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\section{Model} |
\section{Model Formulation} |
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\label{sec:model} |
\label{sec:model} |
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The MITgcm sea ice model (MITsim) is based on a variant of the |
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viscous-plastic (VP) dynamic-thermodynamic sea ice model |
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\citep{zhang97} first introduced by \citet{hibler79, hibler80}. In |
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order to adapt this model to the requirements of coupled |
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ice-ocean simulations, many important aspects of the original code have |
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been modified and improved: |
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\begin{itemize} |
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\item the code has been rewritten for an Arakawa C-grid, both B- and |
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C-grid variants are available; the C-grid code allows for no-slip |
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and free-slip lateral boundary conditions; |
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\item two different solution methods for solving the nonlinear |
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momentum equations have been adopted: LSOR \citep{zha97}, EVP |
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\citep{hunke97}; |
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\item ice-ocean stress can be formulated as in \citet{hibler87}; |
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\item ice variables are advected by sophisticated advection schemes; |
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\item growth and melt parameterizaion have been refined and extended |
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in order to allow for automatic differentiation of the code. |
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\end{itemize} |
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The model equations and their numerical realization are summarized |
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below. |
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\subsection{Dynamics} |
\subsection{Dynamics} |
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\label{sec:dynamics} |
\label{sec:dynamics} |
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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 |
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be related to the ice strain rate and strength by a nonlinear |
be related to the ice strain rate and strength by a nonlinear |
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viscous-plastic (VP) constitutive law \citep{hibler79, zhang98}: |
viscous-plastic (VP) constitutive law \citep{hibler79, zhang97}: |
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\begin{equation} |
\begin{equation} |
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\label{eq:vpequation} |
\label{eq:vpequation} |
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\sigma_{ij}=2\eta(\dot{\epsilon}_{ij},P)\dot{\epsilon}_{ij} |
\sigma_{ij}=2\eta(\dot{\epsilon}_{ij},P)\dot{\epsilon}_{ij} |
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velocity and the ice velocity leading to an inconsistency as the ice |
velocity and the ice velocity leading to an inconsistency as the ice |
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temperature and salinity are different from the oceanic variables. |
temperature and salinity are different from the oceanic variables. |
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Sea ice distributions are characterized by sharp gradients and edges. |
%\subparagraph{boundary conditions: no-slip, free-slip, half-slip} |
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For this reason, we employ positive, multidimensional 2nd and 3rd-order |
%\begin{itemize} |
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advection scheme with flux limiter \citep{roe85, hundsdorfer94} for the |
%\item transition from existing B-Grid to C-Grid |
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thermodynamic variables discussed in the next section. |
%\item boundary conditions: no-slip, free-slip, half-slip |
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%\item fancy (multi dimensional) advection schemes |
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\subparagraph{boundary conditions: no-slip, free-slip, half-slip} |
%\item VP vs.\ EVP \citep{hunke97} |
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%\item ocean stress formulation \citep{hibler87} |
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\begin{itemize} |
%\end{itemize} |
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\item transition from existing B-Grid to C-Grid |
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\item boundary conditions: no-slip, free-slip, half-slip |
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\item fancy (multi dimensional) advection schemes |
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\item VP vs.\ EVP \citep{hunke97} |
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\item ocean stress formulation \citep{hibler87} |
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\end{itemize} |
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\subsection{Thermodynamics} |
\subsection{Thermodynamics} |
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\label{sec:thermodynamics} |
\label{sec:thermodynamics} |
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computations, the mean ice thickness $h$ is split into seven thickness |
computations, the mean ice thickness $h$ is split into seven thickness |
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categories $H_{n}$ that are equally distributed between $2h$ and |
categories $H_{n}$ that are equally distributed between $2h$ and |
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minimum imposed ice thickness of $5\text{\,cm}$ by $H_n= |
minimum imposed ice thickness of $5\text{\,cm}$ by $H_n= |
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\frac{2n-1}{7}\,h$ for $n\in[1,7]$. The heat flux for all thickness |
\frac{2n-1}{7}\,h$ for $n\in[1,7]$. The heat fluxes computed for each |
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categories is averaged to give the total heat flux. |
thickness category area averaged to give the total heat flux. \ml{[I |
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don't have citation for that, anyone?]} |
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The atmospheric heat flux is balanced by an oceanic heat flux from |
The atmospheric heat flux is balanced by an oceanic heat flux from |
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below. The oceanic flux is proportional to |
below. The oceanic flux is proportional to |
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Effective ich thickness (ice volume per unit area, |
Effective ich thickness (ice volume per unit area, |
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$c\cdot{h}$), concentration $c$ and effective snow thickness |
$c\cdot{h}$), concentration $c$ and effective snow thickness |
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($c\cdot{h}_{snow}$) are advected by ice velocities as described in |
($c\cdot{h}_{s}$) are advected by ice velocities: |
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\refsec{dynamics}. From the various advection scheme that are |
\begin{align} |
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available in the MITgcm \citep{mitgcm02}, we choose flux-limited |
\frac{\partial(c\,{h})}{\partial{t}} &= - \nabla\left(\vek{u}\,c\,{h}\right) + |
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schemes to preserve sharp gradients and edges and to rule out |
\Gamma_{h} + D_{h} \\ |
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unphysical over- and undershoots (negative thickness or |
\frac{\partial{c}}{\partial{t}} &= - \nabla\left(\vek{u}\,c\right) + |
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concentration). These scheme conserve volume and horizontal area. |
\Gamma_{c} + D_{c} \\ |
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\ml{[do we need to proove that? can we proove that? citation?]} |
\frac{\partial(c\,{h}_{s})}{\partial{t}} &= - \nabla\left(\vek{u}\,c\,{h}_{s}\right) + |
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\Gamma_{h_{s}} + D_{h_{s}} |
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\end{align} |
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where $\Gamma_X$ are the thermodynamic source terms and $D_{X}$ the |
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diffusive terms for quantity $X=h, c, h_{s}$. |
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% |
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From the various advection scheme that are available in the MITgcm |
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\citep{mitgcm02}, we choose flux-limited schemes |
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\citep[multidimensional 2nd and 3rd-order advection scheme with flux |
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limiter][]{roe85, hundsdorfer94} to preserve sharp gradients and edges |
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that are typical of sea ice distributions and to rule out unphysical |
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over- and undershoots (negative thickness or concentration). These |
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scheme conserve volume and horizontal area and are unconditionally |
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stable, so that we can set $D_{X}=0$. \ml{[do we need to proove that? |
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can we proove that? citation?]} |
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There is considerable doubt about the reliability of such a simple |
There is considerable doubt about the reliability of such a simple |
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thermodynamic model---\citet{semtner84} found significant errors in |
thermodynamic model---\citet{semtner84} found significant errors in |
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