| 170 |
thermodynamic model has been reformulated by Winton \cite{win00}. The |
% thermodynamic model has been reformulated by Winton \cite{win00}. The |
| 171 |
reformulation improves model physics by representing the brine content |
% reformulation improves model physics by representing the brine content |
| 172 |
of the upper ice with a variable heat capacity. It also improves |
% of the upper ice with a variable heat capacity. It also improves |
| 173 |
model numerics and consumes less computer time and memory. The Winton |
% model numerics and consumes less computer time and memory. The Winton |
| 174 |
sea-ice thermodynamics have been ported to the MIT GCM; they currently |
% sea-ice thermodynamics have been ported to the MIT GCM; they currently |
| 175 |
reside under pkg/thsice. At present pkg/thsice is not fully |
% reside under pkg/thsice. The package pkg/thsice is fully |
| 176 |
compatible with pkg/seaice and with pkg/exf. But the eventual |
% compatible with pkg/seaice and with pkg/exf. When turned on togeter |
| 177 |
objective is to have fully compatible and interchangeable |
% with pkg/seaice, the zero-layer thermodynamics are replaced by the by |
| 178 |
thermodynamic packages for sea-ice, so that it becomes possible to use |
% Winton thermodynamics |
| 179 |
Hibler dynamics with Winton thermodyanmics. |
|
| 180 |
|
The MITgcm sea ice model (MITgcm/sim) is based on a variant of the |
| 181 |
|
viscous-plastic (VP) dynamic-thermodynamic sea ice model \citep{zhang97} |
| 182 |
|
first introduced by \citet{hib79, hib80}. In order to adapt this model |
| 183 |
|
to the requirements of coupled ice-ocean state estimation, many |
| 184 |
|
important aspects of the original code have been modified and |
| 185 |
|
improved: |
| 186 |
|
\begin{itemize} |
| 187 |
|
\item the code has been rewritten for an Arakawa C-grid, both B- and |
| 188 |
|
C-grid variants are available; the C-grid code allows for no-slip |
| 189 |
|
and free-slip lateral boundary conditions; |
| 190 |
|
\item two different solution methods for solving the nonlinear |
| 191 |
|
momentum equations have been adopted: LSOR \citep{zhang97}, and EVP |
| 192 |
|
\citep{hun97}; |
| 193 |
|
\item ice-ocean stress can be formulated as in \citet{hibler87} or as in |
| 194 |
|
\citet{cam08}; |
| 195 |
|
\item ice variables are advected by sophisticated, conservative |
| 196 |
|
advection schemes with flux limiting; |
| 197 |
|
\item growth and melt parameterizations have been refined and extended |
| 198 |
|
in order to allow for more stable automatic differentiation of the code. |
| 199 |
|
\end{itemize} |
| 200 |
|
The sea ice model is tightly coupled to the ocean compontent of the |
| 201 |
|
MITgcm. Heat, fresh water fluxes and surface stresses are computed |
| 202 |
|
from the atmospheric state and -- by default -- modified by the ice |
| 203 |
|
model at every time step. |
| 204 |
|
|
| 205 |
The ice dynamics models that are most widely used for large-scale |
The ice dynamics models that are most widely used for large-scale |
| 206 |
climate studies are the viscous-plastic (VP) model \cite{hib79}, the |
climate studies are the viscous-plastic (VP) model \citep{hib79}, the |
| 207 |
cavitating fluid (CF) model \cite{fla92}, and the |
cavitating fluid (CF) model \citep{fla92}, and the |
| 208 |
elastic-viscous-plastic (EVP) model \cite{hun97}. Compared to the VP |
elastic-viscous-plastic (EVP) model \citep{hun97}. Compared to the VP |
| 209 |
model, the CF model does not allow ice shear in calculating ice |
model, the CF model does not allow ice shear in calculating ice |
| 210 |
motion, stress, and deformation. EVP models approximate VP by adding |
motion, stress, and deformation. EVP models approximate VP by adding |
| 211 |
an elastic term to the equations for easier adaptation to parallel |
an elastic term to the equations for easier adaptation to parallel |
| 212 |
computers. Because of its higher accuracy in plastic solution and |
computers. Because of its higher accuracy in plastic solution and |
| 213 |
relatively simpler formulation, compared to the EVP model, we decided |
relatively simpler formulation, compared to the EVP model, we decided |
| 214 |
to use the VP model as the dynamic component of our ice model. To do |
to use the VP model as the default dynamic component of our ice |
| 215 |
this we extended the alternating-direction-implicit (ADI) method of |
model. To do this we extended the line successive over relaxation |
| 216 |
Zhang and Rothrock \cite{zha00} for use in a parallel configuration. |
(LSOR) method of \citet{zhang97} for use in a parallel |
| 217 |
|
configuration. An EVP model and a free-drift implemtation can be |
| 218 |
|
selected with runtime flags. |
| 219 |
|
|
| 220 |
|
\paragraph{Compatibility with ice-thermodynamics package \code{thsice}\label{sec:pkg:seaice:thsice}}~\\ |
| 221 |
|
% |
| 222 |
|
Note, that by default the \code{seaice}-package includes the orginial |
| 223 |
|
so-called zero-layer thermodynamics following \citet{hib80} with a |
| 224 |
|
snow cover as in \citet{zha98a}. The zero-layer thermodynamic model |
| 225 |
|
assumes that ice does not store heat and, therefore, tends to |
| 226 |
|
