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revision 1.5 by mlosch, Mon Jan 14 15:46:54 2008 UTC revision 1.14 by dimitri, Tue Feb 26 00:13:20 2008 UTC
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1    % $Header$
2    % $Name$
3  \documentclass[12pt]{article}  \documentclass[12pt]{article}
4    
5  \usepackage[]{graphicx}  \usepackage[]{graphicx}
# Line 50  Line 52 
52  \maketitle  \maketitle
53    
54  \begin{abstract}  \begin{abstract}
55    Some blabla  As part of ongoing efforts to obtain a best possible synthesis of most
56    available, global-scale, ocean and sea ice data, a dynamic and thermodynamic
57    sea-ice model has been coupled to the Massachusetts Institute of Technology
58    general circulation model (MITgcm).  Ice mechanics follow a viscous plastic
59    rheology and the ice momentum equations are solved numerically using either
60    line successive relaxation (LSR) or elastic-viscous-plastic (EVP) dynamic
61    models.  Ice thermodynamics are represented using either a zero-heat-capacity
62    formulation or a two-layer formulation that conserves enthalpy.  The model
63    includes prognostic variables for snow and for sea-ice salinity.  The above
64    sea ice model components were borrowed from current-generation climate models
65    but they were reformulated on an Arakawa C-grid in order to match the MITgcm
66    oceanic grid and they were modified in many ways to permit efficient and
67    accurate automatic differentiation.  This paper describes the MITgcm sea ice
68    model; it presents example Arctic and Antarctic results from a realistic,
69    eddy-permitting, global ocean and sea-ice configuration; it compares B-grid
70    and C-grid dynamic solvers in a regional Arctic configuration; and it presents
71    example results from coupled ocean and sea-ice adjoint-model integrations.
72    
73  \end{abstract}  \end{abstract}
74    
75  \section{Introduction}  \section{Introduction}
76  \label{sec:intro}  \label{sec:intro}
77    
78  more blabla  The availability of an adjoint model as a powerful research
79    tool complementary to an ocean model was a major design
80  \section{Model}  requirement early on in the development of the MIT general
81  \label{sec:model}  circulation model (MITgcm) [Marshall et al. 1997a,
82    Marotzke et al. 1999, Adcroft et al. 2002]. It was recognized
83    that the adjoint permitted very efficient computation
84    of gradients of various scalar-valued model diagnostics,
85    norms or, generally, objective functions with respect
86    to external or independent parameters. Such gradients
87    arise in at least two major contexts. If the objective function
88    is the sum of squared model vs. obervation differences
89    weighted by e.g. the inverse error covariances, the gradient
90    of the objective function can be used to optimize this measure
91    of model vs. data misfit in a least-squares sense. One
92    is then solving a problem of statistical state estimation.
93    If the objective function is a key oceanographic quantity
94    such as meridional heat or volume transport, ocean heat
95    content or mean surface temperature index, the gradient
96    provides a complete set of sensitivities of this quantity
97    with respect to all independent variables simultaneously.
98    
99    References to existing sea-ice adjoint models, explaining that they are either
100    for simplified configurations, for ice-only studies, or for short-duration
101    studies to motivate the present work.
102    
103  Traditionally, probably for historical reasons and the ease of  Traditionally, probably for historical reasons and the ease of
104  treating the Coriolis term, most standard sea-ice models are  treating the Coriolis term, most standard sea-ice models are
105  discretized on Arakawa-B-grids \citep[e.g.,][]{hibler79, harder99,  discretized on Arakawa-B-grids \citep[e.g.,][]{hibler79, harder99,
106    kreyscher00, zhang98, hunke97}. From the perspective of coupling a  kreyscher00, zhang98, hunke97}. From the perspective of coupling a
107  sea ice-model to a C-grid ocean model, the exchange of fluxes of heat  sea ice-model to a C-grid ocean model, the exchange of fluxes of heat
108  and fresh-water pose no difficulty for a B-grid sea-ice model  and fresh-water pose no difficulty for a B-grid sea-ice model
109  \citep[e.g.,][]{timmermann02a}. However, surface stress is defined at  \citep[e.g.,][]{timmermann02a}. However, surface stress is defined at
# Line 72  velocities points and thus needs to be i Line 111  velocities points and thus needs to be i
111  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
112  associated with this interpolation may mask grid scale noise, it may  associated with this interpolation may mask grid scale noise, it may
113  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
114  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
115  altogether and render the stress coupling as consistent as the  altogether and render the stress coupling as consistent as the
116  buoyancy coupling.  buoyancy coupling.
