4DVAR Assimilation of ADCP Data with the Navy Coastal Ocean Model using the Cycling Representer Method

4D-variational assimilation (4DVAR) is used to combine ADCP velocity observations with the Navy Coastal Ocean model (NCOM) to obtain an optimal solution that minimizes a cost function containing the weighted squared errors of velocity measurements, initial conditions, boundary conditions, and model dynamics. However, in order to converge to the global minimum of this cost function, the ocean model (and its adjoint) must be linear. Ocean models, especially those that are designed to resolve baroclinic and mesoscale processes, are typically highly-nonlinear and must be linearized. Tangent linearization is a linearization method that is performed by expanding the nonlinear dynamics about a background field using the first order approximation of Taylor's expansion. The accuracy and stability of this tangent linearized model (TLM) is a very sensitive function of the background accuracy, the level of nonlinearity of the model, complexity of the bathymetry, and the complexity of the flow field. Therefore, in high-resolution coastal domains, the TLM is only going to be stable for a relatively short period of time. In this paper, assimilation experiments are performed in a high-resolution Mississippi Bight coastal domain. The TLM of NCOM for this domain is only accurate for about 1 day. The representer method is used to solve this highly nonlinear, weak-constraint, 4DVAR problem. However, due to the short stability time period of this assimilation problem, the representer method is cycled by splitting the time period of the assimilation problem into smaller cycles, therefore ensuring TLM stability and proper data assimilation. The cycle time period needs to be such that it is short enough for the TLM to be stable, but long enough to minimize the loss of information due to reducing the temporal correlation of the dynamic error. We have found that for the Mississippi Bight experiments presented in this paper that a cycle length of 1 day works best. For each new cycle, a background is first created as a nonlinear forecast from the previous cycle's assimilated solution. Then, data that falls within the time period of the new cycle is used to calculate a new assimilated solution using the previous cycle's forecast as the background. The cycling representer method has been previously demonstrated to drastically improve assimilation accuracy with simpler nonlinear models. Now, this assimilation method is being applied to NCOM. This assimilation system is demonstrated in the Mississippi Bight by assimilating velocity measurements from an array of 14 ADCP moorings for the month of June, 2004. The initial condition for the first cycle, the boundary conditions, and the background around which the TLM is expanded comes from an operational global NCOM The weak-constraint cycling representer method corrects the velocity components of initial conditions, boundary conditions, and dynamics. This paper will demonstrate the improvement of assimilation accuracy as the time window of the cycles is reduced to 1 day, but when 12-hour cycles are used, the system begins to lose skill. It will also be demonstrated that the forecast skill will be improved as the assimilation system progresses through the cycles.