exaggerate the seasonal variability in ice thickness. This |
| 227 |
|
exaggeration can be significantly reduced by using |
| 228 |
|
\citeauthor{sem76}'s~[\citeyear{sem76}] three-layer thermodynamic |
| 229 |
|
model that permits heat storage in ice. Recently, the three-layer thermodynamic model has been reformulated by |
| 230 |
|
\citet{win00}. The reformulation improves model physics by |
| 231 |
|
representing the brine content of the upper ice with a variable heat |
| 232 |
|
capacity. It also improves model numerics and consumes less computer |
| 233 |
|
time and memory. |
| 234 |
|
|
| 235 |
|
The Winton sea-ice thermodynamics have been ported to the MIT GCM; |
| 236 |
|
they currently reside under \code{pkg/thsice}. The package |
| 237 |
|
\code{thsice} is described in section~\ref{sec:pkg:thsice}; it is |
| 238 |
|
fully compatible with the packages \code{seaice} and \code{exf}. When |
| 239 |
|
turned on together with \code{seaice}, the zero-layer thermodynamics |
| 240 |
|
are replaced by the Winton thermodynamics. In order to use the |
| 241 |
|
\code{seaice}-package with the thermodynamics of \code{thsice}, |
| 242 |
|
compile both packages and turn both package on in \code{data.pkg}; see |
| 243 |
|
an example in \code{global\_ocean.cs32x15/input.icedyn}. Note, that |
| 244 |
|
once \code{thsice} is turned on, the variables and diagnostics |
| 245 |
|
associated to the default thermodynamics are meaningless, and the |
| 246 |
|
diagnostics of \code{thsice} have to be used instead. |
| 247 |
|
|
| 248 |
|
\paragraph{Surface forcing\label{sec:pkg:seaice:surfaceforcing}}~\\ |
| 249 |
|
% |
| 250 |
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 |
| 251 |
air temperature and specific humidity, downward longwave and shortwave |
air temperature and specific humidity, downward longwave and shortwave |
| 252 |
radiations, precipitation, evaporation, and river and glacier runoff. |
radiations, precipitation, evaporation, and river and glacier runoff. |
| 253 |
The sea ice model also requires surface temperature from the ocean |
The sea ice model also requires surface temperature from the ocean |
| 254 |
model and third level horizontal velocity which is used as a proxy for |
model and the top level horizontal velocity. Output fields are |
| 255 |
surface geostrophic velocity. Output fields are surface wind stress, |
surface wind stress, evaporation minus precipitation minus runoff, net |
| 256 |
evaporation minus precipitation minus runoff, net surface heat flux, |
surface heat flux, and net shortwave flux. The sea-ice model is |
| 257 |
and net shortwave flux. The sea-ice model is global: in ice-free |
global: in ice-free regions bulk formulae are used to estimate oceanic |
| 258 |
regions bulk formulae are used to estimate oceanic forcing from the |
forcing from the atmospheric fields. |
|
atmospheric fields. |
|
| 259 |
|
|
| 260 |
|
\paragraph{Dynamics\label{sec:pkg:seaice:dynamics}}~\\ |
| 261 |
|
% |
| 262 |
|
\newcommand{\vek}[1]{\ensuremath{\vec{\mathbf{#1}}}} |
| 263 |
|
\newcommand{\vtau}{\vek{\mathbf{\tau}}} |
| 264 |
|
The momentum equation of the sea-ice model is |
| 265 |
|
\begin{equation} |
| 266 |
|
\label{eq:momseaice} |
| 267 |
|
m \frac{D\vek{u}}{Dt} = -mf\vek{k}\times\vek{u} + \vtau_{air} + |
| 268 |
|
\vtau_{ocean} - m \nabla{\phi(0)} + \vek{F}, |
| 269 |
|
\end{equation} |
| 270 |
|
where $m=m_{i}+m_{s}$ is the ice and snow mass per unit area; |
| 271 |
|
$\vek{u}=u\vek{i}+v\vek{j}$ is the ice velocity vector; |
| 272 |
|
$\vek{i}$, $\vek{j}$, and $\vek{k}$ are unit vectors in the $x$, $y$, and $z$ |
| 273 |
|
directions, respectively; |
| 274 |
|
$f$ is the Coriolis parameter; |
| 275 |
|
$\vtau_{air}$ and $\vtau_{ocean}$ are the wind-ice and ocean-ice stresses, |
| 276 |
|
respectively; |
| 277 |
|
$g$ is the gravity accelation; |
| 278 |
|
$\nabla\phi(0)$ is the gradient (or tilt) of the sea surface height; |
| 279 |
|
$\phi(0) = g\eta + p_{a}/\rho_{0} + mg/\rho_{0}$ is the sea surface |
| 280 |
|
height potential in response to ocean dynamics ($g\eta$), to |
| 281 |
|
atmospheric pressure loading ($p_{a}/\rho_{0}$, where $\rho_{0}$ is a |
| 282 |
|
reference density) and a term due to snow and ice loading \citep{cam08}; |
| 283 |
|
and $\vek{F}=\nabla\cdot\sigma$ is the divergence of the internal ice |
| 284 |
|
stress tensor $\sigma_{ij}$. % |
| 285 |
|
Advection of sea-ice momentum is neglected. The wind and ice-ocean stress |
| 286 |
|
terms are given by |
| 287 |
|
\begin{align*} |
| 288 |
|
\vtau_{air} = & \rho_{air} C_{air} |\vek{U}_{air} -\vek{u}| |
| 289 |
|
R_{air} (\vek{U}_{air} -\vek{u}), \\ |
| 290 |
|
\vtau_{ocean} = & \rho_{ocean}C_{ocean} |\vek{U}_{ocean}-\vek{u}| |
| 291 |
|
R_{ocean}(\vek{U}_{ocean}-\vek{u}), |
| 292 |
|
\end{align*} |
| 293 |
|
where $\vek{U}_{air/ocean}$ are the surface winds of the atmosphere |
| 294 |
|
and surface currents of the ocean, respectively; $C_{air/ocean}$ are |
| 295 |
|
air and ocean drag coefficients; $\rho_{air/ocean}$ are reference |
| 296 |
|
densities; and $R_{air/ocean}$ are rotation matrices that act on the |
| 297 |
|
wind/current vectors. |
| 298 |
|
|
| 299 |
|
\paragraph{Viscous-Plastic (VP) Rheology and LSOR solver \label{sec:pkg:seaice:VPdynamics}}~\\ |
