The FTLE fields exhibited significant negative values that were of the same order of mag- nitude as the positive FTLE values. The blue and purple spectral clusters from high tide at spring tide, at 06:00, were located within the areas of highly negative FTLE values. Ad- vection of the spectral clusters also showed that their area shrunk over time. The spectral clustering analysis for the spring tide event was reiterated with a 12-hour integration window to further investigate the shrinkage. The final step of the protocol, the noise coherence cal- culations, was carried out for the peak in gap ratio with the average distance function, which corresponds to the r value of 0.01, as opposed to r = 0.0175 for the 6-hour analysis. The detected clusters match the 6-hour results almost exactly and are plotted in figure 3-20.a on page 121.
The previous analysis had shown that after 6 hours, the clusters had considerably shrunk in area; now after 12 hours, they vanished. To investigate the role played by the surface convergence, the shrinkage in cluster area was compared to the vertical fluxes from the divergence of the surface velocity field. The area was calculated in two ways: using the embedded MATLAB boundary.m function and using the grid resolution and multiplying it by the number of nodes. The area calculations were validated against polygons computed in QGIS, an open-source geographic information system, as shown in figure 3-20.b on page
Figure 3-19: Step 3 of the spectral clustering protocol with coherence metrics for the Scott Reef channel. Analysis done on the 2016 dataset (a) at high tide around neap tide on September 27, 2016, (b) at high tide around spring tide on October 3, 2016 (c) at the time at which all drifters were released in water. The r values correspond to the peaks from figure 3-17.
(a)
(b)
(c)
Figure 3-20: (a) Step 3 of the spectral clustering protocol for the Scott Reef high tide event with a 12-hour window. The initial (t0) and final (tf) cluster positions are plotted. (b) Area
121. The vertical fluxes F at each time step were measured from the divergence ∇ · V of the surface velocities V = (u, v) output by the SUNTANS model, which used the rigid-lid approximation, over the area A of the cluster:
∇ · V = ∂u ∂x + ∂v ∂y = − ∂w ∂z F = Z T Z A wdA · dtdz (3.1)
The comparison is plotted in figure 3-20.c.: the change in area is qualitatively similar to the total vertical flux out of the area. Quantitatively, there are some differences at the first four time steps, but the changes are comparable in magnitude and likely due to numerical resolution. Between 06:00 and 07:30, the area of the cluster is halved, which explains in part why the recalculated clusters at 07:30 were so different from the 06:00 clusters. The vertical fluxes due to surface convergence are also most important before 07:30.
With the anticipation that the cluster’s shape was dictated by the surface convergence, the velocity field’s divergence component was removed via the least-square estimate method to analyze a non-divergent velocity field during the neap tide event and the spring tide event. The resulting fields, plotted in figure 3-21 on page 123, are virtually divergence-free, with the remaining values being noise. At neap tide, the original velocity fields exhibited divergence one order of magnitude smaller than at spring tide. In both cases, the divergence was maximum inside the channel. At spring tide, the negative divergence was maximum along the channel, somewhat following the shape of the attracting backward FTLE field. This suggests that the high rates of attraction were due, at least in part, to downward convergence within the channel. The two clusters resulting from the parameter-free spectral clustering method were formed around the maximum divergence values, inside the channel and east of Sandy Islet, further suggesting that convergence dictated the spectral clustering results.
(a)
(b)
Figure 3-21: Divergence in the Scott Reef channel before and after removing the divergent component via least-squares estimates for (a) the neap tide event and (b) the spring tide event. Coordinates in UTM Zone 51S.
(a) (b)
Figure 3-22: Forward FTLE fields in the Scott Reef channel for the non-divergent velocity fields around (a) neap tide on September 27 and (b) spring tide on October 3. The period of integration was 6 hours in all cases.
fields with the same parameters. The FTLE field at neap tide, plotted in figure 3-22.a on page 124, resembles the original FTLE field in figure 3-15a. At spring tide, the pronounced FTLE ridge partially disappears, although high values of FTLE are still found around the tip of Scott Reef South. In both cases, the FTLE values are almost exclusively positive, as they should be for a non-divergent flow. Step 1 of the spectral clustering protocol, the sweep of r parameters, yielded very similar results for the neap tide event, as shown in figure 3-23.a on page 125. The peak gap ratio occurred at r = 0.025, same value as for the original, divergent field. This resulted in 18 clusters as well, although their shape and coherent values were a bit different in the non-divergent case, as shown in figure 3-24 on page 125. The similarity in the shape of the gap ratio curve indicates a relative insensitivity to the removal of the divergent component, which is consistent with the relatively small vertical velocity during neap tide as compared to spring tide. For the spring tide event, the shape of the sweep of r parameters, shown in figure 3-23.b, was very different for the non-divergent field. No peak in gap ratio was detected and the values were much lower: this means that there was no strongly connected component within the domain, unlike in the divergent case in which
(a)
(b)
Figure 3-23: Step 1 of the spectral clustering protocol for the non-divergent velocity fields: high tide during (a) neap tide and (b) spring tide. The original sweeps of parameters corre- sponding to the divergent flow field is plotted for comparison. At neap tide, the peak occurs at the same value of r = 0.0025. At spring tide, no peak is detected.
Figure 3-24: Step 3 of the spectral clustering protocol for the non-divergent velocity field at high tide around neap tide: results with coherence metrics for the peak at r = 0.025.
the two clusters were very distinct from the incoherent background. High tide around spring tide was when the vertical velocities and the divergence values were the highest, one order of magnitude greater than during neap tide.
The removal of the divergent component in the velocity field led to large changes in the FTLE and spectral clustering results. The differences between divergent and non-divergent fields were greater for the spring tide event, which was to be expected as the divergence values were an order of magnitude greater. The key structures identified during the experiment are absent in the non-divergent field. This suggests that the LCS detected during the experiment were to a large degree driven by the three-dimensionality of the flow. The experimental drifters were likely attracted to a convergence zone, but constrained by buoyancy, thus clustering into a small area as the cluster shrunk in size. Over the 07:30 to 13:30 experiment, the ebb tide had turned into flood and the vertical velocities throughout the 6-hour window were much lower than in the 06:00 to 12:00 window, where the tide went from high to low. The influence of the surface convergence also explains why the 06:00 and the 07:30 spectral clustering results are so different.