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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, 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 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 intercative sea ice.  This limits the ability of ECCO 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 ECCO ocean state estimates for describing and  constraints over sea-ice covered regions.  This also limits the usefulness of
17  studying polar-subpolar interactions.  the derived ocean state estimates for describing and studying polar-subpolar
18    interactions.  This paper is a first step towards adding sea-ice capability to
19  The availability of an adjoint model as a powerful research tool  the ECCO estimates.  That is, we describe a dynamic and thermodynamic sea ice
20  complementary to an ocean model was a major design requirement early  model that has been coupled to the MITgcm and that has been modified to permit
21  on in the development of the MIT general circulation model (MITgcm)  efficient and accurate forward integration and automatic differentiation.
22  [Marshall et al. 1997a, Marotzke et al. 1999, Adcroft et al. 2002]. It  
23  was recognized that the adjoint model permitted computing the  Although the ECCO2 optimization problem can be expressed succinctly in
24  gradients of various scalar-valued model diagnostics, norms or,  algebra, its numerical implementation for planetary scale problems is
25  generally, objective functions with respect to external or independent  enormously demanding.  First, multiple forward integrations are required to
26  parameters very efficiently. The information associtated with these  derive approximate filter/smoothers and to compute model Green's functions.
27  gradients is useful in at least two major contexts. First, for state  Second, the derivation of the adjoint model, even with the availability of
28  estimation problems, the objective function is the sum of squared  automatic differentiation tools, is a challenging technical task, which
29  differences between observations and model results weighted by the  requires reformulation of some of the model physics to insure
30  inverse error covariances. The gradient of such an objective function  differentiability and the addition of numerous adjoint compiler directives to
31  can be used to reduce this measure of model-data misfit to find the  improve efficiency \citep{marotzke99}.  The MITgcm adjoint typically requires
32  optimal model solution in a least-squares sense.  Second, the  5--10 times more computations and 10--100 times more storage than the forward
33  objective function can be a key oceanographic quantity such as  model.  Third, every evaluation of the cost function entails a full forward
34  meridional heat or volume transport, ocean heat content or mean  integration of the assimilation model and multiple forwards (and adjoint for
35  surface temperature index. In this case the gradient provides a  the adjoint method) iterations are required to achieve satisfactorily
36  complete set of sensitivities of this quantity to all independent  converged solutions.  Finally, evaluating the cost function also requires
37  variables simultaneously. These sensitivities can be used to address  estimating the error statistics associated with unresolved physics in the
38  the cause of, say, changing net transports accurately.  model and with incompatibilities between observed quantities and numerical
39    model variables.  These statistics are obtained from simulations at even
40  References to existing sea-ice adjoint models, explaining that they are either  higher resolutions than the assimilation model.  For all the above reasons, it
41  for simplified configurations, for ice-only studies, or for short-duration  was decided early on that the MITgcm sea ice model would be tightly coupled
42  studies to motivate the present work.  with the ocean component as opposed to loosely coupled via a flux coupler.
43    
44    
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
48  discretized on Arakawa-B-grids \citep[e.g.,][]{hibler79, harder99,  discretized on Arakawa-B-grids \citep[e.g.,][]{hibler79, harder99,
49    kreyscher00, zhang98, hunke97}\ml{, although there are sea ice only    kreyscher00, zhang98, hunke97}, although there are sea ice models
50    models diretized on a C-grid \citep[e.g.,][]{ip91, tremblay97,  diretized on a C-grid \citep[e.g.,][]{ip91, tremblay97,
51      lemieux09}}. From the perspective of coupling a sea ice-model to a    lemieux09}. %
52  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:
53  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}]}
54  \citep[e.g.,][]{timmermann02a}. However, surface stress is defined at  From the perspective of coupling a sea ice-model to a C-grid ocean
55  velocities points and thus needs to be interpolated between a B-grid  model, the exchange of fluxes of heat and fresh-water pose no
56  sea-ice model and a C-grid ocean model. Smoothing implicitly  difficulty for a B-grid sea-ice model \citep[e.g.,][]{timmermann02a}.
57  associated with this interpolation may mask grid scale noise and may  However, surface stress is defined at velocities points and thus needs
58  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
59  smoothing the stress signals are damped which could lead to reduced  model. Smoothing implicitly associated with this interpolation may
60  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.
61  we circumvent this difficulty altogether and render the stress  On the other hand, by smoothing the stress signals are damped which
62  coupling as consistent as the buoyancy coupling.  could lead to reduced variability of the system. By choosing a C-grid
63    for the sea-ice model, we circumvent this difficulty altogether and
64    render the stress coupling as consistent as the buoyancy coupling.
65    
66  A further advantage of the C-grid formulation is apparent in narrow  A further advantage of the C-grid formulation is apparent in narrow
67  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
68  flux allowed for a B-grid (with no-slip lateral boundary counditions),  flux allowed for a B-grid (with no-slip lateral boundary counditions),
69  and models have used topographies artificially widened straits to  and models have used topographies with artificially widened straits to
70  avoid this problem \citep{holloway07}. The C-grid formulation on the  avoid this problem \citep{holloway07}. The C-grid formulation on the
71  other hand allows a flux of sea-ice through narrow passages if  other hand allows a flux of sea-ice through narrow passages if
72  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 75  in the Candian archipelago.
75  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
76  changes in the code that reduce these sensitivities.  changes in the code that reduce these sensitivities.
77    
78  This paper describes the MITgcm sea ice  This paper describes the MITgcm sea ice model; it presents example
79  model; it presents example Arctic and Antarctic results from a realistic,  Arctic and Antarctic results from a realistic, eddy-permitting, global
80  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
81  and C-grid dynamic solvers in a regional Arctic configuration; and it presents  solvers and investigates further aspects of sea ice modeling in a
82  example results from coupled ocean and sea-ice adjoint-model integrations.  regional Arctic configuration; and it presents example results from
83    coupled ocean and sea-ice adjoint-model integrations.
84    
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