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--- MITgcm_contrib/articles/ceaice/ceaice.tex 2007/11/07 17:26:13 1.3
+++ MITgcm_contrib/articles/ceaice/ceaice.tex 2008/01/10 15:47:32 1.4
@@ -1,7 +1,6 @@
\documentclass[12pt]{article}
-\usepackage{epsfig}
-\usepackage{graphics}
-\usepackage{subfigure}
+
+\usepackage{graphicx,subfigure}
\usepackage[round,comma]{natbib}
\bibliographystyle{bib/agu04}
@@ -35,7 +34,10 @@
\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}
@@ -128,13 +130,16 @@
\frac{\partial{u_{j}}}{\partial{x_{i}}}\right).
\end{equation*}
The pressure $P$, a measure of ice strength, depends on both thickness
-$h$ and compactness (concentration) $c$: \[P =
-P^{*}c\,h\,e^{[C^{*}\cdot(1-c)]},\] with the constants $P^{*}$ and
-$C^{*}$. The nonlinear bulk and shear 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:
+$h$ and compactness (concentration) $c$:
+\begin{equation}
+ P = P^{*}c\,h\,e^{[C^{*}\cdot(1-c)]},
+\label{icestrength}
+\end{equation}
+with the constants $P^{*}$ and $C^{*}$. The nonlinear bulk and shear
+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} \\
\eta =& \frac{P}{2\Delta{e}^2} \\
@@ -287,6 +292,96 @@
\section{Funnel Experiments}
\label{sec:funnel}
+For a first/detailed comparison between the different variants of the
+MIT sea ice model an idealized geometry of a periodic channel,
+1000\,km long and 500\,m wide on a non-rotating plane, with converging
+walls forming a symmetric funnel and a narrow strait of 40\,km width
+is used. The horizontal resolution is 5\,km throughout the domain
+making the narrow strait 8 grid points wide. The ice model is
+initialized with a complete ice cover of 50\,cm uniform thickness. The
+ice model is driven by a constant along channel eastward ocean current
+of 25\,cm/s that does not see the walls in the domain. All other
+ice-ocean-atmosphere interactions are turned off, in particular there
+is no feedback of ice dynamics on the ocean current. All thermodynamic
+processes are turned off so that ice thickness variations are only
+caused by convergent or divergent ice flow. Ice volume (effective
+thickness) and concentration are advected with a third-order scheme
+with a flux limiter \citep{hundsdorfer94} to avoid undershoots. This
+scheme is unconditionally stable and does not require additional
+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
+25\,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) or by regularizing the bulk
+and shear viscosities. We revisit the latter option by reproducing the
+results of \citet{hunke01} for the C-grid no-slip cases.
+\begin{figure*}[htbp]
+ \includegraphics[width=\widefigwidth]{\fpath/hun12days}
+ \caption{Hunke's test case.}
+ \label{fig:hunke01}
+\end{figure*}
+
\begin{itemize}
\item B-grid LSR no-slip
\item C-grid LSR no-slip
@@ -594,21 +689,21 @@
\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.
@@ -619,21 +714,21 @@
\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}}
@@ -670,3 +765,54 @@
%%% 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.
+ 0-layer Hibler salinity + snow
+ 3-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
+ 1/2 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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