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1  \section{Introduction}  \section{Introduction}
2  \label{sec:intro}  \label{sec:intro}
3    
4  The availability of an adjoint model as a powerful research tool  In recent years, ocean state estimation has matured to the extent that
5  complementary to an ocean model was a major design requirement early  estimates of the time-evolving ocean circulation, constrained by a multitude
6  on in the development of the MIT general circulation model (MITgcm)  of in-situ and remotely sensed global observations, are now routinely
7  [Marshall et al. 1997a, Marotzke et al. 1999, Adcroft et al. 2002]. It  available and being applied to myriad scientific problems \citep[and
8  was recognized that the adjoint model permitted computing the  references therein]{wun07}.  As formulated by the consortium for Estimating
9  gradients of various scalar-valued model diagnostics, norms or,  the Circulation and Climate of the Ocean (ECCO), least-squares methods, i.e.,
10  generally, objective functions with respect to external or independent  filter/smoother \citep{fuk02}, Green's functions \citep{men05}, and adjoint
11  parameters very efficiently. The information associtated with these  \citep{sta02a}, are used to fit the Massachusetts Institute of Technology
12  gradients is useful in at least two major contexts. First, for state  general circulation model
13  estimation problems, the objective function is the sum of squared  \citep[MITgcm;][]{marshall97:_finit_volum_incom_navier_stokes} to the
14  differences between observations and model results weighted by the  available data.  Much has been achieved but the existing ECCO estimates lack
15  inverse error covariances. The gradient of such an objective function  interactive sea ice.  This limits the ability to utilize satellite data
16  can be used to reduce this measure of model-data misfit to find the  constraints over sea-ice covered regions.  This also limits the usefulness of
17  optimal model solution in a least-squares sense.  Second, the  the derived ocean state estimates for describing and studying polar-subpolar
18  objective function can be a key oceanographic quantity such as  interactions.  This paper is a first step towards adding sea-ice capability to
19  meridional heat or volume transport, ocean heat content or mean  the ECCO estimates.  That is, we describe a dynamic and thermodynamic sea ice
20  surface temperature index. In this case the gradient provides a  model that has been coupled to the MITgcm and that has been modified to permit
21  complete set of sensitivities of this quantity to all independent  efficient and accurate forward integration and automatic differentiation.
22  variables simultaneously. These sensitivities can be used to address  
23  the cause of, say, changing net transports accurately.  Although the ECCO2 optimization problem can be expressed succinctly in
24    algebra, its numerical implementation for planetary scale problems is
25  References to existing sea-ice adjoint models, explaining that they are either  enormously demanding.  First, multiple forward integrations are required to
26  for simplified configurations, for ice-only studies, or for short-duration  derive approximate filter/smoothers and to compute model Green's functions.
27  studies to motivate the present work.  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    
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}. From the perspective of coupling a    kreyscher00, zhang98, hunke97}, although there are sea ice models
50  sea ice-model to a C-grid ocean model, the exchange of fluxes of heat  diretized on a C-grid \citep[e.g.,][]{ip91, tremblay97,
51  and fresh-water pose no difficulty for a B-grid sea-ice model    lemieux09}. %
52  \citep[e.g.,][]{timmermann02a}. However, surface stress is defined at  \ml{[there is also MI-IM, but I only found this as a reference:
53  velocities points and thus needs to be interpolated between a B-grid    \url{http://retro.met.no/english/r_and_d_activities/method/num_mod/MI-IM-Documentation.pdf}]}
54  sea-ice model and a C-grid ocean model. Smoothing implicitly  From the perspective of coupling a sea ice-model to a C-grid ocean
55  associated with this interpolation may mask grid scale noise and may  model, the exchange of fluxes of heat and fresh-water pose no
56  contribute to stabilizing the solution. On the other hand, by  difficulty for a B-grid sea-ice model \citep[e.g.,][]{timmermann02a}.
57  smoothing the stress signals are damped which could lead to reduced  However, surface stress is defined at velocities points and thus needs
58  variability of the system. By choosing a C-grid for the sea-ice model,  to be interpolated between a B-grid sea-ice model and a C-grid ocean
59  we circumvent this difficulty altogether and render the stress  model. Smoothing implicitly associated with this interpolation may
60  coupling as consistent as the buoyancy coupling.  mask grid scale noise and may contribute to stabilizing the solution.
61    On the other hand, by smoothing the stress signals are damped which
62    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 54  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    
85    %%% Local Variables:
86    %%% mode: latex
87    %%% TeX-master: "ceaice"
88    %%% End:
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