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revision 1.4 by mlosch, Thu Jan 10 15:47:32 2008 UTC revision 1.5 by mlosch, Mon Jan 14 15:46:54 2008 UTC
# Line 1  Line 1 
1  \documentclass[12pt]{article}  \documentclass[12pt]{article}
2    
3  \usepackage{graphicx,subfigure}  \usepackage[]{graphicx}
4    \usepackage{subfigure}
5    
6  \usepackage[round,comma]{natbib}  \usepackage[round,comma]{natbib}
7  \bibliographystyle{bib/agu04}  \bibliographystyle{bib/agu04}
# Line 129  The ice strain rate is given by Line 130  The ice strain rate is given by
130      \frac{\partial{u_{i}}}{\partial{x_{j}}} +      \frac{\partial{u_{i}}}{\partial{x_{j}}} +
131      \frac{\partial{u_{j}}}{\partial{x_{i}}}\right).      \frac{\partial{u_{j}}}{\partial{x_{i}}}\right).
132  \end{equation*}  \end{equation*}
133  The pressure $P$, a measure of ice strength, depends on both thickness  The maximum ice pressure $P_{\max}$, a measure of ice strength, depends on
134  $h$ and compactness (concentration) $c$:  both thickness $h$ and compactness (concentration) $c$:
135  \begin{equation}  \begin{equation}
136    P = P^{*}c\,h\,e^{[C^{*}\cdot(1-c)]},    P_{\max} = P^{*}c\,h\,e^{[C^{*}\cdot(1-c)]},
137  \label{icestrength}  \label{icestrength}
138  \end{equation}  \end{equation}
139  with the constants $P^{*}$ and $C^{*}$. The nonlinear bulk and shear  with the constants $P^{*}$ and $C^{*}$. The nonlinear bulk and shear
# Line 141  invariants and ice strength such that th Line 142  invariants and ice strength such that th
142  stress lie on an elliptical yield curve with the ratio of major to  stress lie on an elliptical yield curve with the ratio of major to
143  minor axis $e$ equal to $2$; they are given by:  minor axis $e$ equal to $2$; they are given by:
144  \begin{align*}  \begin{align*}
145    \zeta =& \frac{P}{2\Delta} \\    \zeta =& \min\left(\frac{P_{\max}}{2\max(\Delta,\Delta_{\min})},
146    \eta =& \frac{P}{2\Delta{e}^2} \\     \zeta_{\max}\right) \\
147      \eta =& \frac{\zeta}{e^2} \\
148    \intertext{with the abbreviation}    \intertext{with the abbreviation}
149    \Delta = & \left[    \Delta = & \left[
150      \left(\dot{\epsilon}_{11}^2+\dot{\epsilon}_{22}^2\right)      \left(\dot{\epsilon}_{11}^2+\dot{\epsilon}_{22}^2\right)
# Line 150  minor axis $e$ equal to $2$; they are gi Line 152  minor axis $e$ equal to $2$; they are gi
152      2\dot{\epsilon}_{11}\dot{\epsilon}_{22} (1-e^{-2})      2\dot{\epsilon}_{11}\dot{\epsilon}_{22} (1-e^{-2})
153    \right]^{-\frac{1}{2}}    \right]^{-\frac{1}{2}}
154  \end{align*}  \end{align*}
155    The bulk viscosities are bounded above by imposing both a minimum
156    $\Delta_{\min}=10^{-11}\text{\,s}^{-1}$ (for numerical reasons) and a
157    maximum $\zeta_{\max} = P_{\max}/\Delta^*$, where
158    $\Delta^*=(5\times10^{12}/2\times10^4)\text{\,s}^{-1}$. For stress
159    tensor compuation the replacement pressure $P = 2\,\Delta\zeta$
160    \citep{hibler95} is used so that the stress state always lies on the
161    elliptic yield curve by definition.
162    
163  In the current implementation, the VP-model is integrated with the  In the current implementation, the VP-model is integrated with the
164  semi-implicit line successive over relaxation (LSOR)-solver of  semi-implicit line successive over relaxation (LSOR)-solver of
165  \citet{zhang98}, which allows for long time steps that, in our case,  \citet{zhang98}, which allows for long time steps that, in our case,
# Line 373  The difference between LSR and EVP solut Line 383  The difference between LSR and EVP solut
383  effective thickness and meridional velocity fields. The velocity field  effective thickness and meridional velocity fields. The velocity field
384  appears to be a little noisy. This noise has been address by  appears to be a little noisy. This noise has been address by
385  \citet{hunke01}. It can be dealt with by reducing EVP's internal time  \citet{hunke01}. It can be dealt with by reducing EVP's internal time
386  step (increasing the number of iterations) or by regularizing the bulk  step (increasing the number of iterations along with the computational
387  and shear viscosities. We revisit the latter option by reproducing the  cost) or by regularizing the bulk and shear viscosities. We revisit
388  results of \citet{hunke01} for the C-grid no-slip cases.  the latter option by reproducing some of the results of
389    \citet{hunke01}, namely the experiment described in her section~4, for
390    our C-grid no-slip cases: in a square domain with a few islands the
391    ice model is initialized with constant ice thickness and linearly
392    increasing ice concentration to the east. The model dynamics are
393    forced with a constant anticyclonic ocean gyre and variable
394    atmospheric wind whose mean directed diagnonally to the north-east
395    corner of the domain; ice volume and concentration are held constant
396    (no advection by ice velocity).  \reffig{hunke01} shows the ice
397    velocity field, its divergence, and the bulk viscosity $\zeta$ for the
398    cases C-LRSns and C-EVPns, and for a C-EVPns case, where
399    \citet{hunke01}'s regularization has been implemented; compare to
400    Fig.\,4 in \citet{hunke01}. The regularization contraint limits ice
401    strength and viscosities as a function of damping time scale,
402    resolution and EVP-time step, effectively allowing the elastic waves to
403    damp out more quickly \citep{hunke01}.
404  \begin{figure*}[htbp]  \begin{figure*}[htbp]
405    \includegraphics[width=\widefigwidth]{\fpath/hun12days}    \includegraphics[width=\widefigwidth]{\fpath/hun12days}
406    \caption{Hunke's test case.}    \caption{Hunke's test case.}
407    \label{fig:hunke01}    \label{fig:hunke01}
408  \end{figure*}  \end{figure*}
409    
410  \begin{itemize}  In the far right (``east'') side of the domain the ice concentration
411  \item B-grid LSR no-slip  is close to one and the ice should be nearly rigid. The applied wind
412  \item C-grid LSR no-slip  tends to push ice toward the upper right corner. Because the highly
413  \item C-grid LSR slip  compact ice is confinded by the boundary, it resists any further
414  \item C-grid EVP no-slip  compression and exhibits little motion in the rigid region on the
415  \item C-grid EVP slip  right hand side. The C-LSRns solution (top row) allows high
416  \end{itemize}  viscosities in the rigid region suppressing nearly all flow.
417    \citet{hunke01}'s regularization for the C-EVPns solution (bottom row)
418  \subsection{B-grid vs.\ C-grid}  clearly suppresses the noise present in $\nabla\cdot\vek{u}$ in the
419  Comparison between:  unregularized case (middle row), at the cost of reduced viscosities
420  \begin{itemize}  These reduced viscosities lead to small but finite ice velocities
421  \item B-grid, lsr, no-slip  which in turn can have a strong effect on solutions in the limit of
422  \item C-grid, lsr, no-slip  nearly rigid regimes (arching and blocking, not shown).
423  \item C-grid, evp, no-slip  
424  \end{itemize}  
425  all without ice-ocean stress, because ice-ocean stress does not work  %\begin{itemize}
426  for B-grid.  %\item B-grid LSR no-slip
427    %\item C-grid LSR no-slip
428    %\item C-grid LSR slip
429    %\item C-grid EVP no-slip
430    %\item C-grid EVP slip
431    %\end{itemize}
432    
433    %\subsection{B-grid vs.\ C-grid}
434    %Comparison between:
435    %\begin{itemize}
436    %\item B-grid, lsr, no-slip
437    %\item C-grid, lsr, no-slip
438    %\item C-grid, evp, no-slip
439    %\end{itemize}
440    %all without ice-ocean stress, because ice-ocean stress does not work
441    %for B-grid.
442    
443  \subsection{C-grid}  \subsection{C-grid}
444  \begin{itemize}  \begin{itemize}
# Line 757  parameters that we use here. What about Line 797  parameters that we use here. What about
797  We thank Jinlun Zhang for providing the original B-grid code and many  We thank Jinlun Zhang for providing the original B-grid code and many
798  helpful discussions.  helpful discussions.
799    
800  \bibliography{bib/journal_abrvs,bib/seaice,bib/genocean,bib/maths,bib/mitgcmuv,bib/fram}  %\bibliography{bib/journal_abrvs,bib/seaice,bib/genocean,bib/maths,bib/mitgcmuv,bib/fram}
801    \bibliography{journal_abrvs,seaice,genocean,maths,mitgcmuv,bib/fram}
802    
803  \end{document}  \end{document}
804    
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