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\newcommand{\ml}[1]{\textsf{\slshape #1}} |
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\title{A Dynamic-Thermodynamic Sea ice Model for Ocean Climate |
\title{A Dynamic-Thermodynamic Sea ice Model for Ocean Climate |
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Estimation on an Arakawa C-Grid} |
Estimation on an Arakawa C-Grid} |
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\frac{\partial{u_{j}}}{\partial{x_{i}}}\right). |
\frac{\partial{u_{j}}}{\partial{x_{i}}}\right). |
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\end{equation*} |
\end{equation*} |
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The pressure $P$, a measure of ice strength, depends on both thickness |
The pressure $P$, a measure of ice strength, depends on both thickness |
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$h$ and compactness (concentration) $c$: \[P = |
$h$ and compactness (concentration) $c$: |
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P^{*}c\,h\,e^{[C^{*}\cdot(1-c)]},\] with the constants $P^{*}$ and |
\begin{equation} |
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$C^{*}$. The nonlinear bulk and shear viscosities $\eta$ and $\zeta$ |
P = P^{*}c\,h\,e^{[C^{*}\cdot(1-c)]}, |
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are functions of ice strain rate invariants and ice strength such that |
\label{icestrength} |
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the principal components of the stress lie on an elliptical yield |
\end{equation} |
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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 |
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given by: |
viscosities $\eta$ and $\zeta$ are functions of ice strain rate |
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invariants and ice strength such that the principal components of the |
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stress lie on an elliptical yield curve with the ratio of major to |
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minor axis $e$ equal to $2$; they are given by: |
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\begin{align*} |
\begin{align*} |
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\zeta =& \frac{P}{2\Delta} \\ |
\zeta =& \frac{P}{2\Delta} \\ |
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\eta =& \frac{P}{2\Delta{e}^2} \\ |
\eta =& \frac{P}{2\Delta{e}^2} \\ |
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\section{Funnel Experiments} |
\section{Funnel Experiments} |
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\label{sec:funnel} |
\label{sec:funnel} |
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For a first/detailed comparison between the different variants of the |
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MIT sea ice model an idealized geometry of a periodic channel, |
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1000\,km long and 500\,m wide on a non-rotating plane, with converging |
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walls forming a symmetric funnel and a narrow strait of 40\,km width |
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is used. The horizontal resolution is 5\,km throughout the domain |
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making the narrow strait 8 grid points wide. The ice model is |
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initialized with a complete ice cover of 50\,cm uniform thickness. The |
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ice model is driven by a constant along channel eastward ocean current |
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of 25\,cm/s that does not see the walls in the domain. All other |
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ice-ocean-atmosphere interactions are turned off, in particular there |
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is no feedback of ice dynamics on the ocean current. All thermodynamic |
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processes are turned off so that ice thickness variations are only |
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caused by convergent or divergent ice flow. Ice volume (effective |
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thickness) and concentration are advected with a third-order scheme |
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with a flux limiter \citep{hundsdorfer94} to avoid undershoots. This |
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scheme is unconditionally stable and does not require additional |
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diffusion. The time step used here is 1\,h. |
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\reffig{funnelf0} compares the dynamic fields ice concentration $c$, |
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effective thickness $h_{eff} = h\cdot{c}$, and velocities $(u,v)$ for |
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five different cases at steady state (after 10\,years of integration): |
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\begin{description} |
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\item[B-LSRns:] LSR solver with no-slip boundary conditions on a B-grid, |
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\item[C-LSRns:] LSR solver with no-slip boundary conditions on a C-grid, |
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\item[C-LSRfs:] LSR solver with free-slip boundary conditions on a C-grid, |
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\item[C-EVPns:] EVP solver with no-slip boundary conditions on a C-grid, |
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\item[C-EVPfs:] EVP solver with free-slip boundary conditions on a C-grid, |
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\end{description} |
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\ml{[We have not implemented the EVP solver on a B-grid.]} |
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\begin{figure*}[htbp] |
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\includegraphics[width=\widefigwidth]{\fpath/all_086280} |
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\caption{Ice concentration, effective thickness [m], and ice |
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velocities [m/s] |
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for 5 different numerical solutions.} |
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\label{fig:funnelf0} |
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\end{figure*} |
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At a first glance, the solutions look similar. This is encouraging as |
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the details of discretization and numerics should not affect the |
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solutions to first order. In all cases the ice-ocean stress pushes the |
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ice cover eastwards, where it converges in the funnel. In the narrow |