| 300 |
|
% |
| 301 |
|
For an isotropic system the stress tensor $\sigma_{ij}$ ($i,j=1,2$) can |
| 302 |
|
be related to the ice strain rate and strength by a nonlinear |
| 303 |
|
viscous-plastic (VP) constitutive law \citep{hib79, zhang97}: |
| 304 |
|
\begin{equation} |
| 305 |
|
\label{eq:vpequation} |
| 306 |
|
\sigma_{ij}=2\eta(\dot{\epsilon}_{ij},P)\dot{\epsilon}_{ij} |
| 307 |
|
+ \left[\zeta(\dot{\epsilon}_{ij},P) - |
| 308 |
|
\eta(\dot{\epsilon}_{ij},P)\right]\dot{\epsilon}_{kk}\delta_{ij} |
| 309 |
|
- \frac{P}{2}\delta_{ij}. |
| 310 |
|
\end{equation} |
| 311 |
|
The ice strain rate is given by |
| 312 |
|
\begin{equation*} |
| 313 |
|
\dot{\epsilon}_{ij} = \frac{1}{2}\left( |
| 314 |
|
\frac{\partial{u_{i}}}{\partial{x_{j}}} + |
| 315 |
|
\frac{\partial{u_{j}}}{\partial{x_{i}}}\right). |
| 316 |
|
\end{equation*} |
| 317 |
|
The maximum ice pressure $P_{\max}$, a measure of ice strength, depends on |
| 318 |
|
both thickness $h$ and compactness (concentration) $c$: |
| 319 |
|
\begin{equation} |
| 320 |
|
P_{\max} = P^{*}c\,h\,e^{[C^{*}\cdot(1-c)]}, |
| 321 |
|
\label{eq:icestrength} |
| 322 |
|
\end{equation} |
| 323 |
|
with the constants $P^{*}$ (run-time parameter \code{SEAICE\_strength}) and |
| 324 |
|
$C^{*}=20$. The nonlinear bulk and shear |
| 325 |
|
viscosities $\eta$ and $\zeta$ are functions of ice strain rate |
| 326 |
|
invariants and ice strength such that the principal components of the |
| 327 |
|
stress lie on an elliptical yield curve with the ratio of major to |
| 328 |
|
minor axis $e$ equal to $2$; they are given by: |
| 329 |
|
\begin{align*} |
| 330 |
|
\zeta =& \min\left(\frac{P_{\max}}{2\max(\Delta,\Delta_{\min})}, |
| 331 |
|
\zeta_{\max}\right) \\ |
| 332 |
|
\eta =& \frac{\zeta}{e^2} \\ |
| 333 |
|
\intertext{with the abbreviation} |
| 334 |
|
\Delta = & \left[ |
| 335 |
|
\left(\dot{\epsilon}_{11}^2+\dot{\epsilon}_{22}^2\right) |
| 336 |
|
(1+e^{-2}) + 4e^{-2}\dot{\epsilon}_{12}^2 + |
| 337 |
|
2\dot{\epsilon}_{11}\dot{\epsilon}_{22} (1-e^{-2}) |
| 338 |
|
\right]^{\frac{1}{2}}. |
| 339 |
|
\end{align*} |
| 340 |
|
The bulk viscosities are bounded above by imposing both a minimum |
| 341 |
|
$\Delta_{\min}$ (for numerical reasons, run-time parameter |
| 342 |
|
\code{SEAICE\_EPS} with a default value of |
| 343 |
|
$10^{-10}\text{\,s}^{-1}$) and a maximum $\zeta_{\max} = |
| 344 |
|
P_{\max}/\Delta^*$, where |
| 345 |
|
$\Delta^*=(5\times10^{12}/2\times10^4)\text{\,s}^{-1}$. (There is also |
| 346 |
|
the option of bounding $\zeta$ from below by setting run-time |
| 347 |
|
parameter \code{SEAICE\_zetaMin} $>0$, but this is generally not |
| 348 |
|
recommended). For stress tensor computation the replacement pressure $P |
| 349 |
|
= 2\,\Delta\zeta$ \citep{hibler95} is used so that the stress state |
| 350 |
|
always lies on the elliptic yield curve by definition. |
| 351 |
|
|
| 352 |
|
In the current implementation, the VP-model is integrated with the |
| 353 |
|
semi-implicit line successive over relaxation (LSOR)-solver of |
| 354 |
|
\citet{zhang97}, which allows for long time steps that, in our case, |
| 355 |
|
are limited by the explicit treatment of the Coriolis term. The |
| 356 |
|
explicit treatment of the Coriolis term does not represent a severe |
| 357 |
|
limitation because it restricts the time step to approximately the |
| 358 |
|
same length as in the ocean model where the Coriolis term is also |
| 359 |
|
treated explicitly. |
| 360 |
|
|
| 361 |
|
\paragraph{Elastic-Viscous-Plastic (EVP) Dynamics\label{sec:pkg:seaice:EVPdynamics}}~\\ |
| 362 |
|
% |
| 363 |
|
\citet{hun97}'s introduced an elastic contribution to the strain |
| 364 |
|
rate in order to regularize Eq.~\ref{eq:vpequation} in such a way that |
| 365 |
|
the resulting elastic-viscous-plastic (EVP) and VP models are |
| 366 |
|
identical at steady state, |
| 367 |
|
\begin{equation} |
| 368 |
|
\label{eq:evpequation} |
| 369 |
|
\frac{1}{E}\frac{\partial\sigma_{ij}}{\partial{t}} + |
| 370 |
|
\frac{1}{2\eta}\sigma_{ij} |
| 371 |
|
+ \frac{\eta - \zeta}{4\zeta\eta}\sigma_{kk}\delta_{ij} |
| 372 |
|
+ \frac{P}{4\zeta}\delta_{ij} |
| 373 |
|
= \dot{\epsilon}_{ij}. |
| 374 |
|
\end{equation} |
| 375 |
|
%In the EVP model, equations for the components of the stress tensor |
| 376 |
|
%$\sigma_{ij}$ are solved explicitly. Both model formulations will be |
| 377 |
|
%used and compared the present sea-ice model study. |
| 378 |
|
The EVP-model uses an explicit time stepping scheme with a short |
| 379 |
|
timestep. According to the recommendation of \citet{hun97}, the |
| 380 |
|
EVP-model should be stepped forward in time 120 times |
| 381 |
|
($\code{SEAICE\_deltaTevp} = \code{SEAICIE\_deltaTdyn}/120$) within |
| 382 |
|
the physical ocean model time step (although this parameter is under |
| 383 |
|
debate), to allow for elastic waves to disappear. Because the scheme |
| 384 |
|
does not require a matrix inversion it is fast in spite of the small |
| 385 |
|
internal timestep and simple to implement on parallel computers |
| 386 |
|
\citep{hun97}. For completeness, we repeat the equations for the |
| 387 |
|
components of the stress tensor $\sigma_{1} = |
| 388 |
|
\sigma_{11}+\sigma_{22}$, $\sigma_{2}= \sigma_{11}-\sigma_{22}$, and |
