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% $Header$ |
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\documentclass[12pt]{article} |
\documentclass[12pt]{article} |
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\usepackage{graphicx,subfigure} |
\usepackage[]{graphicx} |
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\usepackage{subfigure} |
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\usepackage[round,comma]{natbib} |
\usepackage[round,comma]{natbib} |
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\bibliographystyle{bib/agu04} |
\bibliographystyle{bib/agu04} |
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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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\frac{\partial{u_{i}}}{\partial{x_{j}}} + |
\frac{\partial{u_{i}}}{\partial{x_{j}}} + |
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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 maximum ice pressure $P_{\max}$, a measure of ice strength, depends on |
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$h$ and compactness (concentration) $c$: |
both thickness $h$ and compactness (concentration) $c$: |
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\begin{equation} |
\begin{equation} |
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P = 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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stress lie on an elliptical yield curve with the ratio of major to |
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: |
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 =& \min\left(\frac{P_{\max}}{2\max(\Delta,\Delta_{\min})}, |
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\eta =& \frac{P}{2\Delta{e}^2} \\ |
\zeta_{\max}\right) \\ |
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\eta =& \frac{\zeta}{e^2} \\ |
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\intertext{with the abbreviation} |
\intertext{with the abbreviation} |
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\Delta = & \left[ |
\Delta = & \left[ |
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\left(\dot{\epsilon}_{11}^2+\dot{\epsilon}_{22}^2\right) |
\left(\dot{\epsilon}_{11}^2+\dot{\epsilon}_{22}^2\right) |
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2\dot{\epsilon}_{11}\dot{\epsilon}_{22} (1-e^{-2}) |
2\dot{\epsilon}_{11}\dot{\epsilon}_{22} (1-e^{-2}) |
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\right]^{-\frac{1}{2}} |
\right]^{-\frac{1}{2}} |
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\end{align*} |
\end{align*} |
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The bulk viscosities are bounded above by imposing both a minimum |
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$\Delta_{\min}=10^{-11}\text{\,s}^{-1}$ (for numerical reasons) and a |
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maximum $\zeta_{\max} = P_{\max}/\Delta^*$, where |
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$\Delta^*=(5\times10^{12}/2\times10^4)\text{\,s}^{-1}$. For stress |
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tensor compuation the replacement pressure $P = 2\,\Delta\zeta$ |
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\citep{hibler95} is used so that the stress state always lies on the |
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elliptic yield curve by definition. |
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In the so-called truncated ellipse method the shear viscosity $\eta$ |
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is capped to suppress any tensile stress \citep{hibler97, geiger98}: |
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\begin{equation} |
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\label{eq:etatem} |
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\eta = \min(\frac{\zeta}{e^2} |
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\frac{\frac{P}{2}-\zeta(\dot{\epsilon}_{11}+\dot{\epsilon}_{22})} |
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{\sqrt{(\dot{\epsilon}_{11}+\dot{\epsilon}_{22})^2 |
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+4\dot{\epsilon}_{12}^2}} |
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\end{equation} |
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In the current implementation, the VP-model is integrated with the |
In the current implementation, the VP-model is integrated with the |
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semi-implicit line successive over relaxation (LSOR)-solver of |
semi-implicit line successive over relaxation (LSOR)-solver of |
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\citet{zhang98}, which allows for long time steps that, in our case, |
\citet{zhang98}, which allows for long time steps that, in our case, |
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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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\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} |
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\item B-grid LSR no-slip |
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\item C-grid LSR no-slip |
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\item C-grid LSR slip |
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\item C-grid EVP no-slip |
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\item C-grid EVP slip |
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\end{itemize} |
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\subsection{B-grid vs.\ C-grid} |
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Comparison between: |
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\begin{itemize} |
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\item B-grid, lsr, no-slip |
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\item C-grid, lsr, no-slip |
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\item C-grid, evp, no-slip |
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\end{itemize} |
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all without ice-ocean stress, because ice-ocean stress does not work |
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for B-grid. |
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\subsection{C-grid} |
\subsection{C-grid} |
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\begin{itemize} |
\begin{itemize} |
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\subsection{Arctic Domain with Open Boundaries} |
\subsection{Arctic Domain with Open Boundaries} |
