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1  \section{Introduction}  \section{Introduction}
2  \label{sec:intro}  \label{sec:intro}
3    
4  In the past five years, oceanographic state estimation has matured to the  Ocean state estimation has matured to the extent that estimates of the
5  extent that estimates of the evolving circulation of the ocean constrained by  time-evolving ocean circulation, constrained by a multitude of in-situ and
6  in-situ and remotely sensed global observations are now routinely available  remotely sensed global observations, are now routinely available and being
7  and being applied to myriad scientific problems \citep{wun07}.  Ocean state  applied to myriad scientific problems \citep[and references therein]{wun07}.
8  estimation is the process of fitting an ocean general circulation model (GCM)  As formulated by the consortium for Estimating the Circulation and Climate of
9  to a multitude of observations.  As formulated by the consortium Estimating  the Ocean (ECCO), least-squares methods are used to fit the Massachusetts
10  the Circulation and Climate of the Ocean (ECCO), an automatic differentiation  Institute of Technology general circulation model \citep[MITgcm;][]{mar97a} to
11  tool is used to calculate the so-called adjoint code of a GCM.  The method of  the available data.  Much has been achieved but the existing ECCO estimates
12  Lagrange multipliers is then used to render the problem one of unconstrained  lack interactive sea ice.  This limits the ability to utilize satellite data
13  least-squares minimization.  Although much has been achieved, the existing  constraints over sea-ice covered regions.  This also limits the usefulness of
14  ECCO estimates lack intercative sea ice.  This limits the ability of ECCO to  the derived ocean state estimates for describing and studying polar-subpolar
15  utilize satellite data constraints over sea-ice covered regions.  This also  interactions.  In this paper we describe a dynamic and thermodynamic sea ice
16  limits the usefulness of the ECCO ocean state estimates for describing and  model that has been coupled to the MITgcm and that has been modified to permit
17  studying polar-subpolar interactions.  efficient and accurate forward and adjoint integration.  The forward model
18    borrows many components from current-generation sea ice models but these
19  The availability of an adjoint model as a powerful research tool  components are reformulated on an Arakawa C grid in order to match the MITgcm
20  complementary to an ocean model was a major design requirement early  oceanic grid and they are modified in many ways to permit efficient and
21  on in the development of the MIT general circulation model (MITgcm)  accurate automatic differentiation.  To illustrate how the use of the forward and
22  [Marshall et al. 1997a, Marotzke et al. 1999, Adcroft et al. 2002]. It  adjoint parts together can help give insight into discrete model dynamics, we
23  was recognized that the adjoint model permitted computing the  study the interaction between littoral regions in the Canadian Arctic
24  gradients of various scalar-valued model diagnostics, norms or,  Archipelago and sea-ice model dynamics.
25  generally, objective functions with respect to external or independent  
26  parameters very efficiently. The information associtated with these  Because early numerical ocean models were formulated on the Arakawa-B grid and
27  gradients is useful in at least two major contexts. First, for state  because of the easier treatment of the Coriolis term, most standard sea-ice
28  estimation problems, the objective function is the sum of squared  models are discretized on Arakawa-B grids \citep[e.g.,][]{hibler79, harder99,
29  differences between observations and model results weighted by the    kreyscher00, zhang98, hunke97}.  As model resolution increases, more and
30  inverse error covariances. The gradient of such an objective function  more ocean and sea ice models are being formulated on the Arakawa-C grid
31  can be used to reduce this measure of model-data misfit to find the  \citep[e.g.,][]{mar97a,ip91,tremblay97,lemieux09}.
