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revision 1.5 by mlosch, Mon Jan 14 15:46:54 2008 UTC revision 1.16 by mlosch, Tue Feb 26 19:14:36 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 tool
79    complementary to an ocean model was a major design requirement early
80  \section{Model}  on in the development of the MIT general circulation model (MITgcm)
81  \label{sec:model}  [Marshall et al. 1997a, Marotzke et al. 1999, Adcroft et al. 2002]. It
82    was recognized that the adjoint model permitted computing the
83    gradients of various scalar-valued model diagnostics, norms or,
84    generally, objective functions with respect to external or independent
85    parameters very efficiently. The information associtated with these
86    gradients is useful in at least two major contexts. First, for state
87    estimation problems, the objective function is the sum of squared
88    differences between observations and model results weighted by the
89    inverse error covariances. The gradient of such an objective function
90    can be used to reduce this measure of model-data misfit to find the
91    optimal model solution in a least-squares sense.  Second, the
92    objective function can be a key oceanographic quantity such as
93    meridional heat or volume transport, ocean heat content or mean
94    surface temperature index. In this case the gradient provides a
95    complete set of sensitivities of this quantity to all independent
96    variables simultaneously. These sensitivities can be used to address
97    the cause of, say, changing net transports accurately.
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
# Line 69  sea ice-model to a C-grid ocean model, t Line 108  sea ice-model to a C-grid ocean model, t
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
110  velocities points and thus needs to be interpolated between a B-grid  velocities points and thus needs to be interpolated between a B-grid
111  sea-ice model and a C-grid ocean model. While the smoothing implicitly  sea-ice model and a C-grid ocean model. Smoothing implicitly
112  associated with this interpolation may mask grid scale noise, it may  associated with this interpolation may mask grid scale noise and may
113  in two-way coupling lead to a computational mode as will be shown. By  contribute to stabilizing the solution. On the other hand, by
114  choosing a C-grid for the sea-ice model, we circumvene this difficulty  smoothing the stress signals are damped which could lead to reduced
115  altogether and render the stress coupling as consistent as the  variability of the system. By choosing a C-grid for the sea-ice model,
116  buoyancy coupling.  we circumvent this difficulty altogether and render the stress
117    coupling as consistent as the buoyancy coupling.
118    
119  A further advantage of the C-grid formulation is apparent in narrow  A further advantage of the C-grid formulation is apparent in narrow
120  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
121  flux allowed for a B-grid (with no-slip lateral boundary counditions),  flux allowed for a B-grid (with no-slip lateral boundary counditions),
122  whereas the C-grid formulation allows a flux of sea-ice through this  and models have used topographies artificially widened straits to
123  passage for all types of lateral boundary conditions. We (will)  avoid this problem \citep{holloway07}. The C-grid formulation on the
124  demonstrate this effect in the Candian archipelago.  other hand allows a flux of sea-ice through narrow passages if
125    free-slip along the boundaries is allowed. We demonstrate this effect
126    in the Candian archipelago.
127    
128    Talk about problems that make the sea-ice-ocean code very sensitive and
129    changes in the code that reduce these sensitivities.
130    
131    This paper describes the MITgcm sea ice
132    model; it presents example Arctic and Antarctic results from a realistic,
133    eddy-permitting, global ocean and sea-ice configuration; it compares B-grid
134    and C-grid dynamic solvers in a regional Arctic configuration; and it presents
135    example results from coupled ocean and sea-ice adjoint-model integrations.
