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revision 1.4 by mlosch, Thu Jan 10 15:47:32 2008 UTC revision 1.13 by dimitri, Mon Feb 25 23:45:46 2008 UTC
# Line 1  Line 1 
1    % $Header$
2    % $Name$
3  \documentclass[12pt]{article}  \documentclass[12pt]{article}
4    
5  \usepackage{graphicx,subfigure}  \usepackage[]{graphicx}
6    \usepackage{subfigure}
7    
8  \usepackage[round,comma]{natbib}  \usepackage[round,comma]{natbib}
9  \bibliographystyle{bib/agu04}  \bibliographystyle{bib/agu04}
# Line 49  Line 52 
52  \maketitle  \maketitle
53    
54  \begin{abstract}  \begin{abstract}
55    Some blabla  
56    As part of ongoing efforts to obtain a best possible synthesis of most
57    available, global-scale, ocean and sea ice data, a dynamic and thermodynamic
58    sea-ice model has been coupled to the Massachusetts Institute of Technology
59    general circulation model (MITgcm).  Ice mechanics follow a viscous plastic
60    rheology and the ice momentum equations are solved numerically using either
61    line successive relaxation (LSR) or elastic-viscous-plastic (EVP) dynamic
62    models.  Ice thermodynamics are represented using either a zero-heat-capacity
63    formulation or a two-layer formulation that conserves enthalpy.  The model
64    includes prognostic variables for snow and for sea-ice salinity.  The above
65    sea ice model components were borrowed from current-generation climate models
66    but they were reformulated on an Arakawa C-grid in order to match the MITgcm
67    oceanic grid and they were modified in many ways to permit efficient and
68    accurate automatic differentiation.  This paper describes the MITgcm sea ice
69    model; it presents example Arctic and Antarctic results from a realistic,
70    eddy-permitting, global ocean and sea-ice configuration; it compares B-grid
71    and C-grid dynamic solvers in a regional Arctic configuration; and it presents
72    example results from coupled ocean and sea-ice adjoint-model integrations.
73    
74  \end{abstract}  \end{abstract}
75    
76  \section{Introduction}  \section{Introduction}
77  \label{sec:intro}  \label{sec:intro}
78    
 more blabla  
   
 \section{Model}  
 \label{sec:model}  
   
79  Traditionally, probably for historical reasons and the ease of  Traditionally, probably for historical reasons and the ease of
80  treating the Coriolis term, most standard sea-ice models are  treating the Coriolis term, most standard sea-ice models are
81  discretized on Arakawa-B-grids \citep[e.g.,][]{hibler79, harder99,  discretized on Arakawa-B-grids \citep[e.g.,][]{hibler79, harder99,
# Line 71  velocities points and thus needs to be i Line 87  velocities points and thus needs to be i
87  sea-ice model and a C-grid ocean model. While the smoothing implicitly  sea-ice model and a C-grid ocean model. While the smoothing implicitly
88  associated with this interpolation may mask grid scale noise, it may  associated with this interpolation may mask grid scale noise, it may
89  in two-way coupling lead to a computational mode as will be shown. By  in two-way coupling lead to a computational mode as will be shown. By
90  choosing a C-grid for the sea-ice model, we circumvene this difficulty  choosing a C-grid for the sea-ice model, we circumvent this difficulty
91  altogether and render the stress coupling as consistent as the  altogether and render the stress coupling as consistent as the
92  buoyancy coupling.  buoyancy coupling.
93    
# Line 82  whereas the C-grid formulation allows a Line 98  whereas the C-grid formulation allows a
98  passage for all types of lateral boundary conditions. We (will)  passage for all types of lateral boundary conditions. We (will)
99  demonstrate this effect in the Candian archipelago.  demonstrate this effect in the Candian archipelago.
