Representer-based variational data assimilation in a nonlinear model of nearshore circulation

A representer‐based variational data assimilation (DA) method is implemented with a shallow‐water model of circulation in the nearshore surf zone and tested with synthetic data. The behavior of the DA system is evaluated over a 1‐hour time interval that is large compared to timescales characteristic of instability growth and eddy interactions. True reference solutions, from which the synthetic data are sampled, correspond to fully developed unsteady nonlinear flows driven by a steady spatially varying forcing representing the effect of breaking waves. Forcing and initial conditions are adjusted to fit the data. The convergence of the nonlinear optimization algorithm and the accuracy of the forcing and state estimates depend on the choice of the forcing error covariance C. In a weakly nonlinear (equilibrated waves) regime, using C that allows only a steady forcing correction yields a convergent and accurate solution. In a more strongly nonlinear regime, the DA system cannot find sufficient degrees of freedom in the steady forcing to control eddy variability. Implementing a bell‐shaped temporal correlation function in C with the 1‐min decorrelation scale yields a convergent linearized inverse solution that describes correctly the spatiotemporal variability in the eddy field. The corresponding estimate of forcing, however, is not satisfactory. Accurate estimates of both the flow and the forcing can be achieved by implementing a composite C with a temporal correlation separated into an O(1) steady and small amplitude time‐variable parts.