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\label{sec:arctic} |
\label{sec:arctic} |
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The Arctic domain of integration is illustrated in Fig.~\ref{???}. It is |
The Arctic domain of integration is illustrated in Fig.~\ref{???}. It |
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carved out from, and obtains open boundary conditions from, the global |
is carved out from, and obtains open boundary conditions from, the |
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cubed-sphere configuration of the Estimating the Circulation and Climate of |
global cubed-sphere configuration of the Estimating the Circulation |
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the Ocean, Phase II (ECCO2) project \cite{men05a}. The domain size is 420 by |
and Climate of the Ocean, Phase II (ECCO2) project |
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384 grid boxes horizontally with mean horizontal grid spacing of 18 km. |
\citet{menemenlis05}. The domain size is 420 by 384 grid boxes |
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horizontally with mean horizontal grid spacing of 18 km. |
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There are 50 vertical levels ranging in thickness from 10 m near the surface |
There are 50 vertical levels ranging in thickness from 10 m near the surface |
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to approximately 450 m at a maximum model depth of 6150 m. Bathymetry is from |
to approximately 450 m at a maximum model depth of 6150 m. Bathymetry is from |
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the National Geophysical Data Center (NGDC) 2-minute gridded global relief |
the National Geophysical Data Center (NGDC) 2-minute gridded global relief |
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data (ETOPO2) and the model employs the partial-cell formulation of |
data (ETOPO2) and the model employs the partial-cell formulation of |
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\cite{adc97}, which permits accurate representation of the bathymetry. The |
\citet{adcroft97:_shaved_cells}, which permits accurate representation of the bathymetry. The |
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model is integrated in a volume-conserving configuration using a finite volume |
model is integrated in a volume-conserving configuration using a finite volume |
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discretization with C-grid staggering of the prognostic variables. In the |
discretization with C-grid staggering of the prognostic variables. In the |
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ocean, the non-linear equation of state of \cite{jac95}. The ocean model is |
ocean, the non-linear equation of state of \citet{jackett95}. The ocean model is |
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coupled to a sea-ice model described hereinabove. |
coupled to a sea-ice model described hereinabove. |
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This particular ECCO2 simulation is initialized from rest using the January |
This particular ECCO2 simulation is initialized from rest using the |
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temperature and salinity distribution from the World Ocean Atlas 2001 (WOA01) |
January temperature and salinity distribution from the World Ocean |
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[Conkright et al., 2002] and it is integrated for 32 years prior to the |
Atlas 2001 (WOA01) [Conkright et al., 2002] and it is integrated for |
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1996-2001 period discussed in the study. Surface boundary conditions are from |
32 years prior to the 1996--2001 period discussed in the study. Surface |
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the National Centers for Environmental Prediction and the National Center for |
boundary conditions are from the National Centers for Environmental |
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Atmospheric Research (NCEP/NCAR) atmospheric reanalysis [Kistler et al., |
Prediction and the National Center for Atmospheric Research |
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2001]. Six-hourly surface winds, temperature, humidity, downward short- and |
(NCEP/NCAR) atmospheric reanalysis [Kistler et al., 2001]. Six-hourly |
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long-wave radiations, and precipitation are converted to heat, freshwater, and |
surface winds, temperature, humidity, downward short- and long-wave |
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wind stress fluxes using the Large and Pond [1981, 1982] bulk |
radiations, and precipitation are converted to heat, freshwater, and |
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formulae. Shortwave radiation decays exponentially as per Paulson and Simpson |
wind stress fluxes using the \citet{large81, large82} bulk formulae. |
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[1977]. Additionally the time-mean river run-off from Large and Nurser [2001] |
Shortwave radiation decays exponentially as per Paulson and Simpson |
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is applied and there is a relaxation to the monthly-mean climatological sea |
[1977]. Additionally the time-mean river run-off from Large and Nurser |
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surface salinity values from WOA01 with a relaxation time scale of 3 |
[2001] is applied and there is a relaxation to the monthly-mean |
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months. Vertical mixing follows Large et al. [1994] with background vertical |
climatological sea surface salinity values from WOA01 with a |
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diffusivity of 1.5 × 10-5 m2 s-1 and viscosity of 10-3 m2 s-1. A third order, |
relaxation time scale of 3 months. Vertical mixing follows |
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direct-space-time advection scheme with flux limiter is employed and there is |
\citet{large94} with background vertical diffusivity of |
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no explicit horizontal diffusivity. Horizontal viscosity follows Leith [1996] |
$1.5\times10^{-5}\text{\,m$^{2}$\,s$^{-1}$}$ and viscosity of |
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but modified to sense the divergent flow as per Fox-Kemper and Menemenlis [in |
$10^{-3}\text{\,m$^{2}$\,s$^{-1}$}$. A third order, direct-space-time |
