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\section{Dynamics\label{app:dynamics}} |
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% \newcommand{\vek}[1]{\ensuremath{\vec{\mathbf{#1}}}} |
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% \newcommand{\vtau}{\vek{\mathbf{\tau}}} |
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For completeness we provide more details on the ice dynamics of the |
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sea-ice model. The momentum equations are |
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\begin{linenomath*}\begin{equation} |
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\label{eq:momseaice} |
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m \frac{D\vek{u}}{Dt} = -mf\vek{k}\times\vek{u} + \vtau_{air} + |
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\vtau_{ocean} - m \nabla{\phi(0)} + \vek{F}, |
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\end{equation}\end{linenomath*} |
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where $m=m_{i}+m_{s}$ is the ice and snow mass per unit area; |
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$\vek{u}=u\vek{i}+v\vek{j}$ is the ice velocity vector; |
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$\vek{i}$, $\vek{j}$, and $\vek{k}$ are unit vectors in the $x$, $y$, and $z$ |
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directions, respectively; |
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$f$ is the Coriolis parameter; |
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$\vtau_{air}$ and $\vtau_{ocean}$ are the wind-ice and ocean-ice stresses, |
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respectively; |
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$g$ is the gravity acceleration; |
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$\nabla\phi(0)$ is the gradient (or tilt) of the sea surface height; |
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$\phi(0) = g\eta + p_{a}/\rho_{0} + mg/\rho_{0}$ is the sea surface |
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height potential in response to ocean dynamics ($g\eta$), to |
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atmospheric pressure loading ($p_{a}/\rho_{0}$, where $\rho_{0}$ is a |
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reference density) and a term due to snow and ice loading \citep{campin08}; |
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and $\vek{F}=\nabla\cdot\sigma$ is the divergence of the internal ice |
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stress tensor $\sigma_{ij}$. % |
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Advection of sea-ice momentum is neglected. The wind and ice-ocean stress |
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terms are given by |
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\begin{linenomath*}\begin{align*} |
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\vtau_{air} = & \rho_{air} C_{air} |\vek{U}_{air} -\vek{u}| |
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R_{air} (\vek{U}_{air} -\vek{u}), \\ |
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\vtau_{ocean} = & \rho_{ocean}C_{ocean} |\vek{U}_{ocean}-\vek{u}| |
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R_{ocean}(\vek{U}_{ocean}-\vek{u}), |
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\end{align*}\end{linenomath*} |
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where $\vek{U}_{air/ocean}$ are the surface winds of the atmosphere |
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and surface currents of the ocean, respectively; $C_{air/ocean}$ are |
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air and ocean drag coefficients; $\rho_{air/ocean}$ are reference |
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densities; and $R_{air/ocean}$ are rotation matrices that act on the |
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wind/current vectors. In this paper both rotation angles are set to |
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zero. |
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|
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For an isotropic system the stress tensor $\sigma_{ij}$ ($i,j=1,2$) can |
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be related to the ice strain rate and strength by a nonlinear |
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viscous-plastic (VP) constitutive law \citep{hibler79, zhang97}: |
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\begin{linenomath*}\begin{equation} |
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\label{eq:vpequation} |
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\sigma_{ij}=2\eta(\dot{\epsilon}_{ij},P)\dot{\epsilon}_{ij} |
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+ \left[\zeta(\dot{\epsilon}_{ij},P) - |
