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-revision 1.1 by dimitri,
Wed Nov  7 14:35:09 2007 UTC
+revision 1.5 by mlosch,
Mon Jan 14 15:46:54 2008 UTC
 Line 1
  \documentclass[12pt]{article}
- \usepackage{epsfig}
- \usepackage{graphics}
+ \usepackage[]{graphicx}
  \usepackage{subfigure}
  \usepackage[round,comma]{natbib}
- \bibliographystyle{agu04}
+ \bibliographystyle{bib/agu04}
  \usepackage{amsmath,amssymb}
  \newcommand\bmmax{10} \newcommand\hmmax{10}
 Line 35
  \newlength{\mediumfigwidth}\setlength{\mediumfigwidth}{39pc}
  %\newlength{\widefigwidth}\setlength{\widefigwidth}{39pc}
  \newlength{\widefigwidth}\setlength{\widefigwidth}{\textwidth}
- \newcommand{\fpath}{.}
+ \newcommand{\fpath}{figs}
+ % commenting scheme
+ \newcommand{\ml}[1]{\textsf{\slshape #1}}
  \title{A Dynamic-Thermodynamic Sea ice Model for Ocean Climate
    Estimation on an Arakawa C-Grid}
-Line 127 
 The ice strain rate is given by
+Line 130 
 The ice strain rate is given by
      \frac{\partial{u_{i}}}{\partial{x_{j}}} +
      \frac{\partial{u_{j}}}{\partial{x_{i}}}\right).
  \end{equation*}
- 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
- $h$ and compactness (concentration) $c$: \[P =
+ both thickness $h$ and compactness (concentration) $c$:
- P^{*}c\,h\,e^{[C^{*}\cdot(1-c)]},\] with the constants $P^{*}$ and
+ \begin{equation}
- $C^{*}$. The nonlinear bulk and shear viscosities $\eta$ and $\zeta$
+   P_{\max} = P^{*}c\,h\,e^{[C^{*}\cdot(1-c)]},
- are functions of ice strain rate invariants and ice strength such that
+ \label{icestrength}
- the principal components of the stress lie on an elliptical yield
+ \end{equation}
- curve with the ratio of major to minor axis $e$ equal to $2$; they are
+ with the constants $P^{*}$ and $C^{*}$. The nonlinear bulk and shear
- given by:
+ viscosities $\eta$ and $\zeta$ are functions of ice strain rate
+ invariants and ice strength such that the principal components of the
+ stress lie on an elliptical yield curve with the ratio of major to
+ minor axis $e$ equal to $2$; they are given by:
  \begin{align*}
-   \zeta =& \frac{P}{2\Delta} \\
+   \zeta =& \min\left(\frac{P_{\max}}{2\max(\Delta,\Delta_{\min})},
-   \eta =& \frac{P}{2\Delta{e}^2} \\
+    \zeta_{\max}\right) \\
+   \eta =& \frac{\zeta}{e^2} \\
    \intertext{with the abbreviation}
    \Delta = & \left[
      \left(\dot{\epsilon}_{11}^2+\dot{\epsilon}_{22}^2\right)
-Line 145 
 given by:
+Line 152 
 given by:
 \dot{\epsilon}_{11}\dot{\epsilon}_{22} (1-e^{-2})
    \right]^{-\frac{1}{2}}
  \end{align*}
+ The bulk viscosities are bounded above by imposing both a minimum
+ $\Delta_{\min}=10^{-11}\text{\,s}^{-1}$ (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 $P = 2\,\Delta\zeta$
+ \citep{hibler95} is used so that the stress state always lies on the
+ elliptic yield curve by definition.
