Development And Application Of An Isopycnic-Coordinate Numerical Model For Simulation Of Internal Tides With Adjoint Assimilation Method

Internal waves at tidal frequencies, or internal tides, are generated by theinteraction of barotropic tidal flows with sloping topography in a density-stratifiedocean. The internal tides may cause the interior seawater to fluctuate withpeak-to-peak amplitudes as large as tens or hundreds of meters, and the mixing causedby the wave breaking of internal tides can exert a direct influence on the transports ofmass, momentum and energy of the seawater. Thus, the internal tides play animportant role in the balance of kinetic energy and heat content in the global oceans.Moreover, the internal tides also have a remarkable contribution to the dissipation ofglobal barotropic tidal energy and to the deep-sea mixing, and then exert an importantinfluence on the ocean circulation. Among all data assimilation methods, thefour-dimensional variational (4D-Var) data assimilation is one of the most effectiveand powerful approaches developed over the past two decades. It is an advanced dataassimilation method which involves the adjoint method and has the advantage ofdirectly assimilating various observations distributed in time and space into thenumerical model while maintaining dynamical and physical consistency with themodel. By assimilating the interior observations, the adjoint method can effectivelyoptimize the model control variables, and then makes the numerical model organicallycombined with observation data and improves the simulation precision. This paperpresents our efforts on the application of adjoint assimilation method to the numericalsimulation of internal tides.In this paper, by using spherical coordinates in the horizontal direction andisopycnic coordinates in the vertical, a numerical model for simulation of internaltides with adjoint assimilation method is constructed based on the nonlinear,time-dependent, free-surface hydrodynamic equations, subject to the hydrostatic approximations. This model consists of two sub-models: the forward model is usedfor simulation of internal tides; the adjoint model is used for optimization of modalcontrol variables. Mode splitting technique is employed in both forward and adjointmodels. This model is expected to make an improvement in the precision of internaltidal simulation through an organic combination of the numerical model and theobservation data. Based on this newly constructed model, some preliminary studiesare also conducted in this paper, centering on optimization of the model controlvariables.The model ability of inversion of its control variables is tested though a series ofideal numerical experiments in which the feasibility and rationality of this model arealso calibrated. The experimental results indicate that: all the prescribed openboundary conditions (OBCs) can be inverted successfully on the consideredtopographies; both the tidal elevations and currents can be simulated very accuratelyby assimilating the surface currents at several observation points; the assimilationprecision as well as the cost function can be reliable and is able to reflect both of theinversion and simulation results in the whole field; with the help of the ‘independentpoint scheme’(IPS), the spatially varying coefficients of the bottom friction, interfacefriction and the horizontal eddy viscosity can also be inverted precisely. Theperformance and results of ideal experiments give a preliminary indication that theconstruction of this adjoint assimilation model is rational and successful.Taking the inversion of the OBCs as an example, further ideal experiments arecarried out to study the performance of optimization methods and artifices whenapplied to this adjoint assimilation model. The study indicates that: the classicaladvanced optimization methods still need to be improved when applied to thepractical cases such as the model constructed in this paper; the simplified gradientdescent method, which is controllable and easy to implement, should be regardedseriously as a choice, especially when the classical advanced optimization techniquesfail or perform poorly; the IPS can improve the well-posedness of the inverse problemby reducing the number of the independent control variables and can make the valueof the control variable varying continuously and smoothly in order to guarantee a rational physical meaning, and as a result, the precisions of the parameter estimationand numerical simulation are both improved.As a preliminary application of this model to the real case, some actualnumerical experiments are conducted to simulate the M2tides around the Luzon Straitin the northeast part of the South China Sea, by assimilating the TOPEX/Poseidon(T/P) altimeter data. The experimental results indicate that: by optimizing the modelcontrol variables, the simulation results can be consistent with a majority ofobservations very well, whereby the M2tides in the area of interest has beensimulated with a high precision; the optimization of OBCs dominates the simulationresult of the internal tides in this area and is responsible for a great majority of thereduction of the cost function during assimilation; the optimization of the horizontaleddy viscosity coefficient has an important impact on the assimilation result, but stillfar less than that of the OBCs; the bottom friction coefficient can influence the resultslightly more than the interface friction coefficient, but still far less than the horizontaleddy viscosity coefficient.