32  optimal model solution in a least-squares sense.  Second, the  %\ml{[there is also MI-IM, but I only found this as a reference:
33  objective function can be a key oceanographic quantity such as  %  \url{http://retro.met.no/english/r_and_d_activities/method/num_mod/MI-IM-Documentation.pdf}]}
34  meridional heat or volume transport, ocean heat content or mean  From the perspective of coupling a sea ice-model to a C-grid ocean model, the
35  surface temperature index. In this case the gradient provides a  exchange of fluxes of heat and fresh-water pose no difficulty for a B-grid
36  complete set of sensitivities of this quantity to all independent  sea-ice model \citep[e.g.,][]{timmermann02a}.  However, surface stress is
37  variables simultaneously. These sensitivities can be used to address  defined at velocities points and thus needs to be interpolated between a
38  the cause of, say, changing net transports accurately.  B-grid sea-ice model and a C-grid ocean model. Smoothing implicitly associated
39    with this interpolation may mask grid scale noise and may contribute to
40  References to existing sea-ice adjoint models, explaining that they are either  stabilizing the solution.  On the other hand, by smoothing the stress signals
41  for simplified configurations, for ice-only studies, or for short-duration  are damped which could lead to reduced variability of the system. By choosing
42  studies to motivate the present work.  a C-grid for the sea-ice model, we circumvent this difficulty altogether and
43    render the stress coupling as consistent as the buoyancy coupling.
 Traditionally, probably for historical reasons and the ease of  
 treating the Coriolis term, most standard sea-ice models are  
 discretized on Arakawa-B-grids \citep[e.g.,][]{hibler79, harder99,  
   kreyscher00, zhang98, hunke97}\ml{, although there are sea ice only  
   models diretized on a C-grid \citep[e.g.,][]{tremblay97,  
     lemieux09}}. From the perspective of coupling a sea ice-model to a  
 C-grid ocean model, the exchange of fluxes of heat and fresh-water  
 pose no difficulty for a B-grid sea-ice model  
 \citep[e.g.,][]{timmermann02a}. However, surface stress is defined at  
 velocities points and thus needs to be interpolated between a B-grid  
 sea-ice model and a C-grid ocean model. Smoothing implicitly  
 associated with this interpolation may mask grid scale noise and may  
 contribute to stabilizing the solution. On the other hand, by  
 smoothing the stress signals are damped which could lead to reduced  
 variability of the system. By choosing a C-grid for the sea-ice model,  
 we circumvent this difficulty altogether and render the stress  
 coupling as consistent as the buoyancy coupling.  
44    
45  A further advantage of the C-grid formulation is apparent in narrow  A further advantage of the C-grid formulation is apparent in narrow
46  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
47  flux allowed for a B-grid (with no-slip lateral boundary counditions),  flux allowed for a B-grid (with no-slip lateral boundary counditions),
48  and models have used topographies artificially widened straits to  and models have used topographies with artificially widened straits to
49  avoid this problem \citep{holloway07}. The C-grid formulation on the  avoid this problem \citep{holloway07}. The C-grid formulation on the
50  other hand allows a flux of sea-ice through narrow passages if  other hand allows a flux of sea-ice through narrow passages if
51  free-slip along the boundaries is allowed. We demonstrate this effect  free-slip along the boundaries is allowed. We demonstrate this effect
52  in the Candian archipelago.  in the Candian Arctic Archipelago (CAA).
53    
54  Talk about problems that make the sea-ice-ocean code very sensitive and  Talk about problems that make the sea-ice-ocean code very sensitive and
55  changes in the code that reduce these sensitivities.  changes in the code that reduce these sensitivities.
56    
57  This paper describes the MITgcm sea ice  This paper describes the MITgcm sea ice model; it presents example
58  model; it presents example Arctic and Antarctic results from a realistic,  Arctic and Antarctic results from a realistic, eddy-permitting, global
59  eddy-permitting, global ocean and sea-ice configuration; it compares B-grid  ocean and sea-ice configuration; it compares B-grid and C-grid dynamic
60  and C-grid dynamic solvers in a regional Arctic configuration; and it presents  solvers and investigates further aspects of sea ice modeling in a
61  example results from coupled ocean and sea-ice adjoint-model integrations.  regional Arctic configuration; and it presents example results from
62    coupled ocean and sea-ice adjoint-model integrations.
63    
64  %%% Local Variables:  %%% Local Variables:
65  %%% mode: latex  %%% mode: latex
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