136    
137    \section{Model}
138    \label{sec:model}
139    
140  \subsection{Dynamics}  \subsection{Dynamics}
141  \label{sec:dynamics}  \label{sec:dynamics}
142    
143  The momentum equations of the sea-ice model are standard with  The momentum equation of the sea-ice model is
144  \begin{equation}  \begin{equation}
145    \label{eq:momseaice}    \label{eq:momseaice}
146    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} +
147    \vtau_{ocean} - m \nabla{\phi(0)} + \vek{F},    \vtau_{ocean} - m \nabla{\phi(0)} + \vek{F},
148  \end{equation}  \end{equation}
149  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;
150  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;
151  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 $z$
152  surface height potential beneath the ice. $\phi$ is the sum of  directions, respectively;
153  atmpheric pressure $p_{a}$ and loading due to ice and snow  $f$ is the Coriolis parameter;
154  $(m_{i}+m_{s})g$. $\vtau_{air}$ and $\vtau_{ocean}$ are the wind and  $\vtau_{air}$ and $\vtau_{ocean}$ are the wind-ice and ocean-ice stresses,
155  ice-ocean stresses, respectively.  $\vek{F}$ is the interaction force  respectively;
156  and $\vek{i}$, $\vek{j}$, and $\vek{k}$ are the unit vectors in the  $g$ is the gravity accelation;
157  $x$, $y$, and $z$ directions.  Advection of sea-ice momentum is  $\nabla\phi(0)$ is the gradient (or tilt) of the sea surface height;
158  neglected. The wind and ice-ocean stress terms are given by  $\phi(0) = g\eta + p_{a}/\rho_{0}$ is the sea surface height potential
159    in response to ocean dynamics ($g\eta$) and to atmospheric pressure
160    loading ($p_{a}/\rho_{0}$, where $\rho_{0}$ is a reference density);
161    and $\vek{F}=\nabla\cdot\sigma$ is the divergence of the internal ice stress
162    tensor $\sigma_{ij}$.
163    When using the rescaled vertical coordinate system, z$^\ast$, of
164    \citet{cam08}, $\phi(0)$ also includes a term due to snow and ice
165    loading, $mg/\rho_{0}$.
166    Advection of sea-ice momentum is neglected. The wind and ice-ocean stress
167    terms are given by
168  \begin{align*}  \begin{align*}
169    \vtau_{air} =& \rho_{air} |\vek{U}_{air}|R_{air}(\vek{U}_{air}) \\    \vtau_{air}   = & \rho_{air}  C_{air}   |\vek{U}_{air}  -\vek{u}|
170    \vtau_{ocean} =& \rho_{ocean} |\vek{U}_{ocean}-\vek{u}|                     R_{air}  (\vek{U}_{air}  -\vek{u}), \\
171      \vtau_{ocean} = & \rho_{ocean}C_{ocean} |\vek{U}_{ocean}-\vek{u}|
172                     R_{ocean}(\vek{U}_{ocean}-\vek{u}), \\                     R_{ocean}(\vek{U}_{ocean}-\vek{u}), \\
173  \end{align*}  \end{align*}
174  where $\vek{U}_{air/ocean}$ are the surface winds of the atmosphere  where $\vek{U}_{air/ocean}$ are the surface winds of the atmosphere
175  and surface currents of the ocean, respectively. $C_{air/ocean}$ are  and surface currents of the ocean, respectively; $C_{air/ocean}$ are
176  air and ocean drag coefficients, $\rho_{air/ocean}$ reference  air and ocean drag coefficients; $\rho_{air/ocean}$ are reference
177  densities, and $R_{air/ocean}$ rotation matrices that act on the  densities; and $R_{air/ocean}$ are rotation matrices that act on the
178  wind/current vectors. $\vek{F} = \nabla\cdot\sigma$ is the divergence  wind/current vectors.
179  of the interal stress tensor $\sigma_{ij}$.  
180    For an isotropic system the stress tensor $\sigma_{ij}$ ($i,j=1,2$) can
181  For an isotropic system this stress tensor can be related to the ice  be related to the ice strain rate and strength by a nonlinear
182  strain rate and strength by a nonlinear viscous-plastic (VP)  viscous-plastic (VP) constitutive law \citep{hibler79, zhang98}:
 constitutive law \citep{hibler79, zhang98}:  
183  \begin{equation}  \begin{equation}
184    \label{eq:vpequation}    \label{eq:vpequation}
185    \sigma_{ij}=2\eta(\dot{\epsilon}_{ij},P)\dot{\epsilon}_{ij}    \sigma_{ij}=2\eta(\dot{\epsilon}_{ij},P)\dot{\epsilon}_{ij}
# Line 134  The maximum ice pressure $P_{\max}$, a m Line 197  The maximum ice pressure $P_{\max}$, a m
197  both thickness $h$ and compactness (concentration) $c$:  both thickness $h$ and compactness (concentration) $c$:
198  \begin{equation}  \begin{equation}
199    P_{\max} = P^{*}c\,h\,e^{[C^{*}\cdot(1-c)]},    P_{\max} = P^{*}c\,h\,e^{[C^{*}\cdot(1-c)]},
200  \label{icestrength}  \label{eq:icestrength}
201  \end{equation}  \end{equation}
202  with the constants $P^{*}$ and $C^{*}$. The nonlinear bulk and shear  with the constants $P^{*}$ and $C^{*}$. The nonlinear bulk and shear
203  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 219  The bulk viscosities are bounded above b
219  $\Delta_{\min}=10^{-11}\text{\,s}^{-1}$ (for numerical reasons) and a  $\Delta_{\min}=10^{-11}\text{\,s}^{-1}$ (for numerical reasons) and a
220  maximum $\zeta_{\max} = P_{\max}/\Delta^*$, where  maximum $\zeta_{\max} = P_{\max}/\Delta^*$, where
221  $\Delta^*=(5\times10^{12}/2\times10^4)\text{\,s}^{-1}$. For stress  $\Delta^*=(5\times10^{12}/2\times10^4)\text{\,s}^{-1}$. For stress
222  tensor compuation the replacement pressure $P = 2\,\Delta\zeta$  tensor computation the replacement pressure $P = 2\,\Delta\zeta$
223  \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
224  elliptic yield curve by definition.  elliptic yield curve by definition.