  In the current implementation, the VP-model is integrated with the
  semi-implicit line successive over relaxation (LSOR)-solver of
  \citet{zhang98}, which allows for long time steps that, in our case,
-Line 287 
 the two ice layers and the thickness of
+Line 302 
 the two ice layers and the thickness of
  \section{Funnel Experiments}
  \label{sec:funnel}
- \begin{itemize}
+ For a first/detailed comparison between the different variants of the
- \item B-grid LSR no-slip
+ MIT sea ice model an idealized geometry of a periodic channel,
- \item C-grid LSR no-slip
+\,km long and 500\,m wide on a non-rotating plane, with converging
- \item C-grid LSR slip
+ walls forming a symmetric funnel and a narrow strait of 40\,km width
- \item C-grid EVP no-slip
+ is used. The horizontal resolution is 5\,km throughout the domain
- \item C-grid EVP slip
+ making the narrow strait 8 grid points wide. The ice model is
- \end{itemize}
+ initialized with a complete ice cover of 50\,cm uniform thickness. The
+ ice model is driven by a constant along channel eastward ocean current
- \subsection{B-grid vs.\ C-grid}
+ of 25\,cm/s that does not see the walls in the domain. All other
- Comparison between:
+ ice-ocean-atmosphere interactions are turned off, in particular there
- \begin{itemize}
+ is no feedback of ice dynamics on the ocean current. All thermodynamic
- \item B-grid, lsr, no-slip
+ processes are turned off so that ice thickness variations are only
- \item C-grid, lsr, no-slip
+ caused by convergent or divergent ice flow. Ice volume (effective
- \item C-grid, evp, no-slip
+ thickness) and concentration are advected with a third-order scheme
- \end{itemize}
+ with a flux limiter \citep{hundsdorfer94} to avoid undershoots. This
- all without ice-ocean stress, because ice-ocean stress does not work
+ scheme is unconditionally stable and does not require additional
- for B-grid.
+ diffusion. The time step used here is 1\,h.
+ \reffig{funnelf0} compares the dynamic fields ice concentration $c$,
+ effective thickness $h_{eff} = h\cdot{c}$, and velocities $(u,v)$ for
+ five different cases at steady state (after 10\,years of integration):
+ \begin{description}
+ \item[B-LSRns:] LSR solver with no-slip boundary conditions on a B-grid,
+ \item[C-LSRns:] LSR solver with no-slip boundary conditions on a C-grid,
+ \item[C-LSRfs:] LSR solver with free-slip boundary conditions on a C-grid,
+ \item[C-EVPns:] EVP solver with no-slip boundary conditions on a C-grid,
+ \item[C-EVPfs:] EVP solver with free-slip boundary conditions on a C-grid,
+ \end{description}
+ \ml{[We have not implemented the EVP solver on a B-grid.]}
+ \begin{figure*}[htbp]
+   \includegraphics[width=\widefigwidth]{\fpath/all_086280}
+   \caption{Ice concentration, effective thickness [m], and ice
+     velocities [m/s]
+     for 5 different numerical solutions.}
+   \label{fig:funnelf0}
+ \end{figure*}
+ At a first glance, the solutions look similar. This is encouraging as
+ the details of discretization and numerics should not affect the
+ solutions to first order. In all cases the ice-ocean stress pushes the
+ ice cover eastwards, where it converges in the funnel. In the narrow
+ channel the ice moves quickly (nearly free drift) and leaves the
+ channel as narrow band.
+ A close look reveals interesting differences between the B- and C-grid
+ results. The zonal velocity in the narrow channel is nearly the free
+ drift velocity ( = ocean velocity) of 25\,cm/s for the C-grid
+ solutions, regardless of the boundary conditions, while it is just
+ above 20\,cm/s for the B-grid solution. The ice accelerates to
+\,cm/s after it exits the channel. Concentrating on the solutions
+ B-LSRns and C-LSRns, the ice volume (effective thickness) along the
+ boundaries in the narrow channel is larger in the B-grid case although
+ the ice concentration is reduces in the C-grid case. The combined
+ effect leads to a larger actual ice thickness at smaller
+ concentrations in the C-grid case. However, since the effective
+ thickness determines the ice strength $P$ in Eq\refeq{icestrength},
+ the ice strength and thus the bulk and shear viscosities are larger in
+ the B-grid case leading to more horizontal friction. This circumstance
+ might explain why the no-slip boundary conditions in the B-grid case
+ appear to be more effective in reducing the flow within the narrow
+ channel, than in the C-grid case. Further, the viscosities are also
+ sensitive to details of the velocity gradients. Via $\Delta$, these
+ gradients enter the viscosities in the denominator so that large
+ gradients tend to reduce the viscosities. This again favors more flow
+ along the boundaries in the C-grid case: larger velocities
+ (\reffig{funnelf0}) on grid points that are closer to the boundary by
+ a factor $\frac{1}{2}$ than in the B-grid case because of the stagger
+ nature of the C-grid lead numerically to larger tangential gradients
+ across the boundary; these in turn make the viscosities smaller for
+ less tangential friction and allow more tangential flow along the
+ boundaries.
+ The above argument can also be invoked to explain the small
+ differences between the free-slip and no-slip solutions on the C-grid.
+ Because of the non-linearities in the ice viscosities, in particular
+ along the boundaries, the no-slip boundary conditions has only a small
+ impact on the solution.