117    
# Line 80  A further advantage of the C-grid formul Line 119  A further advantage of the C-grid formul
119  straits. In the limit of only one grid cell between coasts there is no  straits. In the limit of only one grid cell between coasts there is no
120  flux allowed for a B-grid (with no-slip lateral boundary counditions),  flux allowed for a B-grid (with no-slip lateral boundary counditions),
121  whereas the C-grid formulation allows a flux of sea-ice through this  whereas the C-grid formulation allows a flux of sea-ice through this
122  passage for all types of lateral boundary conditions. We (will)  passage for all types of lateral boundary conditions. We
123  demonstrate this effect in the Candian archipelago.  demonstrate this effect in the Candian archipelago.
124    
125    Talk about problems that make the sea-ice-ocean code very sensitive and
126    changes in the code that reduce these sensitivities.
127    
128    This paper describes the MITgcm sea ice
129    model; it presents example Arctic and Antarctic results from a realistic,
130    eddy-permitting, global ocean and sea-ice configuration; it compares B-grid
131    and C-grid dynamic solvers in a regional Arctic configuration; and it presents
132    example results from coupled ocean and sea-ice adjoint-model integrations.
133    
134    \section{Model}
135    \label{sec:model}
136    
137  \subsection{Dynamics}  \subsection{Dynamics}
138  \label{sec:dynamics}  \label{sec:dynamics}
139    
140  The momentum equations of the sea-ice model are standard with  The momentum equation of the sea-ice model is
141  \begin{equation}  \begin{equation}
142    \label{eq:momseaice}    \label{eq:momseaice}
143    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} +
144    \vtau_{ocean} - m \nabla{\phi(0)} + \vek{F},    \vtau_{ocean} - mg \nabla{\phi(0)} + \vek{F},
145  \end{equation}  \end{equation}
146  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;
147  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;
148  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 $z$
149  surface height potential beneath the ice. $\phi$ is the sum of  directions, respectively;
150  atmpheric pressure $p_{a}$ and loading due to ice and snow  $f$ is the Coriolis parameter;
151  $(m_{i}+m_{s})g$. $\vtau_{air}$ and $\vtau_{ocean}$ are the wind and  $\vtau_{air}$ and $\vtau_{ocean}$ are the wind-ice and ocean-ice stresses,
152  ice-ocean stresses, respectively.  $\vek{F}$ is the interaction force  respectively;
153  and $\vek{i}$, $\vek{j}$, and $\vek{k}$ are the unit vectors in the  $g$ is the gravity accelation;
154  $x$, $y$, and $z$ directions.  Advection of sea-ice momentum is  $\nabla\phi(0)$ is the gradient (or tilt) of the sea surface height;
155  neglected. The wind and ice-ocean stress terms are given by  $\phi(0)$ is the sea surface height potential in response to ocean dynamics
156    and to atmospheric pressure loading;
157    and $\vek{F}=\nabla\cdot\sigma$ is the divergence of the internal ice stress
158    tensor $\sigma_{ij}$.
159    When using the rescaled vertical coordinate system, z$^\ast$, of
160    \citet{cam08}, $\phi(0)$ also includes a term due to snow and ice loading, $mg$.
161    Advection of sea-ice momentum is neglected. The wind and ice-ocean stress
162    terms are given by
163  \begin{align*}  \begin{align*}
164    \vtau_{air} =& \rho_{air} |\vek{U}_{air}|R_{air}(\vek{U}_{air}) \\    \vtau_{air}   = & \rho_{air}  C_{air}   |\vek{U}_{air}  -\vek{u}|
165    \vtau_{ocean} =& \rho_{ocean} |\vek{U}_{ocean}-\vek{u}|                     R_{air}  (\vek{U}_{air}  -\vek{u}), \\
166      \vtau_{ocean} = & \rho_{ocean}C_{ocean} |\vek{U}_{ocean}-\vek{u}|
167                     R_{ocean}(\vek{U}_{ocean}-\vek{u}), \\                     R_{ocean}(\vek{U}_{ocean}-\vek{u}), \\
168  \end{align*}  \end{align*}
169  where $\vek{U}_{air/ocean}$ are the surface winds of the atmosphere  where $\vek{U}_{air/ocean}$ are the surface winds of the atmosphere
170  and surface currents of the ocean, respectively. $C_{air/ocean}$ are  and surface currents of the ocean, respectively; $C_{air/ocean}$ are
171  air and ocean drag coefficients, $\rho_{air/ocean}$ reference  air and ocean drag coefficients; $\rho_{air/ocean}$ are reference
172  densities, and $R_{air/ocean}$ rotation matrices that act on the  densities; and $R_{air/ocean}$ are rotation matrices that act on the
173  wind/current vectors. $\vek{F} = \nabla\cdot\sigma$ is the divergence  wind/current vectors.
