Generalized Inverse of a Reduced Gravity Primitive Equation Ocean Model and Tropical Atmosphere–Ocean Data

A nonlinear 2½-layer reduced gravity primitive equations (PE) ocean model is used to assimilate sea surface\ud temperature (SST) data from the Tropical Atmosphere–Ocean (TAO) moored buoys in the tropical Pacific. The\ud aim of this project is to hindcast cool and warm events of this part of the ocean, on seasonal to interannual\ud timescales.\ud The work extends that of Bennett et al., who used a modified Zebiak–Cane coupled model. They were able\ud to fit a year of 30-day averaged TAO data to within measurement errors, albeit with significant initial and\ud dynamical residuals. They assumed a 100-day decorrelation timescale for the dynamical residuals. This long\ud timescale for the residuals reflects the neglect of resolvable processes in the intermediate coupled model, such\ud as horizontal advection of momentum. However, the residuals in the nonlinear PE model should be relatively\ud short timescale errors in parameterizations. The scales for these residuals are crudely estimated from the upper\ud ocean turbulence studies of Peters et al. and Moum.\ud The assimilation is performed by minimizing a weighted least squares functional expressing the misfits to the\ud data and to the model throughout the tropical Pacific and for 18 months. It is known that the minimum lies in\ud the ‘‘data subspace’’ of the state or solution space. The minimum is therefore sought in the data subspace, by\ud using the representer method to solve the Euler–Lagrange (EL) system. Although the vector space decomposition\ud and solution method assume a linear EL system, the concept and technique are applied to the nonlinear EL\ud system (resulting from the nonlinear PE model), by iterating with linear approximations to the nonlinear EL\ud system. As a first step, the authors verify that sequences of solutions of linear iterates of the forward PE model\ud do converge. The assimilation is also used as a significance test of the hypothesized means and covariances of\ud the errors in the initial conditions, dynamics, and data. A ‘‘strong constraint’’ inverse solution is computed.\ud However, it is outperformed by the ‘‘weak constraint’’ inverse.\ud A cross validation by withheld data is presented, as well as an inversion with the model forced by the Florida\ud State University winds, in place of a climatological wind forcing used in the former inversions