| 389 |
|
$\sigma_{12}$. Introducing the divergence $D_D = |
| 390 |
|
\dot{\epsilon}_{11}+\dot{\epsilon}_{22}$, and the horizontal tension |
| 391 |
|
and shearing strain rates, $D_T = |
| 392 |
|
\dot{\epsilon}_{11}-\dot{\epsilon}_{22}$ and $D_S = |
| 393 |
|
2\dot{\epsilon}_{12}$, respectively, and using the above |
| 394 |
|
abbreviations, the equations~\ref{eq:evpequation} can be written as: |
| 395 |
|
\begin{align} |
| 396 |
|
\label{eq:evpstresstensor1} |
| 397 |
|
\frac{\partial\sigma_{1}}{\partial{t}} + \frac{\sigma_{1}}{2T} + |
| 398 |
|
\frac{P}{2T} &= \frac{P}{2T\Delta} D_D \\ |
| 399 |
|
\label{eq:evpstresstensor2} |
| 400 |
|
\frac{\partial\sigma_{2}}{\partial{t}} + \frac{\sigma_{2} e^{2}}{2T} |
| 401 |
|
&= \frac{P}{2T\Delta} D_T \\ |
| 402 |
|
\label{eq:evpstresstensor12} |
| 403 |
|
\frac{\partial\sigma_{12}}{\partial{t}} + \frac{\sigma_{12} e^{2}}{2T} |
| 404 |
|
&= \frac{P}{4T\Delta} D_S |
| 405 |
|
\end{align} |
| 406 |
|
Here, the elastic parameter $E$ is redefined in terms of a damping |
| 407 |
|
timescale $T$ for elastic waves \[E=\frac{\zeta}{T}.\] |
| 408 |
|
$T=E_{0}\Delta{t}$ with the tunable parameter $E_0<1$ and the external |
| 409 |
|
(long) timestep $\Delta{t}$. $E_{0} = \frac{1}{3}$ is the default |
| 410 |
|
value in the code and close to what \citet{hun97} and |
| 411 |
|
\citet{hun01} recommend. |
| 412 |
|
|
| 413 |
|
To use the EVP solver, make sure that both \code{SEAICE\_CGRID} and |
| 414 |
|
\code{SEAICE\_ALLOW\_EVP} are defined in \code{SEAICE\_OPTIONS.h} |
| 415 |
|
(default). The solver is turned on by setting the sub-cycling time |
| 416 |
|
step \code{SEAICE\_deltaTevp} to a value larger than zero. The |
| 417 |
|
choice of this time step is under debate. \citet{hun97} recommend |
| 418 |
|
order(120) time steps for the EVP solver within one model time step |
| 419 |
|
$\Delta{t}$ (\code{deltaTmom}). One can also choose order(120) time |
| 420 |
|
steps within the forcing time scale, but then we recommend adjusting |
| 421 |
|
the damping time scale $T$ accordingly, by setting either |
| 422 |
|
\code{SEAICE\_elasticParm} ($E_{0}$), so that |
| 423 |
|
$E_{0}\Delta{t}=\mbox{forcing time scale}$, or directly |
| 424 |
|
\code{SEAICE\_evpTauRelax} ($T$) to the forcing time scale. |
| 425 |
|
|
| 426 |
|
\paragraph{Truncated ellipse method (TEM) for yield curve \label{sec:pkg:seaice:TEM}}~\\ |
| 427 |
|
% |
| 428 |
|
In the so-called truncated ellipse method the shear viscosity $\eta$ |
| 429 |
|
is capped to suppress any tensile stress \citep{hibler97, geiger98}: |
| 430 |
|
\begin{equation} |
| 431 |
|
\label{eq:etatem} |
| 432 |
|
\eta = \min\left(\frac{\zeta}{e^2}, |
| 433 |
|
\frac{\frac{P}{2}-\zeta(\dot{\epsilon}_{11}+\dot{\epsilon}_{22})} |
| 434 |
|
{\sqrt{(\dot{\epsilon}_{11}+\dot{\epsilon}_{22})^2 |
| 435 |
|
+4\dot{\epsilon}_{12}^2}}\right). |
| 436 |
|
\end{equation} |
| 437 |
|
To enable this method, set \code{\#define SEAICE\_ALLOW\_TEM} in |
| 438 |
|
\code{SEAICE\_OPTIONS.h} and turn it on with |
| 439 |
|
\code{SEAICEuseTEM} in \code{data.seaice}. |
| 440 |
|
|
| 441 |
|
\paragraph{Ice-Ocean stress \label{sec:pkg:seaice:iceoceanstress}}~\\ |
| 442 |
|
% |
| 443 |
|
Moving sea ice exerts a stress on the ocean which is the opposite of |
| 444 |
|
the stress $\vtau_{ocean}$ in Eq.~\ref{eq:momseaice}. This stess is |
| 445 |
|
applied directly to the surface layer of the ocean model. An |
| 446 |
|
alternative ocean stress formulation is given by \citet{hibler87}. |
| 447 |
|
Rather than applying $\vtau_{ocean}$ directly, the stress is derived |
| 448 |
|
from integrating over the ice thickness to the bottom of the oceanic |
| 449 |
|
surface layer. In the resulting equation for the \emph{combined} |
| 450 |
|
ocean-ice momentum, the interfacial stress cancels and the total |
| 451 |
|
stress appears as the sum of windstress and divergence of internal ice |
| 452 |
|
stresses: $\delta(z) (\vtau_{air} + \vek{F})/\rho_0$, \citep[see also |
| 453 |
|
Eq.\,2 of][]{hibler87}. The disadvantage of this formulation is that |
| 454 |
|
now the velocity in the surface layer of the ocean that is used to |
| 455 |
|
advect tracers, is really an average over the ocean surface |
| 456 |
|
velocity and the ice velocity leading to an inconsistency as the ice |
| 457 |
|
temperature and salinity are different from the oceanic variables. |
| 458 |
|
To turn on the stress formulation of \citet{hibler87}, set |
| 459 |
|
\code{useHB87StressCoupling=.TRUE.} in \code{data.seaice}. |
| 460 |
|
|
| 461 |
|
|
| 462 |
|
% Our discretization differs from \citet{zhang97, zhang03} in the |
| 463 |
|
% underlying grid, namely the Arakawa C-grid, but is otherwise |
| 464 |
|
% straightforward. The EVP model, in particular, is discretized |
| 465 |
|
% naturally on the C-grid with $\sigma_{1}$ and $\sigma_{2}$ on the |
| 466 |
|
% center points and $\sigma_{12}$ on the corner (or vorticity) points of |
| 467 |
|
% the grid. With this choice all derivatives are discretized as central |
| 468 |
|
% differences and averaging is only involved in computing $\Delta$ and |
| 469 |
|
% $P$ at vorticity points. |
| 470 |
|
|
| 471 |
|
\paragraph{Finite-volume discretization of the stress tensor |
