--- MITgcm_contrib/articles/ceaice/ceaice.tex 2008/02/25 22:06:17 1.12 +++ MITgcm_contrib/articles/ceaice/ceaice.tex 2008/02/25 23:45:46 1.13 @@ -1,4 +1,4 @@ -% $Header: /home/ubuntu/mnt/e9_copy/MITgcm_contrib/articles/ceaice/ceaice.tex,v 1.12 2008/02/25 22:06:17 dimitri Exp $ +% $Header: /home/ubuntu/mnt/e9_copy/MITgcm_contrib/articles/ceaice/ceaice.tex,v 1.13 2008/02/25 23:45:46 dimitri Exp $ % $Name: $ \documentclass[12pt]{article} @@ -76,9 +76,6 @@ \section{Introduction} \label{sec:intro} -\section{Model} -\label{sec:model} - Traditionally, probably for historical reasons and the ease of treating the Coriolis term, most standard sea-ice models are discretized on Arakawa-B-grids \citep[e.g.,][]{hibler79, harder99, @@ -90,7 +87,7 @@ sea-ice model and a C-grid ocean model. While the smoothing implicitly associated with this interpolation may mask grid scale noise, it may in two-way coupling lead to a computational mode as will be shown. By -choosing a C-grid for the sea-ice model, we circumvene this difficulty +choosing a C-grid for the sea-ice model, we circumvent this difficulty altogether and render the stress coupling as consistent as the buoyancy coupling. @@ -101,36 +98,46 @@ passage for all types of lateral boundary conditions. We (will) demonstrate this effect in the Candian archipelago. +\section{Model} +\label{sec:model} + \subsection{Dynamics} \label{sec:dynamics} -The momentum equations of the sea-ice model are standard with +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}, + \vtau_{ocean} - mg \nabla{\phi(0)} + \vek{F}, \end{equation} -where $\vek{u} = u\vek{i}+v\vek{j}$ is the ice velocity vectory, $m$ -the ice mass per unit area, $f$ the Coriolis parameter, $g$ is the -gravity accelation, $\nabla\phi$ is the gradient (tilt) of the sea -surface height potential beneath the ice. $\phi$ is the sum of -atmpheric pressure $p_{a}$ and loading due to ice and snow -$(m_{i}+m_{s})g$. $\vtau_{air}$ and $\vtau_{ocean}$ are the wind and -ice-ocean stresses, respectively. $\vek{F}$ is the interaction force -and $\vek{i}$, $\vek{j}$, and $\vek{k}$ are the unit vectors in the -$x$, $y$, and $z$ directions. Advection of sea-ice momentum is -neglected. The wind and ice-ocean stress terms are given by +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; +$\phi(0)$ is the sea surface height potential in response to ocean dynamics +and to atmospheric pressure loading; +and $\vek{F}=\nabla\cdot\sigma$ is the divergence of the internal ice stress +tensor $\sigma_{ij}$. +When using the rescaled vertical coordinate system, z$^\ast$, of +\citet{cam08}, $\phi(0)$ also includes a term due to snow and ice loading, $mg$. +Advection of sea-ice momentum is neglected. The wind and ice-ocean stress +terms are given by \begin{align*} - \vtau_{air} =& \rho_{air} |\vek{U}_{air}|R_{air}(\vek{U}_{air}) \\ - \vtau_{ocean} =& \rho_{ocean} |\vek{U}_{ocean}-\vek{u}| + \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, $\rho_{air/ocean}$ reference -densities, and $R_{air/ocean}$ rotation matrices that act on the -wind/current vectors. $\vek{F} = \nabla\cdot\sigma$ is the divergence -of the interal stress tensor $\sigma_{ij}$. +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. For an isotropic system this stress tensor can be related to the ice strain rate and strength by a nonlinear viscous-plastic (VP) @@ -174,7 +181,7 @@ $\Delta_{\min}=10^{-11}\text{\,s}^{-1}$ (for numerical reasons) and a maximum $\zeta_{\max} = P_{\max}/\Delta^*$, where $\Delta^*=(5\times10^{12}/2\times10^4)\text{\,s}^{-1}$. For stress -tensor compuation the replacement pressure $P = 2\,\Delta\zeta$ +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. @@ -198,9 +205,9 @@ treated explicitly. \citet{hunke97}'s introduced an elastic contribution to the strain -rate elatic-viscous-plastic in order to regularize +rate elastic-viscous-plastic in order to regularize Eq.\refeq{vpequation} in such a way that the resulting -elatic-viscous-plastic (EVP) and VP models are identical at steady +elastic-viscous-plastic (EVP) and VP models are identical at steady state, \begin{equation} \label{eq:evpequation} @@ -226,7 +233,7 @@ \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, +2\dot{\epsilon}_{12}$, respectively, and using the above abbreviations, the equations can be written as: \begin{align} \label{eq:evpstresstensor1} @@ -256,8 +263,8 @@ $P$ at vorticity points. For a general curvilinear grid, one needs in principle to take metric -terms into account that arise in the transformation a curvilinear grid -on the sphere. However, for now we can neglect these metric terms +terms into account that arise in the transformation of a curvilinear grid +on the sphere. For now, however, we can neglect these metric terms because they are very small on the cubed sphere grids used in this paper; in particular, only near the edges of the cubed sphere grid, we expect them to be non-zero, but these edges are at approximately @@ -266,7 +273,7 @@ cartesian. However, for last-glacial-maximum or snowball-earth-like simulations the question of metric terms needs to be reconsidered. Either, one includes these terms as in \citet{zhang03}, or one finds a -vector-invariant formulation fo the sea-ice internal stress term that +vector-invariant formulation for the sea-ice internal stress term that does not require any metric terms, as it is done in the ocean dynamics of the MITgcm \citep{adcroft04:_cubed_sphere}.