--- manual/s_phys_pkgs/text/seaice.tex 2011/08/13 15:50:05 1.18 +++ manual/s_phys_pkgs/text/seaice.tex 2014/03/31 11:30:21 1.20 @@ -1,4 +1,4 @@ -% $Header: /home/ubuntu/mnt/e9_copy/manual/s_phys_pkgs/text/seaice.tex,v 1.18 2011/08/13 15:50:05 heimbach Exp $ +% $Header: /home/ubuntu/mnt/e9_copy/manual/s_phys_pkgs/text/seaice.tex,v 1.20 2014/03/31 11:30:21 mlosch Exp $ % $Name: $ %%EH3 Copied from "MITgcm/pkg/seaice/seaice_description.tex" @@ -61,9 +61,9 @@ (see Section \ref{sec:buildingCode}). Parts of the SEAICE code can be enabled or disabled at compile time -via CPP preprocessor flags. These options are set in either -\code{SEAICE\_OPTIONS.h} or in \code{ECCO\_CPPOPTIONS.h}. -Table \ref{tab:pkg:seaice:cpp} summarizes these options. +via CPP preprocessor flags. These options are set in +\code{SEAICE\_OPTIONS.h}. +Table \ref{tab:pkg:seaice:cpp} summarizes the most important ones. \begin{table}[!ht] \centering @@ -80,22 +80,22 @@ \code{SEAICE\_CGRID} & LSR solver on C-grid (rather than original B-grid) \\ \code{SEAICE\_ALLOW\_EVP} & - use EVP rather than LSR rheology solver \\ + enable use of EVP rheology solver \\ + \code{SEAICE\_ALLOW\_JFNK} & + enable use of JFNK rheology solver \\ \code{SEAICE\_EXTERNAL\_FLUXES} & use EXF-computed fluxes as starting point \\ - \code{SEAICE\_MULTICATEGORY} & - enable 8-category thermodynamics (by default undefined)\\ + \code{SEAICE\_ZETA\_SMOOTHREG} & + use differentialable regularization for viscosities \\ \code{SEAICE\_VARIABLE\_FREEZING\_POINT} & enable linear dependence of the freezing point on salinity (by default undefined)\\ \code{ALLOW\_SEAICE\_FLOODING} & enable snow to ice conversion for submerged sea-ice \\ - \code{SEAICE\_SALINITY} & - enable "salty" sea-ice (by default undefined) \\ - \code{SEAICE\_AGE} & - enable "age tracer" sea-ice (by default undefined) \\ - \code{SEAICE\_CAP\_HEFF} & - enable capping of sea-ice thickness to MAX\_HEFF \\ \hline + \code{SEAICE\_VARIABLE\_SALINITY} & + enable sea-ice with variable salinity (by default undefined) \\ + \code{SEAICE\_SITRACER} & + enable sea-ice tracer package (by default undefined) \\ \code{SEAICE\_BICE\_STRESS} & B-grid only for backward compatiblity: turn on ice-stress on ocean\\ @@ -105,7 +105,8 @@ \hline \end{tabular} } - \caption{~} + \caption{Some of the most relevant CPP preprocessor flags in the + \code{seaice}-package.} \end{table} %---------------------------------------------------------------------- @@ -139,8 +140,6 @@ over grid cell in meters; initializes variable \code{HSNOW}; \item[\code{HsaltFile}:] Initial salinity of sea ice averaged over grid cell in g/m$^2$; initializes variable \code{HSALT}; -\item[\code{IceAgeFile}:] Initial ice age of sea ice averaged over grid - cell in seconds; initializes variable \code{ICEAGE}; \end{description} %---------------------------------------------------------------------- @@ -187,9 +186,9 @@ \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{hun97}; +\item three different solution methods for solving the nonlinear + momentum equations have been adopted: LSOR \citep{zhang97}, EVP + \citep{hun97}, JFNK \citep{lemieux10,losch14:_jfnk}; \item ice-ocean stress can be formulated as in \citet{hibler87} or as in \citet{cam08}; \item ice variables are advected by sophisticated, conservative @@ -296,7 +295,7 @@ densities; and $R_{air/ocean}$ are rotation matrices that act on the wind/current vectors. -\paragraph{Viscous-Plastic (VP) Rheology and LSOR solver \label{sec:pkg:seaice:VPdynamics}}~\\ +\paragraph{Viscous-Plastic (VP) Rheology\label{sec:pkg:seaice:VPrheology}}~\\ % 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 @@ -317,7 +316,7 @@ The maximum ice pressure $P_{\max}$, a measure of ice strength, depends on both thickness $h$ and compactness (concentration) $c$: \begin{equation} - P_{\max} = P^{*}c\,h\,e^{[C^{*}\cdot(1-c)]}, + P_{\max} = P^{*}c\,h\,\exp\{-C^{*}\cdot(1-c)\}, \label{eq:icestrength} \end{equation} with the constants $P^{*}$ (run-time parameter \code{SEAICE\_strength}) and @@ -349,14 +348,184 @@ = 2\,\Delta\zeta$ \citep{hibler95} is used so that the stress state always lies on the elliptic yield curve by definition. -In the current implementation, the VP-model is integrated with the -semi-implicit line successive over relaxation (LSOR)-solver of -\citet{zhang97}, which allows for long time steps that, in our case, -are limited by the explicit treatment of the Coriolis term. The -explicit treatment of the Coriolis term does not represent a severe -limitation because it restricts the time step to approximately the -same length as in the ocean model where the Coriolis term is also -treated explicitly. +Defining the CPP-flag \code{SEAICE\_ZETA\_SMOOTHREG} in +\code{SEAICE\_OPTIONS.h} before compiling replaces the method for +bounding $\zeta$ by a smooth (differentiable) expression: +\begin{equation} + \label{eq:zetaregsmooth} + \begin{split} + \zeta &= \zeta_{\max}\tanh\left(\frac{P}{2\,\min(\Delta,\Delta_{\min}) + \,\zeta_{\max}}\right)\\ + &= \frac{P}{2\Delta^*} + \tanh\left(\frac{\Delta^*}{\min(\Delta,\Delta_{\min})}\right) + \end{split} +\end{equation} +where $\Delta_{\min}=10^{-20}\text{\,s}^{-1}$ is chosen to avoid divisions +by zero. + +\paragraph{LSR and JFNK solver \label{sec:pkg:seaice:LSRJFNK}}~\\ +% +% By default, the VP-model is integrated by a Pickwith the +% semi-implicit line successive over relaxation (LSOR)-solver of +% \citet{zhang97}, which allows for long time steps that, in our case, +% are limited by the explicit treatment of the Coriolis term. The +% explicit treatment of the Coriolis term does not represent a severe +% limitation because it restricts the time step to approximately the +% same length as in the ocean model where the Coriolis term is also +% treated explicitly. + +\newcommand{\mat}[1]{\ensuremath{\mathbf{#1}}} +% +In the matrix notation, the discretized momentum equations can be +written as +\begin{equation} + \label{eq:matrixmom} + \mat{A}(\vek{x})\,\vek{x} = \vek{b}(\vek{x}). +\end{equation} +The solution vector $\vek{x}$ consists of the two velocity components +$u$ and $v$ that contain the velocity variables at all grid points and +at one time level. The standard (and default) method for solving +Eq.\,(\ref{eq:matrixmom}) in the sea ice component of the +\mbox{MITgcm}, as in many sea ice models, is an iterative Picard +solver: in the $k$-th iteration a linearized form +$\mat{A}(\vek{x}^{k-1})\,\vek{x}^{k} = \vek{b}(\vek{x}^{k-1})$ is +solved (in the case of the MITgcm it is a Line Successive (over) +Relaxation (LSR) algorithm \citep{zhang97}). Picard solvers converge +slowly, but generally the iteration is terminated after only a few +non-linear steps \citep{zhang97, lemieux09} and the calculation +continues with the next time level. This method is the default method +in the MITgcm. The number of non-linear iteration steps