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% $Header$ |
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% $Name$ |
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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} |
| 142 |
\frac{\partial{u_{i}}}{\partial{x_{j}}} + |
\frac{\partial{u_{i}}}{\partial{x_{j}}} + |
| 143 |
\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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|
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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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|
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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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The above argument can also be invoked to explain the small |
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. |
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 |
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 |
along the boundaries, the no-slip boundary conditions have only a small |
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impact on the solution. |
impact on the solution. |
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|
|
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The difference between LSR and EVP solutions is largest in the |
The difference between LSR and EVP solutions is largest in the |
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effective thickness and meridional velocity fields. The velocity field |
effective thickness and meridional velocity fields. The EVP velocity |
| 406 |
appears to be a little noisy. This noise has been address by |
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 |
\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 |
step (increasing the number of iterations along with the computational |
| 409 |
and shear viscosities. We revisit the latter option by reproducing the |
cost) or by regularizing the bulk and shear viscosities. We revisit |
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results of \citet{hunke01} for the C-grid no-slip cases. |
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] |
\begin{figure*}[htbp] |
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\includegraphics[width=\widefigwidth]{\fpath/hun12days} |
\includegraphics[width=\widefigwidth]{\fpath/hun12days} |
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\caption{Hunke's test case.} |
\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} |
\label{fig:hunke01} |
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\end{figure*} |
\end{figure*} |
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|
| 435 |
\begin{itemize} |
In the far right (``east'') side of the domain the ice concentration |
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\item B-grid LSR no-slip |
is close to one and the ice should be nearly rigid. The applied wind |
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\item C-grid LSR no-slip |
tends to push ice toward the upper right corner. Because the highly |
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\item C-grid LSR slip |
compact ice is confined by the boundary, it resists any further |
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\item C-grid EVP no-slip |
compression and exhibits little motion in the rigid region on the |
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\item C-grid EVP slip |
right hand side. The C-LSRns solution (top row) allows high |
| 441 |
\end{itemize} |
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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\subsection{B-grid vs.\ C-grid} |
clearly suppresses the noise present in $\nabla\cdot\vek{u}$ and |
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Comparison between: |
$\log_{10}\zeta$ in the |
| 445 |
\begin{itemize} |
unregularized case (middle row), at the cost of reduced viscosities. |
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\item B-grid, lsr, no-slip |
These reduced viscosities lead to small but finite ice velocities |
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\item C-grid, lsr, no-slip |
which in turn can have a strong effect on solutions in the limit of |
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\item C-grid, evp, no-slip |
nearly rigid regimes (arching and blocking, not shown). |
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\end{itemize} |
|
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all without ice-ocean stress, because ice-ocean stress does not work |
\ml{[Say something about performance? This is tricky, as the |
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for B-grid. |
perfomance depends strongly on the configuration. A run with slowly |
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changing forcing is favorable for LSR, because then only very few |
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iterations are required for convergences while EVP uses its fixed |
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number of internal timesteps. If the forcing in changing fast, LSR |
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needs far more iterations while EVP still uses the fixed number of |
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internal timesteps. I have produces runs where for slow forcing LSR |
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is much faster than EVP and for fast forcing, LSR is much slower |
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than EVP. EVP is certainly more efficient in terms of vectorization |
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and MFLOPS on our SX8, but is that a criterion?]} |
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|
|
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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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|
|
| 506 |
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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|