100    
101    \section{Model}
102    \label{sec:model}
103    
104  \subsection{Dynamics}  \subsection{Dynamics}
105  \label{sec:dynamics}  \label{sec:dynamics}
106    
107  The momentum equations of the sea-ice model are standard with  The momentum equation of the sea-ice model is
108  \begin{equation}  \begin{equation}
109    \label{eq:momseaice}    \label{eq:momseaice}
110    m \frac{D\vek{u}}{Dt} = -mf\vek{k}\times\vek{u} + \vtau_{air} +    m \frac{D\vek{u}}{Dt} = -mf\vek{k}\times\vek{u} + \vtau_{air} +
111    \vtau_{ocean} - m \nabla{\phi(0)} + \vek{F},    \vtau_{ocean} - mg \nabla{\phi(0)} + \vek{F},
112  \end{equation}  \end{equation}
113  where $\vek{u} = u\vek{i}+v\vek{j}$ is the ice velocity vectory, $m$  where $m=m_{i}+m_{s}$ is the ice and snow mass per unit area;
114  the ice mass per unit area, $f$ the Coriolis parameter, $g$ is the  $\vek{u}=u\vek{i}+v\vek{j}$ is the ice velocity vector;
115  gravity accelation, $\nabla\phi$ is the gradient (tilt) of the sea  $\vek{i}$, $\vek{j}$, and $\vek{k}$ are unit vectors in the $x$, $y$, and $z$
116  surface height potential beneath the ice. $\phi$ is the sum of  directions, respectively;
117  atmpheric pressure $p_{a}$ and loading due to ice and snow  $f$ is the Coriolis parameter;
118  $(m_{i}+m_{s})g$. $\vtau_{air}$ and $\vtau_{ocean}$ are the wind and  $\vtau_{air}$ and $\vtau_{ocean}$ are the wind-ice and ocean-ice stresses,
119  ice-ocean stresses, respectively.  $\vek{F}$ is the interaction force  respectively;
120  and $\vek{i}$, $\vek{j}$, and $\vek{k}$ are the unit vectors in the  $g$ is the gravity accelation;
121  $x$, $y$, and $z$ directions.  Advection of sea-ice momentum is  $\nabla\phi(0)$ is the gradient (or tilt) of the sea surface height;
122  neglected. The wind and ice-ocean stress terms are given by  $\phi(0)$ is the sea surface height potential in response to ocean dynamics
123    and to atmospheric pressure loading;
124    and $\vek{F}=\nabla\cdot\sigma$ is the divergence of the internal ice stress
125    tensor $\sigma_{ij}$.
126    When using the rescaled vertical coordinate system, z$^\ast$, of
127    \citet{cam08}, $\phi(0)$ also includes a term due to snow and ice loading, $mg$.
128    Advection of sea-ice momentum is neglected. The wind and ice-ocean stress
129    terms are given by
130  \begin{align*}  \begin{align*}
131    \vtau_{air} =& \rho_{air} |\vek{U}_{air}|R_{air}(\vek{U}_{air}) \\    \vtau_{air}   = & \rho_{air}  C_{air}   |\vek{U}_{air}  -\vek{u}|
132    \vtau_{ocean} =& \rho_{ocean} |\vek{U}_{ocean}-\vek{u}|                     R_{air}  (\vek{U}_{air}  -\vek{u}), \\
133      \vtau_{ocean} = & \rho_{ocean}C_{ocean} |\vek{U}_{ocean}-\vek{u}|
134                     R_{ocean}(\vek{U}_{ocean}-\vek{u}), \\                     R_{ocean}(\vek{U}_{ocean}-\vek{u}), \\
135  \end{align*}  \end{align*}
136  where $\vek{U}_{air/ocean}$ are the surface winds of the atmosphere  where $\vek{U}_{air/ocean}$ are the surface winds of the atmosphere
137  and surface currents of the ocean, respectively. $C_{air/ocean}$ are  and surface currents of the ocean, respectively; $C_{air/ocean}$ are
138  air and ocean drag coefficients, $\rho_{air/ocean}$ reference  air and ocean drag coefficients; $\rho_{air/ocean}$ are reference
139  densities, and $R_{air/ocean}$ rotation matrices that act on the  densities; and $R_{air/ocean}$ are rotation matrices that act on the
140  wind/current vectors. $\vek{F} = \nabla\cdot\sigma$ is the divergence  wind/current vectors.
 of the interal stress tensor $\sigma_{ij}$.  
141    
142  For an isotropic system this stress tensor can be related to the ice  For an isotropic system this stress tensor can be related to the ice
143  strain rate and strength by a nonlinear viscous-plastic (VP)  strain rate and strength by a nonlinear viscous-plastic (VP)
# Line 129  The ice strain rate is given by Line 155  The ice strain rate is given by
155      \frac{\partial{u_{i}}}{\partial{x_{j}}} +      \frac{\partial{u_{i}}}{\partial{x_{j}}} +
156      \frac{\partial{u_{j}}}{\partial{x_{i}}}\right).      \frac{\partial{u_{j}}}{\partial{x_{i}}}\right).
