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# Line 52  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 74  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 82  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 158  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    
# Line 182  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 210  $\sigma_{12}$. 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 240  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 250  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 311  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 have 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 EVP velocity  
 fields 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 by variable  
 atmospheric wind whose mean direction is diagnonal to the north-east  
 corner of the domain; ice volume and concentration are held constant  
 (no thermodynamics and 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{Ice flow, divergence and bulk viscosities of three  
     experiments with \citet{hunke01}'s test case: C-LSRns (top),  
     C-EVPns (middle), and C-EVPns with damping described in  
     \citet{hunke01} (bottom).}  
   \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 confined 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 $\nabla\cdot\vek{u}$ and  
 $\log_{10}\zeta$ 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).  
   
 \ml{[Say something about performance? This is tricky, as the  
   perfomance depends strongly on the configuration. A run with slowly  
   changing forcing is favorable for LSR, because then only very few  
   iterations are required for convergences while EVP uses its fixed  
   number of internal timesteps. If the forcing in changing fast, LSR  
   needs far more iterations while EVP still uses the fixed number of  
   internal timesteps. I have produces runs where for slow forcing LSR  
   is much faster than EVP and for fast forcing, LSR is much slower  
   than EVP. EVP is certainly more efficient in terms of vectorization  
   and MFLOPS on our SX8, but is that a criterion?]}  
370    
371  \subsection{C-grid}  \subsection{C-grid}
372  \begin{itemize}  \begin{itemize}
# Line 493  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  The Arctic domain of integration is illustrated in Fig.~\ref{fig:arctic1}.  It
417  is carved out from, and obtains open boundary conditions from, the  is carved out from, and obtains open boundary conditions from, the
418  global cubed-sphere configuration of the Estimating the Circulation  global cubed-sphere configuration of the Estimating the Circulation
419  and Climate of the Ocean, Phase II (ECCO2) project  and Climate of the Ocean, Phase II (ECCO2) project
420  \citet{menemenlis05}.  The domain size is 420 by 384 grid boxes  \citet{menemenlis05}.  The domain size is 420 by 384 grid boxes
421  horizontally with mean horizontal grid spacing of 18 km.  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
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