% $Header: /home/ubuntu/mnt/e9_copy/manual/s_phys_pkgs/text/seaice.tex,v 1.19 2011/12/14 11:22:42 mlosch Exp $ % $Name: $ %%EH3 Copied from "MITgcm/pkg/seaice/seaice_description.tex" %%EH3 which was written by Dimitris M. \subsection{SEAICE Package} \label{sec:pkg:seaice} \begin{rawhtml} \end{rawhtml} Authors: Martin Losch, Dimitris Menemenlis, An Nguyen, Jean-Michel Campin, Patrick Heimbach, Chris Hill and Jinlun Zhang %---------------------------------------------------------------------- \subsubsection{Introduction \label{sec:pkg:seaice:intro}} Package ``seaice'' provides a dynamic and thermodynamic interactive sea-ice model. CPP options enable or disable different aspects of the package (Section \ref{sec:pkg:seaice:config}). Run-Time options, flags, filenames and field-related dates/times are set in \code{data.seaice} (Section \ref{sec:pkg:seaice:runtime}). A description of key subroutines is given in Section \ref{sec:pkg:seaice:subroutines}. Input fields, units and sign conventions are summarized in Section \ref{sec:pkg:seaice:fields_units}, and available diagnostics output is listed in Section \ref{sec:pkg:seaice:diagnostics}. %---------------------------------------------------------------------- \subsubsection{SEAICE configuration, compiling \& running} \paragraph{Compile-time options \label{sec:pkg:seaice:config}} ~ As with all MITgcm packages, SEAICE can be turned on or off at compile time % \begin{itemize} % \item using the \code{packages.conf} file by adding \code{seaice} to it, % \item or using \code{genmake2} adding \code{-enable=seaice} or \code{-disable=seaice} switches % \item \textit{required packages and CPP options}: \\ SEAICE requires the external forcing package \code{exf} to be enabled; no additional CPP options are required. % \end{itemize} (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. \begin{table}[!ht] \centering \label{tab:pkg:seaice:cpp} {\footnotesize \begin{tabular}{|l|p{10cm}|} \hline \textbf{CPP option} & \textbf{Description} \\ \hline \hline \code{SEAICE\_DEBUG} & Enhance STDOUT for debugging \\ \code{SEAICE\_ALLOW\_DYNAMICS} & sea-ice dynamics code \\ \code{SEAICE\_CGRID} & LSR solver on C-grid (rather than original B-grid) \\ \code{SEAICE\_ALLOW\_EVP} & use EVP rather than LSR 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\_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\_BICE\_STRESS} & B-grid only for backward compatiblity: turn on ice-stress on ocean\\ \code{EXPLICIT\_SSH\_SLOPE} & B-grid only for backward compatiblity: use ETAN for tilt computations rather than geostrophic velocities \\ \hline \end{tabular} } \caption{~} \end{table} %---------------------------------------------------------------------- \subsubsection{Run-time parameters \label{sec:pkg:seaice:runtime}} Run-time parameters (see Table~\ref{tab:pkg:seaice:runtimeparms}) are set in files \code{data.pkg} (read in \code{packages\_readparms.F}), and \code{data.seaice} (read in \code{seaice\_readparms.F}). \paragraph{Enabling the package} ~ \\ % A package is switched on/off at run-time by setting (e.g. for SEAICE) \code{useSEAICE = .TRUE.} in \code{data.pkg}. \paragraph{General flags and parameters} ~ \\ % Table~\ref{tab:pkg:seaice:runtimeparms} lists most run-time parameters. \input{s_phys_pkgs/text/seaice-parms.tex} \paragraph{Input fields and units\label{sec:pkg:seaice:fields_units}} \begin{description} \item[\code{HeffFile}:] Initial sea ice thickness averaged over grid cell in meters; initializes variable \code{HEFF}; \item[\code{AreaFile}:] Initial fractional sea ice cover, range $[0,1]$; initializes variable \code{AREA}; \item[\code{HsnowFile}:] Initial snow thickness on sea ice averaged 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} %---------------------------------------------------------------------- \subsubsection{Description \label{sec:pkg:seaice:descr}} [TO BE CONTINUED/MODIFIED] % Sea-ice model thermodynamics are based on Hibler % \cite{hib80}, that is, a 2-category model that simulates ice thickness % and concentration. Snow is simulated as per Zhang et al. % \cite{zha98a}. Although recent years have seen an increased use of % multi-category thickness distribution sea-ice models for climate % studies, the Hibler 2-category ice model is still the most widely used % model and has resulted in realistic simulation of sea-ice variability % on regional and global scales. Being less complicated, compared to % multi-category models, the 2-category model permits easier application % of adjoint model optimization methods. % Note, however, that the Hibler 2-category model and its variants use a % so-called zero-layer thermodynamic model to estimate ice growth and % decay. The zero-layer thermodynamic model assumes that ice does not % store heat and, therefore, tends to exaggerate the seasonal % variability in ice thickness. This exaggeration can be significantly % reduced by using Semtner's \cite{sem76} three-layer thermodynamic % model that permits heat storage in ice. Recently, the three-layer % thermodynamic model has been reformulated by Winton \cite{win00}. The % reformulation improves model physics by representing the brine content % of the upper ice with a variable heat capacity. It also improves % model numerics and consumes less computer time and memory. The Winton % sea-ice thermodynamics have been ported to the MIT GCM; they currently % reside under pkg/thsice. The package pkg/thsice is fully % compatible with pkg/seaice and with pkg/exf. When turned on togeter % with pkg/seaice, the zero-layer thermodynamics are replaced by the by % Winton thermodynamics The MITgcm sea ice model (MITgcm/sim) is based on a variant of the viscous-plastic (VP) dynamic-thermodynamic sea ice model \citep{zhang97} 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: \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{hun97}; \item ice-ocean stress can be formulated as in \citet{hibler87} or as in \citet{cam08}; \item ice variables are advected by sophisticated, conservative advection schemes 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. Heat, fresh water fluxes and surface stresses are computed from the atmospheric state and -- by default -- modified by the ice model at every time step. The ice dynamics models that are most widely used for large-scale climate studies are the viscous-plastic (VP) model \citep{hib79}, the cavitating fluid (CF) model \citep{fla92}, and the elastic-viscous-plastic (EVP) model \citep{hun97}. Compared to the VP model, the CF model does not allow ice shear in calculating ice motion, stress, and deformation. EVP models approximate VP by adding an elastic term to the equations for easier adaptation to parallel computers. Because of its higher accuracy in plastic solution and relatively simpler formulation, compared to the EVP model, we decided to use the VP model as the default dynamic component of our ice model. To do this we extended the line successive over relaxation (LSOR) method of \citet{zhang97} for use in a parallel configuration. An EVP model and a free-drift implemtation can be selected with runtime flags. \paragraph{Compatibility with ice-thermodynamics package \code{thsice}\label{sec:pkg:seaice:thsice}}~\\ % Note, that by default the \code{seaice}-package includes the orginial so-called zero-layer thermodynamics following \citet{hib80} with a snow cover as in \citet{zha98a}. The zero-layer thermodynamic model assumes that ice does not store heat and, therefore, tends to exaggerate the seasonal variability in ice thickness. This exaggeration can be significantly