+ The difference between LSR and EVP solutions is largest in the
+ effective thickness and meridional velocity fields. The velocity field
+ appears to be a little noisy. This noise has been address by
+ \citet{hunke01}. It can be dealt with by reducing EVP's internal time
+ step (increasing the number of iterations along with the computational
+ cost) or by regularizing the bulk and shear viscosities. We revisit
+ the latter option by reproducing some of the results of
+ \citet{hunke01}, namely the experiment described in her section~4, for
+ our C-grid no-slip cases: in a square domain with a few islands the
+ ice model is initialized with constant ice thickness and linearly
+ increasing ice concentration to the east. The model dynamics are
+ forced with a constant anticyclonic ocean gyre and variable
+ atmospheric wind whose mean directed diagnonally to the north-east
+ corner of the domain; ice volume and concentration are held constant
+ (no advection by ice velocity).  \reffig{hunke01} shows the ice
+ velocity field, its divergence, and the bulk viscosity $\zeta$ for the
+ cases C-LRSns and C-EVPns, and for a C-EVPns case, where
+ \citet{hunke01}'s regularization has been implemented; compare to
+ Fig.\,4 in \citet{hunke01}. The regularization contraint limits ice
+ strength and viscosities as a function of damping time scale,
+ resolution and EVP-time step, effectively allowing the elastic waves to
+ damp out more quickly \citep{hunke01}.
+ \begin{figure*}[htbp]
+   \includegraphics[width=\widefigwidth]{\fpath/hun12days}
+   \caption{Hunke's test case.}
+   \label{fig:hunke01}
+ \end{figure*}
+ In the far right (``east'') side of the domain the ice concentration
+ is close to one and the ice should be nearly rigid. The applied wind
+ tends to push ice toward the upper right corner. Because the highly
+ compact ice is confinded by the boundary, it resists any further
+ compression and exhibits little motion in the rigid region on the
+ right hand side. The C-LSRns solution (top row) allows high
+ viscosities in the rigid region suppressing nearly all flow.
+ \citet{hunke01}'s regularization for the C-EVPns solution (bottom row)
+ clearly suppresses the noise present in $\nabla\cdot\vek{u}$ in the
+ unregularized case (middle row), at the cost of reduced viscosities
+ These reduced viscosities lead to small but finite ice velocities
+ which in turn can have a strong effect on solutions in the limit of
+ nearly rigid regimes (arching and blocking, not shown).
+ %\begin{itemize}
+ %\item B-grid LSR no-slip
+ %\item C-grid LSR no-slip
+ %\item C-grid LSR slip
+ %\item C-grid EVP no-slip
+ %\item C-grid EVP slip
+ %\end{itemize}
+ %\subsection{B-grid vs.\ C-grid}
+ %Comparison between:
+ %\begin{itemize}
+ %\item B-grid, lsr, no-slip
+ %\item C-grid, lsr, no-slip
+ %\item C-grid, evp, no-slip
+ %\end{itemize}
+ %all without ice-ocean stress, because ice-ocean stress does not work
+ %for B-grid.
  \subsection{C-grid}
  \begin{itemize}
-Line 434 
 We anticipate small differences between
+Line 569 
 We anticipate small differences between
  \item ocean stress: different water mass properties beneath the ice
  \end{itemize}
- \section{Adjoint sensitivity experiment}
- \label{sec:adjoint}
- Adjoint sensitivity experiment on 1/2-res setup
-  Sensitivity of sea ice volume flow through Fram Strait
  \section{Adjoint sensiivities of the MITsim}
  \subsection{The adjoint of MITsim}
-Line 600 
 the circulation around Svalbard, and ...
+Line 729 
 the circulation around Svalbard, and ...
  \begin{figure}[t!]
