Spectral-clustering approach to Lagrangian vortex detection

One of the ubiquitous features of real-life turbulent flows is the existence and persistence of coherent vortices. Here we show that such coherent vortices can be extracted as clusters of Lagrangian trajectories. We carry out the clustering on a weighted graph, with the weights measuring pairwise distances of fluid trajectories in the extended phase space of positions and time. We then extract coherent vortices from the graph using tools from spectral graph theory. Our method locates all coherent vortices in the flow simultaneously, thereby showing high potential for automated vortex tracking. We illustrate the performance of this technique by identifying coherent Lagrangian vortices in several two- and three-dimensional flows.

Figure 1

Undirected graph partitioning. The dashed line shows the solution of the problem of finding a decomposition of the graph into two size-balanced groups with a minimal number of edges connecting nodes from distinct groups.

Figure 2

Interpretation of the transition matrix constructed from tracer advection as a tight similarity matrix $Z$ of a bipartite graph.

Figure 3

Partitioning of a bipartite graph $G_{\mathcal{B}}=(V_{\mathcal{B}},E_{\mathcal{B}},W_{\mathcal{B}})$ whose set of nodes $V_{\mathcal{B}}$ is divided into two disjoint sets $A$ and $B$ such that $V_{\mathcal{B}}=A\cup B$. The dashed line shows the solution of normalized graph cut yielding a simultaneous decomposition of $A$ and $B$.

Figure 4

(a) Second and (c) third largest (left) singular vectors of the normalized transition matrix for the Bickley jet flow. (b) and (d) Corresponding coherent sets that are obtained by searching through all possible cuts [48]. To compute the transition matrix, we subdivided the domain into a grid of $400\times 120$ identical boxes and released 400 particles in each box. We then advected particles from $t_0=0$ to $t=40$ days.

Figure 5

(a) Ten clusters extracted by $K$-means clustering from the first nine generalized eigenvectors of the graph Laplacian $L$ for the periodically forced pendulum. The tenth cluster corresponds to the chaotic sea filling the space between elliptic regions. (b) Eight hundred iterations of the Poincaré map for the periodically forced pendulum. (c) Computed clusters, compared with the Poincaré map computed for the same integration time (800 iterates).

Figure 6

Comparison of three different diagnostic fields for the periodically forced pendulum. The scalar fields are constructed for the same integration time $T=800\times 2\pi$. (a) Forward-time connectivity field. (b) Forward-time FTLE field. (c) Forward-time FSLE field.

Figure 7

(a) Sorted generalized eigenvalues for the graph Laplacian $L$ for the periodically forced pendulum. (b) First and (c) ninth generalized eigenvectors of the graph Laplacian $L$. Isolated points resulting from the graph sparsification are shown in white.

Figure 8

(a) Run times of Algorithm 1 for six different resolutions for the periodically forced pendulum. The run times represent the average CPU times for 300 processors used in parallel in these computations. The computations are performed on a supercomputer with $2.7\text{−}\mathrm{GHz}$ Intel Xeon CPUs. (b) Clustering sensitivity with respect to the sparsification radius. The plot shows the average within-class similarities (for nine coherent sets) relative to the $\epsilon$-nearest-neighbor radius used to sparsify the pairwise distances $r_{ij}$.

Figure 9

Sorted generalized eigenvalues for the graph Laplacian $L$ for the quasiperiodic Bickley jet flow.

Figure 10

Leading generalized eigenvectors of the graph Laplacian $L$ for the Bickley jet flow.

Figure 11

(a) Seven clusters extracted by $K$-means clustering from the first six generalized eigenvectors of the graph Laplacian $L$ at initial time $t_0=0$. The seventh cluster corresponds to the mixing region filling the space between the coherent clusters. (b) Same clusters advected passively to the final time $t=40$ days. The complete advection sequence over 40 days is illustrated in the Supplemental Material, movie M1 [92].

Figure 12

Comparison of three different diagnostic fields for the ocean data set. The scalar fields are constructed for the same integration time $T=168$ days. (a) Forward-time connectivity field. (b) Forward-time FTLE field. (c) Forward-time FSLE field.

Figure 13

(a) Sorted generalized eigenvalues for the graph Laplacian $L$ for the ocean data set. (b) and (c) First two generalized eigenvectors.

Figure 14

(a) Coherent vortices at initial time $t_0=11\mathrm{January}2006$. (b) Advected image of the vortices at the final time $t=28\mathrm{June}2006$. (c) Magnification of the blue cluster shown in the first panel. The figure shows some isolated points located far from the cluster core. (d) Corresponding cluster obtained from a higher resolution computation, revealing that the previously detected isolated points are part of a narrow fingering emanating from the cluster core. The complete advection sequence over 168 days is illustrated in the Supplemental Material, movies M2 and M3 [92].

Figure 15

Coherent vortices are captured at initial time $t_0=11$ January 2006, with a sparse trajectory data set. Figure 14 is shown in the background for comparison.

Figure 16

(a) Seven clusters extracted by $K$-means clustering $\left(k=7\right)$ from the first six eigenvectors of $L$. The first six clusters correspond to six coherent vortices that were identified earlier in [94]. The chaotic sea between coherent vortices is the seventh cluster and appears as the void between them. (b) Seventh cluster that appears as the chaotic sea between coherent vortices. (c) and (d) Three-dimensional vortices are reconstructed by putting together the coherent cluster pieces.

Figure 17

Coherent vortices extracted by Algorithm 2 are compared with the Poincaré map constructed for integration time $T=3000$.