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\section{Model} |
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\label{sec:model} |
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|
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\subsection{Dynamics} |
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\label{sec:dynamics} |
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|
| 7 |
The momentum equation of the sea-ice model is |
| 8 |
\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} + |
| 11 |
\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, |
| 19 |
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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$\phi(0) = g\eta + p_{a}/\rho_{0}$ is the sea surface height potential |
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in response to ocean dynamics ($g\eta$) and to atmospheric pressure |
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loading ($p_{a}/\rho_{0}$, where $\rho_{0}$ is a reference density); |
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and $\vek{F}=\nabla\cdot\sigma$ is the divergence of the internal ice stress |
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tensor $\sigma_{ij}$. |
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When using the rescaled vertical coordinate system, z$^\ast$, of |
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\citet{cam08}, $\phi(0)$ also includes a term due to snow and ice |
| 29 |
loading, $mg/\rho_{0}$. |
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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 |
| 39 |
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 |
| 41 |
densities; and $R_{air/ocean}$ are rotation matrices that act on the |
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wind/current vectors. |
| 43 |
|
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For an isotropic system the stress tensor $\sigma_{ij}$ ($i,j=1,2$) can |
| 45 |
be related to the ice strain rate and strength by a nonlinear |
| 46 |
viscous-plastic (VP) constitutive law \citep{hibler79, zhang98}: |
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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). |
| 59 |
\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 |
| 68 |
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 |
| 70 |
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 timestep |
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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 abbreviations, |
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the equations 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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For details of the spatial discretization, the reader is referred to |
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\citet{zhang98, zhang03}. Our discretization differs only (but |
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importantly) in the underlying grid, namely the Arakawa C-grid, but is |
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otherwise 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}. |
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Rather than applying $\vtau_{ocean}$ directly, the stress is derived |
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from integrating over the ice thickness to the bottom of the oceanic |
| 198 |
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 |
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advect tracers, is really an average over the ocean surface |
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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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Sea ice distributions are characterized by sharp gradients and edges. |
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For this reason, we employ positive, multidimensional 2nd and 3rd-order |
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advection scheme with flux limiter \citep{roe85, hundsdorfer94} for the |
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thermodynamic variables discussed in the next section. |
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|
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\subparagraph{boundary conditions: no-slip, free-slip, half-slip} |
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|
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\begin{itemize} |
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\item transition from existing B-Grid to C-Grid |
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\item boundary conditions: no-slip, free-slip, half-slip |
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\item fancy (multi dimensional) advection schemes |
| 219 |
\item VP vs.\ EVP \citep{hunke97} |
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\item ocean stress formulation \citep{hibler87} |
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\end{itemize} |
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|
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\subsection{Thermodynamics} |
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\label{sec:thermodynamics} |
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|
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In the original formulation the sea ice model \citep{menemenlis05} |
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uses simple thermodynamics following the appendix of |
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\citet{semtner76}. This formulation does not allow storage of heat |
| 229 |
(heat capacity of ice is zero, and this type of model is often refered |
| 230 |
to as a ``zero-layer'' model). Upward heat flux is parameterized |
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assuming a linear temperature profile and together with a constant ice |
| 232 |
conductivity. It is expressed as $(K/h)(T_{w}-T_{0})$, where $K$ is |
| 233 |
the ice conductivity, $h$ the ice thickness, and $T_{w}-T_{0}$ the |
| 234 |
difference between water and ice surface temperatures. The surface |
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heat budget is computed in a similar way to that of |
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\citet{parkinson79} and \citet{manabe79}. |
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|
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There is considerable doubt about the reliability of such a simple |
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thermodynamic model---\citet{semtner84} found significant errors in |
| 240 |
phase (one month lead) and amplitude ($\approx$50\%\,overestimate) in |
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such models---, so that today many sea ice models employ more complex |
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thermodynamics. A popular thermodynamics model of \citet{winton00} is |
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based on the 3-layer model of \citet{semtner76} and treats brine |
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content by means of enthalphy conservation. This model requires in |
| 245 |
addition to ice-thickness and compactness (fractional area) additional |
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state variables to be advected by ice velocities, namely enthalphy of |
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the two ice layers and the thickness of the overlying snow layer. |
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\ml{[Jean-Michel, your turf: ]Care must be taken in advecting these |
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quantities in order to ensure conservation of enthalphy. Currently |
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this can only be accomplished with a 2nd-order advection scheme with |
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flux limiter \citep{roe85}.} |
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|
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|
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\subsection{C-grid} |
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\begin{itemize} |
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\item no-slip vs. free-slip for both lsr and evp; |
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"diagnostics" to look at and use for comparison |
| 258 |
\begin{itemize} |
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\item ice transport through Fram Strait/Denmark Strait/Davis |
| 260 |
Strait/Bering strait (these are general) |
| 261 |
\item ice transport through narrow passages, e.g.\ Nares-Strait |
| 262 |
\end{itemize} |
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\item compare different advection schemes (if lsr turns out to be more |
| 264 |
effective, then with lsr otherwise I prefer evp), eg. |
| 265 |
\begin{itemize} |
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\item default 2nd-order with diff1=0.002 |
| 267 |
\item 1st-order upwind with diff1=0. |
| 268 |
\item DST3FL (SEAICEadvScheme=33 with diff1=0., should work, works for me) |
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\item 2nd-order wit flux limiter (SEAICEadvScheme=77 with diff1=0.) |
| 270 |
\end{itemize} |
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That should be enough. Here, total ice mass and location of ice edge |
| 272 |
is interesting. However, this comparison can be done in an idealized |
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domain, may not require full Arctic Domain? |
| 274 |
\item |
| 275 |
Do a little study on the parameters of LSR and EVP |
| 276 |
\begin{enumerate} |
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\item convergence of LSR, how many iterations do you need to get a |
| 278 |
true elliptic yield curve |
| 279 |
\item vary deltaTevp and the relaxation parameter for EVP and see when |
| 280 |
the EVP solution breaks down (relative to the forcing time scale). |
| 281 |
For this, it is essential that the evp solver gives use "stripeless" |
| 282 |
solutions, that is your dtevp = 1sec solutions/or 10sec solutions |
| 283 |
with SEAICE\_evpDampC = 615. |
| 284 |
\end{enumerate} |
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\end{itemize} |
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