reduced by using \citeauthor{sem76}'s~[\citeyear{sem76}] three-layer thermodynamic model that permits heat storage in ice. Recently, the three-layer thermodynamic model has been reformulated by \citet{win00}. The reformulation improves model physics by representing the brine content of the upper ice with a variable heat capacity. It also improves model numerics and consumes less computer time and memory. The Winton sea-ice thermodynamics have been ported to the MIT GCM; they currently reside under \code{pkg/thsice}. The package \code{thsice} is described in section~\ref{sec:pkg:thsice}; it is fully compatible with the packages \code{seaice} and \code{exf}. When turned on together with \code{seaice}, the zero-layer thermodynamics are replaced by the Winton thermodynamics. In order to use the \code{seaice}-package with the thermodynamics of \code{thsice}, compile both packages and turn both package on in \code{data.pkg}; see an example in \code{global\_ocean.cs32x15/input.icedyn}. Note, that once \code{thsice} is turned on, the variables and diagnostics associated to the default thermodynamics are meaningless, and the diagnostics of \code{thsice} have to be used instead. \paragraph{Surface forcing\label{sec:pkg:seaice:surfaceforcing}}~\\ % The sea ice model requires the following input fields: 10-m winds, 2-m air temperature and specific humidity, downward longwave and shortwave radiations, precipitation, evaporation, and river and glacier runoff. The sea ice model also requires surface temperature from the ocean model and the top level horizontal velocity. Output fields are surface wind stress, evaporation minus precipitation minus runoff, net surface heat flux, and net shortwave flux. The sea-ice model is global: in ice-free regions bulk formulae are used to estimate oceanic forcing from the atmospheric fields. \paragraph{Dynamics\label{sec:pkg:seaice:dynamics}}~\\ % \newcommand{\vek}[1]{\ensuremath{\vec{\mathbf{#1}}}} \newcommand{\vtau}{\vek{\mathbf{\tau}}} The momentum equation of the sea-ice model is \begin{equation} \label{eq:momseaice} m \frac{D\vek{u}}{Dt} = -mf\vek{k}\times\vek{u} + \vtau_{air} + \vtau_{ocean} - m \nabla{\phi(0)} + \vek{F}, \end{equation} where $m=m_{i}+m_{s}$ is the ice and snow mass per unit area; $\vek{u}=u\vek{i}+v\vek{j}$ is the ice velocity vector; $\vek{i}$, $\vek{j}$, and $\vek{k}$ are unit vectors in the $x$, $y$, and $z$ directions, respectively; $f$ is the Coriolis parameter; $\vtau_{air}$ and $\vtau_{ocean}$ are the wind-ice and ocean-ice stresses, respectively; $g$ is the gravity accelation; $\nabla\phi(0)$ is the gradient (or tilt) of the sea surface height; $\phi(0) = g\eta + p_{a}/\rho_{0} + mg/\rho_{0}$ 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{cam08}; 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*} \vtau_{air} = & \rho_{air} C_{air} |\vek{U}_{air} -\vek{u}| R_{air} (\vek{U}_{air} -\vek{u}), \\ \vtau_{ocean} = & \rho_{ocean}C_{ocean} |\vek{U}_{ocean}-\vek{u}| R_{ocean}(\vek{U}_{ocean}-\vek{u}), \end{align*} where $\vek{U}_{air/ocean}$ are the surface winds of the atmosphere and surface currents of the ocean, respectively; $C_{air/ocean}$ are air and ocean drag coefficients; $\rho_{air/ocean}$ are reference 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}}~\\ % 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{hib79, zhang97}: \begin{equation} \label{eq:vpequation} \sigma_{ij}=2\eta(\dot{\epsilon}_{ij},P)\dot{\epsilon}_{ij} + \left[\zeta(\dot{\epsilon}_{ij},P) - \eta(\dot{\epsilon}_{ij},P)\right]\dot{\epsilon}_{kk}\delta_{ij} - \frac{P}{2}\delta_{ij}. \end{equation} The ice strain rate is given by \begin{equation*} \dot{\epsilon}_{ij} = \frac{1}{2}\left( \frac{\partial{u_{i}}}{\partial{x_{j}}} + \frac{\partial{u_{j}}}{\partial{x_{i}}}\right). \end{equation*} 