Assimilation of simulated altimeter data in a two-layer linear Rossby wave model using the variational method

The variational assimilation method has been examined for ability of reconstructing mesoscale features in altimeter data using a simple dynamic model. A one-dimensional, two-layer Rossby wave model in a cross-track channel has been chosen. The simulated data are constructed from a theoretical solution, which is composed of any combination of two normal vertical (barotropic and baroclinic) modes. The data are collected along tracks and with repeat periods similar to those of the Geosat altimeter. The phase space of control variables is composed of initial and boundary conditions. A cost function is defined to measure differences between the simulated data and the model solution. Regularization (smoothing) terms are also included in the cost function in the form of secon-order spatial and time derivatives of the solution. In this paper, two potential problems existing in the altimeter data assimilation are addressed: one is low cross-track resolution, and the other is vertical projection of the data measured at the sea surface. A succesful metho is developed for reconstructing Rossby waves with wavelengths as short as twice the track intervals for any combination of two vertical modes. A key component to efficient assimilation is a preparation step prior to the actual variational assimilation: a uniform ratio of pressure amplitudes in the two layers is included as an optimization parameter. Starting with the first guess from the preparation step, the variational method is carried out based on adjoint equations without such constraint. Separation of the control variables into the two subsets of the initial and the boundary conditions is found useful. Characteristics of the Hessian matrix are related to the performance of this technique. The method developed for the linear system implies steps to be included in data assimilation for nonlinear meanders and eddies in a major current system as well.