157  \end{equation*}  \end{equation*}
158  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
159  $h$ and compactness (concentration) $c$:  both thickness $h$ and compactness (concentration) $c$:
160  \begin{equation}  \begin{equation}
161    P = P^{*}c\,h\,e^{[C^{*}\cdot(1-c)]},    P_{\max} = P^{*}c\,h\,e^{[C^{*}\cdot(1-c)]},
162  \label{icestrength}  \label{eq:icestrength}
163  \end{equation}  \end{equation}
164  with the constants $P^{*}$ and $C^{*}$. The nonlinear bulk and shear  with the constants $P^{*}$ and $C^{*}$. The nonlinear bulk and shear
165  viscosities $\eta$ and $\zeta$ are functions of ice strain rate  viscosities $\eta$ and $\zeta$ are functions of ice strain rate
# Line 141  invariants and ice strength such that th Line 167  invariants and ice strength such that th
167  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
168  minor axis $e$ equal to $2$; they are given by:  minor axis $e$ equal to $2$; they are given by:
169  \begin{align*}  \begin{align*}
170    \zeta =& \frac{P}{2\Delta} \\    \zeta =& \min\left(\frac{P_{\max}}{2\max(\Delta,\Delta_{\min})},
171    \eta =& \frac{P}{2\Delta{e}^2} \\     \zeta_{\max}\right) \\
172      \eta =& \frac{\zeta}{e^2} \\
173    \intertext{with the abbreviation}    \intertext{with the abbreviation}
174    \Delta = & \left[    \Delta = & \left[
175      \left(\dot{\epsilon}_{11}^2+\dot{\epsilon}_{22}^2\right)      \left(\dot{\epsilon}_{11}^2+\dot{\epsilon}_{22}^2\right)
# Line 150  minor axis $e$ equal to $2$; they are gi Line 177  minor axis $e$ equal to $2$; they are gi
177      2\dot{\epsilon}_{11}\dot{\epsilon}_{22} (1-e^{-2})      2\dot{\epsilon}_{11}\dot{\epsilon}_{22} (1-e^{-2})
178    \right]^{-\frac{1}{2}}    \right]^{-\frac{1}{2}}
179  \end{align*}  \end{align*}
180    The bulk viscosities are bounded above by imposing both a minimum
181    $\Delta_{\min}=10^{-11}\text{\,s}^{-1}$ (for numerical reasons) and a
182    maximum $\zeta_{\max} = P_{\max}/\Delta^*$, where
183    $\Delta^*=(5\times10^{12}/2\times10^4)\text{\,s}^{-1}$. For stress
184    tensor computation the replacement pressure $P = 2\,\Delta\zeta$
185    \citep{hibler95} is used so that the stress state always lies on the
186    elliptic yield curve by definition.
187    
188    In the so-called truncated ellipse method the shear viscosity $\eta$
189    is capped to suppress any tensile stress \citep{hibler97, geiger98}:
190    \begin{equation}
191      \label{eq:etatem}
192      \eta = \min(\frac{\zeta}{e^2}
193      \frac{\frac{P}{2}-\zeta(\dot{\epsilon}_{11}+\dot{\epsilon}_{22})}
194      {\sqrt{(\dot{\epsilon}_{11}+\dot{\epsilon}_{22})^2
195          +4\dot{\epsilon}_{12}^2}}
196    \end{equation}
197    
198  In the current implementation, the VP-model is integrated with the  In the current implementation, the VP-model is integrated with the
199  semi-implicit line successive over relaxation (LSOR)-solver of  semi-implicit line successive over relaxation (LSOR)-solver of
200  \citet{zhang98}, which allows for long time steps that, in our case,  \citet{zhang98}, which allows for long time steps that, in our case,
# Line 160  same length as in the ocean model where Line 205  same length as in the ocean model where
205  treated explicitly.  treated explicitly.
