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Relative Dispersion

We will use relative dispersion as a Lagrangian diagnostic for distinguishing
material curves. Relative dispersion simply measures the distance
between initially nearby tracers after some finite integration time.
Its calculation requires only the initial and final positions of an ensemble
of tracers, and may therefore be computed easily.
We begin with a regularly spaced grid of initial conditions. The
grid is advected both forward and backward in time for eight days.
A scalar value for each of the forward and backward time relative dispersion
is assigned to each interior gridpoint by measuring the distance between
the final positions of neighboring trajectories in the grid:

For initial conditions in the interior of a coherent structure (such
as a vortex), relative dispersion is expected to be low. On the other
hand, boundaries of coherent structures are characterized by large values
of relative dispersion. In relation to the chaotic advection paradigm
we wish to apply, local maxima of the forward time relative dispersion
occur along material curves analogous to stable manifolds. In contrast,
local maxima in backward time occur along material curves analogous to
unstable manifolds. Similar ideas using a different Lagrangian diagnostic
may be found at George Haller's web site here
.

As an example, consider the forced, damped Duffing oscillator.
The stable and unstable manifolds of the hyperbolic fixed point near the
origin may be calculated using standard straddling methods, as we know
a priori where to initialize the invariant manifold calculation (i.e.
we know the approximate location of the hyperbolic fixed point):

However, insight into this manifold geometry may be obtained via relative
dispersion. In image that follows, the green curve corresponds to
local maxima of R in forward time, whereas the red curve corresponds to
local maxima of R in backward time.

As expected, the relative dispersion approach is able to identify portions
of the stable and unstable manifolds, with no a priori knowledge of the
flow geometry. This information is obtained by simply advecting a
grid of tracers under the flow. In particular, we see that we can
identify the approximate location of the hyperbolic fixed point for the
Duffing oscillator by the intersection of the forward and backward time
maxima of R.