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\eta(\dot{\epsilon}_{ij},P)\right]\dot{\epsilon}_{kk}\delta_{ij} |
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- \frac{P}{2}\delta_{ij}. |
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\end{equation}\end{linenomath*} |
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The ice strain rate is given by |
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\begin{linenomath*}\begin{equation*} |
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\dot{\epsilon}_{ij} = \frac{1}{2}\left( |
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\frac{\partial{u_{i}}}{\partial{x_{j}}} + |
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\frac{\partial{u_{j}}}{\partial{x_{i}}}\right). |
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\end{equation*}\end{linenomath*} |
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The maximum ice pressure $P_{\max}$, a measure of ice strength, depends on |
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both thickness $h$ and compactness (concentration) $c$: |
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\begin{linenomath*}\begin{equation} |
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P_{\max} = P^{*}c\,h\,e^{[C^{*}\cdot(1-c)]}, |
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\label{eq:icestrength} |
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\end{equation}\end{linenomath*} |
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with the constants $P^{*}$ and $C^{*}$; we use |
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$P^{*}=27\,500\text{~N\,m$^{-2}$}$ and $C^{*}=20$. The nonlinear bulk |
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and shear viscosities $\eta$ and $\zeta$ are functions of ice strain |
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rate invariants and ice strength such that the principal components of |
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the stress lie on an elliptical yield curve with the ratio of major to |
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minor axis $e$ equal to $2$; they are given by: |
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\begin{align*} |
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\zeta =& \min\left(\frac{P_{\max}}{2\max(\Delta,\Delta_{\min})}, |
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\zeta_{\max}\right) \\ |
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\eta =& \frac{\zeta}{e^2} \\ |
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\intertext{with the abbreviation} |
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\Delta = & \left[ |
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\left(\dot{\epsilon}_{11}^2+\dot{\epsilon}_{22}^2\right) |
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(1+e^{-2}) + 4e^{-2}\dot{\epsilon}_{12}^2 + |
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2\dot{\epsilon}_{11}\dot{\epsilon}_{22} (1-e^{-2}) |
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\right]^{\frac{1}{2}}. |
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\end{align*} |
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In the simulations of this paper, the bulk viscosities are bounded |
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above by imposing both a minimum |
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$\Delta_{\min}=10^{-11}\text{\,s}^{-1}$ and a maximum $\zeta_{\max} = |
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P_{\max}/\Delta^*$, where |
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$\Delta^*=(5\times10^{12}/2\times10^4)\text{\,s}^{-1}$. For stress |
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tensor computation the replacement pressure $P = 2\,\Delta\zeta$ |
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\citep{hibler95} is used so that the stress state always lies on the |
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elliptic yield curve by definition. |
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In the so-called truncated ellipse method (experiment TEM) the shear |
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viscosity $\eta$ is capped to suppress any tensile stress |
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\citep{hibler97, geiger98}: |
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\begin{linenomath*}\begin{equation} |
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\label{eq:etatem} |
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\eta = \min\left(\frac{\zeta}{e^2}, |
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\frac{\frac{P}{2}-\zeta(\dot{\epsilon}_{11}+\dot{\epsilon}_{22})} |
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{\sqrt{(\dot{\epsilon}_{11}+\dot{\epsilon}_{22})^2 |
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+4\dot{\epsilon}_{12}^2}}\right). |
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\end{equation}\end{linenomath*} |
