--- MITgcm_contrib/articles/ceaice/ceaice.tex 2007/11/07 14:35:09 1.1 +++ MITgcm_contrib/articles/ceaice/ceaice.tex 2008/01/14 15:46:54 1.5 @@ -1,10 +1,10 @@ \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} @@ -35,7 +35,10 @@ \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} @@ -127,17 +130,21 @@ \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 -$h$ 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: +The maximum ice pressure $P_{\max}$, a measure of ice strength, depends on +both thickness $h$ and compactness (concentration) $c$: +\begin{equation} + P_{\max} = 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} \\ + \zeta =& \min\left(\frac{P_{\max}}{2\max(\Delta,\Delta_{\min})}, + \zeta_{\max}\right) \\ + \eta =& \frac{\zeta}{e^2} \\ \intertext{with the abbreviation} \Delta = & \left[ \left(\dot{\epsilon}_{11}^2+\dot{\epsilon}_{22}^2\right) @@ -145,6 +152,14 @@ 2\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, @@ -287,23 +302,143 @@ \section{Funnel Experiments} \label{sec:funnel} -\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. +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 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} @@ -434,12 +569,6 @@ \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} @@ -600,21 +729,21 @@ \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. @@ -625,21 +754,21 @@ \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}} @@ -668,7 +797,8 @@ 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} @@ -676,3 +806,54 @@ %%% 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