Numerical Decomposition of the Governing Equation for Lagrangian Residual Velocity Under Nonlinear Eddy Viscosity and Stratification

Abstract Previous analytical investigations of the first‐order Lagrangian residual velocity (LRV) governing equations have typically assumed no‐slip bottom boundary conditions, constant horizontal density gradients, and constant eddy viscosity coefficients, limiting their applicability to realistic, time‐varying stratified environments. To address these limitations, this study systematically investigates the first‐order LRV dynamics within a weakly nonlinear framework, wherein the residual flow is decomposed into five components: tidal body force, bottom friction, baroclinic pressure gradient, wind stress, and river runoff. The research advances current theoretical understanding in three key aspects: First, this study demonstrates that the dominant tidal body force component varies with bay length and the dimensionless parameter (the ratio of eddy viscosity term to local acceleration) under unstratified conditions. Second, the conventional no‐slip bottom boundary condition is replaced by a more realistic slip condition in our numerical framework, revealing that the increasing bottom friction coefficients reduce the relative importance of the bottom friction component compared to the tidal body force component. Furthermore, this research eliminates the constraint of constant eddy viscosity by incorporating the MY‐2.5 turbulence closure model. This refinement allows for a detailed subdivision of the tidal body force into its constituent mechanisms: vertical eddy viscosity term of Stokes' drift, nonlinear advection, and tidal straining. Under unstratified conditions, tidal body force is primarily driven by the nonlinear advection, with tidal straining exerting a damping feedback effect. Conversely, under stratified conditions, tidal body force is predominantly influenced by the vertical eddy viscosity term of Stokes' drift.