 of the interal stress tensor $\sigma_{ij}$.  
174    
175  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
176  strain rate and strength by a nonlinear viscous-plastic (VP)  strain rate and strength by a nonlinear viscous-plastic (VP)
# Line 134  The maximum ice pressure $P_{\max}$, a m Line 192  The maximum ice pressure $P_{\max}$, a m
192  both thickness $h$ and compactness (concentration) $c$:  both thickness $h$ and compactness (concentration) $c$:
193  \begin{equation}  \begin{equation}
194    P_{\max} = P^{*}c\,h\,e^{[C^{*}\cdot(1-c)]},    P_{\max} = P^{*}c\,h\,e^{[C^{*}\cdot(1-c)]},
195  \label{icestrength}  \label{eq:icestrength}
196  \end{equation}  \end{equation}
197  with the constants $P^{*}$ and $C^{*}$. The nonlinear bulk and shear  with the constants $P^{*}$ and $C^{*}$. The nonlinear bulk and shear
198  viscosities $\eta$ and $\zeta$ are functions of ice strain rate  viscosities $\eta$ and $\zeta$ are functions of ice strain rate
# Line 156  The bulk viscosities are bounded above b Line 214  The bulk viscosities are bounded above b
214  $\Delta_{\min}=10^{-11}\text{\,s}^{-1}$ (for numerical reasons) and a  $\Delta_{\min}=10^{-11}\text{\,s}^{-1}$ (for numerical reasons) and a
215  maximum $\zeta_{\max} = P_{\max}/\Delta^*$, where  maximum $\zeta_{\max} = P_{\max}/\Delta^*$, where
216  $\Delta^*=(5\times10^{12}/2\times10^4)\text{\,s}^{-1}$. For stress  $\Delta^*=(5\times10^{12}/2\times10^4)\text{\,s}^{-1}$. For stress
217  tensor compuation the replacement pressure $P = 2\,\Delta\zeta$  tensor computation the replacement pressure $P = 2\,\Delta\zeta$
218  \citep{hibler95} is used so that the stress state always lies on the  \citep{hibler95} is used so that the stress state always lies on the
219  elliptic yield curve by definition.  elliptic yield curve by definition.
220    
221    In the so-called truncated ellipse method the shear viscosity $\eta$
222    is capped to suppress any tensile stress \citep{hibler97, geiger98}:
223    \begin{equation}
224      \label{eq:etatem}
225      \eta = \min(\frac{\zeta}{e^2}
226      \frac{\frac{P}{2}-\zeta(\dot{\epsilon}_{11}+\dot{\epsilon}_{22})}
227      {\sqrt{(\dot{\epsilon}_{11}+\dot{\epsilon}_{22})^2
228          +4\dot{\epsilon}_{12}^2}}
229    \end{equation}
230    
231  In the current implementation, the VP-model is integrated with the  In the current implementation, the VP-model is integrated with the
232  semi-implicit line successive over relaxation (LSOR)-solver of  semi-implicit line successive over relaxation (LSOR)-solver of
233  \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 170  same length as in the ocean model where Line 238  same length as in the ocean model where
238  treated explicitly.  treated explicitly.