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\,\exp\{-C^{*}\cdot(1-c)\}, \label{eq:icestrength} \end{equation} with the constants $P^{*}$ (run-time parameter \code{SEAICE\_strength}) and $C^{*}=20$. The nonlinear bulk and shear viscosities $\eta$ and $\zeta$ are functions of ice strain rate invariants and ice strength such that the principal components of the stress lie on an elliptical yield curve with the ratio of major to minor axis $e$ equal to $2$; they are given by: \begin{align*} \zeta =& \min\left(\frac{P_{\max}}{2\max(\Delta,\Delta_{\min})}, \zeta_{\max}\right) \\ \eta =& \frac{\zeta}{e^2} \\ \intertext{with the abbreviation} \Delta = & \left[ \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}}. \end{align*} The bulk viscosities are bounded above by imposing both a minimum $\Delta_{\min}$ (for numerical reasons, run-time parameter \code{SEAICE\_EPS} with a default value of $10^{-10}\text{\,s}^{-1}$) and a maximum $\zeta_{\max} = P_{\max}/\Delta^*$, where $\Delta^*=(5\times10^{12}/2\times10^4)\text{\,s}^{-1}$. (There is also the option of bounding $\zeta$ from below by setting run-time parameter \code{SEAICE\_zetaMin} $>0$, but this is generally not recommended). For stress tensor computation the replacement pressure $P = 2\,\Delta\zeta$ \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. \paragraph{Elastic-Viscous-Plastic (EVP) Dynamics\label{sec:pkg:seaice:EVPdynamics}}~\\ % \citet{hun97}'s introduced an elastic contribution to the strain rate in order to regularize Eq.~\ref{eq:vpequation} in such a way that the resulting elastic-viscous-plastic (EVP) and VP models are identical at steady state, \begin{equation} \label{eq:evpequation} \frac{1}{E}\frac{\partial\sigma_{ij}}{\partial{t}} + \frac{1}{2\eta}\sigma_{ij} + \frac{\eta - \zeta}{4\zeta\eta}\sigma_{kk}\delta_{ij} + \frac{P}{4\zeta}\delta_{ij} = \dot{\epsilon}_{ij}. \end{equation} %In the EVP model, equations for the components of the stress tensor %$\sigma_{ij}$ are solved explicitly. Both model formulations will be %used and compared the present sea-ice model study. The EVP-model uses an explicit time stepping scheme with a short timestep. According to the recommendation of \citet{hun97}, the EVP-model should be stepped forward in time 120 times ($\code{SEAICE\_deltaTevp} = \code{SEAICIE\_deltaTdyn}/120$) 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 internal timestep and simple to implement on parallel computers \citep{hun97}. 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 $\sigma_{12}$. Introducing the divergence $D_D = \dot{\epsilon}_{11}+\dot{\epsilon}_{22}$, 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~\ref{eq:evpequation} can be written as: \begin{align} \label{eq:evpstresstensor1} \frac{\partial\sigma_{1}}{\partial{t}} + \frac{\sigma_{1}}{2T} + \frac{P}{2T} &= \frac{P}{2T\Delta} D_D \\ \label{eq:evpstresstensor2} \frac{\partial\sigma_{2}}{\partial{t}} + \frac{\sigma_{2} e^{2}}{2T} &= \frac{P}{2T\Delta} D_T \\ \label{eq:evpstresstensor12} \frac{\partial\sigma_{12}}{\partial{t}} + \frac{\sigma_{12} e^{2}}{2T} &= \frac{P}{4T\Delta} D_S \end{align} Here, the elastic parameter $E$ is redefined in terms of a damping timescale $T$ for elastic waves \[E=\frac{\zeta}{T}.