225    
226    In the so-called truncated ellipse method the shear viscosity $\eta$
227    is capped to suppress any tensile stress \citep{hibler97, geiger98}:
228    \begin{equation}
229      \label{eq:etatem}
230      \eta = \min\left(\frac{\zeta}{e^2},
231      \frac{\frac{P}{2}-\zeta(\dot{\epsilon}_{11}+\dot{\epsilon}_{22})}
232      {\sqrt{(\dot{\epsilon}_{11}+\dot{\epsilon}_{22})^2
233          +4\dot{\epsilon}_{12}^2}}\right).
234    \end{equation}
235    
236  In the current implementation, the VP-model is integrated with the  In the current implementation, the VP-model is integrated with the
237  semi-implicit line successive over relaxation (LSOR)-solver of  semi-implicit line successive over relaxation (LSOR)-solver of
238  \citet{zhang98}, which allows for long time steps that, in our case,  \citet{zhang98}, which allows for long time steps that, in our case,
239  is limited by the explicit treatment of the Coriolis term. The  are limited by the explicit treatment of the Coriolis term. The
240  explicit treatment of the Coriolis term does not represent a severe  explicit treatment of the Coriolis term does not represent a severe
241  limitation because it restricts the time step to approximately the  limitation because it restricts the time step to approximately the
242  same length as in the ocean model where the Coriolis term is also  same length as in the ocean model where the Coriolis term is also
243  treated explicitly.  treated explicitly.
244    
245  \citet{hunke97}'s introduced an elastic contribution to the strain  \citet{hunke97}'s introduced an elastic contribution to the strain
246  rate elatic-viscous-plastic in order to regularize  rate in order to regularize Eq.\refeq{vpequation} in such a way that
247  Eq.\refeq{vpequation} in such a way that the resulting  the resulting elastic-viscous-plastic (EVP) and VP models are
248  elatic-viscous-plastic (EVP) and VP models are identical at steady  identical at steady state,
 state,  
249  \begin{equation}  \begin{equation}
250    \label{eq:evpequation}    \label{eq:evpequation}
251    \frac{1}{E}\frac{\partial\sigma_{ij}}{\partial{t}} +    \frac{1}{E}\frac{\partial\sigma_{ij}}{\partial{t}} +
# Line 198  $\sigma_{12}$. Introducing the divergenc Line 270  $\sigma_{12}$. Introducing the divergenc
270  \dot{\epsilon}_{11}+\dot{\epsilon}_{22}$, and the horizontal tension  \dot{\epsilon}_{11}+\dot{\epsilon}_{22}$, and the horizontal tension
271  and shearing strain rates, $D_T =  and shearing strain rates, $D_T =
272  \dot{\epsilon}_{11}-\dot{\epsilon}_{22}$ and $D_S =  \dot{\epsilon}_{11}-\dot{\epsilon}_{22}$ and $D_S =
273  2\dot{\epsilon}_{12}$, respectively and using the above abbreviations,  2\dot{\epsilon}_{12}$, respectively, and using the above abbreviations,
274  the equations can be written as:  the equations can be written as:
275  \begin{align}  \begin{align}
276    \label{eq:evpstresstensor1}    \label{eq:evpstresstensor1}
# Line 220  $E_{0} = \frac{1}{3}$. Line 292  $E_{0} = \frac{1}{3}$.
292  For details of the spatial discretization, the reader is referred to  For details of the spatial discretization, the reader is referred to
293  \citet{zhang98, zhang03}. Our discretization differs only (but  \citet{zhang98, zhang03}. Our discretization differs only (but
294  importantly) in the underlying grid, namely the Arakawa C-grid, but is  importantly) in the underlying grid, namely the Arakawa C-grid, but is
295  otherwise straightforward. The EVP model in particular is discretized  otherwise straightforward. The EVP model, in particular, is discretized
296  naturally on the C-grid with $\sigma_{1}$ and $\sigma_{2}$ on the  naturally on the C-grid with $\sigma_{1}$ and $\sigma_{2}$ on the
297  center points and $\sigma_{12}$ on the corner (or vorticity) points of  center points and $\sigma_{12}$ on the corner (or vorticity) points of
298  the grid. With this choice all derivatives are discretized as central  the grid. With this choice all derivatives are discretized as central
# Line 228  differences and averaging is only involv Line 300  differences and averaging is only involv
300  $P$ at vorticity points.  $P$ at vorticity points.
301    
302  For a general curvilinear grid, one needs in principle to take metric  For a general curvilinear grid, one needs in principle to take metric
303  terms into account that arise in the transformation a curvilinear grid  terms into account that arise in the transformation of a curvilinear
304  on the sphere. However, for now we can neglect these metric terms  grid on the sphere. For now, however, we can neglect these metric
305  because they are very small on the cubed sphere grids used in this  terms because they are very small on the \ml{[modify following
306  paper; in particular, only near the edges of the cubed sphere grid, we    section3:] cubed sphere grids used in this paper; in particular,
307  expect them to be non-zero, but these edges are at approximately  only near the edges of the cubed sphere grid, we expect them to be
308  35\degS\ or 35\degN\ which are never covered by sea-ice in our  non-zero, but these edges are at approximately 35\degS\ or 35\degN\
309  simulations.  Everywhere else the coordinate system is locally nearly  which are never covered by sea-ice in our simulations.  Everywhere
310  cartesian.  However, for last-glacial-maximum or snowball-earth-like  else the coordinate system is locally nearly cartesian.}  However, for
311  simulations the question of metric terms needs to be reconsidered.  last-glacial-maximum or snowball-earth-like simulations the question
312  Either, one includes these terms as in \citet{zhang03}, or one finds a  of metric terms needs to be reconsidered.  Either, one includes these
313  vector-invariant formulation fo the sea-ice internal stress term that  terms as in \citet{zhang03}, or one finds a vector-invariant
314  does not require any metric terms, as it is done in the ocean dynamics  formulation for the sea-ice internal stress term that does not require
315  of the MITgcm \citep{adcroft04:_cubed_sphere}.  any metric terms, as it is done in the ocean dynamics of the MITgcm
316    \citep{adcroft04:_cubed_sphere}.
317    
318    Lateral boundary conditions are naturally ``no-slip'' for B-grids, as
319    the tangential velocities points lie on the boundary. For C-grids, the
320    lateral boundary condition for tangential velocities is realized via
321    ``ghost points'', allowing alternatively no-slip or free-slip
322    conditions. In ocean models free-slip boundary conditions in
323    conjunction with piecewise-constant (``castellated'') coastlines have
324    been shown to reduce in effect to no-slip boundary conditions
325    \citep{adcroft98:_slippery_coast}; for sea-ice models the effects of
326    lateral boundary conditions have not yet been studied.
327    
328  Moving sea ice exerts a stress on the ocean which is the opposite of  Moving sea ice exerts a stress on the ocean which is the opposite of
329  the stress $\vtau_{ocean}$ in Eq.\refeq{momseaice}. This stess is  the stress $\vtau_{ocean}$ in Eq.\refeq{momseaice}. This stess is
# Line 259  velocity and the ice velocity leading to Line 342  velocity and the ice velocity leading to
342  temperature and salinity are different from the oceanic variables.  temperature and salinity are different from the oceanic variables.
343    
344  Sea ice distributions are characterized by sharp gradients and edges.  Sea ice distributions are characterized by sharp gradients and edges.
345  For this reason, we employ a positive 3rd-order advection scheme  For this reason, we employ positive, multidimensional 2nd and 3rd-order
346  \citep{hundsdorfer94} for the thermodynamic variables discussed in the  advection scheme with flux limiter \citep{roe85, hundsdorfer94} for the
347  next section.  thermodynamic variables discussed in the next section.