206    
207  \citet{hunke97}'s introduced an elastic contribution to the strain  \citet{hunke97}'s introduced an elastic contribution to the strain
208  rate elatic-viscous-plastic in order to regularize  rate elastic-viscous-plastic in order to regularize
209  Eq.\refeq{vpequation} in such a way that the resulting  Eq.\refeq{vpequation} in such a way that the resulting
210  elatic-viscous-plastic (EVP) and VP models are identical at steady  elastic-viscous-plastic (EVP) and VP models are identical at steady
211  state,  state,
212  \begin{equation}  \begin{equation}
213    \label{eq:evpequation}    \label{eq:evpequation}
# Line 188  $\sigma_{12}$. Introducing the divergenc Line 233  $\sigma_{12}$. Introducing the divergenc
233  \dot{\epsilon}_{11}+\dot{\epsilon}_{22}$, and the horizontal tension  \dot{\epsilon}_{11}+\dot{\epsilon}_{22}$, and the horizontal tension
234  and shearing strain rates, $D_T =  and shearing strain rates, $D_T =
235  \dot{\epsilon}_{11}-\dot{\epsilon}_{22}$ and $D_S =  \dot{\epsilon}_{11}-\dot{\epsilon}_{22}$ and $D_S =
236  2\dot{\epsilon}_{12}$, respectively and using the above abbreviations,  2\dot{\epsilon}_{12}$, respectively, and using the above abbreviations,
237  the equations can be written as:  the equations can be written as:
238  \begin{align}  \begin{align}
239    \label{eq:evpstresstensor1}    \label{eq:evpstresstensor1}
# Line 218  differences and averaging is only involv Line 263  differences and averaging is only involv
263  $P$ at vorticity points.  $P$ at vorticity points.
264    
265  For a general curvilinear grid, one needs in principle to take metric  For a general curvilinear grid, one needs in principle to take metric
266  terms into account that arise in the transformation a curvilinear grid  terms into account that arise in the transformation of a curvilinear grid
267  on the sphere. However, for now we can neglect these metric terms  on the sphere. For now, however, we can neglect these metric terms
268  because they are very small on the cubed sphere grids used in this  because they are very small on the cubed sphere grids used in this
269  paper; in particular, only near the edges of the cubed sphere grid, we  paper; in particular, only near the edges of the cubed sphere grid, we
270  expect them to be non-zero, but these edges are at approximately  expect them to be non-zero, but these edges are at approximately
# Line 228  simulations.  Everywhere else the coordi Line 273  simulations.  Everywhere else the coordi
273  cartesian.  However, for last-glacial-maximum or snowball-earth-like  cartesian.  However, for last-glacial-maximum or snowball-earth-like
274  simulations the question of metric terms needs to be reconsidered.  simulations the question of metric terms needs to be reconsidered.
275  Either, one includes these terms as in \citet{zhang03}, or one finds a  Either, one includes these terms as in \citet{zhang03}, or one finds a
276  vector-invariant formulation fo the sea-ice internal stress term that  vector-invariant formulation for the sea-ice internal stress term that
277  does not require any metric terms, as it is done in the ocean dynamics  does not require any metric terms, as it is done in the ocean dynamics
278  of the MITgcm \citep{adcroft04:_cubed_sphere}.  of the MITgcm \citep{adcroft04:_cubed_sphere}.
279    
# Line 289  addition to ice-thickness and compactnes Line 334  addition to ice-thickness and compactnes
334  state variables to be advected by ice velocities, namely enthalphy of  state variables to be advected by ice velocities, namely enthalphy of
335  the two ice layers and the thickness of the overlying snow layer.  the two ice layers and the thickness of the overlying snow layer.
336    
 \section{Funnel Experiments}  
 \label{sec:funnel}  
   
 For a first/detailed comparison between the different variants of the  
 MIT sea ice model an idealized geometry of a periodic channel,  
 1000\,km long and 500\,m wide on a non-rotating plane, with converging  
 walls forming a symmetric funnel and a narrow strait of 40\,km width  
 is used. The horizontal resolution is 5\,km throughout the domain  
 making the narrow strait 8 grid points wide. The ice model is  
 initialized with a complete ice cover of 50\,cm uniform thickness. The  
 ice model is driven by a constant along channel eastward ocean current  
 of 25\,cm/s that does not see the walls in the domain. All other  
 ice-ocean-atmosphere interactions are turned off, in particular there  
 is no feedback of ice dynamics on the ocean current. All thermodynamic  
 processes are turned off so that ice thickness variations are only  
 caused by convergent or divergent ice flow. Ice volume (effective  
 thickness) and concentration are advected with a third-order scheme  
 with a flux limiter \citep{hundsdorfer94} to avoid undershoots. This  
 scheme is unconditionally stable and does not require additional  
 diffusion. The time step used here is 1\,h.  