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In the current implementation, the VP-model is integrated with the |
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semi-implicit line successive over relaxation (LSOR)-solver of |
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\citet{zhang97}, which allows for long time steps that, in our case, |
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are limited by the explicit treatment of the Coriolis term. The |
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explicit treatment of the Coriolis term does not represent a severe |
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limitation because it restricts the time step to approximately the |
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same length as in the ocean model where the Coriolis term is also |
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treated explicitly. |
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\citet{hunke97} introduced an elastic contribution to the strain |
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rate in order to regularize Eq.~\ref{eq:vpequation} in such a way that |
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the resulting elastic-viscous-plastic (EVP) and VP models are |
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identical at steady state, |
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\begin{linenomath*}\begin{equation} |
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\label{eq:evpequation} |
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\frac{1}{E}\frac{\partial\sigma_{ij}}{\partial{t}} + |
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\frac{1}{2\eta}\sigma_{ij} |
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+ \frac{\eta - \zeta}{4\zeta\eta}\sigma_{kk}\delta_{ij} |
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+ \frac{P}{4\zeta}\delta_{ij} |
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= \dot{\epsilon}_{ij}. |
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\end{equation}\end{linenomath*} |
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The EVP-model uses an explicit time stepping scheme with a short |
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time step. According to the recommendation of \citet{hunke97}, the |
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EVP-model is stepped forward in time O(120) times within the physical |
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ocean model time step, to |
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allow for elastic waves to disappear. Because the scheme does not |
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require a matrix inversion it is fast in spite of the small internal |
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time step and simple to implement on parallel computers |
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\citep{hunke97}. For completeness, we repeat the equations for the |
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components of the stress tensor $\sigma_{1} = |
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\sigma_{11}+\sigma_{22}$, $\sigma_{2}= \sigma_{11}-\sigma_{22}$, and |
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$\sigma_{12}$. Introducing the divergence $D_D = |
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\dot{\epsilon}_{11}+\dot{\epsilon}_{22}$, and the horizontal tension |
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and shearing strain rates, $D_T = |
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\dot{\epsilon}_{11}-\dot{\epsilon}_{22}$ and $D_S = |
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2\dot{\epsilon}_{12}$, respectively, and using the above |
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abbreviations, the equations~\ref{eq:evpequation} can be written as: |
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\begin{linenomath*}\begin{align} |
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\label{eq:evpstresstensor1} |
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\frac{\partial\sigma_{1}}{\partial{t}} + \frac{\sigma_{1}}{2T} + |
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\frac{P}{2T} &= \frac{P}{2T\Delta} D_D \\ |
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\label{eq:evpstresstensor2} |
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\frac{\partial\sigma_{2}}{\partial{t}} + \frac{\sigma_{2} e^{2}}{2T} |
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&= \frac{P}{2T\Delta} D_T \\ |
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\label{eq:evpstresstensor12} |
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\frac{\partial\sigma_{12}}{\partial{t}} + \frac{\sigma_{12} e^{2}}{2T} |
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&= \frac{P}{4T\Delta} D_S |
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\end{align}\end{linenomath*} |