or pseudo-time +steps can be controlled by the runtime parameter +\code{NPSEUDOTIMESTEPS} (default is 2). + +In order to overcome the poor convergence of the Picard-solver, +\citet{lemieux10} introduced a Jacobian-free Newton-Krylov solver for +the sea ice momentum equations. This solver is also implemented in the +MITgcm \citep{losch14:_jfnk}. The Newton method transforms minimizing +the residual $\vek{F}(\vek{x}) = \mat{A}(\vek{x})\,\vek{x} - +\vek{b}(\vek{x})$ to finding the roots of a multivariate Taylor +expansion of the residual \vek{F} around the previous ($k-1$) estimate +$\vek{x}^{k-1}$: +\begin{equation} + \label{eq:jfnktaylor} + \vek{F}(\vek{x}^{k-1}+\delta\vek{x}^{k}) = + \vek{F}(\vek{x}^{k-1}) + \vek{F}'(\vek{x}^{k-1})\,\delta\vek{x}^{k} +\end{equation} +with the Jacobian $\mat{J}\equiv\vek{F}'$. The root +$\vek{F}(\vek{x}^{k-1}+\delta\vek{x}^{k})=0$ is found by solving +\begin{equation} + \label{eq:jfnklin} + \mat{J}(\vek{x}^{k-1})\,\delta\vek{x}^{k} = -\vek{F}(\vek{x}^{k-1}) +\end{equation} +for $\delta\vek{x}^{k}$. The next ($k$-th) estimate is given by +$\vek{x}^{k}=\vek{x}^{k-1}+a\,\delta\vek{x}^{k}$. In order to avoid +overshoots the factor $a$ is iteratively reduced in a line search +($a=1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \ldots$) until +$\|\vek{F}(\vek{x}^k)\| < \|\vek{F}(\vek{x}^{k-1})\|$, where +$\|\cdot\|=\int\cdot\,dx^2$ is the $L_2$-norm. In practice, the line +search is stopped at $a=\frac{1}{8}$. The line search starts after +$\code{SEAICE\_JFNK\_lsIter}$ non-linear Newton iterations (off by +default). + + +Forming the Jacobian $\mat{J}$ explicitly is often avoided as ``too +error prone and time consuming'' \citep{knoll04:_jfnk}. Instead, +Krylov methods only require the action of \mat{J} on an arbitrary +vector \vek{w} and hence allow a matrix free algorithm for solving +Eq.\,(\ref{eq:jfnklin}) \citep{knoll04:_jfnk}. The action of \mat{J} +can be approximated by a first-order Taylor series expansion: +\begin{equation} + \label{eq:jfnkjacvecfd} + \mat{J}(\vek{x}^{k-1})\,\vek{w} \approx + \frac{\vek{F}(\vek{x}^{k-1}+\epsilon\vek{w}) - \vek{F}(\vek{x}^{k-1})} + {\epsilon} +\end{equation} +or computed exactly with the help of automatic differentiation (AD) +tools. \code{SEAICE\_JFNKepsilon} sets the step size +$\epsilon$. + +We use the Flexible Generalized Minimum RESidual method +\citep[FGMRES,][]{saad93:_fgmres} with right-hand side preconditioning +to solve Eq.\,(\ref{eq:jfnklin}) iteratively starting from a first +guess of $\delta\vek{x}^{k}_{0} = 0$. For the preconditioning matrix +\mat{P} we choose a simplified form of the system matrix +$\mat{A}(\vek{x}^{k-1})$ \citep{lemieux10} where $\vek{x}^{k-1}$ is +the estimate of the previous Newton step $k-1$. The transformed +equation\,(\ref{eq:jfnklin}) becomes +\begin{equation} + \label{eq:jfnklinpc} + \mat{J}(\vek{x}^{k-1})\,\mat{P}^{-1}\delta\vek{z} = + -\vek{F}(\vek{x}^{k-1}), + \quad\text{with}\quad \delta\vek{z}=\mat{P}\delta\vek{x}^{k}. +\end{equation} +The Krylov method iteratively improves the approximate solution +to~(\ref{eq:jfnklinpc}) in subspace ($\vek{r}_0$, +$\mat{J}\mat{P}^{-1}\vek{r}_0$, $(\mat{J}\mat{P}^{-1})^2\vek{r}_0$, +\ldots, $(\mat{J}\mat{P}^{-1})^m\vek{r}_0$) with increasing $m$; +$\vek{r}_0 = -\vek{F}(\vek{x}^{k-1}) +-\mat{J}(\vek{x}^{k-1})\,\delta\vek{x}^{k}_{0}$ +%-\vek{F}(\vek{x}^{k-1}) +%-\mat{J}(\vek{x}^{k-1})\,\mat{P}^{-1}\delta\vek{z}$ +is the initial residual of +(\ref{eq:jfnklin}); $\vek{r}_0=-\vek{F}(\vek{x}^{k-1})$ with the first +guess $\delta\vek{x}^{k}_{0}=0$. We allow a Krylov-subspace of +dimension~$m=50$ and we do not use restarts. The preconditioning operation +involves applying $\mat{P}^{-1}$ to the basis vectors $\vek{v}_0, +\vek{v}_1, \vek{v}_2, \ldots, \vek{v}_m$ of the Krylov subspace. This +operation is approximated by solving the linear system +$\mat{P}\,\vek{w}=\vek{v}_i$. Because $\mat{P} \approx +\mat{A}(\vek{x}^{k-1})$, we can use the LSR-algorithm \citep{zhang97} +already implemented in the Picard solver. Each preconditioning +operation uses a fixed number of 10~LSR-iterations avoiding any +termination criterion. More details and results can be found in +\citet{lemieux10, losch14:_jfnk}. + +To use the JFNK-solver set \code{SEAICEuseJFNK = .TRUE.} in the +namelist file \code{data.seaice}; \code{SEAICE\_ALLOW\_JFNK} needs to +be defined in \code{SEAICE\_OPTIONS.h} and we recommend using a smooth +regularization of $\zeta$ by defining \code{SEAICE\_ZETA\_SMOOTHREG} +(see above) for better convergence. The non-linear Newton iteration +is terminated when the $L_2$-norm of the residual is reduced by +$\gamma_{\mathrm{nl}}$ (runtime parameter \code{JFNKgamma\_nonlin = + 1.e-4} will already lead to expensive simulations) with respect to +the initial norm: $\|\vek{F}(\vek{x}^k)\| < +\gamma_{\mathrm{nl}}\|\vek{F}(\vek{x}^0)\|$. Within a non-linear +iteration, the linear FGMRES solver is terminated when the residual is +smaller than $\gamma_k\|\vek{F}(\vek{x}^{k-1})\|$ where $\gamma_k$ is +determined by +\begin{equation} + \label{eq:jfnkgammalin} + \gamma_k = + \begin{cases} + \gamma_0 &\text{for $\|\vek{F}(\vek{x}^{k-1})\| \geq r$}, \\ + \max\left(\gamma_{\min}, + \frac{\|\vek{F}(\vek{x}^{k-1})\|}{\|\vek{F}(\vek{x}^{k-2})\|}\right) +% \phi\left(\frac{\|\vek{F}(\vek{x}^{k-1})\|}{\|\vek{F}(\vek{x}^{k-2})\|}\right)^\alpha\right) + &\text{for $\|\vek{F}(\vek{x}^{k-1})\| < r$,} + \end{cases} +\end{equation} +so that the linear tolerance parameter $\gamma_k$ decreases with the +non-linear Newton step as the non-linear solution is approached. This +inexact Newton method is generally more robust and computationally +more efficient than exact methods \citep[e.g.,][]{knoll04:_jfnk}. +% \footnote{The general idea behind +% inexact Newton methods is this: The Krylov solver is ``only'' +% used to find an intermediate solution of the linear +% equation\,(\ref{eq:jfnklin}) that is used to improve the approximation of +% the actual equation\,(\ref{eq:matrixmom}). With the choice of a +% relatively weak lower limit for FGMRES convergence +% $\gamma_{\min}$ we make sure that the time spent in the FGMRES +% solver is reduced at the cost of more Newton iterations. Newton +% iterations are cheaper than Krylov iterations so that this choice +% improves the overall efficiency.} +Typical parameter choices are +$\gamma_0=\code{JFNKgamma\_lin\_max}=0.99$, +$\gamma_{\min}=\code{JFNKgamma\_lin\_min}=0.1$, and $r = +\code{JFNKres\_tFac}\times\|\vek{F}(\vek{x}^{0})\|$ with +$\code{JFNKres\_tFac} = \frac{1}{2}$. We recommend a maximum number of +non-linear iterations $\code{SEAICEnewtonIterMax} = 100$ and a maximum +number of Krylov iterations $\code{SEAICEkrylovIterMax} = 50$, because +the Krylov subspace has a fixed dimension of 50. \paragraph{Elastic-Viscous-Plastic (EVP) Dynamics\label{sec:pkg:seaice:EVPdynamics}}~\\ %