348    
349  \subparagraph{boundary conditions: no-slip, free-slip, half-slip}  \subparagraph{boundary conditions: no-slip, free-slip, half-slip}
350    
# Line 298  content by means of enthalphy conservati Line 381  content by means of enthalphy conservati
381  addition to ice-thickness and compactness (fractional area) additional  addition to ice-thickness and compactness (fractional area) additional
382  state variables to be advected by ice velocities, namely enthalphy of  state variables to be advected by ice velocities, namely enthalphy of
383  the two ice layers and the thickness of the overlying snow layer.  the two ice layers and the thickness of the overlying snow layer.
384    \ml{[Jean-Michel, your turf: ]Care must be taken in advecting these
385      quantities in order to ensure conservation of enthalphy. Currently
386      this can only be accomplished with a 2nd-order advection scheme with
387      flux limiter \citep{roe85}.}
388    
 \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.  
389    
390  \subsection{C-grid}  \subsection{C-grid}
391  \begin{itemize}  \begin{itemize}
# Line 485  differences between the two main options Line 432  differences between the two main options
432  \subsection{Arctic Domain with Open Boundaries}  \subsection{Arctic Domain with Open Boundaries}
433  \label{sec:arctic}  \label{sec:arctic}
434    
435  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
436  carved out from, and obtains open boundary conditions from, the global  is carved out from, and obtains open boundary conditions from, the
437  cubed-sphere configuration of the Estimating the Circulation and Climate of  global cubed-sphere configuration of the Estimating the Circulation
438  the Ocean, Phase II (ECCO2) project \cite{men05a}.  The domain size is 420 by  and Climate of the Ocean, Phase II (ECCO2) project
439  384 grid boxes horizontally with mean horizontal grid spacing of 18 km.    \citet{menemenlis05}.  The domain size is 420 by 384 grid boxes
440    horizontally with mean horizontal grid spacing of 18 km.
441    
442    \begin{figure}
443    %\centerline{{\includegraphics*[width=0.44\linewidth]{\fpath/arctic1.eps}}}
444    \caption{Bathymetry of Arctic Domain.\label{fig:arctic1}}
445    \end{figure}
446    
447  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
448  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
449  the National Geophysical Data Center (NGDC) 2-minute gridded global relief  the National Geophysical Data Center (NGDC) 2-minute gridded global relief
450  data (ETOPO2) and the model employs the partial-cell formulation of  data (ETOPO2) and the model employs the partial-cell formulation of
451  \cite{adc97}, which permits accurate representation of the bathymetry. The  \citet{adcroft97:_shaved_cells}, which permits accurate representation of the bathymetry. The
452  model is integrated in a volume-conserving configuration using a finite volume  model is integrated in a volume-conserving configuration using a finite volume
453  discretization with C-grid staggering of the prognostic variables. In the  discretization with C-grid staggering of the prognostic variables. In the
454  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
455  coupled to a sea-ice model described hereinabove.    coupled to a sea-ice model described hereinabove.  
456    
457  This particular ECCO2 simulation is initialized from rest using the January  This particular ECCO2 simulation is initialized from rest using the
458  temperature and salinity distribution from the World Ocean Atlas 2001 (WOA01)  January temperature and salinity distribution from the World Ocean
459  [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
460  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
461  the National Centers for Environmental Prediction and the National Center for  boundary conditions are from the National Centers for Environmental
462  Atmospheric Research (NCEP/NCAR) atmospheric reanalysis [Kistler et al.,  Prediction and the National Center for Atmospheric Research
463  2001]. Six-hourly surface winds, temperature, humidity, downward short- and  (NCEP/NCAR) atmospheric reanalysis [Kistler et al., 2001]. Six-hourly
464  long-wave radiations, and precipitation are converted to heat, freshwater, and  surface winds, temperature, humidity, downward short- and long-wave
465  wind stress fluxes using the Large and Pond [1981, 1982] bulk  radiations, and precipitation are converted to heat, freshwater, and
466  formulae. Shortwave radiation decays exponentially as per Paulson and Simpson  wind stress fluxes using the \citet{large81, large82} bulk formulae.
