Adjoint Data Assimilation in Coupled Atmosphere-Ocean Models

Adjoint data assimilation has become increasingly popular in recent years (Daley, 1991; Errico, 1997), while the importance of coupled atmosphere-ocean models is also being recognized (Zebiak and Cane, 1987; Ji et al., 1994). We have recently completed a trilogy of papers on applying the adjoint data assimilation method to coupled atmosphere-ocean models (Lu and Hsieh, 1997, 1998a, 1998b; henceforth LH1, LH2, LH3 respectively).

What is complex about applying the adjoint method to coupled models is that the atmosphere and the ocean are two distinct fluids, with very different time scales, thereby causing problems in designing an assimilation procedure - e.g., an optimal window for atmospheric data assimilation may not be at all optimal for the oceanic data. Also intriguing is the fact that data assimilated into either the atmospheric part or the oceanic part would exert influence on both parts of the coupled system. Would the influence on the oceanic part exerted by the assimilated atmospheric data be greater than the influence exerted by the assimilated oceanic data? Clearly many complex and intriguing questions arise when the adjoint assimilation method is applied to coupled models.

In a coupled model, there are many parameters, e.g., coupling parameters and damping parameters, which are not precisely known. LH1 examined the potential of the adjoint method in determining parameters in the coupled model by assimilating atmospheric and oceanic data. In forecasting, one needs to estimate the initial conditions from the assimilated data. LH2 used the adjoint method to optimally estimate the initial conditions in a coupled model, while assuming the model parameters were known. Unifying the two studies, LH3 examined the optimal estimation of initial conditions and model parameters together in coupled models by the adjoint data assimilation method.

While the general mathematical formulation for adjoint data assimilation in coupled models was derived in LH1, LH2 and LH3, the method was illustrated with a simple equatorial coupled model, namely that of Philander et al. (1984), where the atmosphere and the ocean were each represented by a 1-layer shallow water equation model. A small perturbation in the equatorial ocean would then grow from the atmosphere-ocean coupling. With this simple model, a series of identical twin experiments were run, where one numerical experiment (the control) would represent the real coupled system, and the other, a data assimilating numerical model of the coupled system, with the ``data'' obtained from the control. By varying the amount of available data, and the quality of data (by adding noise), we could determine what was required for accurate estimation of model parameters and/or initial conditions. We also tested how well the TOGA TAO array could help guide an equatorial coupled model.

An interesting finding was that with noisy data, the estimation of the initial conditions was far more sensitive to noise in the oceanic data than in the atmospheric data. The reason is that when there is noise in the wind data, the ocean does not adjust quickly enough to the erroneous wind data, and with the atmosphere mainly a slave to the ocean at low frequencies, the wind soon adjusts to the ocean state; hence little damage is done by the noise in the original wind data. In contrast, when there is noise in the ocean data, the ``slave'' atmosphere adjusts rapidly to the erroneous ocean data, hence oceanic noise can cause far more damage to the coupled system.

In general, the estimation of both parameters and initial conditions (LH3) yielded poorer results than just parameter estimation (LH1) or just initial condition estimation (LH2). However in the case of just estimating initial conditions (LH2), the model parameters were assumed to be known perfectly. In practice, one would not know the true parameters. When our model parameters were modestly off from the control, estimating only the initial conditions sometimes yielded very poor results. Hence it seemed best to estimate the initial conditions and the model parameters together as in LH3.

The subject of applying adjoint data assimilation to coupled models is clearly a very complex one, and our studies are but a small step into this area. Recently, we have turned our attention to an intriguing connection between adjoint data assimilation and neural network models (Hsieh and Tang, 1998). For more detailed information on our current researches, please visit our web page at http://www.ocgy.ubc.ca/projects/clim.pred.

References

Daley, R., 1991: Atmospheric Data Analysis. Cambridge, Cambridge Univ. Pr., 457 pp.

Errico, R.M. 1997. What is an adjoint model? Bull. Amer. Meteor. Soc., 78, 2577-2591.

Hsieh, W.W. and B. Tang, 1998: Applying neural network models to prediction and data analysis in meteorology and oceanography. Bull. Amer. Meteor. Soc., in press.

Ji, M., A. Kumar, and A. Leetmaa, 1994: An experimental coupled forecast system at the National Meteorological Center: Some early results. Tellus, 46A, 398-418.

Lu, J., and W.W. Hsieh, 1997: Adjoint data assimilation in coupled atmosphere-ocean models: Determining model parameters in a simple equatorial model. Quart. J. Roy. Met. Soc., 123, 2115-2139.

Lu, J., and W.W. Hsieh, 1998a: Adjoint data assimilation in coupled atmosphere-ocean models: Determining initial conditions in a simple equatorial model. J. Met. Soc. Japan, 76, 737-748.

Lu, J., and W.W. Hsieh, 1998b: On determining initial conditions and parameters in a simple coupled atmosphere-ocean model by adjoint data assimilation. Tellus, 50A, 534-544.

Philander, S. G.H., T. Yamagata and R.C. Pacanowski 1984: Unstable air-sea interactions in the tropics. J. Atmos. Sci., 41, 604-613.

Zebiak, S. E. and M. A. Cane, 1987: A model El Nino-Southern Oscillation. Mon. Wea. Rev., 115, 2262-2278.