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press]. Shortwave radiation decays exponentially as per Paulson and Simpson |
advection scheme with flux limiter is employed \citep{hundsdorfer94} |
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[1977]. Additionally, the time-mean runoff of Large and Nurser [2001] is |
and there is no explicit horizontal diffusivity. Horizontal viscosity |
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applied near the coastline and, where there is open water, there is a |
follows \citet{lei96} but |
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relaxation to monthly-mean WOA01 sea surface salinity with a time constant of |
modified to sense the divergent flow as per Fox-Kemper and Menemenlis |
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45 days. |
[in press]. Shortwave radiation decays exponentially as per Paulson |
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and Simpson [1977]. Additionally, the time-mean runoff of Large and |
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Nurser [2001] is applied near the coastline and, where there is open |
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water, there is a relaxation to monthly-mean WOA01 sea surface |
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salinity with a time constant of 45 days. |
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Open water, dry |
Open water, dry |
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ice, wet ice, dry snow, and wet snow albedo are, respectively, 0.15, 0.85, |
ice, wet ice, dry snow, and wet snow albedo are, respectively, 0.15, 0.85, |
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\item C-grid LSR slip |
\item C-grid LSR slip |
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\item C-grid EVP no-slip |
\item C-grid EVP no-slip |
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\item C-grid EVP slip |
\item C-grid EVP slip |
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|
\item C-grid LSR + TEM (truncated ellipse method, no tensile stress, new flag) |
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\item C-grid LSR no-slip + Winton |
\item C-grid LSR no-slip + Winton |
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\item speed-performance-accuracy (small) |
\item speed-performance-accuracy (small) |
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ice transport through Canadian Archipelago differences |
ice transport through Canadian Archipelago differences |
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\begin{itemize} |
\begin{itemize} |
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\item advection schemes: along the ice-edge and regions with large |
\item advection schemes: along the ice-edge and regions with large |
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gradients |
gradients |
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\item C-grid: more transport through narrow straits for no slip |
\item C-grid: less transport through narrow straits for no slip |
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conditons, less for free slip |
conditons, more for free slip |
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\item VP vs.\ EVP: speed performance, accuracy? |
\item VP vs.\ EVP: speed performance, accuracy? |
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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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checkpointing loop. |
checkpointing loop. |
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Again, an initial code adjustment is required to support TAFs |
Again, an initial code adjustment is required to support TAFs |
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checkpointing capability. |
checkpointing capability. |
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The code adjustments are sufficiently simply so as not to cause |
The code adjustments are sufficiently simple so as not to cause |
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major limitations to the full nonlinear parent model. |
major limitations to the full nonlinear parent model. |
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Once in place, an adjoint model of a new model configuration |
Once in place, an adjoint model of a new model configuration |
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may be derived in about 10 minutes. |
may be derived in about 10 minutes. |
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We demonstrate the power of the adjoint method |
We demonstrate the power of the adjoint method |
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in the context of investigating sea-ice export sensitivities through Fram Strait |
in the context of investigating sea-ice export sensitivities through Fram Strait |
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(for details of this study see Heimbach et al., 2007). |
(for details of this study see Heimbach et al., 2007). |
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|
%\citep[for details of this study see][]{heimbach07}. %Heimbach et al., 2007). |
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The domain chosen is a coarsened version of the Arctic face of the |
The domain chosen is a coarsened version of the Arctic face of the |
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high-resolution cubed-sphere configuration of the ECCO2 project |
high-resolution cubed-sphere configuration of the ECCO2 project |
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(see Menemenlis et al. 2005). It covers the entire Arctic, |
\citep[see][]{menemenlis05}. It covers the entire Arctic, |
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extends into the North Pacific such as to cover the entire |
extends into the North Pacific such as to cover the entire |
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ice-covered regions, and comprises parts of the North Atlantic |
ice-covered regions, and comprises parts of the North Atlantic |
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down to XXN to enable analysis of remote influences of the |
down to XXN to enable analysis of remote influences of the |
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(benchmarks have been performed both on an SGI Altix as well as an |
(benchmarks have been performed both on an SGI Altix as well as an |
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IBM SP5 at NASA/ARC). |
IBM SP5 at NASA/ARC). |
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|
|
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Following a 1-year spinup, the model has been integrated for four years |
