--- MITgcm_contrib/articles/ceaice/ceaice_model.tex	2008/03/06 21:54:55	1.7
+++ MITgcm_contrib/articles/ceaice/ceaice_model.tex	2008/08/15 14:16:35	1.12
@@ -5,25 +5,29 @@
 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
+ice-ocean state estimation, 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
+  momentum equations have been adopted: LSOR \citep{zhang97}, and 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.
+\item ice-ocean stress can be formulated as in \citet{hibler87} or as in \citet{cam08}; 
+\item ice variables \ml{can be} advected by sophisticated, \ml{conservative}
+  advection schemes \ml{with flux limiting};
+\item growth and melt parameterizations have been refined and extended
+  in order to allow for more stable automatic differentiation of the code.
 \end{itemize}
-The model equations and their numerical realization are summarized
-below.
+
+The sea ice model is tightly coupled to the ocean compontent of the MITgcm
+\citep{mar97a}.  Heat, fresh water fluxes and surface stresses are computed
+from the atmospheric state and modified by the ice model at every time step.
+
 
 \subsection{Dynamics}
-\label{sec:dynamics}
+\label{app:dynamics}
 
 The momentum equation of the sea-ice model is
 \begin{equation}
@@ -40,14 +44,12 @@
 respectively;
 $g$ is the gravity accelation;
 $\nabla\phi(0)$ is the gradient (or tilt) of the sea surface height;
-$\phi(0) = g\eta + p_{a}/\rho_{0}$ is the sea surface height potential
-in response to ocean dynamics ($g\eta$) and to atmospheric pressure
-loading ($p_{a}/\rho_{0}$, where $\rho_{0}$ is a reference density);
-and $\vek{F}=\nabla\cdot\sigma$ is the divergence of the internal ice stress
-tensor $\sigma_{ij}$.
-When using the rescaled vertical coordinate system, z$^\ast$, of
-\citet{cam08}, $\phi(0)$ also includes a term due to snow and ice
-loading, $mg/\rho_{0}$. 
+$\phi(0) = g\eta + p_{a}/\rho_{0} + mg/\rho_{0}$ is the sea surface
+height potential in response to ocean dynamics ($g\eta$), to
+atmospheric pressure loading ($p_{a}/\rho_{0}$, where $\rho_{0}$ is a
+reference density) and a term due to snow and ice loading \citep{campin08};
+and $\vek{F}=\nabla\cdot\sigma$ is the divergence of the internal ice
+stress tensor $\sigma_{ij}$. %
 Advection of sea-ice momentum is neglected. The wind and ice-ocean stress
 terms are given by
 \begin{align*}
@@ -98,7 +100,7 @@
     \left(\dot{\epsilon}_{11}^2+\dot{\epsilon}_{22}^2\right)
     (1+e^{-2}) +  4e^{-2}\dot{\epsilon}_{12}^2 + 
     2\dot{\epsilon}_{11}\dot{\epsilon}_{22} (1-e^{-2})
-  \right]^{\frac{1}{2}}
+  \right]^{\frac{1}{2}}.
 \end{align*}
 The bulk viscosities are bounded above by imposing both a minimum
 $\Delta_{\min}=10^{-11}\text{\,s}^{-1}$ (for numerical reasons) and a
@@ -147,7 +149,8 @@
 EVP-model is stepped forward in time 120 times within the physical
 ocean model time step (although this parameter is under debate), to
 allow for elastic waves to disappear.  Because the scheme does not
-require a matrix inversion it is fast in spite of the small timestep
+require a matrix inversion it is fast in spite of the small internal
+timestep and simple to implement on parallel computers
 \citep{hunke97}. For completeness, we repeat the equations for the
 components of the stress tensor $\sigma_{1} =
 \sigma_{11}+\sigma_{22}$, $\sigma_{2}= \sigma_{11}-\sigma_{22}$, and
@@ -155,8 +158,8 @@
 \dot{\epsilon}_{11}+\dot{\epsilon}_{22}$, and the horizontal tension
 and shearing strain rates, $D_T =
 \dot{\epsilon}_{11}-\dot{\epsilon}_{22}$ and $D_S =
-2\dot{\epsilon}_{12}$, respectively, and using the above abbreviations,
-the equations can be written as:
+2\dot{\epsilon}_{12}$, respectively, and using the above
+abbreviations, the equations\refeq{evpequation} can be written as:
 \begin{align}
   \label{eq:evpstresstensor1}
   \frac{\partial\sigma_{1}}{\partial{t}} + \frac{\sigma_{1}}{2T} +
@@ -174,10 +177,9 @@
 the external (long) timestep $\Delta{t}$. \citet{hunke97} recommend
 $E_{0} = \frac{1}{3}$.
 