| 472 |
|
divergence\label{sec:pkg:seaice:discretization}}~\\ |
| 473 |
|
% |
| 474 |
|
On an Arakawa C~grid, ice thickness and concentration and thus ice |
| 475 |
|
strength $P$ and bulk and shear viscosities $\zeta$ and $\eta$ are |
| 476 |
|
naturally defined a C-points in the center of the grid |
| 477 |
|
cell. Discretization requires only averaging of $\zeta$ and $\eta$ to |
| 478 |
|
vorticity or Z-points (or $\zeta$-points, but here we use Z in order |
| 479 |
|
avoid confusion with the bulk viscosity) at the bottom left corner of |
| 480 |
|
the cell to give $\overline{\zeta}^{Z}$ and $\overline{\eta}^{Z}$. In |
| 481 |
|
the following, the superscripts indicate location at Z or C points, |
| 482 |
|
distance across the cell (F), along the cell edge (G), between |
| 483 |
|
$u$-points (U), $v$-points (V), and C-points (C). The control volumes |
| 484 |
|
of the $u$- and $v$-equations in the grid cell at indices $(i,j)$ are |
| 485 |
|
$A_{i,j}^{w}$ and $A_{i,j}^{s}$, respectively. With these definitions |
| 486 |
|
(which follow the model code documentation except that $\zeta$-points |
| 487 |
|
have been renamed to Z-points), the strain rates are discretized as: |
| 488 |
|
\begin{align} |
| 489 |
|
\dot{\epsilon}_{11} &= \partial_{1}{u}_{1} + k_{2}u_{2} \\ \notag |
| 490 |
|
=> (\epsilon_{11})_{i,j}^C &= \frac{u_{i+1,j}-u_{i,j}}{\Delta{x}_{i,j}^{F}} |
| 491 |
|
+ k_{2,i,j}^{C}\frac{v_{i,j+1}+v_{i,j}}{2} \\ |
| 492 |
|
\dot{\epsilon}_{22} &= \partial_{2}{u}_{2} + k_{1}u_{1} \\\notag |
| 493 |
|
=> (\epsilon_{22})_{i,j}^C &= \frac{v_{i,j+1}-v_{i,j}}{\Delta{y}_{i,j}^{F}} |
| 494 |
|
+ k_{1,i,j}^{C}\frac{u_{i+1,j}+u_{i,j}}{2} \\ |
| 495 |
|
\dot{\epsilon}_{12} = \dot{\epsilon}_{21} &= \frac{1}{2}\biggl( |
| 496 |
|
\partial_{1}{u}_{2} + \partial_{2}{u}_{1} - k_{1}u_{2} - k_{2}u_{1} |
| 497 |
|
\biggr) \\ \notag |
| 498 |
|
=> (\epsilon_{12})_{i,j}^Z &= \frac{1}{2} |
| 499 |
|
\biggl( \frac{v_{i,j}-v_{i-1,j}}{\Delta{x}_{i,j}^V} |
| 500 |
|
+ \frac{u_{i,j}-u_{i,j-1}}{\Delta{y}_{i,j}^U} \\\notag |
| 501 |
|
&\phantom{=\frac{1}{2}\biggl(} |
| 502 |
|
- k_{1,i,j}^{Z}\frac{v_{i,j}+v_{i-1,j}}{2} |
| 503 |
|
- k_{2,i,j}^{Z}\frac{u_{i,j}+u_{i,j-1}}{2} |
| 504 |
|
\biggr), |
| 505 |
|
\end{align} |
| 506 |
|
so that the diagonal terms of the strain rate tensor are naturally |
| 507 |
|
defined at C-points and the symmetric off-diagonal term at |
| 508 |
|
Z-points. No-slip boundary conditions ($u_{i,j-1}+u_{i,j}=0$ and |
| 509 |
|
$v_{i-1,j}+v_{i,j}=0$ across boundaries) are implemented via |
| 510 |
|
``ghost-points''; for free slip boundary conditions |
| 511 |
|
$(\epsilon_{12})^Z=0$ on boundaries. |
| 512 |
|
|
| 513 |
|
For a spherical polar grid, the coefficients of the metric terms are |
| 514 |
|
$k_{1}=0$ and $k_{2}=-\tan\phi/a$, with the spherical radius $a$ and |
| 515 |
|
the latitude $\phi$; $\Delta{x}_1 = \Delta{x} = a\cos\phi |
| 516 |
|
\Delta\lambda$, and $\Delta{x}_2 = \Delta{y}=a\Delta\phi$. For a |
| 517 |
|
general orthogonal curvilinear grid, $k_{1}$ and |
| 518 |
|
$k_{2}$ can be approximated by finite differences of the cell widths: |
| 519 |
|
\begin{align} |
| 520 |
|
k_{1,i,j}^{C} &= \frac{1}{\Delta{y}_{i,j}^{F}} |
| 521 |
|
\frac{\Delta{y}_{i+1,j}^{G}-\Delta{y}_{i,j}^{G}}{\Delta{x}_{i,j}^{F}} \\ |
| 522 |
|
k_{2,i,j}^{C} &= \frac{1}{\Delta{x}_{i,j}^{F}} |
| 523 |
|
\frac{\Delta{x}_{i,j+1}^{G}-\Delta{x}_{i,j}^{G}}{\Delta{y}_{i,j}^{F}} \\ |
| 524 |
|
k_{1,i,j}^{Z} &= \frac{1}{\Delta{y}_{i,j}^{U}} |
| 525 |
|
\frac{\Delta{y}_{i,j}^{C}-\Delta{y}_{i-1,j}^{C}}{\Delta{x}_{i,j}^{V}} \\ |
| 526 |
|
k_{2,i,j}^{Z} &= \frac{1}{\Delta{x}_{i,j}^{V}} |
| 527 |
|
\frac{\Delta{x}_{i,j}^{C}-\Delta{x}_{i,j-1}^{C}}{\Delta{y}_{i,j}^{U}} |
| 528 |
|
\end{align} |
| 529 |
|
|
| 530 |
|
The stress tensor is given by the constitutive viscous-plastic |
| 531 |
|
relation $\sigma_{\alpha\beta} = 2\eta\dot{\epsilon}_{\alpha\beta} + |
| 532 |
|
[(\zeta-\eta)\dot{\epsilon}_{\gamma\gamma} - P/2 |
| 533 |
|
]\delta_{\alpha\beta}$ \citep{hib79}. The stress tensor divergence |
| 534 |
|
$(\nabla\sigma)_{\alpha} = \partial_\beta\sigma_{\beta\alpha}$, is |
| 535 |
|
discretized in finite volumes. This conveniently avoids dealing with |
| 536 |
|
further metric terms, as these are ``hidden'' in the differential cell |
| 537 |
|
widths. For the $u$-equation ($\alpha=1$) we have: |
| 538 |
|
\begin{align} |
| 539 |
|
(\nabla\sigma)_{1}: \phantom{=}& |
| 540 |
|
\frac{1}{A_{i,j}^w} |
| 541 |
|
\int_{\mathrm{cell}}(\partial_1\sigma_{11}+\partial_2\sigma_{21})\,dx_1\,dx_2 |
| 542 |
|
\\\notag |
| 543 |
|
=& \frac{1}{A_{i,j}^w} \biggl\{ |
| 544 |
|
\int_{x_2}^{x_2+\Delta{x}_2}\sigma_{11}dx_2\biggl|_{x_{1}}^{x_{1}+\Delta{x}_{1}} |
| 545 |
|
+ \int_{x_1}^{x_1+\Delta{x}_1}\sigma_{21}dx_1\biggl|_{x_{2}}^{x_{2}+\Delta{x}_{2}} |
| 546 |
|
\biggr\} \\ \notag |
| 547 |
|
\approx& \frac{1}{A_{i,j}^w} \biggl\{ |
| 548 |
|
\Delta{x}_2\sigma_{11}\biggl|_{x_{1}}^{x_{1}+\Delta{x}_{1}} |
| 549 |
|
+ \Delta{x}_1\sigma_{21}\biggl|_{x_{2}}^{x_{2}+\Delta{x}_{2}} |
| 550 |
|
\biggr\} \\ \notag |
| 551 |
|
=& \frac{1}{A_{i,j}^w} \biggl\{ |
| 552 |
|
(\Delta{x}_2\sigma_{11})_{i,j}^C - |
| 553 |
|
(\Delta{x}_2\sigma_{11})_{i-1,j}^C |
| 554 |
|
\\\notag |
| 555 |
|