   
 \reffig{funnelf0} compares the dynamic fields ice concentration $c$,  
 effective thickness $h_{eff} = h\cdot{c}$, and velocities $(u,v)$ for  
 five different cases at steady state (after 10\,years of integration):  
 \begin{description}  
 \item[B-LSRns:] LSR solver with no-slip boundary conditions on a B-grid,  
 \item[C-LSRns:] LSR solver with no-slip boundary conditions on a C-grid,  
 \item[C-LSRfs:] LSR solver with free-slip boundary conditions on a C-grid,  
 \item[C-EVPns:] EVP solver with no-slip boundary conditions on a C-grid,  
 \item[C-EVPfs:] EVP solver with free-slip boundary conditions on a C-grid,  
 \end{description}  
 \ml{[We have not implemented the EVP solver on a B-grid.]}  
 \begin{figure*}[htbp]  
   \includegraphics[width=\widefigwidth]{\fpath/all_086280}  
   \caption{Ice concentration, effective thickness [m], and ice  
     velocities [m/s]  
     for 5 different numerical solutions.}  
   \label{fig:funnelf0}  
 \end{figure*}  
 At a first glance, the solutions look similar. This is encouraging as  
 the details of discretization and numerics should not affect the  
 solutions to first order. In all cases the ice-ocean stress pushes the  
 ice cover eastwards, where it converges in the funnel. In the narrow  
 channel the ice moves quickly (nearly free drift) and leaves the  
 channel as narrow band.  
   
 A close look reveals interesting differences between the B- and C-grid  
 results. The zonal velocity in the narrow channel is nearly the free  
 drift velocity ( = ocean velocity) of 25\,cm/s for the C-grid  
 solutions, regardless of the boundary conditions, while it is just  
 above 20\,cm/s for the B-grid solution. The ice accelerates to  
 25\,cm/s after it exits the channel. Concentrating on the solutions  
 B-LSRns and C-LSRns, the ice volume (effective thickness) along the  
 boundaries in the narrow channel is larger in the B-grid case although  
 the ice concentration is reduces in the C-grid case. The combined  
 effect leads to a larger actual ice thickness at smaller  
 concentrations in the C-grid case. However, since the effective  
 thickness determines the ice strength $P$ in Eq\refeq{icestrength},  
 the ice strength and thus the bulk and shear viscosities are larger in  
 the B-grid case leading to more horizontal friction. This circumstance  
 might explain why the no-slip boundary conditions in the B-grid case  
 appear to be more effective in reducing the flow within the narrow  
 channel, than in the C-grid case. Further, the viscosities are also  
 sensitive to details of the velocity gradients. Via $\Delta$, these  
 gradients enter the viscosities in the denominator so that large  
 gradients tend to reduce the viscosities. This again favors more flow  
 along the boundaries in the C-grid case: larger velocities  
 (\reffig{funnelf0}) on grid points that are closer to the boundary by  
 a factor $\frac{1}{2}$ than in the B-grid case because of the stagger  
 nature of the C-grid lead numerically to larger tangential gradients  
 across the boundary; these in turn make the viscosities smaller for  
 less tangential friction and allow more tangential flow along the  
 boundaries.  
   
 The above argument can also be invoked to explain the small  
 differences between the free-slip and no-slip solutions on the C-grid.  
 Because of the non-linearities in the ice viscosities, in particular  
 along the boundaries, the no-slip boundary conditions has only a small  
 impact on the solution.  
   
 The difference between LSR and EVP solutions is largest in the  
 effective thickness and meridional velocity fields. The velocity field  
 appears to be a little noisy. This noise has been address by  
 \citet{hunke01}. It can be dealt with by reducing EVP's internal time  
 step (increasing the number of iterations) or by regularizing the bulk  
 and shear viscosities. We revisit the latter option by reproducing the  
 results of \citet{hunke01} for the C-grid no-slip cases.  
 \begin{figure*}[htbp]  
   \includegraphics[width=\widefigwidth]{\fpath/hun12days}  
   \caption{Hunke's test case.}  
   \label{fig:hunke01}  
 \end{figure*}  
   
 \begin{itemize}  
 \item B-grid LSR no-slip  
 \item C-grid LSR no-slip  
 \item C-grid LSR slip  
 \item C-grid EVP no-slip  
 \item C-grid EVP slip  
 \end{itemize}  
   
 \subsection{B-grid vs.\ C-grid}  
 Comparison between:  
 \begin{itemize}  
 \item B-grid, lsr, no-slip  
 \item C-grid, lsr, no-slip  
 \item C-grid, evp, no-slip  
 \end{itemize}  
 all without ice-ocean stress, because ice-ocean stress does not work  
 for B-grid.  