Following a 1-year spinup, the model has been integrated for four |
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between 1992 and 1995. |
years between 1992 and 1995. It is forced using realistic 6-hourly |
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It is forced using realistic 6-hourly NCEP/NCAR atmospheric state variables. |
NCEP/NCAR atmospheric state variables. Over the open ocean these are |
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Over the open ocean these are converted into |
converted into air-sea fluxes via the bulk formulae of |
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air-sea fluxes via the bulk formulae of Large and Yeager (2004). |
\citet{large04}. Derivation of air-sea fluxes in the presence of |
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Derivation of air-sea fluxes in the presence of sea-ice is handled |
sea-ice is handled by the ice model as described in \refsec{model}. |
|
by the ice model as described in Section XXX. |
|
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The objective function chosen is sea-ice export through Fram Strait |
The objective function chosen is sea-ice export through Fram Strait |
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computed for December 1995 |
computed for December 1995. The adjoint model computes sensitivities |
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The adjoint model computes sensitivities to sea-ice export back in time |
to sea-ice export back in time from 1995 to 1992 along this |
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from 1995 to 1992 along this trajectory. |
trajectory. In principle all adjoint model variable (i.e., Lagrange |
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In principle all adjoint model variable (i.e. Lagrange multipliers) |
multipliers) of the coupled ocean/sea-ice model are available to |
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of the coupled ocean/sea-ice model |
analyze the transient sensitivity behaviour of the ocean and sea-ice |
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are available to analyze the transient sensitivity behaviour |
state. Over the open ocean, the adjoint of the bulk formula scheme |
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of the ocean and sea-ice state. |
computes sensitivities to the time-varying atmospheric state. Over |
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Over the open ocean, the adjoint of the bulk formula scheme |
ice-covered parts, the sea-ice adjoint converts surface ocean |
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computes sensitivities to the time-varying atmospheric state. |
sensitivities to atmospheric sensitivities. |
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Over ice-covered parts, the sea-ice adjoint converts |
|
| 590 |
surface ocean sensitivities to atmospheric sensitivities. |
\reffig{4yradjheff}(a--d) depict sensitivities of sea-ice export |
| 591 |
|
through Fram Strait in December 1995 to changes in sea-ice thickness |
| 592 |
Fig. XXX(a--d) depict sensitivities of sea-ice export through Fram Strait |
12, 24, 36, 48 months back in time. Corresponding sensitivities to |
| 593 |
in December 1995 to changes in sea-ice thickness |
ocean surface temperature are depicted in |
| 594 |
12, 24, 36, 48 months back in time. |
\reffig{4yradjthetalev1}(a--d). The main characteristics is |
| 595 |
Corresponding sensitivities to ocean surface temperature are |
consistency with expected advection of sea-ice over the relevant time |
| 596 |
depicted in Fig. XXX(a--d). |
scales considered. The general positive pattern means that an |
| 597 |
The main characteristics is consistency with expected advection |
increase in sea-ice thickness at location $(x,y)$ and time $t$ will |
| 598 |
of sea-ice over the relevant time scales considered. |
increase sea-ice export through Fram Strait at time $T_e$. Largest |
| 599 |
The general positive pattern means that an increase in |
distances from Fram Strait indicate fastest sea-ice advection over the |
| 600 |
sea-ice thickness at location $(x,y)$ and time $t$ will increase |
time span considered. The ice thickness sensitivities are in close |
| 601 |
sea-ice export through Fram Strait at time $T_e$. |
correspondence to ocean surface sentivitites, but of opposite sign. |
| 602 |
Largest distances from Fram Strait indicate fastest sea-ice advection |
An increase in temperature will incur ice melting, decrease in ice |
| 603 |
over the time span considered. |
thickness, and therefore decrease in sea-ice export at time $T_e$. |
|
The ice thickness sensitivities are in close correspondence to |
|
|
ocean surface sentivitites, but of opposite sign. |
|
|
An increase in temperature will incur ice melting, decrease in ice thickness, |
|
|
and therefore decrease in sea-ice export at time $T_e$. |
|
| 604 |
|
|
| 605 |
The picture is fundamentally different and much more complex |
The picture is fundamentally different and much more complex |
| 606 |
for sensitivities to ocean temperatures away from the surface. |
for sensitivities to ocean temperatures away from the surface. |
| 607 |
Fig. XXX (a--d) depicts ice export sensitivities to |
\reffig{4yradjthetalev10??}(a--d) depicts ice export sensitivities to |
| 608 |
temperatures at roughly 400 m depth. |
temperatures at roughly 400 m depth. |
| 609 |
Primary features are the effect of the heat transport of the North |
Primary features are the effect of the heat transport of the North |
| 610 |
Atlantic current which feeds into the West Spitsbergen current, |
Atlantic current which feeds into the West Spitsbergen current, |
| 654 |
-48 months}] |
-48 months}] |
| 655 |
{\includegraphics*[width=0.44\linewidth]{\fpath/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}} |
| 656 |
} |
} |
| 657 |
\caption{Same as Fig. XXX but for sea surface temperature |
\caption{Same as \reffig{4yradjheff} but for sea surface temperature |
| 658 |
\label{fig:4yradjthetalev1}} |
\label{fig:4yradjthetalev1}} |
| 659 |
\end{figure} |
\end{figure} |
| 660 |
|
|
| 679 |
|
|
| 680 |
\paragraph{Acknowledgements} |
\paragraph{Acknowledgements} |
| 681 |
We thank Jinlun Zhang for providing the original B-grid code and many |
We thank Jinlun Zhang for providing the original B-grid code and many |
| 682 |
helpful discussions. |
helpful discussions. ML thanks Elizabeth Hunke for multiple explanations. |
| 683 |
|
|
| 684 |
\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} |
| 685 |
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