\phantom{=}& \phantom{\frac{1}{A_{i,j}^w} \biggl\{} |
| 556 |
|
+ (\Delta{x}_1\sigma_{21})_{i,j+1}^Z - (\Delta{x}_1\sigma_{21})_{i,j}^Z |
| 557 |
|
\biggr\} |
| 558 |
|
\end{align} |
| 559 |
|
with |
| 560 |
|
\begin{align} |
| 561 |
|
(\Delta{x}_2\sigma_{11})_{i,j}^C =& \phantom{+} |
| 562 |
|
\Delta{y}_{i,j}^{F}(\zeta + \eta)^{C}_{i,j} |
| 563 |
|
\frac{u_{i+1,j}-u_{i,j}}{\Delta{x}_{i,j}^{F}} \\ \notag |
| 564 |
|
&+ \Delta{y}_{i,j}^{F}(\zeta + \eta)^{C}_{i,j} |
| 565 |
|
k_{2,i,j}^C \frac{v_{i,j+1}+v_{i,j}}{2} \\ \notag |
| 566 |
|
\phantom{=}& + \Delta{y}_{i,j}^{F}(\zeta - \eta)^{C}_{i,j} |
| 567 |
|
\frac{v_{i,j+1}-v_{i,j}}{\Delta{y}_{i,j}^{F}} \\ \notag |
| 568 |
|
\phantom{=}& + \Delta{y}_{i,j}^{F}(\zeta - \eta)^{C}_{i,j} |
| 569 |
|
k_{1,i,j}^{C}\frac{u_{i+1,j}+u_{i,j}}{2} \\ \notag |
| 570 |
|
\phantom{=}& - \Delta{y}_{i,j}^{F} \frac{P}{2} \\ |
| 571 |
|
(\Delta{x}_1\sigma_{21})_{i,j}^Z =& \phantom{+} |
| 572 |
|
\Delta{x}_{i,j}^{V}\overline{\eta}^{Z}_{i,j} |
| 573 |
|
\frac{u_{i,j}-u_{i,j-1}}{\Delta{y}_{i,j}^{U}} \\ \notag |
| 574 |
|
& + \Delta{x}_{i,j}^{V}\overline{\eta}^{Z}_{i,j} |
| 575 |
|
\frac{v_{i,j}-v_{i-1,j}}{\Delta{x}_{i,j}^{V}} \\ \notag |
| 576 |
|
& - \Delta{x}_{i,j}^{V}\overline{\eta}^{Z}_{i,j} |
| 577 |
|
k_{2,i,j}^{Z}\frac{u_{i,j}+u_{i,j-1}}{2} \\ \notag |
| 578 |
|
& - \Delta{x}_{i,j}^{V}\overline{\eta}^{Z}_{i,j} |
| 579 |
|
k_{1,i,j}^{Z}\frac{v_{i,j}+v_{i-1,j}}{2} |
| 580 |
|
\end{align} |
| 581 |
|
|
| 582 |
|
Similarly, we have for the $v$-equation ($\alpha=2$): |
| 583 |
|
\begin{align} |
| 584 |
|
(\nabla\sigma)_{2}: \phantom{=}& |
| 585 |
|
\frac{1}{A_{i,j}^s} |
| 586 |
|
\int_{\mathrm{cell}}(\partial_1\sigma_{12}+\partial_2\sigma_{22})\,dx_1\,dx_2 |
| 587 |
|
\\\notag |
| 588 |
|
=& \frac{1}{A_{i,j}^s} \biggl\{ |
| 589 |
|
\int_{x_2}^{x_2+\Delta{x}_2}\sigma_{12}dx_2\biggl|_{x_{1}}^{x_{1}+\Delta{x}_{1}} |
| 590 |
|
+ \int_{x_1}^{x_1+\Delta{x}_1}\sigma_{22}dx_1\biggl|_{x_{2}}^{x_{2}+\Delta{x}_{2}} |
| 591 |
|
\biggr\} \\ \notag |
| 592 |
|
\approx& \frac{1}{A_{i,j}^s} \biggl\{ |
| 593 |
|
\Delta{x}_2\sigma_{12}\biggl|_{x_{1}}^{x_{1}+\Delta{x}_{1}} |
| 594 |
|
+ \Delta{x}_1\sigma_{22}\biggl|_{x_{2}}^{x_{2}+\Delta{x}_{2}} |
| 595 |
|
\biggr\} \\ \notag |
| 596 |
|
=& \frac{1}{A_{i,j}^s} \biggl\{ |
| 597 |
|
(\Delta{x}_2\sigma_{12})_{i+1,j}^Z - (\Delta{x}_2\sigma_{12})_{i,j}^Z |
| 598 |
|
\\ \notag |
| 599 |
|
\phantom{=}& \phantom{\frac{1}{A_{i,j}^s} \biggl\{} |
| 600 |
|
+ (\Delta{x}_1\sigma_{22})_{i,j}^C - (\Delta{x}_1\sigma_{22})_{i,j-1}^C |
| 601 |
|
\biggr\} |
| 602 |
|
\end{align} |
| 603 |
|
with |
| 604 |
|
\begin{align} |
| 605 |
|
(\Delta{x}_1\sigma_{12})_{i,j}^Z =& \phantom{+} |
| 606 |
|
\Delta{y}_{i,j}^{U}\overline{\eta}^{Z}_{i,j} |
| 607 |
|
\frac{u_{i,j}-u_{i,j-1}}{\Delta{y}_{i,j}^{U}} |
| 608 |
|
\\\notag & |
| 609 |
|
+ \Delta{y}_{i,j}^{U}\overline{\eta}^{Z}_{i,j} |
| 610 |
|
\frac{v_{i,j}-v_{i-1,j}}{\Delta{x}_{i,j}^{V}} \\\notag |
| 611 |
|
&- \Delta{y}_{i,j}^{U}\overline{\eta}^{Z}_{i,j} |
| 612 |
|
k_{2,i,j}^{Z}\frac{u_{i,j}+u_{i,j-1}}{2} |
| 613 |
|
\\\notag & |
| 614 |
|
- \Delta{y}_{i,j}^{U}\overline{\eta}^{Z}_{i,j} |
| 615 |
|
k_{1,i,j}^{Z}\frac{v_{i,j}+v_{i-1,j}}{2} \\ \notag |
| 616 |
|
(\Delta{x}_2\sigma_{22})_{i,j}^C =& \phantom{+} |
| 617 |
|
\Delta{x}_{i,j}^{F}(\zeta - \eta)^{C}_{i,j} |
| 618 |
|
\frac{u_{i+1,j}-u_{i,j}}{\Delta{x}_{i,j}^{F}} \\ \notag |
| 619 |
|
&+ \Delta{x}_{i,j}^{F}(\zeta - \eta)^{C}_{i,j} |
| 620 |
|
k_{2,i,j}^{C} \frac{v_{i,j+1}+v_{i,j}}{2} \\ \notag |
| 621 |
|
& + \Delta{x}_{i,j}^{F}(\zeta + \eta)^{C}_{i,j} |
| 622 |
|
\frac{v_{i,j+1}-v_{i,j}}{\Delta{y}_{i,j}^{F}} \\ \notag |
| 623 |
|
& + \Delta{x}_{i,j}^{F}(\zeta + \eta)^{C}_{i,j} |
| 624 |
|
k_{1,i,j}^{C}\frac{u_{i+1,j}+u_{i,j}}{2} \\ \notag |
| 625 |
|
& -\Delta{x}_{i,j}^{F} \frac{P}{2} |
| 626 |
|
\end{align} |
| 627 |
|
|
| 628 |
|
Again, no slip boundary conditions are realized via ghost points and |
| 629 |
|
$u_{i,j-1}+u_{i,j}=0$ and $v_{i-1,j}+v_{i,j}=0$ across boundaries. For |
| 630 |
|
free slip boundary conditions the lateral stress is set to zeros. In |
| 631 |
|
analogy to $(\epsilon_{12})^Z=0$ on boundaries, we set |
| 632 |
|
$\sigma_{21}^{Z}=0$, or equivalently $\eta_{i,j}^{Z}=0$, on boundaries. |
| 633 |
|
|
| 634 |
|
\paragraph{Thermodynamics\label{sec:pkg:seaice:thermodynamics}}~\\ |
| 635 |
|
% |
| 636 |
|
In its original formulation the sea ice model \citep{menemenlis05} |
| 637 |
|
uses simple thermodynamics following the appendix of |
| 638 |
|
\citet{sem76}. This formulation does not allow storage of heat, |
| 639 |
|
that is, the heat capacity of ice is zero. Upward conductive heat flux |
| 640 |
|
is parameterized assuming a linear temperature profile and together |
| 641 |
|
with a constant ice conductivity. It is expressed as |
| 642 |
|
$(K/h)(T_{w}-T_{0})$, where $K$ is the ice conductivity, $h$ the ice |
| 643 |
|
thickness, and $T_{w}-T_{0}$ the difference between water and ice |
| 644 |
|
surface temperatures. This type of model is often refered to as a |
| 645 |
|
``zero-layer'' model. The surface heat flux is computed in a similar |
| 646 |
|
way to that of \citet{parkinson79} and \citet{manabe79}. |
| 647 |
|
|
| 648 |
|
The conductive heat flux depends strongly on the ice thickness $h$. |
