Reconciling Eulerian and Lagrangian Water Exchange Diagnostics in Wind-Driven Coastal Oceanography

A long-standing challenge in coastal oceanography is the quantitative divergence between Eulerian concentration-based formulations and Lagrangian particle-tracking frameworks, which frequently yield inconsistent water exchange diagnostics for the same marine domain due to unaligned physical scales and boundary treatments. This study reconciles this diagnostic discrepancy by quantifying the underlying numerical error budgets and establishing scale-aligned boundary mechanisms. Through idealized benchmarking tests against analytical solutions, we systematically isolate the trade-offs between grid-based numerical dissipation and discrete stochastic sampling noise across diffusion, advection, and boundary reflection processes. Quantitative error analysis yields non-dimensional resolution and sampling criteria capable of constraining theoretical numerical approximation errors below a 1% threshold. Extending these scale-aligned guidelines to long-term three-dimensional transport simulations in a natural semi-enclosed domain (Liaodong Bay) under extreme storm and monsoon forcing, both paradigms achieve unprecedented cross-approach concordance. The frameworks yield remarkably consistent bulk mass exchange timescales, with global turnover times matching within 3.7% (0.81 vs. 0.84 years) and half-exchange times showing minor boundary-sensitive variance (0.62 vs. 0.72 years), while successfully unifying the diagnosed seasonal transport macro-structures. This work demonstrates that the historical divergence in coastal water exchange diagnostics is largely an artifact of unaligned numerical and boundary configurations rather than an intrinsic physical limitation, providing a robust, scale-consistent benchmark for model parameterization in coastal transport modeling applications.