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\section{Sea Ice Model Formulation} |
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\label{app:model} |
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|
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\ml{[All of this (and more) should go into the documentation, but we |
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can remove a large part of the text because it is completely |
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redundant. I leave it in for now \ldots]} |
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|
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\subsection{Dynamics} |
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\label{app:dynamics} |
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|
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The momentum equation of the sea-ice model is |
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\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} |
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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 accelation; |
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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{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*} |
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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. |
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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{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} |
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The ice strain rate is given by |
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\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*} |
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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{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} |
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with the constants $P^{*}$ and $C^{*}$. The nonlinear bulk and shear |
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viscosities $\eta$ and $\zeta$ are functions of ice strain rate |
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invariants and ice strength such that the principal components of the |
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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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The bulk viscosities are bounded above by imposing both a minimum |
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$\Delta_{\min}=10^{-11}\text{\,s}^{-1}$ (for numerical reasons) and a |
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maximum $\zeta_{\max} = P_{\max}/\Delta^*$, 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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|
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In the so-called truncated ellipse method the shear viscosity $\eta$ |
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is capped to suppress any tensile stress \citep{hibler97, geiger98}: |
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\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} |
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|
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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{zhang98}, 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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|
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\citet{hunke97}'s introduced an elastic contribution to the strain |
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rate in order to regularize Eq.\refeq{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{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} |
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%In the EVP model, equations for the components of the stress tensor |
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%$\sigma_{ij}$ are solved explicitly. Both model formulations will be |
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%used and compared the present sea-ice model study. |
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The EVP-model uses an explicit time stepping scheme with a short |
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timestep. According to the recommendation of \citet{hunke97}, the |
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EVP-model is stepped forward in time 120 times within the physical |
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ocean model time step (although this parameter is under debate), 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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timestep 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\refeq{evpequation} can be written as: |
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\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} |
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Here, the elastic parameter $E$ is redefined in terms of a damping timescale |
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$T$ for elastic waves \[E=\frac{\zeta}{T}.\] |
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$T=E_{0}\Delta{t}$ with the tunable parameter $E_0<1$ and |
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the external (long) timestep $\Delta{t}$. \citet{hunke97} recommend |
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$E_{0} = \frac{1}{3}$. |
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|
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Our discretization differs from \citet{zhang98, zhang03} in the |
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underlying grid, namely the Arakawa C-grid, but is otherwise |
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straightforward. The EVP model, in particular, is discretized |
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naturally on the C-grid with $\sigma_{1}$ and $\sigma_{2}$ on the |
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center points and $\sigma_{12}$ on the corner (or vorticity) points of |
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the grid. With this choice all derivatives are discretized as central |
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differences and averaging is only involved in computing $\Delta$ and |
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$P$ at vorticity points. |
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|
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For a general curvilinear grid, one needs in principle to take metric |
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terms into account that arise in the transformation of a curvilinear |
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grid on the sphere. For now, however, we can neglect these metric |
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terms because they are very small on the \ml{[modify following |
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section3:] cubed sphere grids used in this paper; in particular, |
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only near the edges of the cubed sphere grid, we expect them to be |
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non-zero, but these edges are at approximately 35\degS\ or 35\degN\ |
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which are never covered by sea-ice in our simulations. Everywhere |
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else the coordinate system is locally nearly cartesian.} However, for |
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last-glacial-maximum or snowball-earth-like simulations the question |
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of metric terms needs to be reconsidered. Either, one includes these |
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terms as in \citet{zhang03}, or one finds a vector-invariant |
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formulation for the sea-ice internal stress term that does not require |
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any metric terms, as it is done in the ocean dynamics of the MITgcm |
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\citep{adcroft04:_cubed_sphere}. |
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|
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Lateral boundary conditions are naturally ``no-slip'' for B-grids, as |
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the tangential velocities points lie on the boundary. For C-grids, the |
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lateral boundary condition for tangential velocities is realized via |
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``ghost points'', allowing alternatively no-slip or free-slip |
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conditions. In ocean models free-slip boundary conditions in |
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conjunction with piecewise-constant (``castellated'') coastlines have |
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been shown to reduce in effect to no-slip boundary conditions |