337    
338  \subsection{C-grid}  \subsection{C-grid}
339  \begin{itemize}  \begin{itemize}
# Line 445  differences between the two main options Line 380  differences between the two main options
380  \subsection{Arctic Domain with Open Boundaries}  \subsection{Arctic Domain with Open Boundaries}
381  \label{sec:arctic}  \label{sec:arctic}
382    
383  The Arctic domain of integration is illustrated in Fig.~\ref{???}.  It is  The Arctic domain of integration is illustrated in Fig.~\ref{fig:arctic1}.  It
384  carved out from, and obtains open boundary conditions from, the global  is carved out from, and obtains open boundary conditions from, the
385  cubed-sphere configuration of the Estimating the Circulation and Climate of  global cubed-sphere configuration of the Estimating the Circulation
386  the Ocean, Phase II (ECCO2) project \cite{men05a}.  The domain size is 420 by  and Climate of the Ocean, Phase II (ECCO2) project
387  384 grid boxes horizontally with mean horizontal grid spacing of 18 km.    \citet{menemenlis05}.  The domain size is 420 by 384 grid boxes
388    horizontally with mean horizontal grid spacing of 18 km.
389    
390    \begin{figure}
391    %\centerline{{\includegraphics*[width=0.44\linewidth]{\fpath/arctic1.eps}}}
392    \caption{Bathymetry of Arctic Domain.\label{fig:arctic1}}
393    \end{figure}
394    
395  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
396  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
397  the National Geophysical Data Center (NGDC) 2-minute gridded global relief  the National Geophysical Data Center (NGDC) 2-minute gridded global relief
398  data (ETOPO2) and the model employs the partial-cell formulation of  data (ETOPO2) and the model employs the partial-cell formulation of
399  \cite{adc97}, which permits accurate representation of the bathymetry. The  \citet{adcroft97:_shaved_cells}, which permits accurate representation of the bathymetry. The
400  model is integrated in a volume-conserving configuration using a finite volume  model is integrated in a volume-conserving configuration using a finite volume
401  discretization with C-grid staggering of the prognostic variables. In the  discretization with C-grid staggering of the prognostic variables. In the
402  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
403  coupled to a sea-ice model described hereinabove.    coupled to a sea-ice model described hereinabove.  
404    
405  This particular ECCO2 simulation is initialized from rest using the January  This particular ECCO2 simulation is initialized from rest using the
406  temperature and salinity distribution from the World Ocean Atlas 2001 (WOA01)  January temperature and salinity distribution from the World Ocean
407  [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
408  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
409  the National Centers for Environmental Prediction and the National Center for  boundary conditions are from the National Centers for Environmental
410  Atmospheric Research (NCEP/NCAR) atmospheric reanalysis [Kistler et al.,  Prediction and the National Center for Atmospheric Research
411  2001]. Six-hourly surface winds, temperature, humidity, downward short- and  (NCEP/NCAR) atmospheric reanalysis [Kistler et al., 2001]. Six-hourly
412  long-wave radiations, and precipitation are converted to heat, freshwater, and  surface winds, temperature, humidity, downward short- and long-wave
413  wind stress fluxes using the Large and Pond [1981, 1982] bulk  radiations, and precipitation are converted to heat, freshwater, and
414  formulae. Shortwave radiation decays exponentially as per Paulson and Simpson  wind stress fluxes using the \citet{large81, large82} bulk formulae.
415  [1977]. Additionally the time-mean river run-off from Large and Nurser [2001]  Shortwave radiation decays exponentially as per Paulson and Simpson
416  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
417  surface salinity values from WOA01 with a relaxation time scale of 3  [2001] is applied and there is a relaxation to the monthly-mean
418  months. Vertical mixing follows Large et al. [1994] with background vertical  climatological sea surface salinity values from WOA01 with a
419  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
420  direct-space-time advection scheme with flux limiter is employed and there is  \citet{large94} with background vertical diffusivity of
421  no explicit horizontal diffusivity. Horizontal viscosity follows Leith [1996]  $1.5\times10^{-5}\text{\,m$^{2}$\,s$^{-1}$}$ and viscosity of
422  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
423  press].  Shortwave radiation decays exponentially as per Paulson and Simpson  advection scheme with flux limiter is employed \citep{hundsdorfer94}
424  [1977].  Additionally, the time-mean runoff of Large and Nurser [2001] is  and there is no explicit horizontal diffusivity. Horizontal viscosity
425  applied near the coastline and, where there is open water, there is a  follows \citet{lei96} but
426  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
427  45 days.  [in press].  Shortwave radiation decays exponentially as per Paulson
428    and Simpson [1977].  Additionally, the time-mean runoff of Large and
429    Nurser [2001] is applied near the coastline and, where there is open
430    water, there is a relaxation to monthly-mean WOA01 sea surface
431    salinity with a time constant of 45 days.
