/[MITgcm]/manual/s_phys_pkgs/text/seaice.tex

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-revision 1.16 by mlosch,
Wed Mar  2 13:46:38 2011 UTC
+revision 1.20 by mlosch,
Mon Mar 31 11:30:21 2014 UTC
 Line 61 
 no additional CPP options are required.
  (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
+ via CPP preprocessor flags. These options are set in
- \code{SEAICE\_OPTIONS.h} or in \code{ECCO\_CPPOPTIONS.h}.
+ \code{SEAICE\_OPTIONS.h}.
- Table \ref{tab:pkg:seaice:cpp} summarizes these options.
+ Table \ref{tab:pkg:seaice:cpp} summarizes the most important ones.
  \begin{table}[!ht]
  \centering
 Line 80 
 Table \ref{tab:pkg:seaice:cpp} summarize
          \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} &
+         \code{SEAICE\_ZETA\_SMOOTHREG} &
-           enable 8-category thermodynamics (by default undefined)\\
+           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} &
+         \code{SEAICE\_VARIABLE\_SALINITY} &
-           enable "salty" sea-ice (by default undefined) \\
+           enable sea-ice with variable salinity (by default undefined) \\
-         \code{SEAICE\_AGE} &
+         \code{SEAICE\_SITRACER} &
-           enable "age tracer" sea-ice (by default undefined) \\
+           enable sea-ice tracer package (by default undefined) \\
-         \code{SEAICE\_CAP\_HEFF} &
-           enable capping of sea-ice thickness to MAX\_HEFF \\ \hline
          \code{SEAICE\_BICE\_STRESS} &
            B-grid only for backward compatiblity: turn on ice-stress on
            ocean\\
 Line 105 
 Table \ref{tab:pkg:seaice:cpp} summarize
        \hline
      \end{tabular}
    }
-   \caption{~}
+   \caption{Some of the most relevant CPP preprocessor flags in the
+     \code{seaice}-package.}
  \end{table}
  %----------------------------------------------------------------------
-Line 139 
 Table~\ref{tab:pkg:seaice:runtimeparms}
+Line 140 
 Table~\ref{tab:pkg:seaice:runtimeparms}
    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}
  %----------------------------------------------------------------------
-Line 187 
 improved:
+Line 186 
 improved:
  \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
+ \item three different solution methods for solving the nonlinear
-   momentum equations have been adopted: LSOR \citep{zhang97}, and EVP
+   momentum equations have been adopted: LSOR \citep{zhang97}, EVP
-   \citep{hun97};
+   \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
-Line 296 
 air and ocean drag coefficients; $\rho_{
+Line 295 
 air and ocean drag coefficients; $\rho_{
  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
-Line 317 
 The ice strain rate is given by
+Line 316 
 The ice strain rate is given by
  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
-Line 349 
 recommended). For stress tensor computat
+Line 348 
 recommended). For stress tensor computat
  = 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
+ Defining the CPP-flag \code{SEAICE\_ZETA\_SMOOTHREG} in
- semi-implicit line successive over relaxation (LSOR)-solver of
+ \code{SEAICE\_OPTIONS.h} before compiling replaces the method for
- \citet{zhang97}, which allows for long time steps that, in our case,
+ bounding $\zeta$ by a smooth (differentiable) expression:
- are limited by the explicit treatment of the Coriolis term. The
+ \begin{equation}
- explicit treatment of the Coriolis term does not represent a severe
+   \label{eq:zetaregsmooth}
- limitation because it restricts the time step to approximately the
+   \begin{split}
- same length as in the ocean model where the Coriolis term is also
+   \zeta &= \zeta_{\max}\tanh\left(\frac{P}{2\,\min(\Delta,\Delta_{\min})
- treated explicitly.
+       \,\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 =
+.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}}~\\
  %
-Line 801 
 Diagnostics output is available via the
+Line 970 
 Diagnostics output is available via the
  Available output fields are summarized in
  Table \ref{tab:pkg:seaice:diagnostics}.
- \begin{table}[!ht]
+ \input{s_phys_pkgs/text/seaice_diags.tex}
- \centering
- \label{tab:pkg:seaice:diagnostics}
- {\footnotesize
- \begin{verbatim}
- ---------+----+----+----------------+-----------------
-  <-Name->|Levs|grid|<--  Units   -->|<- Tile (max=80c)
- ---------+----+----+----------------+-----------------
-  SIarea  |  1 |SM  |m^2/m^2         |SEAICE fractional ice-covered area [0 to 1]
-  SIheff  |  1 |SM  |m               |SEAICE effective ice thickness
-  SIuice  |  1 |UU  |m/s             |SEAICE zonal ice velocity, >0 from West to East
-  SIvice  |  1 |VV  |m/s             |SEAICE merid. ice velocity, >0 from South to North
-  SIhsnow |  1 |SM  |m               |SEAICE snow thickness
-  SIhsalt |  1 |SM  |g/m^2           |SEAICE effective salinity
-  SIatmFW |  1 |SM  |kg/m^2/s        |Net freshwater flux from the atmosphere (+=down)
-  SIuwind |  1 |SM  |m/s             |SEAICE zonal 10-m wind speed, >0 increases uVel
-  SIvwind |  1 |SM  |m/s             |SEAICE meridional 10-m wind speed, >0 increases uVel
-  SIfu    |  1 |UU  |N/m^2           |SEAICE zonal surface wind stress, >0 increases uVel
-  SIfv    |  1 |VV  |N/m^2           |SEAICE merid. surface wind stress, >0 increases vVel
-  SIempmr |  1 |SM  |kg/m^2/s        |SEAICE upward freshwater flux, > 0 increases salt
-  SIqnet  |  1 |SM  |W/m^2           |SEAICE upward heatflux, turb+rad, >0 decreases theta
-  SIqsw   |  1 |SM  |W/m^2           |SEAICE upward shortwave radiat., >0 decreases theta
-  SIpress |  1 |SM  |m^2/s^2         |SEAICE strength (with upper and lower limit)
-  SIzeta  |  1 |SM  |m^2/s           |SEAICE nonlinear bulk viscosity
-  SIeta   |  1 |SM  |m^2/s           |SEAICE nonlinear shear viscosity
-  SIsigI  |  1 |SM  |no units        |SEAICE normalized principle stress, component one
-  SIsigII |  1 |SM  |no units        |SEAICE normalized principle stress, component two
-  SIthdgrh|  1 |SM  |m/s             |SEAICE thermodynamic growth rate of effective ice thickness
-  SIsnwice|  1 |SM  |m/s             |SEAICE ice formation rate due to flooding
-  SIuheff |  1 |UU  |m^2/s           |Zonal Transport of effective ice thickness
-  SIvheff |  1 |VV  |m^2/s           |Meridional Transport of effective ice thickness
-  ADVxHEFF|  1 |UU  |m.m^2/s         |Zonal      Advective Flux of eff ice thickn
-  ADVyHEFF|  1 |VV  |m.m^2/s         |Meridional Advective Flux of eff ice thickn
-  DFxEHEFF|  1 |UU  |m.m^2/s         |Zonal      Diffusive Flux of eff ice thickn
-  DFyEHEFF|  1 |VV  |m.m^2/s         |Meridional Diffusive Flux of eff ice thickn
-  ADVxAREA|  1 |UU  |m^2/m^2.m^2/s   |Zonal      Advective Flux of fract area
-  ADVyAREA|  1 |VV  |m^2/m^2.m^2/s   |Meridional Advective Flux of fract area
-  DFxEAREA|  1 |UU  |m^2/m^2.m^2/s   |Zonal      Diffusive Flux of fract area
-  DFyEAREA|  1 |VV  |m^2/m^2.m^2/s   |Meridional Diffusive Flux of fract area
-  ADVxSNOW|  1 |UU  |m.m^2/s         |Zonal      Advective Flux of eff snow thickn
-  ADVySNOW|  1 |VV  |m.m^2/s         |Meridional Advective Flux of eff snow thickn
-  DFxESNOW|  1 |UU  |m.m^2/s         |Zonal      Diffusive Flux of eff snow thickn
-  DFyESNOW|  1 |VV  |m.m^2/s         |Meridional Diffusive Flux of eff snow thickn
-  ADVxSSLT|  1 |UU  |psu.m^2/s       |Zonal      Advective Flux of seaice salinity
-  ADVySSLT|  1 |VV  |psu.m^2/s       |Meridional Advective Flux of seaice salinity
-  DFxESSLT|  1 |UU  |psu.m^2/s       |Zonal      Diffusive Flux of seaice salinity
-  DFyESSLT|  1 |VV  |psu.m^2/s       |Meridional Diffusive Flux of seaice salinity
- \end{verbatim}
- }
- \caption{Available diagnostics of the seaice-package}
- \end{table}
  %\subsubsection{Package Reference}

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