239    
240  \citet{hunke97}'s introduced an elastic contribution to the strain  \citet{hunke97}'s introduced an elastic contribution to the strain
241  rate elatic-viscous-plastic in order to regularize  rate elastic-viscous-plastic in order to regularize
242  Eq.\refeq{vpequation} in such a way that the resulting  Eq.\refeq{vpequation} in such a way that the resulting
243  elatic-viscous-plastic (EVP) and VP models are identical at steady  elastic-viscous-plastic (EVP) and VP models are identical at steady
244  state,  state,
245  \begin{equation}  \begin{equation}
246    \label{eq:evpequation}    \label{eq:evpequation}
# Line 198  $\sigma_{12}$. Introducing the divergenc Line 266  $\sigma_{12}$. Introducing the divergenc
266  \dot{\epsilon}_{11}+\dot{\epsilon}_{22}$, and the horizontal tension  \dot{\epsilon}_{11}+\dot{\epsilon}_{22}$, and the horizontal tension
267  and shearing strain rates, $D_T =  and shearing strain rates, $D_T =
268  \dot{\epsilon}_{11}-\dot{\epsilon}_{22}$ and $D_S =  \dot{\epsilon}_{11}-\dot{\epsilon}_{22}$ and $D_S =
269  2\dot{\epsilon}_{12}$, respectively and using the above abbreviations,  2\dot{\epsilon}_{12}$, respectively, and using the above abbreviations,
270  the equations can be written as:  the equations can be written as:
271  \begin{align}  \begin{align}
272    \label{eq:evpstresstensor1}    \label{eq:evpstresstensor1}
# Line 228  differences and averaging is only involv Line 296  differences and averaging is only involv
296  $P$ at vorticity points.  $P$ at vorticity points.
297    
298  For a general curvilinear grid, one needs in principle to take metric  For a general curvilinear grid, one needs in principle to take metric
299  terms into account that arise in the transformation a curvilinear grid  terms into account that arise in the transformation of a curvilinear grid
300  on the sphere. However, for now we can neglect these metric terms  on the sphere. For now, however, we can neglect these metric terms
301  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
302  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
303  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 238  simulations.  Everywhere else the coordi Line 306  simulations.  Everywhere else the coordi
306  cartesian.  However, for last-glacial-maximum or snowball-earth-like  cartesian.  However, for last-glacial-maximum or snowball-earth-like
307  simulations the question of metric terms needs to be reconsidered.  simulations the question of metric terms needs to be reconsidered.
308  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
309  vector-invariant formulation fo the sea-ice internal stress term that  vector-invariant formulation for the sea-ice internal stress term that
310  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
311  of the MITgcm \citep{adcroft04:_cubed_sphere}.  of the MITgcm \citep{adcroft04:_cubed_sphere}.
312    
# Line 299  addition to ice-thickness and compactnes Line 367  addition to ice-thickness and compactnes
367  state variables to be advected by ice velocities, namely enthalphy of  state variables to be advected by ice velocities, namely enthalphy of
368  the two ice layers and the thickness of the overlying snow layer.  the two ice layers and the thickness of the overlying snow layer.
369    
 \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 along with the computational  
 cost) or by regularizing the bulk and shear viscosities. We revisit  
 the latter option by reproducing some of the results of  
 \citet{hunke01}, namely the experiment described in her section~4, for  
 our C-grid no-slip cases: in a square domain with a few islands the  
 ice model is initialized with constant ice thickness and linearly  
 increasing ice concentration to the east. The model dynamics are  
 forced with a constant anticyclonic ocean gyre and variable  
 atmospheric wind whose mean directed diagnonally to the north-east  
 corner of the domain; ice volume and concentration are held constant  
 (no advection by ice velocity).  \reffig{hunke01} shows the ice  
 velocity field, its divergence, and the bulk viscosity $\zeta$ for the  
 cases C-LRSns and C-EVPns, and for a C-EVPns case, where  
 \citet{hunke01}'s regularization has been implemented; compare to  
 Fig.\,4 in \citet{hunke01}. The regularization contraint limits ice  
 strength and viscosities as a function of damping time scale,  
 resolution and EVP-time step, effectively allowing the elastic waves to  
 damp out more quickly \citep{hunke01}.  
 \begin{figure*}[htbp]  
   \includegraphics[width=\widefigwidth]{\fpath/hun12days}  
   \caption{Hunke's test case.}  
   \label{fig:hunke01}  
 \end{figure*}  
   
 In the far right (``east'') side of the domain the ice concentration  
 is close to one and the ice should be nearly rigid. The applied wind  
 tends to push ice toward the upper right corner. Because the highly  
 compact ice is confinded by the boundary, it resists any further  
 compression and exhibits little motion in the rigid region on the  
 right hand side. The C-LSRns solution (top row) allows high  
 viscosities in the rigid region suppressing nearly all flow.  
