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revision 1.12 by dimitri, Mon Feb 25 22:06:17 2008 UTC revision 1.13 by dimitri, Mon Feb 25 23:45:46 2008 UTC
# Line 76  example results from coupled ocean and s Line 76  example results from coupled ocean and s
76  \section{Introduction}  \section{Introduction}
77  \label{sec:intro}  \label{sec:intro}
78    
 \section{Model}  
 \label{sec:model}  
   
79  Traditionally, probably for historical reasons and the ease of  Traditionally, probably for historical reasons and the ease of
80  treating the Coriolis term, most standard sea-ice models are  treating the Coriolis term, most standard sea-ice models are
81  discretized on Arakawa-B-grids \citep[e.g.,][]{hibler79, harder99,  discretized on Arakawa-B-grids \citep[e.g.,][]{hibler79, harder99,
# Line 90  velocities points and thus needs to be i Line 87  velocities points and thus needs to be i
87  sea-ice model and a C-grid ocean model. While the smoothing implicitly  sea-ice model and a C-grid ocean model. While the smoothing implicitly
88  associated with this interpolation may mask grid scale noise, it may  associated with this interpolation may mask grid scale noise, it may
89  in two-way coupling lead to a computational mode as will be shown. By  in two-way coupling lead to a computational mode as will be shown. By
90  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
91  altogether and render the stress coupling as consistent as the  altogether and render the stress coupling as consistent as the
92  buoyancy coupling.  buoyancy coupling.
93    
# Line 101  whereas the C-grid formulation allows a Line 98  whereas the C-grid formulation allows a
98  passage for all types of lateral boundary conditions. We (will)  passage for all types of lateral boundary conditions. We (will)
99  demonstrate this effect in the Candian archipelago.  demonstrate this effect in the Candian archipelago.
100    
101    \section{Model}
102    \label{sec:model}
103    
104  \subsection{Dynamics}  \subsection{Dynamics}
105  \label{sec:dynamics}  \label{sec:dynamics}
106    
107  The momentum equations of the sea-ice model are standard with  The momentum equation of the sea-ice model is
108  \begin{equation}  \begin{equation}
109    \label{eq:momseaice}    \label{eq:momseaice}
110    m \frac{D\vek{u}}{Dt} = -mf\vek{k}\times\vek{u} + \vtau_{air} +    m \frac{D\vek{u}}{Dt} = -mf\vek{k}\times\vek{u} + \vtau_{air} +
111    \vtau_{ocean} - m \nabla{\phi(0)} + \vek{F},    \vtau_{ocean} - mg \nabla{\phi(0)} + \vek{F},
112  \end{equation}  \end{equation}
113  where $\vek{u} = u\vek{i}+v\vek{j}$ is the ice velocity vectory, $m$  where $m=m_{i}+m_{s}$ is the ice and snow mass per unit area;
114  the ice mass per unit area, $f$ the Coriolis parameter, $g$ is the  $\vek{u}=u\vek{i}+v\vek{j}$ is the ice velocity vector;
115  gravity accelation, $\nabla\phi$ is the gradient (tilt) of the sea  $\vek{i}$, $\vek{j}$, and $\vek{k}$ are unit vectors in the $x$, $y$, and $z$
116  surface height potential beneath the ice. $\phi$ is the sum of  directions, respectively;
117  atmpheric pressure $p_{a}$ and loading due to ice and snow  $f$ is the Coriolis parameter;
118  $(m_{i}+m_{s})g$. $\vtau_{air}$ and $\vtau_{ocean}$ are the wind and  $\vtau_{air}$ and $\vtau_{ocean}$ are the wind-ice and ocean-ice stresses,
119  ice-ocean stresses, respectively.  $\vek{F}$ is the interaction force  respectively;
120  and $\vek{i}$, $\vek{j}$, and $\vek{k}$ are the unit vectors in the  $g$ is the gravity accelation;
121  $x$, $y$, and $z$ directions.  Advection of sea-ice momentum is  $\nabla\phi(0)$ is the gradient (or tilt) of the sea surface height;
122  neglected. The wind and ice-ocean stress terms are given by  $\phi(0)$ is the sea surface height potential in response to ocean dynamics
123    and to atmospheric pressure loading;
124    and $\vek{F}=\nabla\cdot\sigma$ is the divergence of the internal ice stress
125    tensor $\sigma_{ij}$.
126    When using the rescaled vertical coordinate system, z$^\ast$, of
127    \citet{cam08}, $\phi(0)$ also includes a term due to snow and ice loading, $mg$.
128    Advection of sea-ice momentum is neglected. The wind and ice-ocean stress
129    terms are given by
130  \begin{align*}  \begin{align*}
131    \vtau_{air} =& \rho_{air} |\vek{U}_{air}|R_{air}(\vek{U}_{air}) \\    \vtau_{air}   = & \rho_{air}  C_{air}   |\vek{U}_{air}  -\vek{u}|
132    \vtau_{ocean} =& \rho_{ocean} |\vek{U}_{ocean}-\vek{u}|                     R_{air}  (\vek{U}_{air}  -\vek{u}), \\
133      \vtau_{ocean} = & \rho_{ocean}C_{ocean} |\vek{U}_{ocean}-\vek{u}|
134                     R_{ocean}(\vek{U}_{ocean}-\vek{u}), \\                     R_{ocean}(\vek{U}_{ocean}-\vek{u}), \\
135  \end{align*}  \end{align*}
136  where $\vek{U}_{air/ocean}$ are the surface winds of the atmosphere  where $\vek{U}_{air/ocean}$ are the surface winds of the atmosphere
137  and surface currents of the ocean, respectively. $C_{air/ocean}$ are  and surface currents of the ocean, respectively; $C_{air/ocean}$ are
138  air and ocean drag coefficients, $\rho_{air/ocean}$ reference  air and ocean drag coefficients; $\rho_{air/ocean}$ are reference
139  densities, and $R_{air/ocean}$ rotation matrices that act on the  densities; and $R_{air/ocean}$ are rotation matrices that act on the
140  wind/current vectors. $\vek{F} = \nabla\cdot\sigma$ is the divergence  wind/current vectors.
 of the interal stress tensor $\sigma_{ij}$.  
