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revision 1.1 by dimitri, Tue Feb 26 19:27:26 2008 UTC revision 1.7 by mlosch, Thu Mar 6 21:54:55 2008 UTC
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1  \section{Model}  \section{Model Formulation}
2  \label{sec:model}  \label{sec:model}
3    
4    The MITgcm sea ice model (MITsim) is based on a variant of the
5    viscous-plastic (VP) dynamic-thermodynamic sea ice model
6    \citep{zhang97} first introduced by \citet{hibler79, hibler80}. In
7    order to adapt this model to the requirements of coupled
8    ice-ocean simulations, many important aspects of the original code have
9    been modified and improved:
10    \begin{itemize}
11    \item the code has been rewritten for an Arakawa C-grid, both B- and
12      C-grid variants are available; the C-grid code allows for no-slip
13      and free-slip lateral boundary conditions;
14    \item two different solution methods for solving the nonlinear
15      momentum equations have been adopted: LSOR \citep{zhang97}, EVP
16      \citep{hunke97};
17    \item ice-ocean stress can be formulated as in \citet{hibler87};
18    \item ice variables are advected by sophisticated advection schemes;
19    \item growth and melt parameterizaion have been refined and extended
20      in order to allow for automatic differentiation of the code.
21    \end{itemize}
22    The model equations and their numerical realization are summarized
23    below.
24    
25  \subsection{Dynamics}  \subsection{Dynamics}
26  \label{sec:dynamics}  \label{sec:dynamics}
27    
# Line 43  wind/current vectors. Line 64  wind/current vectors.
64    
65  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
66  be related to the ice strain rate and strength by a nonlinear  be related to the ice strain rate and strength by a nonlinear
67  viscous-plastic (VP) constitutive law \citep{hibler79, zhang98}:  viscous-plastic (VP) constitutive law \citep{hibler79, zhang97}:
68  \begin{equation}  \begin{equation}
69    \label{eq:vpequation}    \label{eq:vpequation}
70    \sigma_{ij}=2\eta(\dot{\epsilon}_{ij},P)\dot{\epsilon}_{ij}    \sigma_{ij}=2\eta(\dot{\epsilon}_{ij},P)\dot{\epsilon}_{ij}
# Line 77  minor axis $e$ equal to $2$; they are gi Line 98  minor axis $e$ equal to $2$; they are gi
98      \left(\dot{\epsilon}_{11}^2+\dot{\epsilon}_{22}^2\right)      \left(\dot{\epsilon}_{11}^2+\dot{\epsilon}_{22}^2\right)
99      (1+e^{-2}) +  4e^{-2}\dot{\epsilon}_{12}^2 +      (1+e^{-2}) +  4e^{-2}\dot{\epsilon}_{12}^2 +
100      2\dot{\epsilon}_{11}\dot{\epsilon}_{22} (1-e^{-2})      2\dot{\epsilon}_{11}\dot{\epsilon}_{22} (1-e^{-2})
101    \right]^{-\frac{1}{2}}    \right]^{\frac{1}{2}}
102  \end{align*}  \end{align*}
103  The bulk viscosities are bounded above by imposing both a minimum  The bulk viscosities are bounded above by imposing both a minimum
104  $\Delta_{\min}=10^{-11}\text{\,s}^{-1}$ (for numerical reasons) and a  $\Delta_{\min}=10^{-11}\text{\,s}^{-1}$ (for numerical reasons) and a
# Line 205  advect tracers, is really an average ove Line 226  advect tracers, is really an average ove
226  velocity and the ice velocity leading to an inconsistency as the ice  velocity and the ice velocity leading to an inconsistency as the ice
227  temperature and salinity are different from the oceanic variables.  temperature and salinity are different from the oceanic variables.
228    
229  Sea ice distributions are characterized by sharp gradients and edges.  %\subparagraph{boundary conditions: no-slip, free-slip, half-slip}
230  For this reason, we employ positive, multidimensional 2nd and 3rd-order  %\begin{itemize}
231  advection scheme with flux limiter \citep{roe85, hundsdorfer94} for the  %\item transition from existing B-Grid to C-Grid
232  thermodynamic variables discussed in the next section.  %\item boundary conditions: no-slip, free-slip, half-slip
233    %\item fancy (multi dimensional) advection schemes
234  \subparagraph{boundary conditions: no-slip, free-slip, half-slip}  %\item VP vs.\ EVP \citep{hunke97}
235    %\item ocean stress formulation \citep{hibler87}
236  \begin{itemize}  %\end{itemize}
 \item transition from existing B-Grid to C-Grid  
 \item boundary conditions: no-slip, free-slip, half-slip  
 \item fancy (multi dimensional) advection schemes  
 \item VP vs.\ EVP \citep{hunke97}  
 \item ocean stress formulation \citep{hibler87}  
 \end{itemize}  
237    
238  \subsection{Thermodynamics}  \subsection{Thermodynamics}
239  \label{sec:thermodynamics}  \label{sec:thermodynamics}
# Line 227  In the original formulation the sea ice Line 242  In the original formulation the sea ice
242  uses simple thermodynamics following the appendix of  uses simple thermodynamics following the appendix of
243  \citet{semtner76}. This formulation does not allow storage of heat  \citet{semtner76}. This formulation does not allow storage of heat
244  (heat capacity of ice is zero, and this type of model is often refered  (heat capacity of ice is zero, and this type of model is often refered
245  to as a ``zero-layer'' model). Upward heat flux is parameterized  to as a ``zero-layer'' model). Upward conductive heat flux is parameterized
246  assuming a linear temperature profile and together with a constant ice  assuming a linear temperature profile and together with a constant ice
247  conductivity. It is expressed as $(K/h)(T_{w}-T_{0})$, where $K$ is  conductivity. It is expressed as $(K/h)(T_{w}-T_{0})$, where $K$ is
248  the ice conductivity, $h$ the ice thickness, and $T_{w}-T_{0}$ the  the ice conductivity, $h$ the ice thickness, and $T_{w}-T_{0}$ the
249  difference between water and ice surface temperatures. The surface  difference between water and ice surface temperatures. The surface
250  heat budget is computed in a similar way to that of  heat flux is computed in a similar way to that of \citet{parkinson79}
251  \citet{parkinson79} and \citet{manabe79}.  and \citet{manabe79}.