|
| 513 |
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 |
| 515 |
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 |
| 517 |
\cite{adc97}, which permits accurate representation of the bathymetry. The |
\citet{adcroft97:_shaved_cells}, which permits accurate representation of the bathymetry. The |
| 518 |
model is integrated in a volume-conserving configuration using a finite volume |
model is integrated in a volume-conserving configuration using a finite volume |
| 519 |
discretization with C-grid staggering of the prognostic variables. In the |
discretization with C-grid staggering of the prognostic variables. In the |
| 520 |
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 |
| 521 |
coupled to a sea-ice model described hereinabove. |
coupled to a sea-ice model described hereinabove. |
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|
|
| 523 |
This particular ECCO2 simulation is initialized from rest using the January |
This particular ECCO2 simulation is initialized from rest using the |
| 524 |
temperature and salinity distribution from the World Ocean Atlas 2001 (WOA01) |
January temperature and salinity distribution from the World Ocean |
| 525 |
[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 |
| 526 |
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 |
| 527 |
the National Centers for Environmental Prediction and the National Center for |
boundary conditions are from the National Centers for Environmental |
| 528 |
Atmospheric Research (NCEP/NCAR) atmospheric reanalysis [Kistler et al., |
Prediction and the National Center for Atmospheric Research |
| 529 |
2001]. Six-hourly surface winds, temperature, humidity, downward short- and |
(NCEP/NCAR) atmospheric reanalysis [Kistler et al., 2001]. Six-hourly |
| 530 |
long-wave radiations, and precipitation are converted to heat, freshwater, and |
surface winds, temperature, humidity, downward short- and long-wave |
| 531 |
wind stress fluxes using the Large and Pond [1981, 1982] bulk |
radiations, and precipitation are converted to heat, freshwater, and |
| 532 |
formulae. Shortwave radiation decays exponentially as per Paulson and Simpson |
wind stress fluxes using the \citet{large81, large82} bulk formulae. |
| 533 |
[1977]. Additionally the time-mean river run-off from Large and Nurser [2001] |
Shortwave radiation decays exponentially as per Paulson and Simpson |
| 534 |
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 |
| 536 |
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 |
| 538 |
direct-space-time advection scheme with flux limiter is employed and there is |
\citet{large94} with background vertical diffusivity of |
| 539 |
no explicit horizontal diffusivity. Horizontal viscosity follows Leith [1996] |
$1.5\times10^{-5}\text{\,m$^{2}$\,s$^{-1}$}$ and viscosity of |
| 540 |
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 |
| 541 |
press]. Shortwave radiation decays exponentially as per Paulson and Simpson |
advection scheme with flux limiter is employed \citep{hundsdorfer94} |
| 542 |
[1977]. Additionally, the time-mean runoff of Large and Nurser [2001] is |
and there is no explicit horizontal diffusivity. Horizontal viscosity |
| 543 |
applied near the coastline and, where there is open water, there is a |
follows \citet{lei96} but |
| 544 |
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 |
| 545 |
45 days. |
[in press]. Shortwave radiation decays exponentially as per Paulson |
| 546 |
|
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 |
| 549 |
|
salinity with a time constant of 45 days. |
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|
|
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Open water, dry |
Open water, dry |
| 552 |
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 |
| 576 |
\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) |
| 581 |
ice transport through Canadian Archipelago differences |
ice transport through Canadian Archipelago differences |
| 587 |
\begin{itemize} |
\begin{itemize} |
| 588 |
\item advection schemes: along the ice-edge and regions with large |
\item advection schemes: along the ice-edge and regions with large |
| 589 |
gradients |
gradients |
| 590 |
\item C-grid: more transport through narrow straits for no slip |
\item C-grid: less transport through narrow straits for no slip |
| 591 |
conditons, less for free slip |
conditons, more for free slip |
| 592 |
\item VP vs.\ EVP: speed performance, accuracy? |
\item VP vs.\ EVP: speed performance, accuracy? |
| 593 |
\item ocean stress: different water mass properties beneath the ice |
\item ocean stress: different water mass properties beneath the ice |
| 594 |
\end{itemize} |
\end{itemize} |
| 669 |
checkpointing loop. |
checkpointing loop. |
| 670 |
Again, an initial code adjustment is required to support TAFs |
Again, an initial code adjustment is required to support TAFs |
| 671 |
checkpointing capability. |
checkpointing capability. |
| 672 |
The code adjustments are sufficiently simply so as not to cause |
The code adjustments are sufficiently simple so as not to cause |
| 673 |
major limitations to the full nonlinear parent model. |
major limitations to the full nonlinear parent model. |
| 674 |
Once in place, an adjoint model of a new model configuration |
Once in place, an adjoint model of a new model configuration |
| 675 |
may be derived in about 10 minutes. |
may be derived in about 10 minutes. |
| 692 |
We demonstrate the power of the adjoint method |
We demonstrate the power of the adjoint method |
| 693 |
in the context of investigating sea-ice export sensitivities through Fram Strait |
in the context of investigating sea-ice export sensitivities through Fram Strait |
| 694 |
(for details of this study see Heimbach et al., 2007). |
(for details of this study see Heimbach et al., 2007). |
| 695 |
|
%\citep[for details of this study see][]{heimbach07}. %Heimbach et al., 2007). |
| 696 |
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 |
