Variational parameter estimation in a two-equation turbulence model: A case study with a 3D primitive-equation ocean model

A three-dimensional and complete adjoint model of the Princeton Ocean Model with a generalized coordinate system (POMgcs) is developed to construct the 4D-Variational data assimilation (4D-Var) algorithm in this study. Uncertain parameters in the Mellor-Yamada 2.5 turbulence submodel (MY-2.5) which is enclosed in POMgcs, are tentatively estimated via the 4D-VAR algorithm within a biased model framework. Here, the control variables in the biased model are set to two uncertain wave-affected parameters (wave energy factor α and Charnock coefficient β ) in the MY-2.5 turbulence model, which play a crucial role in modulating the heat content distribution in the upper coastal sea. First of all, the ocean temperature and salinity in a typical coastal sea, Bohai Sea, are simulated by the model to validate the rationally of the MY-2.5 parameterization scheme for both constructing the “truth model” and generating the “pseudo-observations” in the data assimilation studies. Then, after thoroughly testing the ability of the 4D-Var to optimize the initial state fields of the POMgcs model, a series of parameter estimation experiments are carried out to investigate whether and to what degree the parameters embedded in high-order turbulence models can be significantly optimized. Results of parameter estimation with perfect initial fields show the two estimated parameters in the MY-2.5 submodel can successfully converge to the “truth” value. The local minimum of the cost function can be effectively and efficiently jumped out once two kinds of optimization algorithms, LBFGS and LMBM, are jointly used. In addition, the estimated parameter will converge to the optimal value rather than the truth one to compensate for the initial field error when the state-parameter are estimated simultaneously. Further, the performance of the parameter estimation is also deeply discussed when the observation samples are noised. Finally, prescribing the initial field and parameter as error source, a forecasting experiment for sea temperature is performed. The experiment results indicate that assimilating “pseudo-observations” to the model based on 4D-Var can significantly improve the sea temperature simulation. Moreover, adjusting the initial field and parameter leads to a better result than the only initial field, and this conclusion is more evident at the surface than in the deeper ocean.