

Re = 70

Re = 237

Figure 1: Curvature fields in the Kolmogorov
flow for Re = 70, where the flow is a hexagonal vortex lattice, and
Re = 237, where the flow is spatiotemporally chaotic. Blue corresponds
to low curvature, while red is high curvature. Elliptic points are
marked with circles, while hyperbolic points are marked with crosses. 
Re = 70

Re = 237

Figure 2: Trajectories of the critical points
for the same two Reynolds numbers. Elliptic point trajectories are
shown in blue and hyperbolic point trajectories are in red. At the
smaller Reynolds number, the critical points are pinned to their lattice
positions, while in the spatiotemporally chaotic state they can
wander. 
Animations 


Re = 70

Re = 237

Figure 3: Animations of the motion of the critical
points superimposed on the curvature fields (top row) and the vorticity
fields (bottom row). 

Figure 4: The mean number of critical points per frame
as a function of Reynolds number. The number drops as the driving
is increased, showing a coarsening of the pattern. 
The number of critical points can change in three ways:
they can enter or leave the measurement volume (the central 10 cm x 10
cm of the flow cell), they can appear or disappear singly (part of which
may be due to experimental difficulties in locating them), or they can
be created or annihilated in hyperbolic/elliptic pairs. Below are the
rates for these processes as a function of the driving:

Figure 5: The rates at which the critical points enter
and leave the measurement area, as a function of Reynolds number. 

Figure 6: The rates at which the critical points appear and disappear singly
within the measurement area. 

Figure 7: The rates at which the critical points are created and annihilated in pairs. 
We have also computed the probability density functions
for the separation of the critical points along the two coordinate axes,
where the y coordinate runs parallel to the Kolmogorov Flow's shear bands
and the x coordinate is transverse.

Figure 8: PDFs of the separation distance r (scaled
by the shear band wavelength λ) between two critical points
for Re = 70 and Re = 237. The peaks in the Re = 70 data clearly
show its lattice structure. For the Re = 237 data, where the flow
is spatiotemporally chaotic, the forcing structure is lost in the
direction parallel to the shear bands, but is still retained in
the transverse direction. 
Finally, we have investigated the movement of these points by computing
their meansquared displacement.
Figure 9: Evolution of the magnitude of the
vector meansquared separation of the critical points in all directions.
As the Reynolds number is increased, the critical points become
unpinned from their lattice sites and begin to wander, collapsing
onto a power law with an exponent of 1.4. 
Figure 10: Evolution of the meansquared
separation of the critical points in the direction transverse
to the shear bands. Even at high Reynolds number, it is still
difficult for the critical points to travel across the forced
shear bands. The meansquared displacement in this direction is
an order of magnitude smaller than in the parallel direction. 