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Here, the elastic parameter $E$ is redefined in terms of a damping |
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time scale $T$ for elastic waves \[E=\frac{\zeta}{T}.\] |
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$T=E_{0}\Delta{t}$ with the tunable parameter $E_0<1$ and the external |
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(long) time step $\Delta{t}$. In experiment C-EVP-10 use $E_{0} = |
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\frac{1}{3}$ which is close to value of 0.36 used by |
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\citet{hunke01}. In experiment C-EVP-03 we use $E_{0} = \frac{1}{10}$ |
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resulting in $T = 120\text{~s}$ for our choice of $\Delta{t}$. |
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1.7 |
|
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1.3 |
\section{Finite-volume discretization of the stress tensor |
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divergence} |
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\label{app:discretization} |
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On an Arakawa C~grid, ice thickness and concentration and thus ice |
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strength $P$ and bulk and shear viscosities $\zeta$ and $\eta$ are |
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1.5 |
naturally defined at C-points in the center of the grid |
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1.4 |
cell. Discretization requires only averaging of $\zeta$ and $\eta$ to |
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vorticity or Z-points at the bottom left corner of the cell to give |
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$\overline{\zeta}^{Z}$ and $\overline{\eta}^{Z}$. In the following, |
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the superscripts indicate location at Z or C points, distance across |
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the cell (F), along the cell edge (G), between $u$-points (U), |
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$v$-points (V), and C-points (C). The control volumes of the $u$- and |
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$v$-equations in the grid cell at indices $(i,j)$ are $A_{i,j}^{w}$ |
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and $A_{i,j}^{s}$, respectively. With these definitions (which follow |
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the model code documentation at \url{http://mitgcm.org} except that |
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vorticity or $\zeta$-points have been renamed to Z-points in order to |
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avoid confusion with the bulk viscosity $\zeta$), the strain |
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rates are discretized as: |
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\begin{linenomath*}\begin{align} |
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\dot{\epsilon}_{11} &= \partial_{1}{u}_{1} + k_{2}u_{2} \\ \notag |
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=> (\epsilon_{11})_{i,j}^C &= \frac{u_{i+1,j}-u_{i,j}}{\Delta{x}_{i,j}^{F}} |
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+ k_{2,i,j}^{C}\frac{v_{i,j+1}+v_{i,j}}{2} \\ |
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\dot{\epsilon}_{22} &= \partial_{2}{u}_{2} + k_{1}u_{1} \\\notag |
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=> (\epsilon_{22})_{i,j}^C &= \frac{v_{i,j+1}-v_{i,j}}{\Delta{y}_{i,j}^{F}} |
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+ k_{1,i,j}^{C}\frac{u_{i+1,j}+u_{i,j}}{2} \\ |
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\dot{\epsilon}_{12} = \dot{\epsilon}_{21} &= \frac{1}{2}\biggl( |
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\partial_{1}{u}_{2} + \partial_{2}{u}_{1} - k_{1}u_{2} - k_{2}u_{1} |
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\biggr) \\ \notag |
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=> (\epsilon_{12})_{i,j}^Z &= \frac{1}{2} |
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\biggl( \frac{v_{i,j}-v_{i-1,j}}{\Delta{x}_{i,j}^V} |
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+ \frac{u_{i,j}-u_{i,j-1}}{\Delta{y}_{i,j}^U} \\\notag |
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&\phantom{=\frac{1}{2}\biggl(} |
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- k_{1,i,j}^{Z}\frac{v_{i,j}+v_{i-1,j}}{2} |
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- k_{2,i,j}^{Z}\frac{u_{i,j}+u_{i,j-1}}{2} |
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\biggr), |
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mlosch |
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\end{align}\end{linenomath*} |