| 649 |
|
However, the ice thickness in the model represents a mean over a |
| 650 |
|
potentially very heterogeneous thickness distribution. In order to |
| 651 |
|
parameterize a sub-grid scale distribution for heat flux |
| 652 |
|
computations, the mean ice thickness $h$ is split into seven thickness |
| 653 |
|
categories $H_{n}$ that are equally distributed between $2h$ and a |
| 654 |
|
minimum imposed ice thickness of $5\text{\,cm}$ by $H_n= |
| 655 |
|
\frac{2n-1}{7}\,h$ for $n\in[1,7]$. The heat fluxes computed for each |
| 656 |
|
thickness category is area-averaged to give the total heat flux |
| 657 |
|
\citep{hibler84}. To use this thickness category parameterization set |
| 658 |
|
\code{\#define SEAICE\_MULTICATEGORY}; note that this requires |
| 659 |
|
different restart files and switching this flag on in the middle of an |
| 660 |
|
integration is not possible. |
| 661 |
|
|
| 662 |
|
The atmospheric heat flux is balanced by an oceanic heat flux from |
| 663 |
|
below. The oceanic flux is proportional to |
| 664 |
|
$\rho\,c_{p}\left(T_{w}-T_{fr}\right)$ where $\rho$ and $c_{p}$ are |
| 665 |
|
the density and heat capacity of sea water and $T_{fr}$ is the local |
| 666 |
|
freezing point temperature that is a function of salinity. This flux |
| 667 |
|
is not assumed to instantaneously melt or create ice, but a time scale |
| 668 |
|
of three days (run-time parameter \code{SEAICE\_gamma\_t}) is used |
| 669 |
|
to relax $T_{w}$ to the freezing point. |
| 670 |
|
% |
| 671 |
|
The parameterization of lateral and vertical growth of sea ice follows |
| 672 |
|
that of \citet{hib79, hib80}; the so-called lead closing parameter |
| 673 |
|
$h_{0}$ (run-time parameter \code{HO}) has a default value of |
| 674 |
|
0.5~meters. |
| 675 |
|
|
| 676 |
|
On top of the ice there is a layer of snow that modifies the heat flux |
| 677 |
|
and the albedo \citep{zha98a}. Snow modifies the effective |
| 678 |
|
conductivity according to |
| 679 |
|
\[\frac{K}{h} \rightarrow \frac{1}{\frac{h_{s}}{K_{s}}+\frac{h}{K}},\] |
| 680 |
|
where $K_s$ is the conductivity of snow and $h_s$ the snow thickness. |
| 681 |
|
If enough snow accumulates so that its weight submerges the ice and |
| 682 |
|
the snow is flooded, a simple mass conserving parameterization of |
| 683 |
|
snowice formation (a flood-freeze algorithm following Archimedes' |
| 684 |
|
principle) turns snow into ice until the ice surface is back at $z=0$ |
| 685 |
|
\citep{leppaeranta83}. The flood-freeze algorithm is enabled with the CPP-flag |
| 686 |
|
\code{SEAICE\_ALLOW\_FLOODING} and turned on with run-time parameter |
| 687 |
|
\code{SEAICEuseFlooding=.true.}. |
| 688 |
|
|
| 689 |
|
\paragraph{Advection of thermodynamic variables\label{sec:pkg:seaice:advection}}~\\ |
| 690 |
|
% |
| 691 |
|
Effective ice thickness (ice volume per unit area, |
| 692 |
|
$c\cdot{h}$), concentration $c$ and effective snow thickness |
| 693 |
|
($c\cdot{h}_{s}$) are advected by ice velocities: |
| 694 |
|
\begin{equation} |
| 695 |
|
\label{eq:advection} |
| 696 |
|
\frac{\partial{X}}{\partial{t}} = - \nabla\cdot\left(\vek{u}\,X\right) + |
| 697 |
|
\Gamma_{X} + D_{X} |
| 698 |
|
\end{equation} |
| 699 |
|
where $\Gamma_X$ are the thermodynamic source terms and $D_{X}$ the |
| 700 |
|
diffusive terms for quantities $X=(c\cdot{h}), c, (c\cdot{h}_{s})$. |
| 701 |
|
% |
| 702 |
|
From the various advection scheme that are available in the MITgcm, we |
| 703 |
|
recommend flux-limited schemes \citep[multidimensional 2nd and |
| 704 |
|
3rd-order advection scheme with flux limiter][]{roe:85, hundsdorfer94} |
| 705 |
|
to preserve sharp gradients and edges that are typical of sea ice |
| 706 |
|
distributions and to rule out unphysical over- and undershoots |
| 707 |
|
(negative thickness or concentration). These schemes conserve volume |
| 708 |
|
and horizontal area and are unconditionally stable, so that we can set |
| 709 |
|
$D_{X}=0$. Run-timeflags: \code{SEAICEadvScheme} (default=2, is the |
| 710 |
|
historic 2nd-order, centered difference scheme), \code{DIFF1} = |
| 711 |
|
$D_{X}/\Delta{x}$ |
| 712 |
|
(default=0.004). |
| 713 |
|
|
| 714 |
|
The MITgcm sea ice model provides the option to use |
| 715 |
|
the thermodynamics model of \citet{win00}, which in turn is based on |
| 716 |
|
the 3-layer model of \citet{sem76} and which treats brine content by |
| 717 |
|
means of enthalpy conservation; the corresponding package |
| 718 |
|
\code{thsice} is described in section~\ref{sec:pkg:thsice}. This |
| 719 |
|
scheme requires additional state variables, namely the enthalpy of the |
| 720 |
|
two ice layers (instead of effective ice salinity), to be advected by |
| 721 |
|
ice velocities. |
| 722 |
|
% |
| 723 |
|
The internal sea ice temperature is inferred from ice enthalpy. To |
| 724 |
|
avoid unphysical (negative) values for ice thickness and |
| 725 |
|
concentration, a positive 2nd-order advection scheme with a SuperBee |
| 726 |
|
flux limiter \citep{roe:85} should be used to advect all |
| 727 |
|
sea-ice-related quantities of the \citet{win00} thermodynamic model |
| 728 |
|
(runtime flag \code{thSIceAdvScheme=77} and |
| 729 |
|
