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\section{Introduction} |
\section{Introduction} |
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\label{sec:intro} |
\label{sec:intro} |
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In the past five years, oceanographic state estimation has matured to the |
In recent years, ocean state estimation has matured to the extent that |
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extent that estimates of the evolving circulation of the ocean constrained by |
estimates of the time-evolving ocean circulation, constrained by a multitude |
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in-situ and remotely sensed global observations are now routinely available |
of in-situ and remotely sensed global observations, are now routinely |
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and being applied to myriad scientific problems \citep{wun07}. Ocean state |
available and being applied to myriad scientific problems \citep[and |
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estimation is the process of fitting an ocean general circulation model (GCM) |
references therein]{wun07}. As formulated by the consortium for Estimating |
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to a multitude of observations. As formulated by the consortium Estimating |
the Circulation and Climate of the Ocean (ECCO), least-squares methods, i.e., |
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the Circulation and Climate of the Ocean (ECCO), an automatic differentiation |
filter/smoother \citep{fuk02}, Green's functions \citep{men05}, and adjoint |
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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 |
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Lagrange multipliers is then used to render the problem one of unconstrained |
general circulation model |
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least-squares minimization. Although much has been achieved, the existing |
\citep[MITgcm;][]{marshall97:_finit_volum_incom_navier_stokes} to the |
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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 |
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utilize satellite data constraints over sea-ice covered regions. This also |
interactive sea ice. This limits the ability to utilize satellite data |
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limits the usefulness of the ECCO ocean state estimates for describing and |
constraints over sea-ice covered regions. This also limits the usefulness of |
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studying polar-subpolar interactions. |
the derived ocean state estimates for describing and studying polar-subpolar |
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interactions. This paper is a first step towards adding sea-ice capability to |
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the ECCO estimates. That is, we describe a dynamic and thermodynamic sea ice |
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model that has been coupled to the MITgcm and that has been modified to permit |
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efficient and accurate forward integration and automatic differentiation. |
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Although the ECCO2 optimization problem can be expressed succinctly in |
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algebra, its numerical implementation for planetary scale problems is |
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enormously demanding. First, multiple forward integrations are required to |
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derive approximate filter/smoothers and to compute model Green's functions. |
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Second, the derivation of the adjoint model, even with the availability of |
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automatic differentiation tools, is a challenging technical task, which |
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requires reformulation of some of the model physics to insure |
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differentiability and the addition of numerous adjoint compiler directives to |
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improve efficiency \citep{marotzke99}. The MITgcm adjoint typically requires |
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5--10 times more computations and 10--100 times more storage than the forward |
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model. Third, every evaluation of the cost function entails a full forward |
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integration of the assimilation model and multiple forwards (and adjoint for |
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the adjoint method) iterations are required to achieve satisfactorily |
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converged solutions. Finally, evaluating the cost function also requires |
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estimating the error statistics associated with unresolved physics in the |
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model and with incompatibilities between observed quantities and numerical |
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model variables. These statistics are obtained from simulations at even |
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higher resolutions than the assimilation model. For all the above reasons, it |
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was decided early on that the MITgcm sea ice model would be tightly coupled |
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with the ocean component as opposed to loosely coupled via a flux coupler. |
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The availability of an adjoint model as a powerful research tool |
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complementary to an ocean model was a major design requirement early |
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on in the development of the MIT general circulation model (MITgcm) |
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[Marshall et al. 1997a, Marotzke et al. 1999, Adcroft et al. 2002]. It |
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was recognized that the adjoint model permitted computing the |
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gradients of various scalar-valued model diagnostics, norms or, |
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generally, objective functions with respect to external or independent |
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parameters very efficiently. The information associtated with these |
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gradients is useful in at least two major contexts. First, for state |
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estimation problems, the objective function is the sum of squared |
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differences between observations and model results weighted by the |
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inverse error covariances. The gradient of such an objective function |
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can be used to reduce this measure of model-data misfit to find the |
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optimal model solution in a least-squares sense. Second, the |
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objective function can be a key oceanographic quantity such as |
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meridional heat or volume transport, ocean heat content or mean |
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surface temperature index. In this case the gradient provides a |
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complete set of sensitivities of this quantity to all independent |
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variables simultaneously. These sensitivities can be used to address |
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the cause of, say, changing net transports accurately. |
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References to existing sea-ice adjoint models, explaining that they are either |
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for simplified configurations, for ice-only studies, or for short-duration |
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studies to motivate the present work. |
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Traditionally, probably for historical reasons and the ease of |
Traditionally, probably for historical reasons and the ease of |
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treating the Coriolis term, most standard sea-ice models are |
treating the Coriolis term, most standard sea-ice models are |
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