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channel the ice moves quickly (nearly free drift) and leaves the |
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channel as narrow band. |
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A close look reveals interesting differences between the B- and C-grid |
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results. The zonal velocity in the narrow channel is nearly the free |
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drift velocity ( = ocean velocity) of 25\,cm/s for the C-grid |
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solutions, regardless of the boundary conditions, while it is just |
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above 20\,cm/s for the B-grid solution. The ice accelerates to |
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25\,cm/s after it exits the channel. Concentrating on the solutions |
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B-LSRns and C-LSRns, the ice volume (effective thickness) along the |
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boundaries in the narrow channel is larger in the B-grid case although |
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the ice concentration is reduces in the C-grid case. The combined |
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effect leads to a larger actual ice thickness at smaller |
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concentrations in the C-grid case. However, since the effective |
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thickness determines the ice strength $P$ in Eq\refeq{icestrength}, |
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the ice strength and thus the bulk and shear viscosities are larger in |
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the B-grid case leading to more horizontal friction. This circumstance |
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might explain why the no-slip boundary conditions in the B-grid case |
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appear to be more effective in reducing the flow within the narrow |
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channel, than in the C-grid case. Further, the viscosities are also |
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sensitive to details of the velocity gradients. Via $\Delta$, these |
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gradients enter the viscosities in the denominator so that large |
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gradients tend to reduce the viscosities. This again favors more flow |
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along the boundaries in the C-grid case: larger velocities |
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(\reffig{funnelf0}) on grid points that are closer to the boundary by |
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a factor $\frac{1}{2}$ than in the B-grid case because of the stagger |
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nature of the C-grid lead numerically to larger tangential gradients |
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across the boundary; these in turn make the viscosities smaller for |
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less tangential friction and allow more tangential flow along the |
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boundaries. |
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The above argument can also be invoked to explain the small |
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differences between the free-slip and no-slip solutions on the C-grid. |
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Because of the non-linearities in the ice viscosities, in particular |
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along the boundaries, the no-slip boundary conditions has only a small |
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impact on the solution. |
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The difference between LSR and EVP solutions is largest in the |
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effective thickness and meridional velocity fields. The velocity field |
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appears to be a little noisy. This noise has been address by |
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\citet{hunke01}. It can be dealt with by reducing EVP's internal time |
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step (increasing the number of iterations) or by regularizing the bulk |
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and shear viscosities. We revisit the latter option by reproducing the |
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results of \citet{hunke01} for the C-grid no-slip cases. |
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\begin{figure*}[htbp] |
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\includegraphics[width=\widefigwidth]{\fpath/hun12days} |
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\caption{Hunke's test case.} |
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\label{fig:hunke01} |
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\end{figure*} |
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\begin{itemize} |
\begin{itemize} |
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\item B-grid LSR no-slip |
\item B-grid LSR no-slip |
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\item C-grid LSR no-slip |
\item C-grid LSR no-slip |
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\item ocean stress: different water mass properties beneath the ice |
\item ocean stress: different water mass properties beneath the ice |
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\end{itemize} |
\end{itemize} |
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\section{Adjoint sensitivity experiment} |
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\label{sec:adjoint} |
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Adjoint sensitivity experiment on 1/2-res setup |
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Sensitivity of sea ice volume flow through Fram Strait |
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\section{Adjoint sensiivities of the MITsim} |
\section{Adjoint sensiivities of the MITsim} |
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\subsection{The adjoint of MITsim} |
\subsection{The adjoint of MITsim} |
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\begin{figure}[t!] |
\begin{figure}[t!] |
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\centerline{ |
\centerline{ |
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\subfigure[{\footnotesize -12 months}] |
\subfigure[{\footnotesize -12 months}] |
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{\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}} |
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%\includegraphics*[width=.3\textwidth]{H_c.bin_res_100_lev1.pdf} |
%\includegraphics*[width=.3\textwidth]{H_c.bin_res_100_lev1.pdf} |
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% |
% |