 \citet{hunke01}'s regularization for the C-EVPns solution (bottom row)  
 clearly suppresses the noise present in $\nabla\cdot\vek{u}$ in the  
 unregularized case (middle row), at the cost of reduced viscosities  
 These reduced viscosities lead to small but finite ice velocities  
 which in turn can have a strong effect on solutions in the limit of  
 nearly rigid regimes (arching and blocking, not shown).  
   
   
 %\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.  
370    
371  \subsection{C-grid}  \subsection{C-grid}
372  \begin{itemize}  \begin{itemize}
# Line 485  differences between the two main options Line 413  differences between the two main options
413  \subsection{Arctic Domain with Open Boundaries}  \subsection{Arctic Domain with Open Boundaries}
414  \label{sec:arctic}  \label{sec:arctic}
415    
416  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
417  carved out from, and obtains open boundary conditions from, the global  is carved out from, and obtains open boundary conditions from, the
418  cubed-sphere configuration of the Estimating the Circulation and Climate of  global cubed-sphere configuration of the Estimating the Circulation
419  the Ocean, Phase II (ECCO2) project \cite{men05a}.  The domain size is 420 by  and Climate of the Ocean, Phase II (ECCO2) project
420  384 grid boxes horizontally with mean horizontal grid spacing of 18 km.    \citet{menemenlis05}.  The domain size is 420 by 384 grid boxes
421    horizontally with mean horizontal grid spacing of 18 km.
422    
423    \begin{figure}
424    %\centerline{{\includegraphics*[width=0.44\linewidth]{\fpath/arctic1.eps}}}
425    \caption{Bathymetry of Arctic Domain.\label{fig:arctic1}}
426    \end{figure}
427    
428  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
429  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
430  the National Geophysical Data Center (NGDC) 2-minute gridded global relief  the National Geophysical Data Center (NGDC) 2-minute gridded global relief
431  data (ETOPO2) and the model employs the partial-cell formulation of  data (ETOPO2) and the model employs the partial-cell formulation of
432  \cite{adc97}, which permits accurate representation of the bathymetry. The  \citet{adcroft97:_shaved_cells}, which permits accurate representation of the bathymetry. The
433  model is integrated in a volume-conserving configuration using a finite volume  model is integrated in a volume-conserving configuration using a finite volume
434  discretization with C-grid staggering of the prognostic variables. In the  discretization with C-grid staggering of the prognostic variables. In the
435  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
436  coupled to a sea-ice model described hereinabove.    coupled to a sea-ice model described hereinabove.  
437    
438  This particular ECCO2 simulation is initialized from rest using the January  This particular ECCO2 simulation is initialized from rest using the
439  temperature and salinity distribution from the World Ocean Atlas 2001 (WOA01)  January temperature and salinity distribution from the World Ocean
440  [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
441  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
442  the National Centers for Environmental Prediction and the National Center for  boundary conditions are from the National Centers for Environmental
443  Atmospheric Research (NCEP/NCAR) atmospheric reanalysis [Kistler et al.,  Prediction and the National Center for Atmospheric Research
444  2001]. Six-hourly surface winds, temperature, humidity, downward short- and  (NCEP/NCAR) atmospheric reanalysis [Kistler et al., 2001]. Six-hourly
445  long-wave radiations, and precipitation are converted to heat, freshwater, and  surface winds, temperature, humidity, downward short- and long-wave
446  wind stress fluxes using the Large and Pond [1981, 1982] bulk  radiations, and precipitation are converted to heat, freshwater, and
447  formulae. Shortwave radiation decays exponentially as per Paulson and Simpson  wind stress fluxes using the \citet{large81, large82} bulk formulae.
448  [1977]. Additionally the time-mean river run-off from Large and Nurser [2001]  Shortwave radiation decays exponentially as per Paulson and Simpson
449  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
450  surface salinity values from WOA01 with a relaxation time scale of 3  [2001] is applied and there is a relaxation to the monthly-mean
451  months. Vertical mixing follows Large et al. [1994] with background vertical  climatological sea surface salinity values from WOA01 with a
452  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
453  direct-space-time advection scheme with flux limiter is employed and there is  \citet{large94} with background vertical diffusivity of
454  no explicit horizontal diffusivity. Horizontal viscosity follows Leith [1996]  $1.5\times10^{-5}\text{\,m$^{2}$\,s$^{-1}$}$ and viscosity of
455  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
456  press].  Shortwave radiation decays exponentially as per Paulson and Simpson  advection scheme with flux limiter is employed \citep{hundsdorfer94}
457  [1977].  Additionally, the time-mean runoff of Large and Nurser [2001] is  and there is no explicit horizontal diffusivity. Horizontal viscosity
458  applied near the coastline and, where there is open water, there is a  follows \citet{lei96} but
459  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
460  45 days.  [in press].  Shortwave radiation decays exponentially as per Paulson
461    and Simpson [1977].  Additionally, the time-mean runoff of Large and
462    Nurser [2001] is applied near the coastline and, where there is open
463    water, there is a relaxation to monthly-mean WOA01 sea surface
464    salinity with a time constant of 45 days.
