Bickley Jet example

Most methods mentioned in sections 1.3.1–1.3.3 are illustrated in figures 1-6–1-7 on pages 37–37. The example is the Bickley Jet flow, which will be further detailed in the following chapter. It consists of a zonal jet that is a barrier to meridional transport with three recircu- lation vortices on each side. The Bickley Jet is a benchmark flow in dynamical system theory because it exhibits transport behaviors that are qualitatively different and are common in realistic ocean flows: the jet, the vortices and the background chaotic zone.

In figure 1-6, panel (a) shows the Poincar´e section, which was computed here for the periodic Bickley jet with the parameters as in Rypina et al. [2007]. The Poincar´e section is a long-established methodology in dynamical system theory to study periodic flows: this stroboscopic mapping plots the trajectory positions at each period. Figure 1-6.(a) reveals how the jet acts as barrier to transport, as well as the presence of vortices corresponding to concentric discretely-sampled closed orbits, which are islands of coherence among the incoherent background. In contrast, the stroboscopic mapping of the particle trajectories outside these closed orbits show clouds of dots corresponding to the chaotic regime of the incoherent background. The six vortices are all of similar sizes.

Panels (b)-(i) in figure 1-6 were taken from Hadjighasem et al. [2017], in which the Bickley Jet was modified from Rypina et al. [2007] to become quasi-periodic. Because the flow is

Figure 1-6: Comparison of Lagrangian methods on the Bickley jet example (forward-time calculations only). (a) Poincar´e section computed for the parameters in Rypina et al. [2007] corresponding to the periodic Bickley Jet flow. (b)-(j) Results of LCS methods on the quasiperiodic Bickley jet example. The panels were adapted from Hadjighasem et al. [2017].

Figure 1-7: Encounter volume for the periodic Bickley jet flow using encounter radius 5×105. Figure adapted from Rypina & Pratt [2017]

not periodic, the Poincar´e section could not be constructed, but the quasi-periodic system is qualitatively similar to the periodic system. The methods that will be used for analysis in chapters 2–4 of this thesis are the FTLE (figure 1-6.c), FCM (figure 1-6.g), spectral clustering (figure 1-6.h), FCM (figure 1-6.g) and encounter volume (figure 1-7). The flow used to calculate the Poincar´e section in figure 1-6.a corresponds to the flow used for the encounter volume method in figure 1-7, the periodic Bickley Jet. The trajectory complexity (figure 1-6.d), the geodesics approach (figure 1-6.e), the LAVD (figure 1-6.f) and the FSLE (figure 1-6.i) are included for completeness and because they are compared with the other methods throughout this thesis, including in section 1.4 and in chapter 2.

Figure 1-6.(b) shows the trajectory length, which corresponds to an arc length computa- tion of the distance covered by the advection particles, as introduced by Mancho et al. [2013]. It is not an LCS method per se: just as trajectory complexity, FSLE or even FTLE, it does not define what the sought coherent structure are, but it provides an illustration of transport within the domain of interest. Figure 1-6.(b) shows the contrast between the meandering jet, with high values of length in red, and the cores of the recirculation vortices, in light blue. The vortices are hard to discern, however, as there are no clear vortex boundaries.

For the Bickley Jet, various methods captured different aspects of transport within the flow. In the Poincar´e section (figure 1-6.a), the closed orbits and the islands of coherence they bound were of similar sizes, suggesting that the vortices detected by LCS method should also be of comparable size. The FTLE scalar field (figure 1-6.c) reveals useful information about the differences in separation rates within the flow, but the individual vortices are hard to discern. The core of the jet shows minimal rates of separation, in blue, consistent with the notion of transport barrier, whereas the boundaries of the jet show maximal rates of separation, in red. The vortices are hardly delimited, however, as the FTLE values smoothly go from blue to green away from the core. While the shape of the jet can be discerned, the vortices are hard to bound. The trajectory complexity (figure 1-6.d) most clearly highlights the meandering jet through its very low values, in dark blue; the vortices are discernible

through relatively low values shown in blue, but the exact boundaries are hard to extract. The geodesic approach (figure 1-6.e) successfully detects the core of the meandering jet as a parabolic LCS, in blue, but only identifies three out of six vortices, the elliptic LCS, in green. The LAVD (figure 1-6.f) method successfully extracts the boundaries of the cores of the vortices, although the cores vary in areas. The jet is also characterized by low values, in blue, but its boundaries are not extractable. The FCM method (figure 1-6.g) can roughly detect the vortices as different clusters, but one vortex is missing its core and the method also includes very distant filaments in the clusters, which contradicts the definition of a coherent structure. These filaments indicate that the coherent sets include trajectories much further than the core of the clusters. The shape and size of the clusters vary highly, in contradiction to the orbits shown in the Poincar´e section, which are of similar size. Spectral clustering (figure 1-6.h) successfully detects each of the cores of the recirculation vortices as individual clusters. Moreover, these vortex cores are of similar sizes. The FSLE field (figure 1-6.i) shows the vortices as having low values, but the discrepancy in shape and size of the low FSLE regions make it nearly impossible to extract the coherent structures present in the flow. The jet is hardly discernible. Lastly, the encounter volume method (figure 1-7) also outputs a field that is a representative depiction of the flow transport: the cores of the recirculation vortices have very low encounter volume, in dark blue, consistent with the low mixing potential inside a coherent vortex. The jet core displays a low encounter volume and is delimited by the regions exhibiting high encounter volumes, in red, which is also consistent with high mixing potential.

To summarize, the output of LCS methods generally provides a picture of flow transport within the domain of interest and explains how transport is organized in the flow by revealing its key structures. Different LCS methods emphasize different properties of the flow; for this reason, it is expected that they capture different types of structures. No method is able to detect all the features of interest: here in figure 1-6, no method was able to rigorously extract the jet and the 6 recirculation vortices. Lastly, most methods depend on several user-input

parameters. In chapter 2, this thesis develops an improved parameter-free spectral clustering method. The applicability of different LCS methods, including the chapter 2 parameter-free spectral clustering, to geophysical flows will be assessed in chapters 3 and 4.