Effects of Simpson number and unsteadiness parameter on the analytical solution of Lagrangian residual velocity under a constant density gradient

Previous investigations of analytical solutions of the Lagrangian residual velocity (LRV) have predominantly assumed a barotropic environment characterized by uniform density. In this study, we derive an analytical solution for LRV explicitly incorporating baroclinic effects, revealing the synergistic regulatory mechanism between baroclinic term and tidal body force. Under the assumption of a constant longitudinal salinity gradient within the bay, we quantify the relationship between baroclinic terms and two dimensionless parameters: the Simpson number (Si) and the unsteadiness parameter (Un). The individual contributions of various dynamical terms to the overall LRV are systematically evaluated. Results demonstrate that under conditions of large Si and small Un, the LRV is predominantly driven by baroclinic effects, yielding a classical estuarine circulation characterized by seaward flow at the surface and landward flow near the bottom. Conversely, when Un is large, tidal body force dominates the LRV, resulting in a pattern opposite to that induced by baroclinic forcing. Our findings highlight a quadratic interaction between Si and Un in modulating both the directional shift and dynamics of the LRV. This work addresses the limitations inherent in classical analytical models that neglect baroclinic processes and provides a rigorous parametric framework for improved analytical representations of coastal material transport dynamics.