465    
466  Open water, dry  Open water, dry
467  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 552  ice, wet ice, dry snow, and wet snow alb Line 490  ice, wet ice, dry snow, and wet snow alb
490  \item C-grid LSR slip  \item C-grid LSR slip
491  \item C-grid EVP no-slip  \item C-grid EVP no-slip
492  \item C-grid EVP slip  \item C-grid EVP slip
493    \item C-grid LSR + TEM (truncated ellipse method, no tensile stress, new flag)
494  \item C-grid LSR no-slip + Winton  \item C-grid LSR no-slip + Winton
495  \item  speed-performance-accuracy (small)  \item  speed-performance-accuracy (small)
496    ice transport through Canadian Archipelago differences    ice transport through Canadian Archipelago differences
# Line 563  We anticipate small differences between Line 502  We anticipate small differences between
502  \begin{itemize}  \begin{itemize}
503  \item advection schemes: along the ice-edge and regions with large  \item advection schemes: along the ice-edge and regions with large
504    gradients    gradients
505  \item C-grid: more transport through narrow straits for no slip  \item C-grid: less transport through narrow straits for no slip
506    conditons, less for free slip    conditons, more for free slip
507  \item VP vs.\ EVP: speed performance, accuracy?  \item VP vs.\ EVP: speed performance, accuracy?
508  \item ocean stress: different water mass properties beneath the ice  \item ocean stress: different water mass properties beneath the ice
509  \end{itemize}  \end{itemize}
# Line 645  storing vs. recomputation of the model s Line 584  storing vs. recomputation of the model s
584  checkpointing loop.  checkpointing loop.
585  Again, an initial code adjustment is required to support TAFs  Again, an initial code adjustment is required to support TAFs
586  checkpointing capability.  checkpointing capability.
587  The code adjustments are sufficiently simply so as not to cause  The code adjustments are sufficiently simple so as not to cause
588  major limitations to the full nonlinear parent model.  major limitations to the full nonlinear parent model.
589  Once in place, an adjoint model of a new model configuration  Once in place, an adjoint model of a new model configuration
590  may be derived in about 10 minutes.  may be derived in about 10 minutes.
# Line 668  may be derived in about 10 minutes. Line 607  may be derived in about 10 minutes.
607  We demonstrate the power of the adjoint method  We demonstrate the power of the adjoint method
608  in the context of investigating sea-ice export sensitivities through Fram Strait  in the context of investigating sea-ice export sensitivities through Fram Strait
609  (for details of this study see Heimbach et al., 2007).  (for details of this study see Heimbach et al., 2007).
610    %\citep[for details of this study see][]{heimbach07}. %Heimbach et al., 2007).
611  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
612  high-resolution cubed-sphere configuration of the ECCO2 project  high-resolution cubed-sphere configuration of the ECCO2 project
613  (see Menemenlis et al. 2005). It covers the entire Arctic,  \citep[see][]{menemenlis05}. It covers the entire Arctic,
614  extends into the North Pacific such as to cover the entire  extends into the North Pacific such as to cover the entire
615  ice-covered regions, and comprises parts of the North Atlantic  ice-covered regions, and comprises parts of the North Atlantic
616  down to XXN to enable analysis of remote influences of the  down to XXN to enable analysis of remote influences of the
# Line 681  The adjoint models run efficiently on 80 Line 621  The adjoint models run efficiently on 80
621  (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
622  IBM SP5 at NASA/ARC).  IBM SP5 at NASA/ARC).
623    
624  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
625  between 1992 and 1995.  years between 1992 and 1995. It is forced using realistic 6-hourly
626  It is forced using realistic 6-hourly NCEP/NCAR atmospheric state variables.  NCEP/NCAR atmospheric state variables. Over the open ocean these are
627  Over the open ocean these are converted into  converted into air-sea fluxes via the bulk formulae of
628  air-sea fluxes via the bulk formulae of Large and Yeager (2004).  \citet{large04}.  Derivation of air-sea fluxes in the presence of
629  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.  