467  [1977]. Additionally the time-mean river run-off from Large and Nurser [2001]  Shortwave radiation decays exponentially as per Paulson and Simpson
468  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
469  surface salinity values from WOA01 with a relaxation time scale of 3  [2001] is applied and there is a relaxation to the monthly-mean
470  months. Vertical mixing follows Large et al. [1994] with background vertical  climatological sea surface salinity values from WOA01 with a
471  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
472  direct-space-time advection scheme with flux limiter is employed and there is  \citet{large94} with background vertical diffusivity of
473  no explicit horizontal diffusivity. Horizontal viscosity follows Leith [1996]  $1.5\times10^{-5}\text{\,m$^{2}$\,s$^{-1}$}$ and viscosity of
474  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
475  press].  Shortwave radiation decays exponentially as per Paulson and Simpson  advection scheme with flux limiter is employed \citep{hundsdorfer94}
476  [1977].  Additionally, the time-mean runoff of Large and Nurser [2001] is  and there is no explicit horizontal diffusivity. Horizontal viscosity
477  applied near the coastline and, where there is open water, there is a  follows \citet{lei96} but
478  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
479  45 days.  [in press].  Shortwave radiation decays exponentially as per Paulson
480    and Simpson [1977].  Additionally, the time-mean runoff of Large and
481    Nurser [2001] is applied near the coastline and, where there is open
482    water, there is a relaxation to monthly-mean WOA01 sea surface
483    salinity with a time constant of 45 days.
484    
485  Open water, dry  Open water, dry
486  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 509  ice, wet ice, dry snow, and wet snow alb
509  \item C-grid LSR slip  \item C-grid LSR slip
510  \item C-grid EVP no-slip  \item C-grid EVP no-slip
511  \item C-grid EVP slip  \item C-grid EVP slip
512    \item C-grid LSR + TEM (truncated ellipse method, no tensile stress, new flag)
513  \item C-grid LSR no-slip + Winton  \item C-grid LSR no-slip + Winton
514  \item  speed-performance-accuracy (small)  \item  speed-performance-accuracy (small)
515    ice transport through Canadian Archipelago differences    ice transport through Canadian Archipelago differences
# Line 563  We anticipate small differences between Line 521  We anticipate small differences between
521  \begin{itemize}  \begin{itemize}
522  \item advection schemes: along the ice-edge and regions with large  \item advection schemes: along the ice-edge and regions with large
523    gradients    gradients
524  \item C-grid: more transport through narrow straits for no slip  \item C-grid: less transport through narrow straits for no slip
525    conditons, less for free slip    conditons, more for free slip
526  \item VP vs.\ EVP: speed performance, accuracy?  \item VP vs.\ EVP: speed performance, accuracy?
527  \item ocean stress: different water mass properties beneath the ice  \item ocean stress: different water mass properties beneath the ice
528  \end{itemize}  \end{itemize}
# Line 645  storing vs. recomputation of the model s Line 603  storing vs. recomputation of the model s
603  checkpointing loop.  checkpointing loop.
604  Again, an initial code adjustment is required to support TAFs  Again, an initial code adjustment is required to support TAFs
605  checkpointing capability.  checkpointing capability.
606  The code adjustments are sufficiently simply so as not to cause  The code adjustments are sufficiently simple so as not to cause
607  major limitations to the full nonlinear parent model.  major limitations to the full nonlinear parent model.
608  Once in place, an adjoint model of a new model configuration  Once in place, an adjoint model of a new model configuration
609  may be derived in about 10 minutes.  may be derived in about 10 minutes.
# Line 668  may be derived in about 10 minutes. Line 626  may be derived in about 10 minutes.
626  We demonstrate the power of the adjoint method  We demonstrate the power of the adjoint method
627  in the context of investigating sea-ice export sensitivities through Fram Strait  in the context of investigating sea-ice export sensitivities through Fram Strait
628  (for details of this study see Heimbach et al., 2007).  (for details of this study see Heimbach et al., 2007).
629    %\citep[for details of this study see][]{heimbach07}. %Heimbach et al., 2007).
630  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
631  high-resolution cubed-sphere configuration of the ECCO2 project  high-resolution cubed-sphere configuration of the ECCO2 project
632  (see Menemenlis et al. 2005). It covers the entire Arctic,  \citep[see][]{menemenlis05}. It covers the entire Arctic,
633  extends into the North Pacific such as to cover the entire  extends into the North Pacific such as to cover the entire
634  ice-covered regions, and comprises parts of the North Atlantic  ice-covered regions, and comprises parts of the North Atlantic
635  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 640  The adjoint models run efficiently on 80
640  (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
641  IBM SP5 at NASA/ARC).  IBM SP5 at NASA/ARC).
642    
643  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
644  between 1992 and 1995.  years between 1992 and 1995. It is forced using realistic 6-hourly
645  It is forced using realistic 6-hourly NCEP/NCAR atmospheric state variables.  NCEP/NCAR atmospheric state variables. Over the open ocean these are
646  Over the open ocean these are converted into  converted into air-sea fluxes via the bulk formulae of
647  air-sea fluxes via the bulk formulae of Large and Yeager (2004).  \citet{large04}.  Derivation of air-sea fluxes in the presence of
648  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.  
