--- MITgcm_contrib/articles/ceaice/ceaice.tex 2008/01/21 08:06:00 1.9 +++ MITgcm_contrib/articles/ceaice/ceaice.tex 2008/02/25 19:30:56 1.11 @@ -1,4 +1,4 @@ -% $Header: /home/ubuntu/mnt/e9_copy/MITgcm_contrib/articles/ceaice/ceaice.tex,v 1.9 2008/01/21 08:06:00 mlosch Exp $ +% $Header: /home/ubuntu/mnt/e9_copy/MITgcm_contrib/articles/ceaice/ceaice.tex,v 1.11 2008/02/25 19:30:56 mlosch Exp $ % $Name: $ \documentclass[12pt]{article} @@ -52,7 +52,17 @@ \maketitle \begin{abstract} - Some blabla + +As part of ongoing efforts to obtain a best possible synthesis of most +available, global-scale, ocean and sea ice data, dynamic and thermodynamic +sea-ice model components have been incorporated in the Massachusetts Institute +of Technology general circulation model (MITgcm). Sea-ice dynamics use either +a visco-plastic rheology solved with a line successive relaxation (LSR) +technique, reformulated on an Arakawa C-grid in order to match the oceanic and +atmospheric grids of the MITgcm, and modified to permit efficient and accurate +automatic differentiation of the coupled ocean and sea-ice model +configurations. + \end{abstract} \section{Introduction} @@ -311,142 +321,6 @@ state variables to be advected by ice velocities, namely enthalphy of the two ice layers and the thickness of the overlying snow layer. -\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 have 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 EVP velocity -fields 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 by variable -atmospheric wind whose mean direction is diagnonal to the north-east -corner of the domain; ice volume and concentration are held constant -(no thermodynamics and 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{Ice flow, divergence and bulk viscosities of three - experiments with \citet{hunke01}'s test case: C-LSRns (top), - C-EVPns (middle), and C-EVPns with damping described in - \citet{hunke01} (bottom).} - \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 confined 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 $\nabla\cdot\vek{u}$ and -$\log_{10}\zeta$ 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). - -\ml{[Say something about performance? This is tricky, as the - perfomance depends strongly on the configuration. A run with slowly - changing forcing is favorable for LSR, because then only very few - iterations are required for convergences while EVP uses its fixed - number of internal timesteps. If the forcing in changing fast, LSR - needs far more iterations while EVP still uses the fixed number of - internal timesteps. I have produces runs where for slow forcing LSR - is much faster than EVP and for fast forcing, LSR is much slower - than EVP. EVP is certainly more efficient in terms of vectorization - and MFLOPS on our SX8, but is that a criterion?]} \subsection{C-grid} \begin{itemize}