--- manual/s_phys_pkgs/text/seaice.tex 2014/03/31 11:30:21 1.20 +++ manual/s_phys_pkgs/text/seaice.tex 2015/01/22 09:10:27 1.23 @@ -1,4 +1,4 @@ -% $Header: /home/ubuntu/mnt/e9_copy/manual/s_phys_pkgs/text/seaice.tex,v 1.20 2014/03/31 11:30:21 mlosch Exp $ +% $Header: /home/ubuntu/mnt/e9_copy/manual/s_phys_pkgs/text/seaice.tex,v 1.23 2015/01/22 09:10:27 mlosch Exp $ % $Name: $ %%EH3 Copied from "MITgcm/pkg/seaice/seaice_description.tex" @@ -63,7 +63,8 @@ Parts of the SEAICE code can be enabled or disabled at compile time via CPP preprocessor flags. These options are set in \code{SEAICE\_OPTIONS.h}. -Table \ref{tab:pkg:seaice:cpp} summarizes the most important ones. +Table \ref{tab:pkg:seaice:cpp} summarizes the most important ones. For +more options see the default \code{pkg/seaice/SEAICE\_OPTIONS.h}. \begin{table}[!ht] \centering @@ -181,7 +182,7 @@ first introduced by \citet{hib79, hib80}. 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: +improved \citep{losch10:_mitsim}: \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 @@ -527,6 +528,15 @@ number of Krylov iterations $\code{SEAICEkrylovIterMax} = 50$, because the Krylov subspace has a fixed dimension of 50. +Setting \code{SEAICEuseStrImpCpl = .TRUE.,} turns on ``strength +implicit coupling'' \citep{hutchings04} in the LSR-solver and in the +LSR-preconditioner for the JFNK-solver. In this mode, the different +contributions of the stress divergence terms are re-ordered in order +to increase the diagonal dominance of the system +matrix. Unfortunately, the convergence rate of the LSR solver is +increased only slightly, while the JFNK-convergence appears to be +unaffected. + \paragraph{Elastic-Viscous-Plastic (EVP) Dynamics\label{sec:pkg:seaice:EVPdynamics}}~\\ % \citet{hun97}'s introduced an elastic contribution to the strain @@ -592,6 +602,63 @@ $E_{0}\Delta{t}=\mbox{forcing time scale}$, or directly \code{SEAICE\_evpTauRelax} ($T$) to the forcing time scale. +\paragraph{More stable variant of Elastic-Viscous-Plastic Dynamics: EVP*\label{sec:pkg:seaice:EVPstar}}~\\ +% +The genuine EVP schemes appears to give noisy solutions \citep{hun01, + lemieux12, bouillon13}. This has lead to a modified EVP or EVP* +\citep{lemieux12, bouillon13, kimmritz15}; here, refer to these +variants by EVP*. The main idea is to modify the ``natural'' +time-discretization of the momentum equations: +\begin{equation} + \label{eq:evpstar} + m\frac{D\vec{u}}{Dt} \approx m\frac{u^{p+1}-u^{n}}{\Delta{t}} + + \beta^{*}\frac{u^{p+1}-u^{p}}{\Delta{t}_{\mathrm{EVP}}} +\end{equation} +where $n$ is the previous time step index, and $p$ is the previous +sub-cycling index. The extra ``intertial'' term +$m\,(u^{p+1}-u^{n})/\Delta{t})$ allows the definition of a residual +$|u^{p+1}-u^{p}|$ that, as $u^{p+1} \rightarrow u^{n+1}$, converges to +$0$. In this way EVP can be re-interpreted as a pure iterative solver +where the sub-cycling has no association with time-relation (through +$\Delta{t}_{\mathrm{EVP}}$) \citep{bouillon13, kimmritz15}. Using the +terminology of \citet{kimmritz15}, the evolution equations of stress +$\sigma_{ij}$ and momentum $\vec{u}$ can be written as: +\begin{align} + \label{eq:evpstarsigma} + \sigma_{ij}^{p+1}&=\sigma_{ij}^p+\frac{1}{\alpha} + \Big(\sigma_{ij}(\vec{u}^p)-\sigma_{ij}^p\Big), + \phantom{\int}\\ + \label{eq:evpstarmom} + \vec{u}^{p+1}&=\vec{u}^p+\frac{1}{\beta} + \Big(\frac{\Delta t}{m}\nabla \cdot{\bf \sigma}^{p+1}+ + \frac{\Delta t}{m}\vec{R}^{p}+\vec{u}_n-\vec{u}^p\Big). +\end{align} +$\vec{R}$ contains all terms in the momentum equations except for the +rheology terms and the time derivative; $\alpha$ and $\beta$ are free +parameters (\code{SEAICE\_evpAlpha}, \code{SEAICE\_evpBeta}) that +replace the time stepping parameters \code{SEAICE\_deltaTevp} +($\Delta{T}_{\mathrm{EVP}}$), \code{SEAICE\_elasticParm} ($E_{0}$), or +\code{SEAICE\_evpTauRelax} ($T$). $\alpha$ and $\beta$ determine the +speed of convergence and the stability. Usually, it makes sense to use +$\alpha = \beta$, and \code{SEAICEnEVPstarSteps} $\gg +(\alpha,\,\beta)$ \citep{kimmritz15}. Currently, there is no +termination criterion and the number of EVP* iterations is fixed to +\code{SEAICEnEVPstarSteps}. + +In order to use EVP* in the MITgcm, set \code{SEAICEuseEVPstar = + .TRUE.,} in \code{data.seaice}. If \code{SEAICEuseEVPrev =.TRUE.,} +the actual form of equations (\ref{eq:evpstarsigma}) and +(\ref{eq:evpstarmom}) is used with fewer implicit terms and the factor +of $e^{2}$ dropped in the stress equations (\ref{eq:evpstresstensor2}) +and (\ref{eq:evpstresstensor12}). Although this modifies the original +EVP-equations, it turns out to improve convergence \citep{bouillon13}. + +Note, that for historical reasons, \code{SEAICE\_deltaTevp} needs to +be set to some (any!) value in order to use also EVP*; this behavoir +many change in the future. Also note, that +probably because of the C-grid staggering of velocities and stresses, +EVP* does not converge as successfully as in \citet{kimmritz15}. + \paragraph{Truncated ellipse method (TEM) for yield curve \label{sec:pkg:seaice:TEM}}~\\ % In the so-called truncated ellipse method the shear viscosity $\eta$ @@ -701,7 +768,8 @@ [(\zeta-\eta)\dot{\epsilon}_{\gamma\gamma} - P/2 ]\delta_{\alpha\beta}$ \citep{hib79}. The stress tensor divergence $(\nabla\sigma)_{\alpha} = \partial_\beta\sigma_{\beta\alpha}$, is -discretized in finite volumes. This conveniently avoids dealing with +discretized in finite volumes \citep[see +also][]{losch10:_mitsim}. This conveniently avoids dealing with further metric terms, as these are ``hidden'' in the differential cell widths. For the $u$-equation ($\alpha=1$) we have: \begin{align} @@ -802,6 +870,7 @@ \paragraph{Thermodynamics\label{sec:pkg:seaice:thermodynamics}}~\\ % +\noindent\textbf{NOTE: THIS SECTION IS TERRIBLY OUT OF DATE}\\ In its original formulation the sea ice model \citep{menemenlis05} uses simple thermodynamics following the appendix of \citet{sem76}. This formulation does not allow storage of heat, @@ -817,16 +886,22 @@ 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}. To use this thickness category parameterization set -\code{\#define SEAICE\_MULTICATEGORY}; note that this requires -different restart files and switching this flag on in the middle of an -integration is not possible. +parameterize a sub-grid scale distribution for heat flux computations, +the mean ice thickness $h$ is split into $N$ 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 $n\in[1,N]$. The heat fluxes computed for each thickness category +is area-averaged to give the total heat flux \citep{hibler84}. To use +this thickness category parameterization set \code{SEAICE\_multDim} to +the number of desired categories (7 is a good guess, for anything +larger than 7 modify \code{SEAICE\_SIZE.h}) in +\code{data.seaice}; note that this requires different restart files +and switching this flag on in the middle of an integration is not +advised. In order to include the same distribution for snow, set +\code{SEAICE\_useMultDimSnow = .TRUE.}; only then, the +parameterization of always having a fraction of thin ice is efficient +and generally thicker ice is produced \citep{castro-morales14}. + The atmospheric heat flux is balanced by an oceanic heat flux from below. The oceanic flux is proportional to