| 697 |
high-resolution cubed-sphere configuration of the ECCO2 project |
high-resolution cubed-sphere configuration of the ECCO2 project |
| 698 |
(see Menemenlis et al. 2005). It covers the entire Arctic, |
\citep[see][]{menemenlis05}. It covers the entire Arctic, |
| 699 |
extends into the North Pacific such as to cover the entire |
extends into the North Pacific such as to cover the entire |
| 700 |
ice-covered regions, and comprises parts of the North Atlantic |
ice-covered regions, and comprises parts of the North Atlantic |
| 701 |
down to XXN to enable analysis of remote influences of the |
down to XXN to enable analysis of remote influences of the |
| 706 |
(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 |
| 707 |
IBM SP5 at NASA/ARC). |
IBM SP5 at NASA/ARC). |
| 708 |
|
|
| 709 |
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 |
| 710 |
between 1992 and 1995. |
years between 1992 and 1995. It is forced using realistic 6-hourly |
| 711 |
It is forced using realistic 6-hourly NCEP/NCAR atmospheric state variables. |
NCEP/NCAR atmospheric state variables. Over the open ocean these are |
| 712 |
Over the open ocean these are converted into |
converted into air-sea fluxes via the bulk formulae of |
| 713 |
air-sea fluxes via the bulk formulae of Large and Yeager (2004). |
\citet{large04}. Derivation of air-sea fluxes in the presence of |
| 714 |
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. |
|
| 715 |
The objective function chosen is sea-ice export through Fram Strait |
The objective function chosen is sea-ice export through Fram Strait |
| 716 |
computed for December 1995 |
computed for December 1995. The adjoint model computes sensitivities |
| 717 |
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 |
| 718 |
from 1995 to 1992 along this trajectory. |
trajectory. In principle all adjoint model variable (i.e., Lagrange |
| 719 |
In principle all adjoint model variable (i.e. Lagrange multipliers) |
multipliers) of the coupled ocean/sea-ice model are available to |
| 720 |
of the coupled ocean/sea-ice model |
analyze the transient sensitivity behaviour of the ocean and sea-ice |
| 721 |
are available to analyze the transient sensitivity behaviour |
state. Over the open ocean, the adjoint of the bulk formula scheme |
| 722 |
of the ocean and sea-ice state. |
computes sensitivities to the time-varying atmospheric state. Over |
| 723 |
Over the open ocean, the adjoint of the bulk formula scheme |
ice-covered parts, the sea-ice adjoint converts surface ocean |
| 724 |
computes sensitivities to the time-varying atmospheric state. |
sensitivities to atmospheric sensitivities. |
| 725 |
Over ice-covered parts, the sea-ice adjoint converts |
|
| 726 |
surface ocean sensitivities to atmospheric sensitivities. |
\reffig{4yradjheff}(a--d) depict sensitivities of sea-ice export |
| 727 |
|
through Fram Strait in December 1995 to changes in sea-ice thickness |
| 728 |
Fig. XXX(a--d) depict sensitivities of sea-ice export through Fram Strait |
12, 24, 36, 48 months back in time. Corresponding sensitivities to |
| 729 |
in December 1995 to changes in sea-ice thickness |
ocean surface temperature are depicted in |
| 730 |
12, 24, 36, 48 months back in time. |
\reffig{4yradjthetalev1}(a--d). The main characteristics is |
| 731 |
Corresponding sensitivities to ocean surface temperature are |
consistency with expected advection of sea-ice over the relevant time |
| 732 |
depicted in Fig. XXX(a--d). |
scales considered. The general positive pattern means that an |
| 733 |
The main characteristics is consistency with expected advection |
increase in sea-ice thickness at location $(x,y)$ and time $t$ will |
| 734 |
of sea-ice over the relevant time scales considered. |
increase sea-ice export through Fram Strait at time $T_e$. Largest |
| 735 |
The general positive pattern means that an increase in |
distances from Fram Strait indicate fastest sea-ice advection over the |
| 736 |
sea-ice thickness at location $(x,y)$ and time $t$ will increase |
time span considered. The ice thickness sensitivities are in close |
| 737 |
sea-ice export through Fram Strait at time $T_e$. |
correspondence to ocean surface sentivitites, but of opposite sign. |
| 738 |
Largest distances from Fram Strait indicate fastest sea-ice advection |
An increase in temperature will incur ice melting, decrease in ice |
| 739 |
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 $T_e$. |
|
| 740 |
|
|
| 741 |
The picture is fundamentally different and much more complex |
The picture is fundamentally different and much more complex |
| 742 |
for sensitivities to ocean temperatures away from the surface. |
for sensitivities to ocean temperatures away from the surface. |
| 743 |
Fig. XXX (a--d) depicts ice export sensitivities to |
\reffig{4yradjthetalev10??}(a--d) depicts ice export sensitivities to |
| 744 |
temperatures at roughly 400 m depth. |
temperatures at roughly 400 m depth. |
| 745 |
Primary features are the effect of the heat transport of the North |
Primary features are the effect of the heat transport of the North |
| 746 |
Atlantic current which feeds into the West Spitsbergen current, |
Atlantic current which feeds into the West Spitsbergen current, |
| 790 |
-48 months}] |
-48 months}] |
| 791 |
{\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}} |
| 792 |
} |
} |
| 793 |
\caption{Same as Fig. XXX but for sea surface temperature |
\caption{Same as \reffig{4yradjheff} but for sea surface temperature |
| 794 |
\label{fig:4yradjthetalev1}} |
\label{fig:4yradjthetalev1}} |
| 795 |
\end{figure} |
\end{figure} |
| 796 |
|
|
| 815 |
|
|
| 816 |
\paragraph{Acknowledgements} |
\paragraph{Acknowledgements} |
| 817 |
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 |
| 818 |
helpful discussions. |
helpful discussions. ML thanks Elizabeth Hunke for multiple explanations. |
| 819 |
|
|
| 820 |
\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} |
| 821 |
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