Yundong Li Guijun Han ^* Wei Li ^* Xiaobo Wu Lige Cao Gongfu Zhou ^*

• Tianjin Key Laboratory for Marine Environmental Research and Service, School of Marine Science and Technology, Tianjin University, Tianjin, China

Salinity is a key variable in the dynamic and thermal balance of the entire climate system. To address the complexities of diverse terrains and fluctuating ocean waves, we commonly use free-surface models with quasi-stationary (e.g. height, pressure, or terrain following) coordinates (Adcroft and Hallberg, 2006) for simulating salinity. In such models, the vertical grid dynamically adjusts with the undulation of seawater. However, this adjustment also occurs when freshwater enters or exits the ocean. Freshwater-induced salinity changes at the ocean's surface are artificially distributed to each vertical layer within a model time step. This means that the freshwater at the ocean surface instantaneously and directly affects the seafloor. This process is different from physical diffusion processes. The diffusion effects caused by the influx and outflow of freshwater have a very small impact on the seafloor. This error leads to salinity non-conservation and disrupts the vertical distribution structure of salinity. Previous studies have also addressed this issue with solutions such as the vertical Lagrangian-remap method. This paper proposes a natural vertical distribution calculation scheme (NVDCS) which is different from the approaches of our predecessors. In the discrete formulation of the original ocean equations, freshwater flux is introduced to ensure salinity conservation. In each model time step, by calculating the seawater volume changes due to freshwater inflow or outflow, as well as the vertical grid changes caused by sea surface undulations, the aforementioned artificial error is eliminated from each vertical layer. This scheme ensures that changes in the vertical coordinates of each layer result solely from internal oceanic dynamic processes, avoiding the instantaneous and directly impact of surface freshwater. Ultimately, the influence of freshwater is confined to the ocean surface. This method is straightforward to implement and user-friendly. Sensitivity experiments indicate that in free-surface models, quasi-stationary coordinates introduce artificial errors. The proposed calculation scheme not only eliminates this error but also achieves a better vertical distribution structure than using virtual salt flux, while ensuring salinity conservation.