\code{thSIce\_diffK}=$D_{X}$=0 in \code{data.ice}, defaults are 0). Because of the |
| 730 |
|
non-linearity of the advection scheme, care must be taken in advecting |
| 731 |
|
these quantities: when simply using ice velocity to advect enthalpy, |
| 732 |
|
the total energy (i.e., the volume integral of enthalpy) is not |
| 733 |
|
conserved. Alternatively, one can advect the energy content (i.e., |
| 734 |
|
product of ice-volume and enthalpy) but then false enthalpy extrema |
| 735 |
|
can occur, which then leads to unrealistic ice temperature. In the |
| 736 |
|
currently implemented solution, the sea-ice mass flux is used to |
| 737 |
|
advect the enthalpy in order to ensure conservation of enthalpy and to |
| 738 |
|
prevent false enthalpy extrema. % |
| 739 |
|
|
| 740 |
%---------------------------------------------------------------------- |
%---------------------------------------------------------------------- |
| 741 |
|
|
| 742 |
\subsubsection{Key subroutines |
\subsubsection{Key subroutines |
| 743 |
\label{sec:pkg:seaice:subroutines}} |
\label{sec:pkg:seaice:subroutines}} |
| 744 |
|
|
| 745 |
Top-level routine: \texttt{exf\_getforcing.F} |
Top-level routine: \code{seaice\_model.F} |
| 746 |
|
|
| 747 |
{\footnotesize |
{\footnotesize |
| 748 |
\begin{verbatim} |
\begin{verbatim} |
| 793 |
|
|
| 794 |
%---------------------------------------------------------------------- |
%---------------------------------------------------------------------- |
| 795 |
|
|
| 796 |
\subsubsection{EXF diagnostics |
\subsubsection{SEAICE diagnostics |
| 797 |
\label{sec:pkg:seaice:diagnostics}} |
\label{sec:pkg:seaice:diagnostics}} |
| 798 |
|
|
| 799 |
Diagnostics output is available via the diagnostics package |
Diagnostics output is available via the diagnostics package |
| 801 |
Available output fields are summarized in |
Available output fields are summarized in |
| 802 |
Table \ref{tab:pkg:seaice:diagnostics}. |
Table \ref{tab:pkg:seaice:diagnostics}. |
| 803 |
|
|
| 804 |
\begin{table}[h!] |
\begin{table}[!ht] |
| 805 |
\centering |
\centering |
| 806 |
\label{tab:pkg:seaice:diagnostics} |
\label{tab:pkg:seaice:diagnostics} |
| 807 |
{\footnotesize |
{\footnotesize |
| 815 |
SIvice | 1 |VV |m/s |SEAICE merid. ice velocity, >0 from South to North |
SIvice | 1 |VV |m/s |SEAICE merid. ice velocity, >0 from South to North |
| 816 |
SIhsnow | 1 |SM |m |SEAICE snow thickness |
SIhsnow | 1 |SM |m |SEAICE snow thickness |
| 817 |
SIhsalt | 1 |SM |g/m^2 |SEAICE effective salinity |
SIhsalt | 1 |SM |g/m^2 |SEAICE effective salinity |
| 818 |
SIatmFW | 1 |SM |m/s |Net freshwater flux from the atmosphere (+=down) |
SIatmFW | 1 |SM |kg/m^2/s |Net freshwater flux from the atmosphere (+=down) |
| 819 |
SIuwind | 1 |SM |m/s |SEAICE zonal 10-m wind speed, >0 increases uVel |
SIuwind | 1 |SM |m/s |SEAICE zonal 10-m wind speed, >0 increases uVel |
| 820 |
SIvwind | 1 |SM |m/s |SEAICE meridional 10-m wind speed, >0 increases uVel |
SIvwind | 1 |SM |m/s |SEAICE meridional 10-m wind speed, >0 increases uVel |
| 821 |
SIfu | 1 |UU |N/m^2 |SEAICE zonal surface wind stress, >0 increases uVel |
SIfu | 1 |UU |N/m^2 |SEAICE zonal surface wind stress, >0 increases uVel |
| 822 |
SIfv | 1 |VV |N/m^2 |SEAICE merid. surface wind stress, >0 increases vVel |
SIfv | 1 |VV |N/m^2 |SEAICE merid. surface wind stress, >0 increases vVel |
| 823 |
SIempmr | 1 |SM |m/s |SEAICE upward freshwater flux, > 0 increases salt |
SIempmr | 1 |SM |kg/m^2/s |SEAICE upward freshwater flux, > 0 increases salt |
| 824 |
SIqnet | 1 |SM |W/m^2 |SEAICE upward heatflux, turb+rad, >0 decreases theta |
SIqnet | 1 |SM |W/m^2 |SEAICE upward heatflux, turb+rad, >0 decreases theta |
| 825 |
SIqsw | 1 |SM |W/m^2 |SEAICE upward shortwave radiat., >0 decreases theta |
SIqsw | 1 |SM |W/m^2 |SEAICE upward shortwave radiat., >0 decreases theta |
| 826 |
SIpress | 1 |SM |m^2/s^2 |SEAICE strength (with upper and lower limit) |
SIpress | 1 |SM |m^2/s^2 |SEAICE strength (with upper and lower limit) |
| 850 |
DFyESSLT| 1 |VV |psu.m^2/s |Meridional Diffusive Flux of seaice salinity |
DFyESSLT| 1 |VV |psu.m^2/s |Meridional Diffusive Flux of seaice salinity |
| 851 |
\end{verbatim} |
\end{verbatim} |
| 852 |
} |
} |
| 853 |
\caption{~} |
\caption{Available diagnostics of the seaice-package} |
| 854 |
\end{table} |
\end{table} |
| 855 |
|
|
| 856 |
|
|
| 860 |
\label{sec:pkg:seaice:experiments} |
\label{sec:pkg:seaice:experiments} |
| 861 |
|
|
| 862 |
\begin{itemize} |
\begin{itemize} |
| 863 |
\item{Labrador Sea experiment in lab\_sea verification directory. } |
\item{Labrador Sea experiment in \code{lab\_sea} verification directory. } |
| 864 |
|
\item \code{seaice\_obcs}, based on \code{lab\_sea} |
| 865 |
|
\item \code{offline\_exf\_seaice/input.seaicetd}, based on \code{lab\_sea} |
| 866 |
|
\item \code{global\_ocean.cs32x15/input.icedyn} and |
| 867 |
|
\code{global\_ocean.cs32x15/input.seaice}, global |
| 868 |
|
cubed-sphere-experiment with combinations of \code{seaice} and |
| 869 |
|
\code{thsice} |
| 870 |
\end{itemize} |
\end{itemize} |
| 871 |
|
|
| 872 |
|
|
| 873 |
|
%%% Local Variables: |
| 874 |
|
%%% mode: latex |
| 875 |
|
%%% TeX-master: "../../manual" |
| 876 |
|
%%% End: |
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