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so that the diagonal terms of the strain rate tensor are naturally |
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defined at C-points and the symmetric off-diagonal term at |
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Z-points. No-slip boundary conditions ($u_{i,j-1}+u_{i,j}=0$ and |
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$v_{i-1,j}+v_{i,j}=0$ across boundaries) are implemented via |
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``ghost-points''; for free slip boundary conditions |
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$(\epsilon_{12})^Z=0$ on boundaries. |
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For a spherical polar grid, the coefficients of the metric terms are |
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$k_{1}=0$ and $k_{2}=-\tan\phi/a$, with the spherical radius $a$ and |
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the latitude $\phi$; $\Delta{x}_1 = \Delta{x} = a\cos\phi |
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\Delta\lambda$, and $\Delta{x}_2 = \Delta{y}=a\Delta\phi$. For a |
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general orthogonal curvilinear grid as used in this paper, $k_{1}$ and |
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$k_{2}$ can be approximated by finite differences of the cell widths: |
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%\enlargethispage{1cm} |
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1.4 |
\begin{linenomath*}\begin{align} |
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k_{1,i,j}^{C} &= \frac{1}{\Delta{y}_{i,j}^{F}} |
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\frac{\Delta{y}_{i+1,j}^{G}-\Delta{y}_{i,j}^{G}}{\Delta{x}_{i,j}^{F}} \\ |
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k_{2,i,j}^{C} &= \frac{1}{\Delta{x}_{i,j}^{F}} |
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\frac{\Delta{x}_{i,j+1}^{G}-\Delta{x}_{i,j}^{G}}{\Delta{y}_{i,j}^{F}} \\ |
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k_{1,i,j}^{Z} &= \frac{1}{\Delta{y}_{i,j}^{U}} |
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\frac{\Delta{y}_{i,j}^{C}-\Delta{y}_{i-1,j}^{C}}{\Delta{x}_{i,j}^{V}} \\ |
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k_{2,i,j}^{Z} &= \frac{1}{\Delta{x}_{i,j}^{V}} |
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\frac{\Delta{x}_{i,j}^{C}-\Delta{x}_{i,j-1}^{C}}{\Delta{y}_{i,j}^{U}} |
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mlosch |
1.4 |
\end{align}\end{linenomath*} |
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heimbach |
1.1 |
|
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mlosch |
1.3 |
The stress tensor is given by the constitutive viscous-plastic |
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relation $\sigma_{\alpha\beta} = 2\eta\dot{\epsilon}_{\alpha\beta} + |
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[(\zeta-\eta)\dot{\epsilon}_{\gamma\gamma} - P/2 |
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]\delta_{\alpha\beta}$ \citep{hibler79}. The stress tensor divergence |
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$(\nabla\sigma)_{\alpha} = \partial_\beta\sigma_{\beta\alpha}$, is |
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discretized in finite volumes. This conveniently avoids dealing with |
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further metric terms, as these are ``hidden'' in the differential cell |
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widths. For the $u$-equation ($\alpha=1$) we have: |
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mlosch |
1.4 |
\begin{linenomath*}\begin{align} |
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mlosch |
1.3 |
(\nabla\sigma)_{1}: \phantom{=}& |
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\frac{1}{A_{i,j}^w} |
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\int_{\mathrm{cell}}(\partial_1\sigma_{11}+\partial_2\sigma_{21})\,dx_1\,dx_2 |
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\\\notag |
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=& \frac{1}{A_{i,j}^w} \biggl\{ |
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\int_{x_2}^{x_2+\Delta{x}_2}\sigma_{11}dx_2\biggl|_{x_{1}}^{x_{1}+\Delta{x}_{1}} |
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+ \int_{x_1}^{x_1+\Delta{x}_1}\sigma_{21}dx_1\biggl|_{x_{2}}^{x_{2}+\Delta{x}_{2}} |
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\biggr\} \\ \notag |
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\approx& \frac{1}{A_{i,j}^w} \biggl\{ |
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\Delta{x}_2\sigma_{11}\biggl|_{x_{1}}^{x_{1}+\Delta{x}_{1}} |
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+ \Delta{x}_1\sigma_{21}\biggl|_{x_{2}}^{x_{2}+\Delta{x}_{2}} |
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\biggr\} \\ \notag |
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=& \frac{1}{A_{i,j}^w} \biggl\{ |
