Marine Ecosystem Dynamics Modeling Laboratory

Case 05. The thermal boundary layer on steep slope bottom

1. Design of the Numerical Experiment

When a model with a -transformation coordinate system is applied to a shallow bank or shelf break, it faces difficulty due to -errors over a steep slope bottom. In a -transformation coordinate system, the baroclinic pressure gradient force is divided into two terms in each of the horizontal momentum equations. These two terms have the same order of magnitude but opposite signs. Small numerical errors in the calculation of these two terms could result in significant numerical biases of the model-predicted currents and density. Since the steep slope bottom is a region where a rapid change of the thickness of -levels occurs, it is a region with the largest -errors (Haney, 1991; Mellor et al., 1994, Chen and Beardsley, 1995; Song et al., 2001).

  Fig. 1: The distribution of the water   temperature at initial in an idealized   circular lake.

The -errors in POM and ECOM-si have been examined both theoretically and numerically by Mellor et al. (1994) and Chen and Beardsley (1995). Mellor et al. pointed out that these errors in POM could retain at a certain level if sufficient high horizontal and vertical resolutions are used. Chen and Beardsley found that at a fixed location over a sloping bottom -errors in ECOM-si increases with water depth. Since the buoyancy velocity error caused by the -transformation always opposes the velocity in the thermal boundary layer on the bottom of the slope, this error can be suppressed if the vertical resolution is high enough to resolve the thermal current near the bottom.

The experiments shown here are focused on the influence of the horizontal resolution used in FVCOM and POM on the thermal boundary layer on steep topography. POM uses the simplified temperature bottom condition of , while FVCOM uses the completed exact temperature bottom condition. To compare FVCOM with POM, we ran FVCOM for the two separate cases: one with the same bottom boundary condition as POM and the other with the exact condition.

Geometry of the circular lake: R = 78 km (radius), H = 300 m (maximum water depth); L = 15 km (width of the shelf), and α= 0.02 (slope at the shelf break).

The initial temperature distribution:


where is the surface water temperature given as 15o C, is the bottom water temperature given as 6oC, and H is the maximum water depth specified as 300 m.

Forcing: Km= 10-4 m2/s (background mixing).

2. Results

For given 31 -levels in the vertical, the structures of the thermal boundary current predicted by POM and FVCOM depend on the horizontal resolution used in numerical computation (Fig. 4). For the case with a horizontal resolution of 4.5 km, for example, both POM and FVCOM develop the multiple layer along-lake boundary currents with time. The scale of these currents are the same as the width of the shelf, and the along-shelf and vertical velocities reach 1 cm/s and 3 X 103 cm/s, respectively after the 10 model days. The only difference is that FVCOM-predicted currents are symmetrically distributed to the center of the lake, while POM-predicted currents do not.

Fig. 2: Comparison of the along-isobath and   vertical velocities at the end of 10th model day   between FVCOM and POM.
There are no physical reasons supporting an asymmetric distribution of the current across the lake for POM, since the numerical grids are composed of squares which are symmetrically distributed around the circular lake, and also the mixing coefficient is the same everywhere. The fact that the asymmetric pattern of the currents mainly occur near the surface around the coast implies that POM has a limitation when applied to a very shallow region. This is probably due to the time-and space-smoothing program used in the code and also to the limitation in geometric fitting.

The pattern of multiple layer currents disappears in both the POM and FVCOM results as horizontal grid size decreases to 2.25 km (Fig. 3). The currents predicted by these two models are characterized with a two-layer flow: one is near the surface and rotates cyclonically along the coast, and the other is limited to a very thin thermal boundary layer above the bottom of the slope and rotates anti-cyclonically around the lake. The maximum speed of the along-lake velocity within the thermal boundary layer is about 0.6 cm/s. An upwelling, with a vertical velocity of about 1 X 103 cm/s is found in the thermal boundary layerThese current patterns are consistent with the theory derived by Wunsch (1970). This theory suggests that a background mixing tends to produce an along-isobath thermal current.Facing downstream in the direction of the current, the lighter water is always on the observer’s left. .

  Fig. 3: Comparison of the along-shelf and vertical   velocities at the end of the 10th model day   between FVCOM and POM for the idealized   circular lake. The horizontal resolution for FVCOM   and POM in this case is 2.25 km. Contour interval   is 0.1 cm s-1 for u and 0.4�10-3 cm s-1 for w.
Also, when the pressure gradient forcing moves the water in the interior into the boundary layer, the thermal diffusion tends to reduce it density and hence cause it to upwell along the slope.

Our numerical experiments imply that both POM and FVCOM could resolve the structure of the thermal boundary layer over the sloping bottom topography at a certain horizontal resolution. In spite of it, FVCOM utilizes unstructured grids which provide more flexibility to increase the local horizontal resolution at the shelf break. Also, the POM-predicted currents are not symmetrically distributed in the shallow water close to the coast, even when the horizontal resolution is increased. This issue needs to be fixed when applying POM to a very shallow water region.

It should be pointed out here that the model results shown above are all based on the bottom boundary of . This simplification may lead to an overestimation of the vertical mixing and hence horizontal and vertical velocities over the steep slope bottom like shelf break. For a given horizontal resolution of 2.25 km, the currents at the 10th model day predicted by FVCOM with an exact no flux normal to the slope are about one order of magnitude weaker than those described above (Chen et al., 2004). These results suggest that an inaccurate use of the bottom boundary condition of temperature exaggerates vertical mixing and thus overestimates both along-isobath and vertical velocities within the thermal boundary layer over the slope.

«PreviousNext» Posted on November 13, 2013