252    
253    The conductive heat flux depends strongly on the ice thickness $h$.
254    However, the ice thickness in the model represents a mean over a
255    potentially very heterogeneous thickness distribution.  In order to
256    parameterize this sub-grid scale distribution for heat flux
257    computations, the mean ice thickness $h$ is split into seven thickness
258    categories $H_{n}$ that are equally distributed between $2h$ and
259    minimum imposed ice thickness of $5\text{\,cm}$ by $H_n=
260    \frac{2n-1}{7}\,h$ for $n\in[1,7]$. The heat fluxes computed for each
261    thickness category area averaged to give the total heat flux. \ml{[I
262      don't have citation for that, anyone?]}
263    
264    The atmospheric heat flux is balanced by an oceanic heat flux from
265    below.  The oceanic flux is proportional to
266    $\rho\,c_{p}\left(T_{w}-T_{fr}\right)$ where $\rho$ and $c_{p}$ are
267    the density and heat capacity of sea water and $T_{fr}$ is the local
268    freezing point temperature that is a function of salinity. Contrary to
269    \citet{menemenlis05}, this flux is not assumed to instantaneously melt
270    or create ice, but a time scale of three days is used to relax $T_{w}$
271    to the freezing point.
272    
273    The parameterization of lateral and vertical growth of sea ice follows
274    that of \citet{hibler79, hibler80}.
275    
276    On top of the ice there is a layer of snow that modifies the heat flux
277    and the albedo \citep{zhang98}. If enough snow accumulates so that its
278    weight submerges the ice and the snow is flooded, a simple mass
279    conserving parameterization of snowice formation (a flood-freeze
280    algorithm following Archimedes' principle) turns snow into ice until
281    the ice surface is back at $z=0$ \citep{leppaeranta83}.
282    
283    Effective ich thickness (ice volume per unit area,
284    $c\cdot{h}$), concentration $c$ and effective snow thickness
285    ($c\cdot{h}_{s}$) are advected by ice velocities:
286    \begin{equation}
287      \label{eq:advection}
288      \frac{\partial{X}}{\partial{t}} = - \nabla\cdot\left(\vek{u}\,X\right) +
289      \Gamma_{X} + D_{X}
290    \end{equation}
291    where $\Gamma_X$ are the thermodynamic source terms and $D_{X}$ the
292    diffusive terms for quantities $X=(c\cdot{h}), c, (c\cdot{h}_{s})$.
293    %
294    From the various advection scheme that are available in the MITgcm
295    \citep{mitgcm02}, we choose flux-limited schemes
296    \citep[multidimensional 2nd and 3rd-order advection scheme with flux
297    limiter][]{roe85, hundsdorfer94} to preserve sharp gradients and edges
298    that are typical of sea ice distributions and to rule out unphysical
299    over- and undershoots (negative thickness or concentration). These
300    scheme conserve volume and horizontal area and are unconditionally
301    stable, so that we can set $D_{X}=0$.  \ml{[do we need to proove that?
302      can we proove that? citation?]}
303    
304  There is considerable doubt about the reliability of such a simple  There is considerable doubt about the reliability of such a simple
305  thermodynamic model---\citet{semtner84} found significant errors in  thermodynamic model---\citet{semtner84} found significant errors in
# Line 282  Do a little study on the parameters of L Line 348  Do a little study on the parameters of L
348    solutions, that is your dtevp = 1sec solutions/or 10sec solutions    solutions, that is your dtevp = 1sec solutions/or 10sec solutions
349    with SEAICE\_evpDampC = 615.    with SEAICE\_evpDampC = 615.
350  \end{enumerate}  \end{enumerate}
351    
352  \end{itemize}  \end{itemize}
353    
354    %%% Local Variables:
355    %%% mode: latex
356    %%% TeX-master: "ceaice"
357    %%% End:
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