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Oregon State University
Pacific Marine Environmental Laboratory
We have smoothed a year of monthly-mean SST, winds and 20° isotherm depths at all TAO moorings, using a modification of the Zebiak-Cane model as a weak or imperfect constraint. The weights for the smoothing are inverses of estimated covariances for data errors, dynamical errors and initial errors. The covariances are estimated using recent scale studies, and elementary g.f.d.
The best fit is found by solving the nonlinear Euler-Lagrange equations iteratively. Each iterate is a linear Euler-Lagrange equation. Imposing it supresses the huge null space at each iterate ( there are 40 million state variables but only 2,600 data). Each linear E-L equation is solved iteratively; 5,200 model integrations would yield the exact solution, but far fewer are needed as an efficient preconditioner may be found in the form of a partial representer matrix. Covariances are efficiently convolved with adjoint variables by solving solving suitably-chosen p.d.es. Nevertheless a large, fast parallel computer is needed.
In addition to providing smoothed fields of all variables in the coupled model, the inversion provides estimates of sources and sinks of heat, layer thickness and atmospheric momentum. These are significant, and consistent with known processes that are unrepresented or unresolved in the simple model
Finally, an analysis of the conditioning of the inverse will be made, in order to assess the efficiency of the TAO observing system.
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