The Applicability of the Integral Method with Variable Limit in Solving the Governing Equations for Temperature and Salinity in an Ocean Circulation Model

Abstract. To address limitations in traditional discretization methods for ocean numerical modeling, this study develops a integral method with variable limit (IMVL) to enhance the simulation accuracy of thermohaline dynamics in ocean models. Under the Arakawa-C grid framework, we propose novel discretization schemes applying variable-limit integration to horizontal advection, horizontal diffusion, and vertical diffusion terms in temperature-salinity equations. For the vertical diffusion term, the variable limit integral scheme is also designed and combined with the time discretization of the difference method to form an implicit fully discrete scheme. Stability analysis based on convection equation principles confirms the numerical robustness of the proposed method. Implementation within the Princeton Ocean Model (POM) demonstrates significant improvements: 1) Strait test cases reveal 40–60 % error reduction in temperature-salinity simulations compared to standard POM; 2) Enhanced topographic sensitivity enables superior representation of overflow dynamics across steep sills; 3) The modified scheme eliminates numerical instabilities in zero-Coriolis scenarios, maintaining physical validity beyond 720 simulation days by preventing artificial water stacking and gradient accumulation. The computational efficiency analysis demonstrates that the introduction of the variable-bound integration method increases the total runtime by merely 25 %.These findings establish the variable-limit integration method as an effective approach for improving the dynamic framework of ocean models, particularly demonstrating outstanding performance in enhancing model stability and resolving dynamic processes under complex topographies. It is noteworthy that the variable-limit integral method designed herein for the thermohaline equations represents a novel and more stable solution approach, which, while implemented in POM within this study, is universally applicable to other ocean numerical models.