William Carlisle Thacker

An Optimal-Control/Adjoint-Equations Approach to Studying the Oceanic General Circulation

Abstract An efficient procedure is presented for analyzing oceanographic observations with the aid of a general circulation model. Poorly known model parameters, such as eddy-mixing coefficients, surface forcing and tracer boundary fluxes, can be calculated by fitting model results to observations. Optimal estimates for all model fields, including the observed ones, can then be computed by running the model with the best-fit values of the calculated parameters. Information about the resolution and the error-covariances of the model parameters can be computed. This information is shown to be very valuable for critically evaluating how well the data determine the parameters values. An adjoint model, similar in structure to the numerical model, uses information on model-data misfit to improve estimates of the unknown model parameters, and improve the fit to observations. The procedure is illustrated using simulated data and a simple, barotropic, nonlinear, quasi-geostrophic model. Examples are discussed in ...

Oceanic Data Analysis Using a General Circulation Model. Part II: A North Atlantic Model

Abstract A general circulation model and North Atlantic climatological data of temperature salinity, wind stress, evaporation minus precipitation, and air–sea heat fluxes are used to examine the possibility of solving inverse problems using a full-scale numerical GCM and real oceanographic data, combined through an optimization approach. In this study several solutions for the model inputs and the structure of the cost function as a function of the model inputs are examined to demonstrate two of the main difficulties confronting such large-case nonlinear inverse problems (about 30 000 unknowns and a similar number of constraints for the problem examined here). The first is the possible existence of local minima of the cost function, which prevents convergence of the optimization to the global minimum representing the desired optimal solution for the model inputs. The second difficulty, which seems the dominant one for many of the problem examined in this part as well as in Part I, is the ill conditioning ...

Oceanic Data Analysis Using a General Circulation Model. Part I: Simulations

Abstract This paper deals with the solution of inverse problems involving complex numerical models of the oceanic general circulation and large datasets. The goal of these inverse problems is to find values for model inputs consistent with a steady circulation and, at the same time, consistent with the available data. They are formulated as optimization problems, seeking values for the models inputs that minimize a cost function measuring departures from steady state and from date. The two main objectives of this work are 1) to examine the feasibility of solving inverse problems involving a realistic numerical model of the oceanic general circulation and 2) to understand how the optimization uses various data to calculate the desired model parameters. The model considered here is similar to the primitive equation model of Bryan and of Cox, the principal difference being that here the horizontal momentum balance is essentially geostrophic. The models inputs calculated by the optimization consist of surfa...

Data assimilation into a numerical equatorial ocean model. II. Assimilation experiments

Abstract A sequence of numerical experiments is conducted using a linear, semi-spectral equatorial ocean model and an advanced data assimilation scheme. The numerical model is based on decomposition of the oceanic fields into Kelvin and Rossby waves belonging to the baroclinic modes of a stratified equatorial ocean. The assimilation procedure finds that solution to the model equations that best fits, in the generalized least-squares sense, all observations made within some specified space-time interval. All experiments are of the ‘identical twin’ type; synthetic data are generated by sampling the observable fields produced by a control run of the model, then the data are assimilated using the same model. The sequence of numerical experiments serves two purposes; to demonstrate the performance of the assimilation procedure in the context of a fully three-dimensional, time-varying equatorial ocean model; and to examine the utility of specified data sets, in particular, observations of sea level, in estimating the state of the equatorial ocean. The results indicate that the assimilation procedure works very well when sufficient data are provided. However, sea-level data alone are not sufficient and must be supplemented with subsurface observations if more than a few baroclinic modes are allowed in the model ocean. The required amount of supplementary subsurface data (in the form of density profiles in these experiments) can be reduced by imposing smoothness contraints on the recovered model solution.

Combining data and a global primitive equation ocean general circulation model using the adjoint method

Abstract A Primitive Equation Ocean General Circulation Model (PE OGCM) in a global configuration similar to that used in coupled ocean-atmosphere models is fitted to climatological data using the adjoint method. The ultimate objective is the use of data assimilation for the improvement of the ocean component of coupled models, and for the calculation of initial conditions for initializing coupled model integrations. It is argued that oceanic models that are used for coupled climate studies are an especially appropriate target for data assimilation using the adjoint method. It is demonstrated that a successful assimilating of data into a fully complex PE OGCM critically depends on a very careful choice of the surface boundary condition formulation, on the optimization problem formulation, and on the initial guess for the optimization solution. The use of restoring rather than fixed surface-flux boundary conditions for the temperature seems to result in significantly improved model results as compared with previous studies using fixed surface-flux boundary conditions. The convergence of the optimization seems very sensitive to the cost formulation in a PE model, and a successful cost formulation is discussed and demonstrated. Finally, the use of simple, suboptimal, assimilation schemes for obtaining an initial guess for the adjoint optimization is advocated and demonstrated.

Data-model-error compatibility

Abstract During data assimilation, differences between observations and their model counterparts should be consistent with the error statistics that govern how the model is to be corrected. The concept of incompatibility distance between observations and their model counterparts is introduced as a way of detecting inconsistencies, and formulae are presented for estimating the probability of encountering greater incompatibility. Observations can be examined one-by-one to insure that their confidence intervals are not widely separated from those of the model counterparts. They can be further examined in pairs to detect whether contrasts across fronts are consistent with assumptions about error correlations.

Computing the steady oceanic circulation using an optimization approach

Abstract The traditional method for computing the steady oceanic circulation has been by stepping an oceanic model forward in time until transients are damped by friction. An alternative method, which has the potential for being more economical is to minimize the sum of the squares of the residuals of the steady model equations. A variety of algorithms might be considered for computing the minimum; attention here is focused on preconditioned conjugate-gradient descent with the gradient computed using an adjoint model. The choice of varibles, i.e. the preconditioning transformation used in the optimization process, is found to be critical to the efficiency of the method. An appropriate preconditioning transformation can be suggested by a heuristic analysis similar to that commonly used to test the stability of numerical models. The method is demonstrated within the context of the barotropic vorticity equation.

Can Hydrographic Data Provide Evidence That the Rate of Oceanic Uptake of Anthropogenic CO2 Is Increasing

Predictions of the rate of accumulation of anthropogenic carbon dioxide in the Pacific Ocean near 32°S and 150°W based on the P16 surveys of 1991 and 2005 and on the P06 surveys of 1992 and 2003 underestimate the amount found in the P06 survey of 2009–2010, suggesting an increasing uptake rate. Assuming the accumulation rate to be constant over the two decades, analyses using all five surveys lead to upward revision of the rates based only on the first four. On the other hand, accumulation rates estimated for 2003–2010 are significantly greater than those for 1991–2003, again suggesting an increasing uptake rate. In addressing this question it is important to acknowledge the limitations of the repeat hydrography and consequent uncertainties of estimated accumulation rates.