Nonlinear Physics Lab


 
Figure 1: Velocity, vorticity, and curvature fields at Re = 32 (left column), 93 (center column), and 245 (right column). In the top line, velocity vectores are shown as arrows, undersampled by a factor of 8 for clarity. The vorticity is shown by color: red corresponds to large negative vorticity (clockwise rotation), and blue corresonds to large positive vorticity (counterclockwise rotation). As Re increases, the flow becomes more disordered. The bottom row shows the logarithm of the curvature; red corresponds to large and blue to small values. Low values of curvature typically form lines, while the highest values appear as points.

 

Figure 2: Trajectories of the topologically special points for the same three Reynolds numbers. Hyperbolic points are plotted in red, while elliptic points are in blue. At Re = 32, the special points are tightly bound to the forced vortex lattice. At Re = 93, they remain bound but make larger excursions. At Re = 245, where the flow is spatiotermporally chaotic, the special points wander over the domain.

 

Figure 3: The radial distribution function g(r), shown for Re = 32 (squares), 93 (circles), and 245 (triangles). The separation r is scaled by the mean magnet spacing L. Dashed vertical lines show the positions of some of the peaks expected for a 2D square lattice (other peaks are not observed due to the finite resolution). As Re increases, the spatial order vanishes and g(r) becomes liquid-like.

 

Figure 4: The maximum value of g(r) (squares, left axis), showing a gradual transition to spatial disorder with increasing Re, and the annihilation rate (circles, right axis) of hyperbolic-elliptic pairs, which shows a well-defined threshold. The solid lines are drawn to guide the eye.

 

Figure 5: PDFs of the number N of topologically special points present in the measurement area at each instant in time, normalized by the number of lattice sites, for four Reynolds numbers. As Re increases, the mean number of special points decreases, while the width of the distribution grows.