  \centerline{
  \subfigure[{\footnotesize -12 months}]
- {\includegraphics*[width=0.44\linewidth]{figs/run_4yr_ADJheff_arc_lev1_tim072_cmax2.0E+02.eps}}
+ {\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJheff_arc_lev1_tim072_cmax2.0E+02.eps}}
  %\includegraphics*[width=.3\textwidth]{H_c.bin_res_100_lev1.pdf}
  %
  \subfigure[{\footnotesize -24 months}]
- {\includegraphics*[width=0.44\linewidth]{figs/run_4yr_ADJheff_arc_lev1_tim145_cmax2.0E+02.eps}}
+ {\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJheff_arc_lev1_tim145_cmax2.0E+02.eps}}
  }
  \centerline{
  \subfigure[{\footnotesize
  -36 months}]
- {\includegraphics*[width=0.44\linewidth]{figs/run_4yr_ADJheff_arc_lev1_tim218_cmax2.0E+02.eps}}
+ {\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJheff_arc_lev1_tim218_cmax2.0E+02.eps}}
  %
  \subfigure[{\footnotesize
  -48 months}]
- {\includegraphics*[width=0.44\linewidth]{figs/run_4yr_ADJheff_arc_lev1_tim292_cmax2.0E+02.eps}}
+ {\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJheff_arc_lev1_tim292_cmax2.0E+02.eps}}
  }
  \caption{Sensitivity of sea-ice export through Fram Strait in December 2005 to
  sea-ice thickness at various prior times.
-Line 625 
 sea-ice thickness at various prior times
+Line 754 
 sea-ice thickness at various prior times
  \begin{figure}[t!]
  \centerline{
  \subfigure[{\footnotesize -12 months}]
- {\includegraphics*[width=0.44\linewidth]{figs/run_4yr_ADJtheta_arc_lev1_tim072_cmax5.0E+01.eps}}
+ {\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJtheta_arc_lev1_tim072_cmax5.0E+01.eps}}
  %\includegraphics*[width=.3\textwidth]{H_c.bin_res_100_lev1.pdf}
  %
  \subfigure[{\footnotesize -24 months}]
- {\includegraphics*[width=0.44\linewidth]{figs/run_4yr_ADJtheta_arc_lev1_tim145_cmax5.0E+01.eps}}
+ {\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJtheta_arc_lev1_tim145_cmax5.0E+01.eps}}
  }
  \centerline{
  \subfigure[{\footnotesize
  -36 months}]
- {\includegraphics*[width=0.44\linewidth]{figs/run_4yr_ADJtheta_arc_lev1_tim218_cmax5.0E+01.eps}}
+ {\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJtheta_arc_lev1_tim218_cmax5.0E+01.eps}}
  %
  \subfigure[{\footnotesize
  -48 months}]
- {\includegraphics*[width=0.44\linewidth]{figs/run_4yr_ADJtheta_arc_lev1_tim292_cmax5.0E+01.eps}}
+ {\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJtheta_arc_lev1_tim292_cmax5.0E+01.eps}}
  }
  \caption{Same as Fig. XXX but for sea surface temperature
  \label{fig:4yradjthetalev1}}
-Line 668 
 parameters that we use here. What about
+Line 797 
 parameters that we use here. What about
  We thank Jinlun Zhang for providing the original B-grid code and many
  helpful discussions.
- \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}
+ \bibliography{journal_abrvs,seaice,genocean,maths,mitgcmuv,bib/fram}
  \end{document}
-Line 676 
 helpful discussions.
+Line 806 
 helpful discussions.
  %%% mode: latex
  %%% TeX-master: t
  %%% End:
+ A Dynamic-Thermodynamic Sea ice Model for Ocean Climate
+   Estimation on an Arakawa C-Grid
+ Introduction
+ Ice Model:
+  Dynamics formulation.
+   B-C, LSR, EVP, no-slip, slip
+   parallellization
+  Thermodynamics formulation.
+-layer Hibler salinity + snow
+-layer Winton
+ Idealized tests
+  Funnel Experiments
+  Downstream Island tests
+   B-grid LSR no-slip
+   C-grid LSR no-slip
+   C-grid LSR slip
+   C-grid EVP no-slip
+   C-grid EVP slip
+ Arctic Setup
+  Configuration
+  OBCS from cube
+  forcing
+/2 and full resolution
+  with a few JFM figs from C-grid LSR no slip
+   ice transport through Canadian Archipelago
+   thickness distribution
+   ice velocity and transport
+ Arctic forward sensitivity experiments
+  B-grid LSR no-slip
+  C-grid LSR no-slip
+  C-grid LSR slip
+  C-grid EVP no-slip
+  C-grid EVP slip
+  C-grid LSR no-slip + Winton
+   speed-performance-accuracy (small)
+   ice transport through Canadian Archipelago differences
+   thickness distribution differences
+   ice velocity and transport differences
+ Adjoint sensitivity experiment on 1/2-res setup
+  Sensitivity of sea ice volume flow through Fram Strait
+ *** Sensitivity of sea ice volume flow through Canadian Archipelago
+ Summary and conluding remarks

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