432    
433  Open water, dry  Open water, dry
434  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,
# Line 512  ice, wet ice, dry snow, and wet snow alb Line 457  ice, wet ice, dry snow, and wet snow alb
457  \item C-grid LSR slip  \item C-grid LSR slip
458  \item C-grid EVP no-slip  \item C-grid EVP no-slip
459  \item C-grid EVP slip  \item C-grid EVP slip
460    \item C-grid LSR + TEM (truncated ellipse method, no tensile stress, new flag)
461  \item C-grid LSR no-slip + Winton  \item C-grid LSR no-slip + Winton
462  \item  speed-performance-accuracy (small)  \item  speed-performance-accuracy (small)
463    ice transport through Canadian Archipelago differences    ice transport through Canadian Archipelago differences
# Line 523  We anticipate small differences between Line 469  We anticipate small differences between
469  \begin{itemize}  \begin{itemize}
470  \item advection schemes: along the ice-edge and regions with large  \item advection schemes: along the ice-edge and regions with large
471    gradients    gradients
472  \item C-grid: more transport through narrow straits for no slip  \item C-grid: less transport through narrow straits for no slip
473    conditons, less for free slip    conditons, more for free slip
474  \item VP vs.\ EVP: speed performance, accuracy?  \item VP vs.\ EVP: speed performance, accuracy?
475  \item ocean stress: different water mass properties beneath the ice  \item ocean stress: different water mass properties beneath the ice
476  \end{itemize}  \end{itemize}
# Line 605  storing vs. recomputation of the model s Line 551  storing vs. recomputation of the model s
551  checkpointing loop.  checkpointing loop.
552  Again, an initial code adjustment is required to support TAFs  Again, an initial code adjustment is required to support TAFs
553  checkpointing capability.  checkpointing capability.
554  The code adjustments are sufficiently simply so as not to cause  The code adjustments are sufficiently simple so as not to cause
555  major limitations to the full nonlinear parent model.  major limitations to the full nonlinear parent model.
556  Once in place, an adjoint model of a new model configuration  Once in place, an adjoint model of a new model configuration
557  may be derived in about 10 minutes.  may be derived in about 10 minutes.
# Line 628  may be derived in about 10 minutes. Line 574  may be derived in about 10 minutes.
574  We demonstrate the power of the adjoint method  We demonstrate the power of the adjoint method
575  in the context of investigating sea-ice export sensitivities through Fram Strait  in the context of investigating sea-ice export sensitivities through Fram Strait
576  (for details of this study see Heimbach et al., 2007).  (for details of this study see Heimbach et al., 2007).
577    %\citep[for details of this study see][]{heimbach07}. %Heimbach et al., 2007).
578  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
579  high-resolution cubed-sphere configuration of the ECCO2 project  high-resolution cubed-sphere configuration of the ECCO2 project
580  (see Menemenlis et al. 2005). It covers the entire Arctic,  \citep[see][]{menemenlis05}. It covers the entire Arctic,
581  extends into the North Pacific such as to cover the entire  extends into the North Pacific such as to cover the entire
582  ice-covered regions, and comprises parts of the North Atlantic  ice-covered regions, and comprises parts of the North Atlantic
583  down to XXN to enable analysis of remote influences of the  down to XXN to enable analysis of remote influences of the
# Line 641  The adjoint models run efficiently on 80 Line 588  The adjoint models run efficiently on 80
588  (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
589  IBM SP5 at NASA/ARC).  IBM SP5 at NASA/ARC).
590    
591  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
592  between 1992 and 1995.  years between 1992 and 1995. It is forced using realistic 6-hourly
593  It is forced using realistic 6-hourly NCEP/NCAR atmospheric state variables.  NCEP/NCAR atmospheric state variables. Over the open ocean these are
594  Over the open ocean these are converted into  converted into air-sea fluxes via the bulk formulae of
595  air-sea fluxes via the bulk formulae of Large and Yeager (2004).  \citet{large04}.  Derivation of air-sea fluxes in the presence of
596  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.  
