Adaptivity

We use the adaptive refinement in order to check the levels of errors in our previous simulations. The fact is that large discontinuities develop in the vorticity field between elements close to Bump 1 and 2. These discontinuities are located close to the tip of the bumps and at the edge of the recirculation. We also noted that some discontinuities are associated with the piecewise boundary parabolas near inflection points along the coastline where third degree polynomials would be more adequate. We therefore redesign a mesh with more points along the western bumps and slightly more in the interior. On the new mesh, the vorticity field is indeed improved but further refinement would be needed to obtain a reasonable vorticity field, especially in proximity of Bump 2 where a strong velocity shear exists. Moreover, strong negative vorticity seems to originate from Bump 3 and is shed in front of Bump 2 with dramatic consequences to the resolution of the vorticity field. This is possibly connected to two anticyclonic eddies trapped between Bump 3-4 and 2-3. Starting from this mesh, we use the adaptive strategy developed in Section 2.4.3. We have a certain level of liberty in the choice of the fields and the parameters controlling the selection of the elements to be refined. In Section 3.4, we used the primitive variables for controlling the level errors. From a geophysical fluid perspective, it would be interesting to control the errors using the vorticity, which is a one order higher field relative to the velocity. If the latter is correctly resolved, it should follow that the other fields are also well resolved. We found that this approach was reliable by testing the adaptive strategy in a simpler experiment. From this experiment, we noted that the velocity, elevation and vorticity fields are indeed well resolved, and that, for the same parameters $\lambda_i$, the vorticity controlling adaptivity induces one additional level of refinement. Unfortunately for Geometry V, we could not afford in terms of computational cost more than one adaptive cycle. Therefore any claim of convergence has to be discarded. After one cycle (Fig. 5.21), the refined elements are concentrated along the tip of Bump 2 and less near Bump 1. Of course, the refinement has a cost. The simulation on the refined mesh is about four times more expensive than that on the original mesh, due to time-step limitations.

  

Figure 5.21: Mesh for the original and the refined runs.

We now compare the two experiments for the Geometry V and $\nu =100$m²/s. Figure 5.22 shows the total (kinetic + potential) energy for the two experiments. The refined and original results are rather similar for the first year, but they depart afterwards. However, we see the same approximate low frequency (about 1000 day period) behavior, which is the signature of a Rossby basin mode. The higher frequencies (small eddies) may be responsible for the most of the discrepancy. What is more intriguing is that the Kelvin wave activity increases significantly for the refined experiment. Between day 4400 and 4800, the amplitude of the Kelvin wave on both meshes at the same location is rather similar, with the amplitude on the refined mesh being slightly larger. Then, after day 5000, the amplitude on the refined mesh quickly doubles relative to the amplitude on the original mesh and this factor then remains more or less constant. Presumably, the production of Kelvin waves is enhanced by the increased resolution in the region of Bump 2 where Kelvin waves are generated. As with the original mesh, the amplitude of the Kelvin waves on the refined mesh are anti-correlated with the total energy. However, the increased amplitude of Kelvin waves on the refined mesh did not lead to a decrease of the total energy between the original and refined meshes. This raises two possibilities. Either the Kelvin waves are only marginal in the dissipation of the energy or, more probably, this could be an artifact of the resolution. On the original mesh, it is possible that processes located near Bump 2 were too dissipative because of the too coarse resolution.

The overall structure of the mean elevation field is rather similar for the original and refined meshes (Fig. 5.24). Although not noticeable in Fig. 5.24, an important improvement lies in the structure of the elevation field close to the tip of Bump 1, 2 and 3, where the elevation shows a rather singular behavior on the original mesh. By contrast, the elevation field at the same locations is much smoother and the amplitude of the peaks in elevation much less on the refined mesh. The mean total energy for the refined mesh tends to be slightly larger than that on the original mesh, although, due to the relative short period of observation (6 years), this may not be significant. Much improvement can be noticed in the mean vorticity field. The strong peak at Bump 2 is better resolved (although the amplitude is not severely modified), as well as the zone of negative vorticity near the same bump. This undershoot seems to be real and not an artifact of the lack of resolution on the original mesh. Some improvement is also noticeable in the interior of the basin, likely due to a slightly improved resolution on the new mesh in the interior. The edge of the recirculation would stand some refinement. However, since the largest discontinuities in the vorticity field are close to the tip of the bumps, the mesh is first refined there. To conclude, we gain, using the adaptive strategy, improvements over the complex processes happening close to the bumps and consequently some improvement of the nonlinear interactions and related energetics (the increased amplitude of the Kelvin modes). However, there are no significant changes in the overall mean circulation.

  

Figure 5.22: Total energy for the last 6 years of simulation for the original and refined meshes. The two curves are very similar for the first year and then depart slowly from each other.

  

Figure 5.23: Amplitude of the fast oscillations at (x=500 km, y=0 km) along the boundary for the original and refined meshes.

  

Figure 5.24: Mean elevation fields for the original mesh and the refined mesh.

  

Figure 5.25: Mean vorticity fields for the original mesh and the refined mesh.

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