141    
142  For an isotropic system this stress tensor can be related to the ice  For an isotropic system this stress tensor can be related to the ice
143  strain rate and strength by a nonlinear viscous-plastic (VP)  strain rate and strength by a nonlinear viscous-plastic (VP)
# Line 174  The bulk viscosities are bounded above b Line 181  The bulk viscosities are bounded above b
181  $\Delta_{\min}=10^{-11}\text{\,s}^{-1}$ (for numerical reasons) and a  $\Delta_{\min}=10^{-11}\text{\,s}^{-1}$ (for numerical reasons) and a
182  maximum $\zeta_{\max} = P_{\max}/\Delta^*$, where  maximum $\zeta_{\max} = P_{\max}/\Delta^*$, where
183  $\Delta^*=(5\times10^{12}/2\times10^4)\text{\,s}^{-1}$. For stress  $\Delta^*=(5\times10^{12}/2\times10^4)\text{\,s}^{-1}$. For stress
184  tensor compuation the replacement pressure $P = 2\,\Delta\zeta$  tensor computation the replacement pressure $P = 2\,\Delta\zeta$
185  \citep{hibler95} is used so that the stress state always lies on the  \citep{hibler95} is used so that the stress state always lies on the
186  elliptic yield curve by definition.  elliptic yield curve by definition.
187    
# Line 198  same length as in the ocean model where Line 205  same length as in the ocean model where
205  treated explicitly.  treated explicitly.
206    
207  \citet{hunke97}'s introduced an elastic contribution to the strain  \citet{hunke97}'s introduced an elastic contribution to the strain
208  rate elatic-viscous-plastic in order to regularize  rate elastic-viscous-plastic in order to regularize
209  Eq.\refeq{vpequation} in such a way that the resulting  Eq.\refeq{vpequation} in such a way that the resulting
210  elatic-viscous-plastic (EVP) and VP models are identical at steady  elastic-viscous-plastic (EVP) and VP models are identical at steady
211  state,  state,
212  \begin{equation}  \begin{equation}
213    \label{eq:evpequation}    \label{eq:evpequation}
# Line 226  $\sigma_{12}$. Introducing the divergenc Line 233  $\sigma_{12}$. Introducing the divergenc
233  \dot{\epsilon}_{11}+\dot{\epsilon}_{22}$, and the horizontal tension  \dot{\epsilon}_{11}+\dot{\epsilon}_{22}$, and the horizontal tension
234  and shearing strain rates, $D_T =  and shearing strain rates, $D_T =
235  \dot{\epsilon}_{11}-\dot{\epsilon}_{22}$ and $D_S =  \dot{\epsilon}_{11}-\dot{\epsilon}_{22}$ and $D_S =
236  2\dot{\epsilon}_{12}$, respectively and using the above abbreviations,  2\dot{\epsilon}_{12}$, respectively, and using the above abbreviations,
237  the equations can be written as:  the equations can be written as:
238  \begin{align}  \begin{align}
239    \label{eq:evpstresstensor1}    \label{eq:evpstresstensor1}
# Line 256  differences and averaging is only involv Line 263  differences and averaging is only involv
263  $P$ at vorticity points.  $P$ at vorticity points.
264    
265  For a general curvilinear grid, one needs in principle to take metric  For a general curvilinear grid, one needs in principle to take metric
266  terms into account that arise in the transformation a curvilinear grid  terms into account that arise in the transformation of a curvilinear grid
267  on the sphere. However, for now we can neglect these metric terms  on the sphere. For now, however, we can neglect these metric terms
268  because they are very small on the cubed sphere grids used in this  because they are very small on the cubed sphere grids used in this
269  paper; in particular, only near the edges of the cubed sphere grid, we  paper; in particular, only near the edges of the cubed sphere grid, we
270  expect them to be non-zero, but these edges are at approximately  expect them to be non-zero, but these edges are at approximately
# Line 266  simulations.  Everywhere else the coordi Line 273  simulations.  Everywhere else the coordi
273  cartesian.  However, for last-glacial-maximum or snowball-earth-like  cartesian.  However, for last-glacial-maximum or snowball-earth-like
274  simulations the question of metric terms needs to be reconsidered.  simulations the question of metric terms needs to be reconsidered.
275  Either, one includes these terms as in \citet{zhang03}, or one finds a  Either, one includes these terms as in \citet{zhang03}, or one finds a
276  vector-invariant formulation fo the sea-ice internal stress term that  vector-invariant formulation for the sea-ice internal stress term that
277  does not require any metric terms, as it is done in the ocean dynamics  does not require any metric terms, as it is done in the ocean dynamics
278  of the MITgcm \citep{adcroft04:_cubed_sphere}.  of the MITgcm \citep{adcroft04:_cubed_sphere}.
279    
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