597  The objective function chosen is sea-ice export through Fram Strait  The objective function chosen is sea-ice export through Fram Strait
598  computed for December 1995  computed for December 1995.  The adjoint model computes sensitivities
599  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
600  from 1995 to 1992 along this trajectory.  trajectory.  In principle all adjoint model variable (i.e., Lagrange
601  In principle all adjoint model variable (i.e. Lagrange multipliers)  multipliers) of the coupled ocean/sea-ice model are available to
602  of the coupled ocean/sea-ice model  analyze the transient sensitivity behaviour of the ocean and sea-ice
603  are available to analyze the transient sensitivity behaviour  state.  Over the open ocean, the adjoint of the bulk formula scheme
604  of the ocean and sea-ice state.  computes sensitivities to the time-varying atmospheric state.  Over
605  Over the open ocean, the adjoint of the bulk formula scheme  ice-covered parts, the sea-ice adjoint converts surface ocean
606  computes sensitivities to the time-varying atmospheric state.  sensitivities to atmospheric sensitivities.
607  Over ice-covered parts, the sea-ice adjoint converts  
608  surface ocean sensitivities to atmospheric sensitivities.  \reffig{4yradjheff}(a--d) depict sensitivities of sea-ice export
609    through Fram Strait in December 1995 to changes in sea-ice thickness
610  Fig. XXX(a--d) depict sensitivities of sea-ice export through Fram Strait  12, 24, 36, 48 months back in time. Corresponding sensitivities to
611  in December 1995 to changes in sea-ice thickness  ocean surface temperature are depicted in
612  12, 24, 36, 48 months back in time.  \reffig{4yradjthetalev1}(a--d).  The main characteristics is
613  Corresponding sensitivities to ocean surface temperature are  consistency with expected advection of sea-ice over the relevant time
614  depicted in Fig. XXX(a--d).  scales considered.  The general positive pattern means that an
615  The main characteristics is consistency with expected advection  increase in sea-ice thickness at location $(x,y)$ and time $t$ will
616  of sea-ice over the relevant time scales considered.  increase sea-ice export through Fram Strait at time $T_e$.  Largest
617  The general positive pattern means that an increase in  distances from Fram Strait indicate fastest sea-ice advection over the
618  sea-ice thickness at location $(x,y)$ and time $t$ will increase  time span considered.  The ice thickness sensitivities are in close
619  sea-ice export through Fram Strait at time $T_e$.  correspondence to ocean surface sentivitites, but of opposite sign.
620  Largest distances from Fram Strait indicate fastest sea-ice advection  An increase in temperature will incur ice melting, decrease in ice
621  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 $T_e$.  
622    
623  The picture is fundamentally different and much more complex  The picture is fundamentally different and much more complex
624  for sensitivities to ocean temperatures away from the surface.  for sensitivities to ocean temperatures away from the surface.
625  Fig. XXX (a--d) depicts ice export sensitivities to  \reffig{4yradjthetalev10??}(a--d) depicts ice export sensitivities to
626  temperatures at roughly 400 m depth.  temperatures at roughly 400 m depth.
627  Primary features are the effect of the heat transport of the North  Primary features are the effect of the heat transport of the North
628  Atlantic current which feeds into the West Spitsbergen current,  Atlantic current which feeds into the West Spitsbergen current,
# Line 730  sea-ice thickness at various prior times Line 672  sea-ice thickness at various prior times
672  -48 months}]  -48 months}]
673  {\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}}
674  }  }
675  \caption{Same as Fig. XXX but for sea surface temperature  \caption{Same as \reffig{4yradjheff} but for sea surface temperature
676  \label{fig:4yradjthetalev1}}  \label{fig:4yradjthetalev1}}
677  \end{figure}  \end{figure}
678    
# Line 755  parameters that we use here. What about Line 697  parameters that we use here. What about
697    
698  \paragraph{Acknowledgements}  \paragraph{Acknowledgements}
699  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
700  helpful discussions.  helpful discussions. ML thanks Elizabeth Hunke for multiple explanations.
701    
702  \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}
703    
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