\] $T=E_{0}\Delta{t}$ with the tunable parameter $E_0<1$ and the external (long) timestep $\Delta{t}$. $E_{0} = \frac{1}{3}$ is the default value in the code and close to what \citet{hun97} and \citet{hun01} recommend. To use the EVP solver, make sure that both \code{SEAICE\_CGRID} and \code{SEAICE\_ALLOW\_EVP} are defined in \code{SEAICE\_OPTIONS.h} (default). The solver is turned on by setting the sub-cycling time step \code{SEAICE\_deltaTevp} to a value larger than zero. The choice of this time step is under debate. \citet{hun97} recommend order(120) time steps for the EVP solver within one model time step $\Delta{t}$ (\code{deltaTmom}). One can also choose order(120) time steps within the forcing time scale, but then we recommend adjusting the damping time scale $T$ accordingly, by setting either \code{SEAICE\_elasticParm} ($E_{0}$), so that $E_{0}\Delta{t}=\mbox{forcing time scale}$, or directly \code{SEAICE\_evpTauRelax} ($T$) to the forcing time scale. \paragraph{Truncated ellipse method (TEM) for yield curve \label{sec:pkg:seaice:TEM}}~\\ % In the so-called truncated ellipse method the shear viscosity $\eta$ is capped to suppress any tensile stress \citep{hibler97, geiger98}: \begin{equation} \label{eq:etatem} \eta = \min\left(\frac{\zeta}{e^2}, \frac{\frac{P}{2}-\zeta(\dot{\epsilon}_{11}+\dot{\epsilon}_{22})} {\sqrt{(\dot{\epsilon}_{11}+\dot{\epsilon}_{22})^2 +4\dot{\epsilon}_{12}^2}}\right). \end{equation} To enable this method, set \code{\#define SEAICE\_ALLOW\_TEM} in \code{SEAICE\_OPTIONS.h} and turn it on with \code{SEAICEuseTEM} in \code{data.seaice}. \paragraph{Ice-Ocean stress \label{sec:pkg:seaice:iceoceanstress}}~\\ % Moving sea ice exerts a stress on the ocean which is the opposite of the stress $\vtau_{ocean}$ in Eq.~\ref{eq:momseaice}. This stess is applied directly to the surface layer of the ocean model. An alternative ocean stress formulation is given by \citet{hibler87}. Rather than applying $\vtau_{ocean}$ directly, the stress is derived from integrating over the ice thickness to the bottom of the oceanic surface layer. In the resulting equation for the \emph{combined} ocean-ice momentum, the interfacial stress cancels and the total stress appears as the sum of windstress and divergence of internal ice stresses: $\delta(z) (\vtau_{air} + \vek{F})/\rho_0$, \citep[see also Eq.\,2 of][]{hibler87}. The disadvantage of this formulation is that now the velocity in the surface layer of the ocean that is used to advect tracers, is really an average over the ocean surface velocity and the ice velocity leading to an inconsistency as the ice temperature and salinity are different from the oceanic variables. To turn on the stress formulation of \citet{hibler87}, set \code{useHB87StressCoupling=.TRUE.} in \code{data.seaice}. % Our discretization differs from \citet{zhang97, 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 % differences and averaging is only involved in computing $\Delta$ and % $P$ at vorticity points. \paragraph{Finite-volume discretization of the stress tensor divergence\label{sec:pkg:seaice:discretization}}~\\ % On an Arakawa C~grid, ice thickness and concentration and thus ice strength $P$ and bulk and shear viscosities $\zeta$ and $\eta$ are naturally defined a C-points in the center of the grid cell. Discretization requires only averaging of $\zeta$ and $\eta$ to vorticity or Z-points (or $\zeta$-points, but here we use Z in order avoid confusion with the bulk viscosity) at the bottom left corner of the cell to give $\overline{\zeta}^{Z}$ and $\overline{\eta}^{Z}$. In the following, the superscripts indicate location at Z or C points, distance across the cell (F), along the cell edge (G), between $u$-points (U), $v$-points (V), and C-points (C). The control volumes of the $u$- and $v$-equations in the grid cell at indices $(i,j)$ are $A_{i,j}^{w}$ and $A_{i,j}^{s}$, respectively. With these definitions (which follow the model code documentation except that $\zeta$-points have been renamed to Z-points), the strain rates are discretized as: \begin{align} \dot{\epsilon}_{11} &= \partial_{1}{u}_{1} + k_{2}u_{2} \\ \notag => (\epsilon_{11})_{i,j}^C &= \frac{u_{i+1,j}-u_{i,j}}{\Delta{x}_{i,j}^{F}} + k_{2,i,j}^{C}\frac{v_{i,j+1}+v_{i,j}}{2} \\ \dot{\epsilon}_{22} &= \partial_{2}{u}_{2} + k_{1}u_{1} \\\notag => (\epsilon_{22})_{i,j}^C &= \frac{v_{i,j+1}-v_{i,j}}{\Delta{y}_{i,j}^{F}} + k_{1,i,j}^{C}\frac{u_{i+1,j}+u_{i,j}}{2} \\ \dot{\epsilon}_{12} = \dot{\epsilon}_{21} &= \frac{1}{2}\biggl( \partial_{1}{u}_{2} + \partial_{2}{u}_{1} - k_{1}u_{2} - k_{2}u_{1} \biggr) \\ \notag => (\epsilon_{12})_{i,j}^Z &= \frac{1}{2} \biggl( \frac{v_{i,j}-v_{i-1,j}}{\Delta{x}_{i,j}^V} + \frac{u_{i,j}-u_{i,j-1}}{\Delta{y}_{i,j}^U} \\\notag &\phantom{=\frac{1}{2}\biggl(} - k_{1,i,j}^{Z}\frac{v_{i,j}+v_{i-1,j}}{2} - k_{2,i,j}^{Z}\frac{u_{i,j}+u_{i,j-1}}{2} \biggr), \end{align} so that the diagonal terms of the strain rate tensor are naturally defined at C-points and the symmetric off-diagonal term at Z-points. No-slip boundary conditions ($u_{i,j-1}+u_{i,j}=0$ and $v_{i-1,j}+v_{i,j}=0$ across boundaries) are implemented via ``ghost-points''; for free slip boundary conditions $(\epsilon_{12})^Z=0$ on boundaries. For a spherical polar grid, the coefficients of the metric terms are $k_{1}=0$ and $k_{2}=-\tan\phi/a$, with the spherical radius $a$ and the latitude $\phi$; $\Delta{x}_1 = \Delta{x} = a\cos\phi \Delta\lambda$, and $\Delta{x}_2 = \Delta{y}=a\Delta\phi$. For a general orthogonal curvilinear grid, $k_{1}$ and $k_{2}$ can be approximated by finite differences of the cell widths: \begin{align} k_{1,i,j}^{C} &= \frac{1}{\Delta{y}_{i,j}^{F}} \frac{\Delta{y}_{i+1,j}^{G}-\Delta{y}_{i,j}^{G}}{\Delta{x}_{i,j}^{F}} \\ k_{2,i,j}^{C} &= \frac{1}{\Delta{x}_{i,j}^{F}} \frac{\Delta{x}_{i,j+1}^{G}-\Delta{x}_{i,j}^{G}}{\Delta{y}_{i,j}^{F}} \\ k_{1,i,j}^{Z} &= \frac{1}{\Delta{y}_{i,j}^{U}} \frac{\Delta{y}_{i,j}^{C}-\Delta{y}_{i-1,j}^{C}}{\Delta{x}_{i,j}^{V}} \\ k_{2,i,j}^{Z} &= \frac{1}{\Delta{x}_{i,j}^{V}} \frac{\Delta{x}_{i,j}^{C}-\Delta{x}_{i,j-1}^{C}}{\Delta{y}_{i,j}^{U}} \end{align} The stress tensor is given by the constitutive viscous-plastic relation $\sigma_{\alpha\beta} = 2\eta\dot{\epsilon}_{\alpha\beta} + [(\zeta-\eta)\dot{\epsilon}_{\gamma\gamma} - P/2 ]\delta_{\alpha\beta}$ \citep{hib79}. The stress tensor divergence $(\nabla\sigma)_{\alpha} = \partial_\beta\sigma_{\beta\alpha}$, is discretized in finite volumes. This conveniently avoids dealing with further metric terms, as these are ``hidden'' in the differential cell widths. For the $u$-equation ($\alpha=1$) we have: \begin{align} (\nabla\sigma)_{1}: \phantom{=}& \frac{1}{A_{i,j}^w} \int_{\mathrm{cell}}(\partial_1\sigma_{11}+\partial_2\sigma_{21})\,dx_1\,dx_2 \\\notag =& \frac{1}{A_{i,j}^w} \biggl\{ \int_{x_2}^{x_2+\Delta{x}_2}\sigma_{11}dx_2\biggl|_{x_{1}}^{x_{1}+\Delta{x}_{1}} + \int_{x_1}^{x_1+\Delta{x}_1}\sigma_{21}dx_1\biggl|_{x_{2}}^{x_{2}+\Delta{x}_{2}} \biggr\} \\ \notag \approx& \frac{1}{A_{i,j}^w} \biggl\{ \Delta{x}_2\sigma_{11}\biggl|_{x_{1}}^{x_{1}+\Delta{x}_{1}} + \Delta{x}_1\sigma_{21}\biggl|_{x_{2}}^{x_{2}+\Delta{x}_{2}} \biggr\} \\ \notag =& \frac{1}{A_{i,j}^w} \biggl\{ (\Delta{x}_2\sigma_{11})_{i,j}^C - (\Delta{x}_2\sigma_{11})_{i-1,j}^C \\\notag \phantom{=}& \phantom{\frac{1}{A_{i,j}^w} \biggl\{} + (\Delta{x}_1\sigma_{21})_{i,j+1}^Z - (\Delta{x}_1\sigma_{21})_{i,j}^Z \biggr\} \end{align} with \begin{align} (\Delta{x}_2\sigma_{11})_{i,j}^C =& \phantom{+} \Delta{y}_{i,j}^{F}(\zeta + \eta)^{C}_{i,j} \frac{u_{i+1,j}-u_{i,j}}{\Delta{x}_{i,j}^{F}} \\ \notag &+ \Delta{y}_{i,j}^{F}(\zeta + \eta)^{C}_{i,j} k_{2,i,j}^C \frac{v_{i,j+1}+v_{i,j}}{2} \\ \notag \phantom{=}& + \Delta{y}_{i,j}^{F}(\zeta - \eta)^{C}_{i,j} \frac{v_{i,j+1}-v_{i,j}}{\Delta{y}_{i,j}^{F}} \\ \notag \phantom{=}& + \Delta{y}_{i,j}^{F}(\zeta - \eta)^{C}_{i,j} k_{1,i,j}^{C}\frac{u_{i+1,j}+u_{i,j}}{2} \\ \notag \phantom{=}& - \Delta{y}_{i,j}^{F} \frac{P}{2} \\ (\Delta{x}_1\sigma_{21})_{i,j}^Z =& \phantom{+} \Delta{x}_{i,j}^{V}\overline{\eta}^{Z}_{i,j} \frac{u_{i,j}-u_{i,j-1}}{\Delta{y}_{i,j}^{U}} \\ \notag & + \Delta{x}_{i,j}^{V}\overline{\eta}^{Z}_{i,j} \frac{v_{i,j}-v_{i-1,j}}{\Delta{x}_{i,j}^{V}} \\ \notag & - \Delta{x}_{i,j}^{V}\overline{\eta}^{Z}_{i,j} k_{2,i,j}^{Z}\frac{u_{i,j}+u_{i,j-1}}{2} \\ \notag & - \Delta{x}_{i,j}^{V}\overline{\eta}^{Z}_{i,j} k_{1,i,j}^{Z}\frac{v_{i,j}+v_{i-1,j}}{2} \end{align} Similarly, we have for the $v$-equation ($\alpha=2$): \begin{align} (\nabla\sigma)_{2}: \phantom{=}& \frac{1}{A_{i,j}^s} \int_{\mathrm{cell}}(\partial_1\sigma_{12}+\partial_2\sigma_{22})\,dx_1\,dx_2 \\\notag =& \frac{1}{A_{i,j}^s} \biggl\{ \int_{x_2}^{x_2+\Delta{x}_2}\sigma_{12}dx_2\biggl|_{x_{1}}^{x_{1}+\Delta{x}_{1}} + \int_{x_1}^{x_1+\Delta{x}_1}\sigma_{22}dx_1\biggl|_{x_{2}}^{x_{2}+\Delta{x}_{2}} \biggr\} \\ \notag \approx& \frac{1}{A_{i,j}^s} \biggl\{ \Delta{x}_2\sigma_{12}\biggl|_{x_{1}}^{x_{1}+\Delta{x}_{1}} + \Delta{x}_1\sigma_{22}\biggl|_{x_{2}}^{x_{2}+\Delta{x}_{2}} \biggr\} \\ \notag =& \frac{1}{A_{i,j}^s} \biggl\{ (\Delta{x}_2\sigma_{12})_{i+1,j}^Z - (\Delta{x}_2\sigma_{12})_{i,j}^Z \\ \notag \phantom{=}& \phantom{\frac{1}{A_{i,j}^s} \biggl\{} + (\Delta{x}_1\sigma_{22})_{i,j}^C - (\Delta{x}_1\sigma_{22})_{i,j-1}^C \biggr\} \end{align} with \begin{align} (\Delta{x}_1\sigma_{12})_{i,j}^Z =& \phantom{+} \Delta{y}_{i,j}^{U}\overline{\eta}^{Z}_{i,j} \frac{u_{i,j}-u_{i,j-1}}{\Delta{y}_{i,j}^{U}} \\\notag & + \Delta{y}_{i,j}^{U}\overline{\eta}^{Z}_{i,j} \frac{v_{i,j}-v_{i-1,j}}{\Delta{x}_{i,j}^{V}} \\\notag &- \Delta{y}_{i,j}^{U}\overline{\eta}^{Z}_{i,j} k_{2,i,j}^{Z}\frac{u_{i,j}+u_{i,j-1}}{2} \\\notag & - \Delta{y}_{i,j}^{U}\overline{\eta}^{Z}_{i,j} k_{1,i,j}^{Z}\frac{v_{i,j}+v_{i-1,j}}{2} \\ \notag (\Delta{x}_2\sigma_{22})_{i,j}^C =& \phantom{+} \Delta{x}_{i,j}^{F}(\zeta - \eta)^{C}_{i,j} \frac{u_{i+1,j}-u_{i,j}}{\Delta{x}_{i,j}^{F}} \\ \notag &+ \Delta{x}_{i,j}^{F}(\zeta - \eta)^{C}_{i,j} k_{2,i,j}^{C} \frac{v_{i,j+1}+v_{i,j}}{2} \\ \notag & + \Delta{x}_{i,j}^{F}(\zeta + \eta)^{C}_{i,j} \frac{v_{i,j+1}-v_{i,j}}{\Delta{y}_{i,j}^{F}} \\ \notag & + \Delta{x}_{i,j}^{F}(\zeta + \eta)^{C}_{i,j} k_{1,i,j}^{C}\frac{u_{i+1,j}+u_{i,j}}{2} \\ \notag & -\Delta{x}_{i,j}^{F} \frac{P}{2} \end{align} Again, no slip boundary conditions are realized via ghost points and $u_{i,j-1}+u_{i,j}=0$ and $v_{i-1,j}+v_{i,j}=0$ across boundaries. For free slip boundary conditions the lateral stress is set to zeros. In analogy to $(\epsilon_{12})^Z=0$ on boundaries, we set $\sigma_{21}^{Z}=0$, or equivalently $\eta_{i,j}^{Z}=0$, on boundaries. \paragraph{Thermodynamics\label{sec:pkg:seaice:thermodynamics}}~\\ % 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, 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}. 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. 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. This flux is not assumed to instantaneously melt or create ice, but a time scale of three days (run-time parameter \code{SEAICE\_gamma\_t}) is used to relax $T_{w}$ to the freezing point. % The parameterization of lateral and vertical growth of sea ice follows that of \citet{hib79, hib80}; the so-called lead closing parameter $h_{0}$ (run-time parameter \code{HO}) has a default value of 0.5~meters. On top of the ice there is a layer of snow that modifies the heat flux and the albedo \citep{zha98a}. 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}. The flood-freeze algorithm is enabled with the CPP-flag \code{SEAICE\_ALLOW\_FLOODING} and turned on with run-time parameter \code{SEAICEuseFlooding=.true.