--- MITgcm_contrib/articles/ceaice/ceaice_model.tex 2008/02/26 19:27:26 1.1 +++ MITgcm_contrib/articles/ceaice/ceaice_model.tex 2008/08/15 14:16:35 1.12 @@ -1,8 +1,33 @@ -\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 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}, and EVP + \citep{hunke97}; +\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 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} @@ -19,14 +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*} @@ -43,7 +66,7 @@ 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} @@ -77,7 +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 @@ -126,7 +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 @@ -134,8 +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} + @@ -153,10 +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 @@ -205,35 +228,86 @@ 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. -For this reason, we employ positive, multidimensional 2nd and 3rd-order -advection scheme with flux limiter \citep{roe85, hundsdorfer94} for the -thermodynamic variables discussed in the next section. - -\subparagraph{boundary conditions: no-slip, free-slip, half-slip} - -\begin{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} +%\subparagraph{boundary conditions: no-slip, free-slip, half-slip} +%\begin{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} +\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 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 -\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 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 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 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 +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}. 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 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} + \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 @@ -250,36 +324,45 @@ 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} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "ceaice" +%%% End: