Case 04. Freshwater discharges over idealized continental shelves
1. Design of the Numerical Experiment The experiments have been conducted for idealized shelves with 1) a linear slope and a straight coastline and 2) a linear slope with a curvature coastline. Case 1: The straight coastline
In FVCOM, when the water elevation at the open boundary is determined by a radiation boundary condition the velocity in the boundary triangular cells could be calculated based on the mass conservation.This approach causes little reflection at the open boundary, and open boundary at 800 km away from the origin is far enough to remain numerically accurate in the computational domain. POM is numerically configured with Cgrids, in which elevation and crossisobath velocity component are at the same boundary line, while the velocity normal to the open boundary is located at cells half a grid from the boundary.Although the velocity component normal to the open boundary could be determined using a simplified liner momentum equation, the elevation and velocity component paralleled to the open boundary must be determined by radiation boundary condition. This approach can not guarantee the mass conservation at the boundary cell, making it difficult to filter out all the reflected waves, especially for the stratified case. To ensure minimum influence of the open boundary on the numerical solution in the computational domain over the time scale in which we are interested, the open boundary for POM is moved to 1400 km away from the origin.


Case 2: The circular coastline
The slope of the shelf is given as
where is the water depth at the distance of r from the origin of the circle, R is the radius of the circular lake, r_{0} is the distance from the origin of the circle to the edge of the shelf, is the water depth at R, H_{d} and is the water depth in the region of << r. Freshwater discharge rate Q = 1000 m^{3}/s, injected into the lake from the mesh at the center point of the southern coast. The background salinity S = 35 PSU. 2. Results a) The straight coastline
Given the same freshwater discharge rate, the distributions of elevation, currents, and salinity computed by POM and FVCOM are significantly different (Fig. 3). The buoyancy current computed by FVCOM is characterized by a relatively strong clockwise circulation out of the mouth of the freshwater source and a coastal trapped plume along the shelf, while the current computed by POM is most equally divided towards the upstream (left) and downstream (right) along the coast: no distinct clockwise circulation forms out of the mouth of the freshwater source. As a result of large backward advection, the salinity plume predicted by POM moves downstream at a slower speed than that shown in FVCOM. Similar differences are also found in the distributions of the water elevations: POM shows a large gradient of the water elevation in the upstream region and a slower downward movement of the trapped wave, which is consistent with the distributions of salinity and currents predicted by this model. For a same given discharge rate, FVCOM tends to produce a larger crossshelf gradient of elevation and salinity in the downstream region (for example, 100 km from the freshwater source) than POM, even though the crossshelf scale of the plume is the same for both models. The comparison between FVCOM and POM for this idealized case clearly shows that the structure of the modelpredicted plume depends on the numerical schemes used in these models. The publicavailable code of POM uses a central difference scheme for advection terms of salinity. Although this method ensures the secondorder accuracy, it produces an artificial backward transport against the direction of the current. This is the reason why POM shows a more significant upstreamward transport and a slower downstreamward propagation speed of the plume. Therefore, the results obtained from POM for the river dischargeinduced buoyancy flow must be interpreted with caution because of the deficiency of a central difference scheme for tracer advection. FVCOM uses a secondorder accurate upwind numerical scheme for salinity advection. This scheme not only ensures the salinity conservation in the individual TCE but also avoids the occurrence of artificial backward transport as that shown in the central difference scheme. Under a condition with the same secondorder cutoff, the upwind scheme is better than the central difference scheme for the tracer simulation. b) The circular coastline
For the case with a horizontal resolution of 4.22 km, the lowsalinity plume predicted by FVCOM occupies the entire shelf with a crossshelf scale as the same as the width of the shelf, while the plume predicted by POM extends over the interior region off the shelf, forming a detached eddylike circulation at the outer edge of the shelf after 10 model days. As the horizontal grid size reduces to 1.78 km, the plumes predicted by FVCOM and POM shift toward the coast. A quasiequilibrium state occurs as the horizontal grid size is smaller or equal to 0.89 km. These results provide us with two important facts. Firstly, attention must be paid to the horizontal resolution in simulating the spatial structure of the lowsalinity plume in the inner shelf of the coastal ocean, no matter which model is used. Secondly, the finitedifference model might lose the shelfcontrolling nature of the lowsalinity plume at a certain lower horizontal resolution and thus produce an artificial eddy formation over the shelf. This issue must be taken into account when applying a finitedifference model to simulate the lowsalinity plume in the realistic inner shelf of the coastal ocean. Unstructured triangular grids used in FVCOM provide an accurate coastline matching with a guarantee of no mass flux on the coastal wall. In the downstream of the freshwater source, the plume water predicted by FVCOM flows along the curvature coastline, with a maximum water level and alongshelf current at the coast. The square grids used in POM, however, result in a steplike coastal boundary in the numerical computational domain. Since no flux condition is applied in a direction normal to the stepshape boundary, the maximum alongshelf current occurs at a distance away from the coast. This stepshape coastal model boundary acts like a drag force to slow down the downstream movement of the lowsalinity plume and to exaggerate the crossshelf secondary current within the plume. Note: as discussed in the case with the straight coastline, the central difference scheme used to calculate the salinity advection in POM can cause a significant backward water transport on the left side of the freshwater source. When a curvature coastline is applied, however, the numerical drags caused by a mismatching of the curvature coastal geometry not only reduces the downstreamward propagation speed of the lowsalinity plume on the right side of the freshwater source, but also decreases the backward buoyancy flow on the left side of the freshwater source. It might provide a good agreement with the observations made with wrong physical reasons. 