}. \paragraph{Advection of thermodynamic variables\label{sec:pkg:seaice:advection}}~\\ % 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, we recommend flux-limited schemes \citep[multidimensional 2nd and 3rd-order advection scheme with flux limiter][]{roe:85, 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 schemes conserve volume and horizontal area and are unconditionally stable, so that we can set $D_{X}=0$. Run-timeflags: \code{SEAICEadvScheme} (default=2, is the historic 2nd-order, centered difference scheme), \code{DIFF1} = $D_{X}/\Delta{x}$ (default=0.004). The MITgcm sea ice model provides the option to use the thermodynamics model of \citet{win00}, which in turn is based on the 3-layer model of \citet{sem76} and which treats brine content by means of enthalpy conservation; the corresponding package \code{thsice} is described in section~\ref{sec:pkg:thsice}. This scheme requires additional state variables, namely the enthalpy of the two ice layers (instead of effective ice salinity), to be advected by ice velocities. % The internal sea ice temperature is inferred from ice enthalpy. To avoid unphysical (negative) values for ice thickness and concentration, a positive 2nd-order advection scheme with a SuperBee flux limiter \citep{roe:85} should be used to advect all sea-ice-related quantities of the \citet{win00} thermodynamic model (runtime flag \code{thSIceAdvScheme=77} and \code{thSIce\_diffK}=$D_{X}$=0 in \code{data.ice}, defaults are 0). Because of the non-linearity of the advection scheme, care must be taken in advecting these quantities: when simply using ice velocity to advect enthalpy, the total energy (i.e., the volume integral of enthalpy) is not conserved. Alternatively, one can advect the energy content (i.e., product of ice-volume and enthalpy) but then false enthalpy extrema can occur, which then leads to unrealistic ice temperature. In the currently implemented solution, the sea-ice mass flux is used to advect the enthalpy in order to ensure conservation of enthalpy and to prevent false enthalpy extrema. % %---------------------------------------------------------------------- \subsubsection{Key subroutines \label{sec:pkg:seaice:subroutines}} Top-level routine: \code{seaice\_model.F} {\footnotesize \begin{verbatim} C !CALLING SEQUENCE: c ... c seaice_model (TOP LEVEL ROUTINE) c | c |-- #ifdef SEAICE_CGRID c | SEAICE_DYNSOLVER c | | c | |-- < compute proxy for geostrophic velocity > c | | c | |-- < set up mass per unit area and Coriolis terms > c | | c | |-- < dynamic masking of areas with no ice > c | | c | | c | #ELSE c | DYNSOLVER c | #ENDIF c | c |-- if ( useOBCS ) c | OBCS_APPLY_UVICE c | c |-- if ( SEAICEadvHeff .OR. SEAICEadvArea .OR. SEAICEadvSnow .OR. SEAICEadvSalt ) c | SEAICE_ADVDIFF c | c |-- if ( usePW79thermodynamics ) c | SEAICE_GROWTH c | c |-- if ( useOBCS ) c | if ( SEAICEadvHeff ) OBCS_APPLY_HEFF c | if ( SEAICEadvArea ) OBCS_APPLY_AREA c | if ( SEAICEadvSALT ) OBCS_APPLY_HSALT c | if ( SEAICEadvSNOW ) OBCS_APPLY_HSNOW c | c |-- < do various exchanges > c | c |-- < do additional diagnostics > c | c o \end{verbatim} } %---------------------------------------------------------------------- \subsubsection{SEAICE diagnostics \label{sec:pkg:seaice:diagnostics}} Diagnostics output is available via the diagnostics package (see Section \ref{sec:pkg:diagnostics}). Available output fields are summarized in Table \ref{tab:pkg:seaice:diagnostics}. \input{s_phys_pkgs/text/seaice_diags.tex} %\subsubsection{Package Reference} \subsubsection{Experiments and tutorials that use seaice} \label{sec:pkg:seaice:experiments} \begin{itemize} \item{Labrador Sea experiment in \code{lab\_sea} verification directory. } \item \code{seaice\_obcs}, based on \code{lab\_sea} \item \code{offline\_exf\_seaice/input.seaicetd}, based on \code{lab\_sea} \item \code{global\_ocean.cs32x15/input.icedyn} and \code{global\_ocean.cs32x15/input.seaice}, global cubed-sphere-experiment with combinations of \code{seaice} and \code{thsice} \end{itemize} %%% Local Variables: %%% mode: latex %%% TeX-master: "../../manual" %%% End: