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revision 1.20 by mlosch, Mon Mar 31 11:30:21 2014 UTC revision 1.23 by mlosch, Thu Jan 22 09:10:27 2015 UTC
# Line 63  no additional CPP options are required. Line 63  no additional CPP options are required.
63  Parts of the SEAICE code can be enabled or disabled at compile time  Parts of the SEAICE code can be enabled or disabled at compile time
64  via CPP preprocessor flags. These options are set in  via CPP preprocessor flags. These options are set in
65  \code{SEAICE\_OPTIONS.h}.  \code{SEAICE\_OPTIONS.h}.
66  Table \ref{tab:pkg:seaice:cpp} summarizes the most important ones.  Table \ref{tab:pkg:seaice:cpp} summarizes the most important ones. For
67    more options see the default \code{pkg/seaice/SEAICE\_OPTIONS.h}.
68    
69  \begin{table}[!ht]  \begin{table}[!ht]
70  \centering  \centering
# Line 181  viscous-plastic (VP) dynamic-thermodynam Line 182  viscous-plastic (VP) dynamic-thermodynam
182  first introduced by \citet{hib79, hib80}. In order to adapt this model  first introduced by \citet{hib79, hib80}. In order to adapt this model
183  to the requirements of coupled ice-ocean state estimation, many  to the requirements of coupled ice-ocean state estimation, many
184  important aspects of the original code have been modified and  important aspects of the original code have been modified and
185  improved:  improved \citep{losch10:_mitsim}:
186  \begin{itemize}  \begin{itemize}
187  \item the code has been rewritten for an Arakawa C-grid, both B- and  \item the code has been rewritten for an Arakawa C-grid, both B- and
188    C-grid variants are available; the C-grid code allows for no-slip    C-grid variants are available; the C-grid code allows for no-slip
# Line 527  non-linear iterations $\code{SEAICEnewto Line 528  non-linear iterations $\code{SEAICEnewto
528  number of Krylov iterations $\code{SEAICEkrylovIterMax} = 50$, because  number of Krylov iterations $\code{SEAICEkrylovIterMax} = 50$, because
529  the Krylov subspace has a fixed dimension of 50.  the Krylov subspace has a fixed dimension of 50.
530    
531    Setting \code{SEAICEuseStrImpCpl = .TRUE.,} turns on ``strength
532    implicit coupling'' \citep{hutchings04} in the LSR-solver and in the
533    LSR-preconditioner for the JFNK-solver. In this mode, the different
534    contributions of the stress divergence terms are re-ordered in order
535    to increase the diagonal dominance of the system
536    matrix. Unfortunately, the convergence rate of the LSR solver is
537    increased only slightly, while the JFNK-convergence appears to be
538    unaffected.
539    
540  \paragraph{Elastic-Viscous-Plastic (EVP) Dynamics\label{sec:pkg:seaice:EVPdynamics}}~\\  \paragraph{Elastic-Viscous-Plastic (EVP) Dynamics\label{sec:pkg:seaice:EVPdynamics}}~\\
541  %  %
542  \citet{hun97}'s introduced an elastic contribution to the strain  \citet{hun97}'s introduced an elastic contribution to the strain
# Line 592  the damping time scale $T$ accordingly, Line 602  the damping time scale $T$ accordingly,
602  $E_{0}\Delta{t}=\mbox{forcing time scale}$, or directly  $E_{0}\Delta{t}=\mbox{forcing time scale}$, or directly
603  \code{SEAICE\_evpTauRelax} ($T$) to the forcing time scale.  \code{SEAICE\_evpTauRelax} ($T$) to the forcing time scale.
604    
605    \paragraph{More stable variant of Elastic-Viscous-Plastic Dynamics:  EVP*\label{sec:pkg:seaice:EVPstar}}~\\
606    %
607    The genuine EVP schemes appears to give noisy solutions \citep{hun01,
608      lemieux12, bouillon13}. This has lead to a modified EVP or EVP*
609    \citep{lemieux12, bouillon13, kimmritz15}; here, refer to these
610    variants by EVP*. The main idea is to modify the ``natural''
611    time-discretization of the momentum equations:
612    \begin{equation}
613      \label{eq:evpstar}
614      m\frac{D\vec{u}}{Dt} \approx m\frac{u^{p+1}-u^{n}}{\Delta{t}}
615      + \beta^{*}\frac{u^{p+1}-u^{p}}{\Delta{t}_{\mathrm{EVP}}}
616    \end{equation}
617    where $n$ is the previous time step index, and $p$ is the previous
618    sub-cycling index. The extra ``intertial'' term
619    $m\,(u^{p+1}-u^{n})/\Delta{t})$ allows the definition of a residual
620    $|u^{p+1}-u^{p}|$ that, as $u^{p+1} \rightarrow u^{n+1}$, converges to
621    $0$. In this way EVP can be re-interpreted as a pure iterative solver
622    where the sub-cycling has no association with time-relation (through
623    $\Delta{t}_{\mathrm{EVP}}$) \citep{bouillon13, kimmritz15}. Using the
624    terminology of \citet{kimmritz15}, the evolution equations of stress
625    $\sigma_{ij}$ and momentum $\vec{u}$ can be written as:
626    \begin{align}
627      \label{eq:evpstarsigma}
628      \sigma_{ij}^{p+1}&=\sigma_{ij}^p+\frac{1}{\alpha}
629      \Big(\sigma_{ij}(\vec{u}^p)-\sigma_{ij}^p\Big),
630      \phantom{\int}\\
631      \label{eq:evpstarmom}
632      \vec{u}^{p+1}&=\vec{u}^p+\frac{1}{\beta}
633      \Big(\frac{\Delta t}{m}\nabla \cdot{\bf \sigma}^{p+1}+
634      \frac{\Delta t}{m}\vec{R}^{p}+\vec{u}_n-\vec{u}^p\Big).
635    \end{align}
636    $\vec{R}$ contains all terms in the momentum equations except for the
637    rheology terms and the time derivative; $\alpha$ and $\beta$ are free
638    parameters (\code{SEAICE\_evpAlpha}, \code{SEAICE\_evpBeta}) that
639    replace the time stepping parameters \code{SEAICE\_deltaTevp}
640    ($\Delta{T}_{\mathrm{EVP}}$), \code{SEAICE\_elasticParm} ($E_{0}$), or
641    \code{SEAICE\_evpTauRelax} ($T$). $\alpha$ and $\beta$ determine the
642    speed of convergence and the stability. Usually, it makes sense to use
643    $\alpha = \beta$, and \code{SEAICEnEVPstarSteps} $\gg
644    (\alpha,\,\beta)$ \citep{kimmritz15}. Currently, there is no
645    termination criterion and the number of EVP* iterations is fixed to
646    \code{SEAICEnEVPstarSteps}.
647    
648    In order to use EVP* in the MITgcm, set \code{SEAICEuseEVPstar =
649      .TRUE.,} in \code{data.seaice}. If \code{SEAICEuseEVPrev =.TRUE.,}
650    the actual form of equations (\ref{eq:evpstarsigma}) and
651    (\ref{eq:evpstarmom}) is used with fewer implicit terms and the factor
652    of $e^{2}$ dropped in the stress equations (\ref{eq:evpstresstensor2})
653    and (\ref{eq:evpstresstensor12}). Although this modifies the original
654    EVP-equations, it turns out to improve convergence \citep{bouillon13}.
655    
656    Note, that for historical reasons, \code{SEAICE\_deltaTevp} needs to
657    be set to some (any!) value in order to use also EVP*; this behavoir
658    many change in the future. Also note, that
659    probably because of the C-grid staggering of velocities and stresses,
660    EVP* does not converge as successfully as in \citet{kimmritz15}.
661    
662  \paragraph{Truncated ellipse method (TEM) for yield curve \label{sec:pkg:seaice:TEM}}~\\  \paragraph{Truncated ellipse method (TEM) for yield curve \label{sec:pkg:seaice:TEM}}~\\
663  %  %
664  In the so-called truncated ellipse method the shear viscosity $\eta$  In the so-called truncated ellipse method the shear viscosity $\eta$
# Line 701  relation $\sigma_{\alpha\beta} = 2\eta\d Line 768  relation $\sigma_{\alpha\beta} = 2\eta\d
768  [(\zeta-\eta)\dot{\epsilon}_{\gamma\gamma} - P/2  [(\zeta-\eta)\dot{\epsilon}_{\gamma\gamma} - P/2
769  ]\delta_{\alpha\beta}$ \citep{hib79}. The stress tensor divergence  ]\delta_{\alpha\beta}$ \citep{hib79}. The stress tensor divergence
770  $(\nabla\sigma)_{\alpha} = \partial_\beta\sigma_{\beta\alpha}$, is  $(\nabla\sigma)_{\alpha} = \partial_\beta\sigma_{\beta\alpha}$, is
771  discretized in finite volumes. This conveniently avoids dealing with  discretized in finite volumes \citep[see
772    also][]{losch10:_mitsim}. This conveniently avoids dealing with
773  further metric terms, as these are ``hidden'' in the differential cell  further metric terms, as these are ``hidden'' in the differential cell
774  widths. For the $u$-equation ($\alpha=1$) we have:  widths. For the $u$-equation ($\alpha=1$) we have:
775  \begin{align}  \begin{align}
# Line 802  $\sigma_{21}^{Z}=0$, or equivalently $\e Line 870  $\sigma_{21}^{Z}=0$, or equivalently $\e
870    
871  \paragraph{Thermodynamics\label{sec:pkg:seaice:thermodynamics}}~\\  \paragraph{Thermodynamics\label{sec:pkg:seaice:thermodynamics}}~\\
872  %  %
873    \noindent\textbf{NOTE: THIS SECTION IS TERRIBLY OUT OF DATE}\\
874  In its original formulation the sea ice model \citep{menemenlis05}  In its original formulation the sea ice model \citep{menemenlis05}
875  uses simple thermodynamics following the appendix of  uses simple thermodynamics following the appendix of
876  \citet{sem76}. This formulation does not allow storage of heat,  \citet{sem76}. This formulation does not allow storage of heat,
# Line 817  way to that of \citet{parkinson79} and \ Line 886  way to that of \citet{parkinson79} and \
886  The conductive heat flux depends strongly on the ice thickness $h$.  The conductive heat flux depends strongly on the ice thickness $h$.
887  However, the ice thickness in the model represents a mean over a  However, the ice thickness in the model represents a mean over a
888  potentially very heterogeneous thickness distribution.  In order to  potentially very heterogeneous thickness distribution.  In order to
889  parameterize a sub-grid scale distribution for heat flux  parameterize a sub-grid scale distribution for heat flux computations,
890  computations, the mean ice thickness $h$ is split into seven thickness  the mean ice thickness $h$ is split into $N$ thickness categories
891  categories $H_{n}$ that are equally distributed between $2h$ and a  $H_{n}$ that are equally distributed between $2h$ and a minimum
892  minimum imposed ice thickness of $5\text{\,cm}$ by $H_n=  imposed ice thickness of $5\text{\,cm}$ by $H_n= \frac{2n-1}{7}\,h$
893  \frac{2n-1}{7}\,h$ for $n\in[1,7]$. The heat fluxes computed for each  for $n\in[1,N]$. The heat fluxes computed for each thickness category
894  thickness category is area-averaged to give the total heat flux  is area-averaged to give the total heat flux \citep{hibler84}. To use
895  \citep{hibler84}. To use this thickness category parameterization set  this thickness category parameterization set \code{SEAICE\_multDim} to
896  \code{\#define SEAICE\_MULTICATEGORY}; note that this requires  the number of desired categories (7 is a good guess, for anything
897  different restart files and switching this flag on in the middle of an  larger than 7 modify \code{SEAICE\_SIZE.h}) in
898  integration is not possible.  \code{data.seaice}; note that this requires different restart files
899    and switching this flag on in the middle of an integration is not
900    advised. In order to include the same distribution for snow, set
901    \code{SEAICE\_useMultDimSnow = .TRUE.}; only then, the
902    parameterization of always having a fraction of thin ice is efficient
903    and generally thicker ice is produced \citep{castro-morales14}.
904    
905    
906  The atmospheric heat flux is balanced by an oceanic heat flux from  The atmospheric heat flux is balanced by an oceanic heat flux from
907  below.  The oceanic flux is proportional to  below.  The oceanic flux is proportional to
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