649  The objective function chosen is sea-ice export through Fram Strait  The objective function chosen is sea-ice export through Fram Strait
650  computed for December 1995  computed for December 1995.  The adjoint model computes sensitivities
651  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
652  from 1995 to 1992 along this trajectory.  trajectory.  In principle all adjoint model variable (i.e., Lagrange
653  In principle all adjoint model variable (i.e. Lagrange multipliers)  multipliers) of the coupled ocean/sea-ice model are available to
654  of the coupled ocean/sea-ice model  analyze the transient sensitivity behaviour of the ocean and sea-ice
655  are available to analyze the transient sensitivity behaviour  state.  Over the open ocean, the adjoint of the bulk formula scheme
656  of the ocean and sea-ice state.  computes sensitivities to the time-varying atmospheric state.  Over
657  Over the open ocean, the adjoint of the bulk formula scheme  ice-covered parts, the sea-ice adjoint converts surface ocean
658  computes sensitivities to the time-varying atmospheric state.  sensitivities to atmospheric sensitivities.
659  Over ice-covered parts, the sea-ice adjoint converts  
660  surface ocean sensitivities to atmospheric sensitivities.  \reffig{4yradjheff}(a--d) depict sensitivities of sea-ice export
661    through Fram Strait in December 1995 to changes in sea-ice thickness
662  Fig. XXX(a--d) depict sensitivities of sea-ice export through Fram Strait  12, 24, 36, 48 months back in time. Corresponding sensitivities to
663  in December 1995 to changes in sea-ice thickness  ocean surface temperature are depicted in
664  12, 24, 36, 48 months back in time.  \reffig{4yradjthetalev1}(a--d).  The main characteristics is
665  Corresponding sensitivities to ocean surface temperature are  consistency with expected advection of sea-ice over the relevant time
666  depicted in Fig. XXX(a--d).  scales considered.  The general positive pattern means that an
667  The main characteristics is consistency with expected advection  increase in sea-ice thickness at location $(x,y)$ and time $t$ will
668  of sea-ice over the relevant time scales considered.  increase sea-ice export through Fram Strait at time $T_e$.  Largest
669  The general positive pattern means that an increase in  distances from Fram Strait indicate fastest sea-ice advection over the
670  sea-ice thickness at location $(x,y)$ and time $t$ will increase  time span considered.  The ice thickness sensitivities are in close
671  sea-ice export through Fram Strait at time $T_e$.  correspondence to ocean surface sentivitites, but of opposite sign.
672  Largest distances from Fram Strait indicate fastest sea-ice advection  An increase in temperature will incur ice melting, decrease in ice
673  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 $T_e$.  
674    
675  The picture is fundamentally different and much more complex  The picture is fundamentally different and much more complex
676  for sensitivities to ocean temperatures away from the surface.  for sensitivities to ocean temperatures away from the surface.
677  Fig. XXX (a--d) depicts ice export sensitivities to  \reffig{4yradjthetalev10??}(a--d) depicts ice export sensitivities to
678  temperatures at roughly 400 m depth.  temperatures at roughly 400 m depth.
679  Primary features are the effect of the heat transport of the North  Primary features are the effect of the heat transport of the North
680  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 724  sea-ice thickness at various prior times
724  -48 months}]  -48 months}]
725  {\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}}
726  }  }
727  \caption{Same as Fig. XXX but for sea surface temperature  \caption{Same as \reffig{4yradjheff} but for sea surface temperature
728  \label{fig:4yradjthetalev1}}  \label{fig:4yradjthetalev1}}
729  \end{figure}  \end{figure}
730    
# Line 795  parameters that we use here. What about Line 749  parameters that we use here. What about
749    
750  \paragraph{Acknowledgements}  \paragraph{Acknowledgements}
751  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
752  helpful discussions.  helpful discussions. ML thanks Elizabeth Hunke for multiple explanations.
753    
754  %\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}  
755    
756  \end{document}  \end{document}
757    
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