630  The objective function chosen is sea-ice export through Fram Strait  The objective function chosen is sea-ice export through Fram Strait
631  computed for December 1995  computed for December 1995.  The adjoint model computes sensitivities
632  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
633  from 1995 to 1992 along this trajectory.  trajectory.  In principle all adjoint model variable (i.e., Lagrange
634  In principle all adjoint model variable (i.e. Lagrange multipliers)  multipliers) of the coupled ocean/sea-ice model are available to
635  of the coupled ocean/sea-ice model  analyze the transient sensitivity behaviour of the ocean and sea-ice
636  are available to analyze the transient sensitivity behaviour  state.  Over the open ocean, the adjoint of the bulk formula scheme
637  of the ocean and sea-ice state.  computes sensitivities to the time-varying atmospheric state.  Over
638  Over the open ocean, the adjoint of the bulk formula scheme  ice-covered parts, the sea-ice adjoint converts surface ocean
639  computes sensitivities to the time-varying atmospheric state.  sensitivities to atmospheric sensitivities.
640  Over ice-covered parts, the sea-ice adjoint converts  
641  surface ocean sensitivities to atmospheric sensitivities.  \reffig{4yradjheff}(a--d) depict sensitivities of sea-ice export
642    through Fram Strait in December 1995 to changes in sea-ice thickness
643  Fig. XXX(a--d) depict sensitivities of sea-ice export through Fram Strait  12, 24, 36, 48 months back in time. Corresponding sensitivities to
644  in December 1995 to changes in sea-ice thickness  ocean surface temperature are depicted in
645  12, 24, 36, 48 months back in time.  \reffig{4yradjthetalev1}(a--d).  The main characteristics is
646  Corresponding sensitivities to ocean surface temperature are  consistency with expected advection of sea-ice over the relevant time
647  depicted in Fig. XXX(a--d).  scales considered.  The general positive pattern means that an
648  The main characteristics is consistency with expected advection  increase in sea-ice thickness at location $(x,y)$ and time $t$ will
649  of sea-ice over the relevant time scales considered.  increase sea-ice export through Fram Strait at time $T_e$.  Largest
650  The general positive pattern means that an increase in  distances from Fram Strait indicate fastest sea-ice advection over the
651  sea-ice thickness at location $(x,y)$ and time $t$ will increase  time span considered.  The ice thickness sensitivities are in close
652  sea-ice export through Fram Strait at time $T_e$.  correspondence to ocean surface sentivitites, but of opposite sign.
653  Largest distances from Fram Strait indicate fastest sea-ice advection  An increase in temperature will incur ice melting, decrease in ice
654  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 $T_e$.  
655    
656  The picture is fundamentally different and much more complex  The picture is fundamentally different and much more complex
657  for sensitivities to ocean temperatures away from the surface.  for sensitivities to ocean temperatures away from the surface.
658  Fig. XXX (a--d) depicts ice export sensitivities to  \reffig{4yradjthetalev10??}(a--d) depicts ice export sensitivities to
659  temperatures at roughly 400 m depth.  temperatures at roughly 400 m depth.
660  Primary features are the effect of the heat transport of the North  Primary features are the effect of the heat transport of the North
661  Atlantic current which feeds into the West Spitsbergen current,  Atlantic current which feeds into the West Spitsbergen current,
# Line 770  sea-ice thickness at various prior times Line 705  sea-ice thickness at various prior times
705  -48 months}]  -48 months}]
706  {\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}}
707  }  }
708  \caption{Same as Fig. XXX but for sea surface temperature  \caption{Same as \reffig{4yradjheff} but for sea surface temperature
709  \label{fig:4yradjthetalev1}}  \label{fig:4yradjthetalev1}}
710  \end{figure}  \end{figure}
711    
# Line 795  parameters that we use here. What about Line 730  parameters that we use here. What about
730    
731  \paragraph{Acknowledgements}  \paragraph{Acknowledgements}
732  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
733  helpful discussions.  helpful discussions. ML thanks Elizabeth Hunke for multiple explanations.
734    
735  %\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}
 \bibliography{journal_abrvs,seaice,genocean,maths,mitgcmuv,bib/fram}  
736    
737  \end{document}  \end{document}
738    
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