-For details of the spatial discretization, the reader is referred to
-\citet{zhang98, zhang03}. Our discretization differs only (but
-importantly) in the underlying grid, namely the Arakawa C-grid, but is
-otherwise straightforward. The EVP model, in particular, is discretized
+Our discretization differs from \citet{zhang98, zhang03} in the
+underlying grid, namely the Arakawa C-grid, but is otherwise
+straightforward. The EVP model, in particular, is discretized
 naturally on the C-grid with $\sigma_{1}$ and $\sigma_{2}$ on the
 center points and $\sigma_{12}$ on the corner (or vorticity) points of
 the grid. With this choice all derivatives are discretized as central
@@ -236,31 +238,33 @@
 %\end{itemize}
 
 \subsection{Thermodynamics}
-\label{sec:thermodynamics}
+\label{app:thermodynamics}
 
-In the original formulation the sea ice model \citep{menemenlis05}
+In its original formulation the sea ice model \citep{menemenlis05}
 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 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 flux is computed in a similar way to that of \citet{parkinson79}
-and \citet{manabe79}.
+\citet{semtner76}. This formulation does not allow storage of heat,
+that is, the heat capacity of ice is zero. 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. This type of model is often refered to as a
+``zero-layer'' model. The surface heat flux is computed in a similar
+way to that of \citet{parkinson79} 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
+parameterize a 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
+categories $H_{n}$ that are equally distributed between $2h$ and a
 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?]}
+thickness category is area-averaged to give the total heat flux
+\citep{hibler84}.
 
+\ml{[This is Ian Fenty's work and we may want to remove this paragraph
+  from the paper]: %
 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
@@ -268,19 +272,23 @@
 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.
-
+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}.
+and the albedo \citep{zhang98}. Snow modifies the effective
+conductivity according to 
+\[\frac{K}{h} \rightarrow \frac{1}{\frac{h_{s}}{K_{s}}+\frac{h}{K}},\]
+where $K_s$ is the conductivity of snow and $h_s$ the snow thickness.
+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,
+Effective ice 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}
@@ -316,40 +324,43 @@
   this can only be accomplished with a 2nd-order advection scheme with
   flux limiter \citep{roe85}.}
 
+Further documentation and model code can be found at
+\url{http://mitgcm.org}.
 
-\subsection{C-grid}
-\begin{itemize}
-\item no-slip vs. free-slip for both lsr and evp; 
-  "diagnostics" to look at and use for comparison
-  \begin{itemize}
-  \item ice transport through Fram Strait/Denmark Strait/Davis
-    Strait/Bering strait (these are general) 
-  \item ice transport through narrow passages, e.g.\ Nares-Strait
-  \end{itemize}
-\item compare different advection schemes (if lsr turns out to be more
-  effective, then with lsr otherwise I prefer evp), eg. 
-  \begin{itemize}
-  \item default 2nd-order with diff1=0.002
-  \item 1st-order upwind with diff1=0.
-  \item DST3FL (SEAICEadvScheme=33 with diff1=0., should work, works for me)
-  \item 2nd-order wit flux limiter (SEAICEadvScheme=77 with diff1=0.)
-  \end{itemize}
-  That should be enough. Here, total ice mass and location of ice edge
-  is interesting. However, this comparison can be done in an idealized
-  domain, may not require full Arctic Domain?
-\item
-Do a little study on the parameters of LSR and EVP
-\begin{enumerate}
-\item convergence of LSR, how many iterations do you need to get a
-  true elliptic yield curve 
-\item vary deltaTevp and the relaxation parameter for EVP and see when
-  the EVP solution breaks down (relative to the forcing time scale).
-  For this, it is essential that the evp solver gives use "stripeless"
-  solutions, that is your dtevp = 1sec solutions/or 10sec solutions
-  with SEAICE\_evpDampC = 615.
-\end{enumerate}
 
-\end{itemize}
+%\subsection{C-grid}
+%\begin{itemize}
+%\item no-slip vs. free-slip for both lsr and evp; 
+%  "diagnostics" to look at and use for comparison
+%  \begin{itemize}
+%  \item ice transport through Fram Strait/Denmark Strait/Davis
+%    Strait/Bering strait (these are general) 
+%  \item ice transport through narrow passages, e.g.\ Nares-Strait
+%  \end{itemize}
+%\item compare different advection schemes (if lsr turns out to be more
+%  effective, then with lsr otherwise I prefer evp), eg. 
+%  \begin{itemize}
+%  \item default 2nd-order with diff1=0.002
+%  \item 1st-order upwind with diff1=0.
+%  \item DST3FL (SEAICEadvScheme=33 with diff1=0., should work, works for me)
+%  \item 2nd-order wit flux limiter (SEAICEadvScheme=77 with diff1=0.)
+%  \end{itemize}
+%  That should be enough. Here, total ice mass and location of ice edge
+%  is interesting. However, this comparison can be done in an idealized
+%  domain, may not require full Arctic Domain?
+%\item
+%Do a little study on the parameters of LSR and EVP
+%\begin{enumerate}
+%\item convergence of LSR, how many iterations do you need to get a
+%  true elliptic yield curve 
+%\item vary deltaTevp and the relaxation parameter for EVP and see when
+%  the EVP solution breaks down (relative to the forcing time scale).
+%  For this, it is essential that the evp solver gives use "stripeless"
+%  solutions, that is your dtevp = 1sec solutions/or 10sec solutions
+%  with SEAICE\_evpDampC = 615.
+%\end{enumerate}
+
+%\end{itemize}
 
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