| 240 |
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(\Delta{x}_2\sigma_{11})_{i,j}^C - (\Delta{x}_2\sigma_{11})_{i-1,j}^C \\\notag |
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\phantom{=}& \phantom{\frac{1}{A_{i,j}^w} \biggl\{} |
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+ (\Delta{x}_1\sigma_{21})_{i,j+1}^Z - (\Delta{x}_1\sigma_{21})_{i,j}^Z |
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\biggr\} |
| 244 |
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\intertext{with} |
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(\Delta{x}_2\sigma_{11})_{i,j}^C =& \phantom{+} |
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\Delta{y}_{i,j}^{F}(\zeta + \eta)^{C}_{i,j} |
| 247 |
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\frac{u_{i+1,j}-u_{i,j}}{\Delta{x}_{i,j}^{F}} \\ \notag |
| 248 |
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&+ \Delta{y}_{i,j}^{F}(\zeta + \eta)^{C}_{i,j} |
| 249 |
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k_{2,i,j}^C \frac{v_{i,j+1}+v_{i,j}}{2} \\ \notag |
| 250 |
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\phantom{=}& + \Delta{y}_{i,j}^{F}(\zeta - \eta)^{C}_{i,j} |
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|
\frac{v_{i,j+1}-v_{i,j}}{\Delta{y}_{i,j}^{F}} \\ \notag |
| 252 |
|
|
\phantom{=}& + \Delta{y}_{i,j}^{F}(\zeta - \eta)^{C}_{i,j} |
| 253 |
|
|
k_{1,i,j}^{C}\frac{u_{i+1,j}+u_{i,j}}{2} \\ \notag |
| 254 |
|
|
\phantom{=}& - \Delta{y}_{i,j}^{F} \frac{P}{2} \\ |
| 255 |
|
|
% |
| 256 |
|
|
(\Delta{x}_1\sigma_{21})_{i,j}^Z =& \phantom{+} |
| 257 |
|
|
\Delta{x}_{i,j}^{V}\overline{\eta}^{Z}_{i,j} |
| 258 |
|
|
\frac{u_{i,j}-u_{i,j-1}}{\Delta{y}_{i,j}^{U}} \\ \notag |
| 259 |
|
|
& + \Delta{x}_{i,j}^{V}\overline{\eta}^{Z}_{i,j} |
| 260 |
|
|
\frac{v_{i,j}-v_{i-1,j}}{\Delta{x}_{i,j}^{V}} \\ \notag |
| 261 |
|
|
& - \Delta{x}_{i,j}^{V}\overline{\eta}^{Z}_{i,j} |
| 262 |
|
|
k_{2,i,j}^{Z}\frac{u_{i,j}+u_{i,j-1}}{2} \\ \notag |
| 263 |
|
|
& - \Delta{x}_{i,j}^{V}\overline{\eta}^{Z}_{i,j} |
| 264 |
|
|
k_{1,i,j}^{Z}\frac{v_{i,j}+v_{i-1,j}}{2} |
| 265 |
mlosch |
1.4 |
\end{align}\end{linenomath*} |
| 266 |
heimbach |
1.1 |
|
| 267 |
mlosch |
1.3 |
Similarly, we have for the $v$-equation ($\alpha=2$): |
| 268 |
mlosch |
1.4 |
\begin{linenomath*}\begin{align} |
| 269 |
mlosch |
1.3 |
(\nabla\sigma)_{2}: \phantom{=}& |
| 270 |
|
|
\frac{1}{A_{i,j}^s} |
| 271 |
|
|
\int_{\mathrm{cell}}(\partial_1\sigma_{12}+\partial_2\sigma_{22})\,dx_1\,dx_2 |
| 272 |
|
|
\\\notag |
| 273 |
|
|
=& \frac{1}{A_{i,j}^s} \biggl\{ |
| 274 |
|
|
\int_{x_2}^{x_2+\Delta{x}_2}\sigma_{12}dx_2\biggl|_{x_{1}}^{x_{1}+\Delta{x}_{1}} |
| 275 |
|
|
+ \int_{x_1}^{x_1+\Delta{x}_1}\sigma_{22}dx_1\biggl|_{x_{2}}^{x_{2}+\Delta{x}_{2}} |
| 276 |
|
|
\biggr\} \\ \notag |
| 277 |
|
|
\approx& \frac{1}{A_{i,j}^s} \biggl\{ |
| 278 |
|
|
\Delta{x}_2\sigma_{12}\biggl|_{x_{1}}^{x_{1}+\Delta{x}_{1}} |
| 279 |
|
|
+ \Delta{x}_1\sigma_{22}\biggl|_{x_{2}}^{x_{2}+\Delta{x}_{2}} |
| 280 |
|
|
\biggr\} \\ \notag |
| 281 |
|
|
=& \frac{1}{A_{i,j}^s} \biggl\{ |
| 282 |
|
|
(\Delta{x}_2\sigma_{12})_{i+1,j}^Z - (\Delta{x}_2\sigma_{12})_{i,j}^Z |
| 283 |
|
|
\\ \notag |
| 284 |
|
|
\phantom{=}& \phantom{\frac{1}{A_{i,j}^s} \biggl\{} |
| 285 |
|
|
+ (\Delta{x}_1\sigma_{22})_{i,j}^C - (\Delta{x}_1\sigma_{22})_{i,j-1}^C |
| 286 |
|
|
\biggr\} |
| 287 |
|
|
\intertext{with} |
| 288 |
|
|
(\Delta{x}_1\sigma_{12})_{i,j}^Z =& \phantom{+} |
| 289 |
|
|
\Delta{y}_{i,j}^{U}\overline{\eta}^{Z}_{i,j} |
| 290 |
|
|
\frac{u_{i,j}-u_{i,j-1}}{\Delta{y}_{i,j}^{U}} \\\notag |
| 291 |
|
|
&+ \Delta{y}_{i,j}^{U}\overline{\eta}^{Z}_{i,j} |
| 292 |
|
|
\frac{v_{i,j}-v_{i-1,j}}{\Delta{x}_{i,j}^{V}} \\ \notag |
| 293 |
|
|
&- \Delta{y}_{i,j}^{U}\overline{\eta}^{Z}_{i,j} |
| 294 |
|
|
k_{2,i,j}^{Z}\frac{u_{i,j}+u_{i,j-1}}{2} \\ \notag |
| 295 |
|
|
&- \Delta{y}_{i,j}^{U}\overline{\eta}^{Z}_{i,j} |
| 296 |
mlosch |
1.11 |
k_{1,i,j}^{Z}\frac{v_{i,j}+v_{i-1,j}}{2} \\ |
| 297 |
mlosch |
1.3 |
% |
| 298 |
|
|
(\Delta{x}_2\sigma_{22})_{i,j}^C =& \phantom{+} |
| 299 |
|
|
\Delta{x}_{i,j}^{F}(\zeta - \eta)^{C}_{i,j} |
| 300 |
|
|
\frac{u_{i+1,j}-u_{i,j}}{\Delta{x}_{i,j}^{F}} \\ \notag |
| 301 |
|
|
&+ \Delta{x}_{i,j}^{F}(\zeta - \eta)^{C}_{i,j} |
| 302 |
|
|
k_{2,i,j}^{C} \frac{v_{i,j+1}+v_{i,j}}{2} \\ \notag |
| 303 |
|
|
& + \Delta{x}_{i,j}^{F}(\zeta + \eta)^{C}_{i,j} |
| 304 |
|
|
\frac{v_{i,j+1}-v_{i,j}}{\Delta{y}_{i,j}^{F}} \\ \notag |
| 305 |
|
|
& + \Delta{x}_{i,j}^{F}(\zeta + \eta)^{C}_{i,j} |
| 306 |
|
|
k_{1,i,j}^{C}\frac{u_{i+1,j}+u_{i,j}}{2} \\ \notag |
| 307 |
|
|
& -\Delta{x}_{i,j}^{F} \frac{P}{2} |
| 308 |
mlosch |
1.4 |
\end{align}\end{linenomath*} |
| 309 |
heimbach |
1.1 |
|
| 310 |
|
|
|
| 311 |
|
|
%%% Local Variables: |
| 312 |
|
|
%%% mode: latex |
| 313 |
mlosch |
1.3 |
%%% TeX-master: "ceaice_part1" |
| 314 |
heimbach |
1.1 |
%%% End: |
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