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\citep{adcroft98:_slippery_coast}; for sea-ice models the effects of |
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lateral boundary conditions have not yet been studied. |
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|
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Moving sea ice exerts a stress on the ocean which is the opposite of |
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the stress $\vtau_{ocean}$ in Eq.\refeq{momseaice}. This stess is |
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applied directly to the surface layer of the ocean model. An |
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alternative ocean stress formulation is given by \citet{hibler87}. |
| 198 |
Rather than applying $\vtau_{ocean}$ directly, the stress is derived |
| 199 |
from integrating over the ice thickness to the bottom of the oceanic |
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surface layer. In the resulting equation for the \emph{combined} |
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ocean-ice momentum, the interfacial stress cancels and the total |
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stress appears as the sum of windstress and divergence of internal ice |
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stresses: $\delta(z) (\vtau_{air} + \vek{F})/\rho_0$, \citep[see also |
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Eq.\,2 of][]{hibler87}. The disadvantage of this formulation is that |
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now the velocity in the surface layer of the ocean that is used to |
| 206 |
advect tracers, is really an average over the ocean surface |
| 207 |
velocity and the ice velocity leading to an inconsistency as the ice |
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temperature and salinity are different from the oceanic variables. |
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|
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%\subparagraph{boundary conditions: no-slip, free-slip, half-slip} |
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%\begin{itemize} |
| 212 |
%\item transition from existing B-Grid to C-Grid |
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%\item boundary conditions: no-slip, free-slip, half-slip |
| 214 |
%\item fancy (multi dimensional) advection schemes |
| 215 |
%\item VP vs.\ EVP \citep{hunke97} |
| 216 |
%\item ocean stress formulation \citep{hibler87} |
| 217 |
%\end{itemize} |
| 218 |
|
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\subsection{Thermodynamics} |
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\label{app:thermodynamics} |
| 221 |
|
| 222 |
In its original formulation the sea ice model \citep{menemenlis05} |
| 223 |
uses simple thermodynamics following the appendix of |
| 224 |
\citet{semtner76}. This formulation does not allow storage of heat, |
| 225 |
that is, the heat capacity of ice is zero. Upward conductive heat flux |
| 226 |
is parameterized assuming a linear temperature profile and together |
| 227 |
with a constant ice conductivity. It is expressed as |
| 228 |
$(K/h)(T_{w}-T_{0})$, where $K$ is the ice conductivity, $h$ the ice |
| 229 |
thickness, and $T_{w}-T_{0}$ the difference between water and ice |
| 230 |
surface temperatures. TThis type of model is often refered to as a |
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``zero-layer'' model. The surface heat flux is computed in a similar |
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way to that of \citet{parkinson79} and \citet{manabe79}. |
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|
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The conductive heat flux depends strongly on the ice thickness $h$. |
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However, the ice thickness in the model represents a mean over a |
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potentially very heterogeneous thickness distribution. In order to |
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parameterize a sub-grid scale distribution for heat flux |
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computations, the mean ice thickness $h$ is split into seven thickness |
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categories $H_{n}$ that are equally distributed between $2h$ and a |
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minimum imposed ice thickness of $5\text{\,cm}$ by $H_n= |
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\frac{2n-1}{7}\,h$ for $n\in[1,7]$. The heat fluxes computed for each |
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thickness category is area-averaged to give the total heat flux |
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\citep{hibler84}. |
| 244 |
|
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\ml{[This is Ian Fenty's work and we may want to remove this paragraph |
| 246 |
from the paper]: % |
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The atmospheric heat flux is balanced by an oceanic heat flux from |
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below. The oceanic flux is proportional to |
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$\rho\,c_{p}\left(T_{w}-T_{fr}\right)$ where $\rho$ and $c_{p}$ are |
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the density and heat capacity of sea water and $T_{fr}$ is the local |
| 251 |
freezing point temperature that is a function of salinity. Contrary to |
| 252 |
\citet{menemenlis05}, this flux is not assumed to instantaneously melt |
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or create ice, but a time scale of three days is used to relax $T_{w}$ |
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to the freezing point.} |
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|
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The parameterization of lateral and vertical growth of sea ice follows |
| 257 |
that of \citet{hibler79, hibler80}. |
| 258 |
|
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On top of the ice there is a layer of snow that modifies the heat flux |
| 260 |
and the albedo \citep{zhang98}. If enough snow accumulates so that its |
| 261 |
weight submerges the ice and the snow is flooded, a simple mass |
| 262 |
conserving parameterization of snowice formation (a flood-freeze |
| 263 |
algorithm following Archimedes' principle) turns snow into ice until |
| 264 |
the ice surface is back at $z=0$ \citep{leppaeranta83}. |
| 265 |
|
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Effective ice thickness (ice volume per unit area, |
| 267 |
$c\cdot{h}$), concentration $c$ and effective snow thickness |
| 268 |
($c\cdot{h}_{s}$) are advected by ice velocities: |
| 269 |
\begin{equation} |
| 270 |
\label{eq:advection} |
| 271 |
\frac{\partial{X}}{\partial{t}} = - \nabla\cdot\left(\vek{u}\,X\right) + |
| 272 |
\Gamma_{X} + D_{X} |
| 273 |
\end{equation} |
| 274 |
where $\Gamma_X$ are the thermodynamic source terms and $D_{X}$ the |
| 275 |
diffusive terms for quantities $X=(c\cdot{h}), c, (c\cdot{h}_{s})$. |
| 276 |
% |
| 277 |
From the various advection scheme that are available in the MITgcm |
| 278 |
\citep{mitgcm02}, we choose flux-limited schemes |
| 279 |
\citep[multidimensional 2nd and 3rd-order advection scheme with flux |
| 280 |
limiter][]{roe85, hundsdorfer94} to preserve sharp gradients and edges |
| 281 |
that are typical of sea ice distributions and to rule out unphysical |
| 282 |
over- and undershoots (negative thickness or concentration). These |
| 283 |
scheme conserve volume and horizontal area and are unconditionally |
| 284 |
stable, so that we can set $D_{X}=0$. \ml{[do we need to proove that? |
| 285 |
can we proove that? citation?]} |
| 286 |
|
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There is considerable doubt about the reliability of such a simple |
| 288 |
thermodynamic model---\citet{semtner84} found significant errors in |
| 289 |
phase (one month lead) and amplitude ($\approx$50\%\,overestimate) in |
| 290 |
such models---, so that today many sea ice models employ more complex |
| 291 |
thermodynamics. A popular thermodynamics model of \citet{winton00} is |
| 292 |
based on the 3-layer model of \citet{semtner76} and treats brine |
| 293 |
content by means of enthalphy conservation. This model requires in |
| 294 |
addition to ice-thickness and compactness (fractional area) additional |
| 295 |
state variables to be advected by ice velocities, namely enthalphy of |
| 296 |
the two ice layers and the thickness of the overlying snow layer. |
| 297 |
\ml{[Jean-Michel, your turf: ]Care must be taken in advecting these |
| 298 |
quantities in order to ensure conservation of enthalphy. Currently |
| 299 |
this can only be accomplished with a 2nd-order advection scheme with |
| 300 |
flux limiter \citep{roe85}.} |
| 301 |
|
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%%% Local Variables: |
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%%% mode: latex |
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%%% TeX-master: "ceaice" |
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%%% End: |
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