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% $Name$ |
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\documentclass[12pt]{article} |
\documentclass[12pt]{article} |
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\usepackage[]{graphicx} |
\usepackage[]{graphicx} |
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\maketitle |
\maketitle |
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\begin{abstract} |
\begin{abstract} |
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Some blabla |
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As part of ongoing efforts to obtain a best possible synthesis of most |
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available, global-scale, ocean and sea ice data, dynamic and thermodynamic |
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sea-ice model components have been incorporated in the Massachusetts Institute |
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of Technology general circulation model (MITgcm). Sea-ice dynamics use either |
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a visco-plastic rheology solved with a line successive relaxation (LSR) |
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technique, reformulated on an Arakawa C-grid in order to match the oceanic and |
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atmospheric grids of the MITgcm, and modified to permit efficient and accurate |
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automatic differentiation of the coupled ocean and sea-ice model |
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configurations. |
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\end{abstract} |
\end{abstract} |
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\section{Introduction} |
\section{Introduction} |
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both thickness $h$ and compactness (concentration) $c$: |
both thickness $h$ and compactness (concentration) $c$: |
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\begin{equation} |
\begin{equation} |
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P_{\max} = P^{*}c\,h\,e^{[C^{*}\cdot(1-c)]}, |
P_{\max} = P^{*}c\,h\,e^{[C^{*}\cdot(1-c)]}, |
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\label{icestrength} |
\label{eq:icestrength} |
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\end{equation} |
\end{equation} |
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with the constants $P^{*}$ and $C^{*}$. The nonlinear bulk and shear |
with the constants $P^{*}$ and $C^{*}$. The nonlinear bulk and shear |
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viscosities $\eta$ and $\zeta$ are functions of ice strain rate |
viscosities $\eta$ and $\zeta$ are functions of ice strain rate |
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state variables to be advected by ice velocities, namely enthalphy of |
state variables to be advected by ice velocities, namely enthalphy of |
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the two ice layers and the thickness of the overlying snow layer. |
the two ice layers and the thickness of the overlying snow layer. |
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\section{Funnel Experiments} |
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\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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%GET \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 have 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 EVP velocity |
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fields 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 along with the computational |
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cost) or by regularizing the bulk and shear viscosities. We revisit |
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the latter option by reproducing some of the results of |
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\citet{hunke01}, namely the experiment described in her section~4, for |
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our C-grid no-slip cases: in a square domain with a few islands the |
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ice model is initialized with constant ice thickness and linearly |
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increasing ice concentration to the east. The model dynamics are |
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forced with a constant anticyclonic ocean gyre and by variable |
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atmospheric wind whose mean direction is diagnonal to the north-east |
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corner of the domain; ice volume and concentration are held constant |
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(no thermodynamics and no advection by ice velocity). |
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\reffig{hunke01} shows the ice velocity field, its divergence, and the |
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bulk viscosity $\zeta$ for the cases C-LRSns and C-EVPns, and for a |
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C-EVPns case, where \citet{hunke01}'s regularization has been |
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implemented; compare to Fig.\,4 in \citet{hunke01}. The regularization |
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contraint limits ice strength and viscosities as a function of damping |
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time scale, resolution and EVP-time step, effectively allowing the |
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elastic waves to damp out more quickly \citep{hunke01}. |
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\begin{figure*}[htbp] |
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%GET \includegraphics[width=\widefigwidth]{\fpath/hun12days} |
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\caption{Ice flow, divergence and bulk viscosities of three |
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experiments with \citet{hunke01}'s test case: C-LSRns (top), |
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C-EVPns (middle), and C-EVPns with damping described in |
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\citet{hunke01} (bottom).} |
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\label{fig:hunke01} |
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\end{figure*} |
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In the far right (``east'') side of the domain the ice concentration |
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is close to one and the ice should be nearly rigid. The applied wind |
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tends to push ice toward the upper right corner. Because the highly |
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compact ice is confined by the boundary, it resists any further |
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compression and exhibits little motion in the rigid region on the |
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right hand side. The C-LSRns solution (top row) allows high |
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viscosities in the rigid region suppressing nearly all flow. |
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\citet{hunke01}'s regularization for the C-EVPns solution (bottom row) |
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clearly suppresses the noise present in $\nabla\cdot\vek{u}$ and |
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$\log_{10}\zeta$ in the |
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unregularized case (middle row), at the cost of reduced viscosities. |
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These reduced viscosities lead to small but finite ice velocities |
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which in turn can have a strong effect on solutions in the limit of |
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nearly rigid regimes (arching and blocking, not shown). |
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\subsection{C-grid} |
\subsection{C-grid} |
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\begin{itemize} |
\begin{itemize} |
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helpful discussions. ML thanks Elizabeth Hunke for multiple explanations. |
helpful discussions. ML thanks Elizabeth Hunke for multiple explanations. |
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\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} |
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%\bibliography{journal_abrvs,seaice,genocean,maths,mixing,mitgcmuv,bib/fram} |
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\end{document} |
\end{document} |
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