--- 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