Fundamental Properties of Adjoint Model and Adjoint Sensitivity Under Fully Nonlinear Hydrostatic Internal Gravity Waves

Abstract Adjoint sensitivity modeling plays an important role in inverse problems, including four‐dimensional variational data assimilation or state estimation, by providing the sensitivity of the objective (or cost) function to the model input data. Although data assimilation has become common in regional ocean modeling, and the model resolution has become high enough to resolve internal tides, it remains unclear whether physically sensible and stable adjoint sensitivity modeling is feasible in the presence of internal waves. As a first step to tackle this problem, this study investigates fundamental properties of the adjoint of a time‐dependent circulation model, and the resulting sensitivity under internal waves. The theoretical and numerical results, based on simple‐internal‐wave theory and MITgcm modeling, show that stable adjoint sensitivity modeling is feasible under fully nonlinear (but stable) hydrostatic internal waves. However, it requires a careful choice of the objective function to obtain sensitivity which has the same propagation properties as (real‐world) internal waves, because the definition primarily controls the directionality and vertical‐mode content of internal‐wave signals in the adjoint model. Also, it should be noted that the internal‐wave signals lack indirect sensitivity through the dependence of the wave‐propagation speed on the model's prognostic variables. An important implication of the results is that the standard form of the objective function, which has been used in data assimilation studies for quasi‐geostrophic flows and barotropic tides, could be problematic for internal tides.