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1  \section{Introduction}  \section{Introduction}
2  \label{sec:intro}  \label{sec:intro}
3    
4  In recent years, oceanographic state estimation has matured to the  In recent years, ocean state estimation has matured to the extent that
5  extent that estimates of the evolving circulation of the ocean constrained by  estimates of the time-evolving ocean circulation, constrained by a multitude
6  in-situ and remotely sensed global observations are now routinely available  of in-situ and remotely sensed global observations, are now routinely
7  and being applied to myriad scientific problems \citep{wun07}.  Ocean state  available and being applied to myriad scientific problems \citep[and
8  estimation is the process of fitting an ocean General Circulation Model (GCM)  references therein]{wun07}.  As formulated by the consortium for Estimating
9  to a multitude of observations.  As formulated by the consortium for Estimating  the Circulation and Climate of the Ocean (ECCO), least-squares methods, i.e.,
10  the Circulation and Climate of the Ocean (ECCO), an automatic differentiation  filter/smoother \citep{fuk02}, Green's functions \citep{men05}, and adjoint
11  tool is used to calculate the so-called adjoint code of a GCM.  The method of  \citep{sta02a}, are used to fit the Massachusetts Institute of Technology
12  Lagrange multipliers is then used to render the problem one of unconstrained  general circulation model
13  least-squares minimization.  Although much has been achieved, the existing  \citep[MITgcm;][]{marshall97:_finit_volum_incom_navier_stokes} to the
14  ECCO estimates lack interactive sea ice.  This limits the ability to  available data.  Much has been achieved but the existing ECCO estimates lack
15  utilize satellite data constraints over sea-ice covered regions.  This also  interactive sea ice.  This limits the ability to utilize satellite data
16  limits the usefulness of the derived ocean state estimates for describing and  constraints over sea-ice covered regions.  This also limits the usefulness of
17  studying polar-subpolar interactions.  This paper is a first step towards  the derived ocean state estimates for describing and studying polar-subpolar
18  adding sea-ice capability to the ECCO estimates.  That is, we describe a  interactions.  This paper is a first step towards adding sea-ice capability to
19  dynamic and thermodynamic sea ice model that has been coupled to the  the ECCO estimates.  That is, we describe a dynamic and thermodynamic sea ice
20  Massachusetts Institute of Technology general circulation model  model that has been coupled to the MITgcm and that has been modified to permit
21  \citep[MITgcm][]{mar97a} and that has been modified to permit efficient and  efficient and accurate forward integration and automatic differentiation.
22  accurate automatic differentiation.  
23    Although the ECCO2 optimization problem can be expressed succinctly in
24    algebra, its numerical implementation for planetary scale problems is
25    enormously demanding.  First, multiple forward integrations are required to
26    derive approximate filter/smoothers and to compute model Green's functions.
27    Second, the derivation of the adjoint model, even with the availability of
28    automatic differentiation tools, is a challenging technical task, which
29    requires reformulation of some of the model physics to insure
30    differentiability and the addition of numerous adjoint compiler directives to
31    improve efficiency \citep{marotzke99}.  The MITgcm adjoint typically requires
32    5--10 times more computations and 10--100 times more storage than the forward
33    model.  Third, every evaluation of the cost function entails a full forward
34    integration of the assimilation model and multiple forwards (and adjoint for
35    the adjoint method) iterations are required to achieve satisfactorily
36    converged solutions.  Finally, evaluating the cost function also requires
37    estimating the error statistics associated with unresolved physics in the
38    model and with incompatibilities between observed quantities and numerical
39    model variables.  These statistics are obtained from simulations at even
40    higher resolutions than the assimilation model.  For all the above reasons, it
41    was decided early on that the MITgcm sea ice model would be tightly coupled
42    with the ocean component as opposed to loosely coupled via a flux coupler.
43    
 The availability of an adjoint model as a powerful research tool  
 complementary to an ocean model was a major design requirement early  
 on in the development of the MITgcm \citep{marotzke99}. It  
 was recognized that the adjoint model permitted computing the  
 gradients of various scalar-valued model diagnostics, norms or,  
 generally, objective functions with respect to external or independent  
 parameters very efficiently. The information associtated with these  
 gradients is useful in at least two major contexts. First, for state  
 estimation problems, the objective function is the sum of squared  
 differences between observations and model results weighted by the  
 inverse error covariances. The gradient of such an objective function  
 can be used to reduce this measure of model-data misfit to find the  
 optimal model solution in a least-squares sense.  Second, the  
 objective function can be a key oceanographic quantity such as  
 meridional heat or volume transport, ocean heat content or mean  
 surface temperature index. In this case the gradient provides a  
 complete set of sensitivities of this quantity to all independent  
 variables simultaneously. These sensitivities can be used to address  
 the cause of, say, changing net transports accurately.  
44    
 References to existing sea-ice adjoint models, explaining that they are either  
 for simplified configurations, for ice-only studies, or for short-duration  
 studies to motivate the present work.  
45    
46  Traditionally, probably for historical reasons and the ease of  Traditionally, probably for historical reasons and the ease of
47  treating the Coriolis term, most standard sea-ice models are  treating the Coriolis term, most standard sea-ice models are
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