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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  In recent years, oceanographic state estimation has matured to the
5  extent that estimates of the evolving circulation of the ocean constrained by  extent that estimates of the evolving circulation of the ocean constrained by
6  in-situ and remotely sensed global observations are now routinely available  in-situ and remotely sensed global observations are now routinely available
7  and being applied to myriad scientific problems \citep{wun07}.  Ocean state  and being applied to myriad scientific problems \citep{wun07}.  Ocean state
8  estimation is the process of fitting an ocean general circulation model (GCM)  estimation is the process of fitting an ocean General Circulation Model (GCM)
9  to a multitude of observations.  As formulated by the consortium Estimating  to a multitude of observations.  As formulated by the consortium for Estimating
10  the Circulation and Climate of the Ocean (ECCO), an automatic differentiation  the Circulation and Climate of the Ocean (ECCO), an automatic differentiation
11  tool is used to calculate the so-called adjoint code of a GCM.  The method of  tool is used to calculate the so-called adjoint code of a GCM.  The method of
12  Lagrange multipliers is then used to render the problem one of unconstrained  Lagrange multipliers is then used to render the problem one of unconstrained
13  least-squares minimization.  Although much has been achieved, the existing  least-squares minimization.  Although much has been achieved, the existing
14  ECCO estimates lack intercative sea ice.  This limits the ability of ECCO to  ECCO estimates lack interactive sea ice.  This limits the ability to
15  utilize satellite data constraints over sea-ice covered regions.  This also  utilize satellite data constraints over sea-ice covered regions.  This also
16  limits the usefulness of the ECCO ocean state estimates for describing and  limits the usefulness of the derived ocean state estimates for describing and
17  studying polar-subpolar interactions.  studying polar-subpolar interactions.  This paper is a first step towards
18    adding sea-ice capability to the ECCO estimates.  That is, we describe a
19    dynamic and thermodynamic sea ice model that has been coupled to the
20    Massachusetts Institute of Technology general circulation model
21    \citep[MITgcm][]{mar97a} and that has been modified to permit efficient and
22    accurate automatic differentiation.
23    
24  The availability of an adjoint model as a powerful research tool  The availability of an adjoint model as a powerful research tool
25  complementary to an ocean model was a major design requirement early  complementary to an ocean model was a major design requirement early
26  on in the development of the MIT general circulation model (MITgcm)  on in the development of the MITgcm \citep{marotzke99}. It
 [Marshall et al. 1997a, Marotzke et al. 1999, Adcroft et al. 2002]. It  
27  was recognized that the adjoint model permitted computing the  was recognized that the adjoint model permitted computing the
28  gradients of various scalar-valued model diagnostics, norms or,  gradients of various scalar-valued model diagnostics, norms or,
29  generally, objective functions with respect to external or independent  generally, objective functions with respect to external or independent
# Line 44  studies to motivate the present work. Line 48  studies to motivate the present work.
48  Traditionally, probably for historical reasons and the ease of  Traditionally, probably for historical reasons and the ease of
49  treating the Coriolis term, most standard sea-ice models are  treating the Coriolis term, most standard sea-ice models are
50  discretized on Arakawa-B-grids \citep[e.g.,][]{hibler79, harder99,  discretized on Arakawa-B-grids \citep[e.g.,][]{hibler79, harder99,
51    kreyscher00, zhang98, hunke97}\ml{, although there are sea ice only    kreyscher00, zhang98, hunke97}, although there are sea ice models
52    models diretized on a C-grid \citep[e.g.,][]{ip91, tremblay97,  diretized on a C-grid \citep[e.g.,][]{ip91, tremblay97,
53      lemieux09}}. From the perspective of coupling a sea ice-model to a    lemieux09}. %
54  C-grid ocean model, the exchange of fluxes of heat and fresh-water  \ml{[there is also MI-IM, but I only found this as a reference:
55  pose no difficulty for a B-grid sea-ice model    \url{http://retro.met.no/english/r_and_d_activities/method/num_mod/MI-IM-Documentation.pdf}]}
56  \citep[e.g.,][]{timmermann02a}. However, surface stress is defined at  From the perspective of coupling a sea ice-model to a C-grid ocean
57  velocities points and thus needs to be interpolated between a B-grid  model, the exchange of fluxes of heat and fresh-water pose no
58  sea-ice model and a C-grid ocean model. Smoothing implicitly  difficulty for a B-grid sea-ice model \citep[e.g.,][]{timmermann02a}.
59  associated with this interpolation may mask grid scale noise and may  However, surface stress is defined at velocities points and thus needs
60  contribute to stabilizing the solution. On the other hand, by  to be interpolated between a B-grid sea-ice model and a C-grid ocean
61  smoothing the stress signals are damped which could lead to reduced  model. Smoothing implicitly associated with this interpolation may
62  variability of the system. By choosing a C-grid for the sea-ice model,  mask grid scale noise and may contribute to stabilizing the solution.
63  we circumvent this difficulty altogether and render the stress  On the other hand, by smoothing the stress signals are damped which
64  coupling as consistent as the buoyancy coupling.  could lead to reduced variability of the system. By choosing a C-grid
65    for the sea-ice model, we circumvent this difficulty altogether and
66    render the stress coupling as consistent as the buoyancy coupling.
67    
68  A further advantage of the C-grid formulation is apparent in narrow  A further advantage of the C-grid formulation is apparent in narrow
69  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
70  flux allowed for a B-grid (with no-slip lateral boundary counditions),  flux allowed for a B-grid (with no-slip lateral boundary counditions),
71  and models have used topographies artificially widened straits to  and models have used topographies with artificially widened straits to
72  avoid this problem \citep{holloway07}. The C-grid formulation on the  avoid this problem \citep{holloway07}. The C-grid formulation on the
73  other hand allows a flux of sea-ice through narrow passages if  other hand allows a flux of sea-ice through narrow passages if
74  free-slip along the boundaries is allowed. We demonstrate this effect  free-slip along the boundaries is allowed. We demonstrate this effect
# Line 71  in the Candian archipelago. Line 77  in the Candian archipelago.
77  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
78  changes in the code that reduce these sensitivities.  changes in the code that reduce these sensitivities.
79    
80  This paper describes the MITgcm sea ice  This paper describes the MITgcm sea ice model; it presents example
81  model; it presents example Arctic and Antarctic results from a realistic,  Arctic and Antarctic results from a realistic, eddy-permitting, global
82  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
83  and C-grid dynamic solvers in a regional Arctic configuration; and it presents  solvers and investigates further aspects of sea ice modeling in a
84  example results from coupled ocean and sea-ice adjoint-model integrations.  regional Arctic configuration; and it presents example results from
85    coupled ocean and sea-ice adjoint-model integrations.
86    
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