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+revision 1.6 by mlosch,
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 Line 1
- \section{Model}
+ \section{Model Formulation}
  \label{sec:model}
+ The MITgcm sea ice model (MITsim) is based on a variant of the
+ viscous-plastic (VP) dynamic-thermodynamic sea ice model
+ \citep{zhang97} first introduced by \citet{hibler79, hibler80}. In
+ order to adapt this model to the requirements of coupled
+ ice-ocean simulations, many important aspects of the original code have
+ been modified and improved:
+ \begin{itemize}
+ \item the code has been rewritten for an Arakawa C-grid, both B- and
+   C-grid variants are available; the C-grid code allows for no-slip
+   and free-slip lateral boundary conditions;
+ \item two different solution methods for solving the nonlinear
+   momentum equations have been adopted: LSOR \citep{zhang97}, EVP
+   \citep{hunke97};
+ \item ice-ocean stress can be formulated as in \citet{hibler87};
+ \item ice variables are advected by sophisticated advection schemes;
+ \item growth and melt parameterizaion have been refined and extended
+   in order to allow for automatic differentiation of the code.
+ \end{itemize}
+ The model equations and their numerical realization are summarized
+ below.
  \subsection{Dynamics}
  \label{sec:dynamics}
-Line 43 
 wind/current vectors.
+Line 64 
 wind/current vectors.
  For an isotropic system the stress tensor $\sigma_{ij}$ ($i,j=1,2$) can
  be related to the ice strain rate and strength by a nonlinear
- viscous-plastic (VP) constitutive law \citep{hibler79, zhang98}:
+ viscous-plastic (VP) constitutive law \citep{hibler79, zhang97}:
  \begin{equation}
    \label{eq:vpequation}
    \sigma_{ij}=2\eta(\dot{\epsilon}_{ij},P)\dot{\epsilon}_{ij}
-Line 205 
 advect tracers, is really an average ove
+Line 226 
 advect tracers, is really an average ove
  velocity and the ice velocity leading to an inconsistency as the ice
  temperature and salinity are different from the oceanic variables.
- Sea ice distributions are characterized by sharp gradients and edges.
+ %\subparagraph{boundary conditions: no-slip, free-slip, half-slip}
- For this reason, we employ positive, multidimensional 2nd and 3rd-order
+ %\begin{itemize}
- advection scheme with flux limiter \citep{roe85, hundsdorfer94} for the
+ %\item transition from existing B-Grid to C-Grid
- thermodynamic variables discussed in the next section.
+ %\item boundary conditions: no-slip, free-slip, half-slip
+ %\item fancy (multi dimensional) advection schemes
- \subparagraph{boundary conditions: no-slip, free-slip, half-slip}
+ %\item VP vs.\ EVP \citep{hunke97}
+ %\item ocean stress formulation \citep{hibler87}
- \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}
  \subsection{Thermodynamics}
  \label{sec:thermodynamics}
-Line 227 
 In the original formulation the sea ice
+Line 242 
 In the original formulation the sea ice
  uses simple thermodynamics following the appendix of
  \citet{semtner76}. This formulation does not allow storage of heat
  (heat capacity of ice is zero, and this type of model is often refered
- to as a ``zero-layer'' model). Upward heat flux is parameterized
+ to as a ``zero-layer'' model). Upward conductive heat flux is parameterized
  assuming a linear temperature profile and together with a constant ice
  conductivity. It is expressed as $(K/h)(T_{w}-T_{0})$, where $K$ is
  the ice conductivity, $h$ the ice thickness, and $T_{w}-T_{0}$ the
  difference between water and ice surface temperatures. The surface
- heat budget is computed in a similar way to that of
+ heat flux is computed in a similar way to that of \citet{parkinson79}
- \citet{parkinson79} and \citet{manabe79}.
+ and \citet{manabe79}.
+ The conductive heat flux depends strongly on the ice thickness $h$.
+ However, the ice thickness in the model represents a mean over a
+ potentially very heterogeneous thickness distribution.  In order to
+ parameterize this sub-grid scale distribution for heat flux
+ computations, the mean ice thickness $h$ is split into seven thickness
+ categories $H_{n}$ that are equally distributed between $2h$ and
+ minimum imposed ice thickness of $5\text{\,cm}$ by $H_n=
+ \frac{2n-1}{7}\,h$ for $n\in[1,7]$. The heat fluxes computed for each
+ thickness category area averaged to give the total heat flux. \ml{[I
+   don't have citation for that, anyone?]}
+ The atmospheric heat flux is balanced by an oceanic heat flux from
+ below.  The oceanic flux is proportional to
+ $\rho\,c_{p}\left(T_{w}-T_{fr}\right)$ where $\rho$ and $c_{p}$ are
+ the density and heat capacity of sea water and $T_{fr}$ is the local
+ freezing point temperature that is a function of salinity. Contrary to
+ \citet{menemenlis05}, this flux is not assumed to instantaneously melt
+ or create ice, but a time scale of three days is used to relax $T_{w}$
+ to the freezing point.
+ The parameterization of lateral and vertical growth of sea ice follows
+ that of \citet{hibler79, hibler80}.
+ On top of the ice there is a layer of snow that modifies the heat flux
+ and the albedo \citep{zhang98}. If enough snow accumulates so that its
+ weight submerges the ice and the snow is flooded, a simple mass
+ conserving parameterization of snowice formation (a flood-freeze
+ algorithm following Archimedes' principle) turns snow into ice until
+ the ice surface is back at $z=0$ \citep{leppaeranta83}.
+ Effective ich thickness (ice volume per unit area,
+ $c\cdot{h}$), concentration $c$ and effective snow thickness
+ ($c\cdot{h}_{s}$) are advected by ice velocities:
+ \begin{equation}
+   \label{eq:advection}
+   \frac{\partial{X}}{\partial{t}} = - \nabla\cdot\left(\vek{u}\,X\right) +
+   \Gamma_{X} + D_{X}
+ \end{equation}
+ where $\Gamma_X$ are the thermodynamic source terms and $D_{X}$ the
+ diffusive terms for quantities $X=(c\cdot{h}), c, (c\cdot{h}_{s})$.
+ %
+ From the various advection scheme that are available in the MITgcm
+ \citep{mitgcm02}, we choose flux-limited schemes
+ \citep[multidimensional 2nd and 3rd-order advection scheme with flux
+ limiter][]{roe85, hundsdorfer94} to preserve sharp gradients and edges
+ that are typical of sea ice distributions and to rule out unphysical
+ over- and undershoots (negative thickness or concentration). These
+ scheme conserve volume and horizontal area and are unconditionally
+ stable, so that we can set $D_{X}=0$.  \ml{[do we need to proove that?
+   can we proove that? citation?]}
  There is considerable doubt about the reliability of such a simple
  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
    solutions, that is your dtevp = 1sec solutions/or 10sec solutions
    with SEAICE\_evpDampC = 615.
  \end{enumerate}
  \end{itemize}
+ %%% Local Variables:
+ %%% mode: latex
+ %%% TeX-master: "ceaice"
+ %%% End:

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