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\subfigure[{\footnotesize -24 months}] |
\subfigure[{\footnotesize -24 months}] |
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{\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}} |
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} |
} |
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\centerline{ |
\centerline{ |
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\subfigure[{\footnotesize |
\subfigure[{\footnotesize |
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-36 months}] |
-36 months}] |
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{\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}} |
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% |
% |
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\subfigure[{\footnotesize |
\subfigure[{\footnotesize |
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-48 months}] |
-48 months}] |
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{\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}} |
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} |
} |
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\caption{Sensitivity of sea-ice export through Fram Strait in December 2005 to |
\caption{Sensitivity of sea-ice export through Fram Strait in December 2005 to |
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sea-ice thickness at various prior times. |
sea-ice thickness at various prior times. |
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\begin{figure}[t!] |
\begin{figure}[t!] |
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\centerline{ |
\centerline{ |
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\subfigure[{\footnotesize -12 months}] |
\subfigure[{\footnotesize -12 months}] |
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{\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}} |
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%\includegraphics*[width=.3\textwidth]{H_c.bin_res_100_lev1.pdf} |
%\includegraphics*[width=.3\textwidth]{H_c.bin_res_100_lev1.pdf} |
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% |
% |
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\subfigure[{\footnotesize -24 months}] |
\subfigure[{\footnotesize -24 months}] |
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{\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}} |
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} |
} |
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\centerline{ |
\centerline{ |
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\subfigure[{\footnotesize |
\subfigure[{\footnotesize |
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-36 months}] |
-36 months}] |
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{\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}} |
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% |
% |
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\subfigure[{\footnotesize |
\subfigure[{\footnotesize |
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-48 months}] |
-48 months}] |
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{\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}} |
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} |
} |
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\caption{Same as Fig. XXX but for sea surface temperature |
\caption{Same as Fig. XXX but for sea surface temperature |
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\label{fig:4yradjthetalev1}} |
\label{fig:4yradjthetalev1}} |
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%%% mode: latex |
%%% mode: latex |
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%%% TeX-master: t |
%%% TeX-master: t |
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A Dynamic-Thermodynamic Sea ice Model for Ocean Climate |
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Estimation on an Arakawa C-Grid |
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Introduction |
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Ice Model: |
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Dynamics formulation. |
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B-C, LSR, EVP, no-slip, slip |
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parallellization |
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Thermodynamics formulation. |
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0-layer Hibler salinity + snow |
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3-layer Winton |
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Idealized tests |
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Funnel Experiments |
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Downstream Island tests |
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B-grid LSR no-slip |
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C-grid LSR no-slip |
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C-grid LSR slip |
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C-grid EVP no-slip |
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C-grid EVP slip |
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Arctic Setup |
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Configuration |
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OBCS from cube |
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forcing |
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1/2 and full resolution |
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with a few JFM figs from C-grid LSR no slip |
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ice transport through Canadian Archipelago |
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thickness distribution |
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ice velocity and transport |
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Arctic forward sensitivity experiments |
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B-grid LSR no-slip |
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C-grid LSR no-slip |
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C-grid LSR slip |
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C-grid EVP no-slip |
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C-grid EVP slip |
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C-grid LSR no-slip + Winton |
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speed-performance-accuracy (small) |
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ice transport through Canadian Archipelago differences |
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thickness distribution differences |
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ice velocity and transport differences |
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Adjoint sensitivity experiment on 1/2-res setup |
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Sensitivity of sea ice volume flow through Fram Strait |
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*** Sensitivity of sea ice volume flow through Canadian Archipelago |
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Summary and conluding remarks |
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