On a Model of Vortex Filament Motion: Closed Solutions and Knot Energies

1
On a Model of Vortex Filament Motion Closed
Solutions and Knot Energies
Introduction
Vorticity is the tendency of a fluid to move in a
circular motion. It is measured by curl(u),
where u is the velocity field of the fluid. A
vortex filament is an approximately
one-dimensional concentration of vorticity.
Current Work
The Residues of the Beta Function
If w curl(u) is the vorticity of a fluid with
velocity field u,
Numerical Evolution 1. Discretize the Initial
Curve. 2. Take the Fast Fourier Transform of each
of the component vectors of the curve. 3.
Multiply these transformed vectors by the
appropriate coefficients to generate the
Fourier transforms of the first and second
derivatives with respect to arclength. 4.
Take the real parts of the Inverse Fast Fourier
Transforms of these vectors 5. Use these
derivative vectors to compute the Cross product
between the first and second derivatives
with respect to arclength in a component-wise
fashion. 6. Send the resulting derivative with
respect to time to an ODE solver. (As a
system of equations, with each component vector
being one of the rows in the system.)
Assume w is nonzero and of constant magnitude in
a thin tubular region of constant cross section
Where dl is tangent to the vortex filament and w
w.
Approximate the filament, near a point O, with an
arc of circle, and write the components of x
(along the filament) and x (near the filament) in
terms of the Frenet frame (T, N, B). For small
s x (the radius of the vortex core),
compute
For
, we have
where L is an endpoint of integration, ? the
curvature of the curve, f as in the figure below
The first term gives the rotational component of
the velocity, which does not induce a
displacement. Ignoring this term and rescaling
time as follows
we have
We call this the Vortex Filament Equation (or VFE
for short). Note that the self induced motion of
the vortex at a point is directed in the binormal
direction with speed proportional to the
curvature at that point.
Knot Energies
The concept of Knot Energy was motivated first by
the desire to generate a canonical form of each
knot, taking inspiration from physical analogs
such as the Coulomb potential. We utilize
several different definitions of knot energy,
the most commonly used is the OHara energy (also
known as the Möbius Energy)
Time
Time
These energies can be seen as expressions of knot
complexity, and can even be used to draw some
general conclusions about knot type. The
following is an example of an initial condition
for a curve (in fact, it is what we call a
minimal energy configuration of the (5,2)-Torus
Knot) and its corresponding OHara energy
evolution under the flow of the VFE.
Continuing Research The table below lists the
first several conserved quantities and the
corresponding residues of the Beta function of a
knot. Note that they are very similar. As the
residue number increases, so does the number of
terms that differ from the corresponding
conserved quantity. We would like to know if
there is a pattern for the additional terms. We
are working on a MATLAB code that computes the
evolution of each of the first few residues.
Does the difference between a residue of the Beta
function and the corresponding VFE conserved
integral remain bounded during the evolution? We
would like to be able to use the relationships
between the Beta Function residues and the
conserved quantities to add correcting terms to
the VFE so that the new evolution preserves the
Brylinski energy, and therefore the knot type of
the evolving curve (Note Knot type is not
preserved under the VFE).
Evolution of the Minimum Energy Trefoil
Evolution of the Minimum Energy Josephine Knot
Conserved Quantities
A conserved quantity for a partial differential
equation is an integral expression that is
preserved under the flow.
We can see that the change in total length is
very small for this particular initial condition,
known as the minimum energy trefoil.
Additionally, we see that the total squared
curvature is also very well preserved. This is
convincing evidence that this evolution is a good
approximation of the analytic evolution of the
curve. Visually, we see that the curve appears
to behave in a tame fashion, which is reasonable
based on the nature of curve evolution under the
flow. When we look at the knot energy, we see
that this curve is only an energy minimizer at
time zero, but interestingly, the energy seems to
periodically dip back toward this minimal energy
state. We are experimenting with ways to run the
evolution for a longer time period to determine
whether this oscillation is periodic, or whether
it eventually converges to some steady state.
We see that the total length and the total
squared curvature are very well preserved for
this initial condition, known as the minimum
energy Josephine knot. Note the corkscrew-like
motion of each of the curves above. If we look
closely, we can see that in addition to this
corkscrew motion, the curve is actually sliding
backwards along itself. This is easiest to see
be following the red\blue border. The evolution
of the knot energy is very interesting, as it
seems to be oscillating around, and possibly
converging to, a steady state energy. However,
in order to test this conjecture, improved
methods of error correction must be sought so
that the evolution can be approximated for longer
intervals. Another numerical concern involves
the amount of memory needed to store the required
large matrices of evolution data. A combination
of coding upgrades in these two areas would allow
for some very interesting experiments.
The VFE has an infinite number of conserved
quantities, expressed in terms of kappa and tau,
and these are a list the first few encountered.
These conserved quantities are very useful as
numerical diagnostics to determine the accuracy
of a simulated evolution. For instance, in our
numerical simulation of the VFE flow, we plot the
change in total length and the change in total
squared curvature in order to check our
simulation accuracy. Additionally, we will see
later that these conserved quantities are
mimicked very closely by the residues of the Beta
Function of a knot. The following is a graph of
the change over time of the total length of the
pictured initial condition
Minimum Energies The table to the left is a
table of minimum energy projections of different
knot types, and their corresponding minimum
energies. These minimum energies provide an
interesting way to classify knot type. In
general, more complicated knots have higher knot
energies, as long as we choose the minimum energy
configuration of each knot type for comparison.
We can make a projection of any knot type have
arbitrarily high knot energy simply by bringing
two of the strands close to one another. We want
to avoid this problem when comparing knot
energies, and we do so by finding the minimum
energy configuration of each knot type. In fact,
this knowledge is very applicable in studying
DNA, as DNA is sometimes knotted. If we apply
gel electrophoresis to a given knotted strand of
DNA, the distance the DNA travels in the gel is
an excellent predictor of the knot type. Knots
with higher knot energy travel farther in the
gel, with some exceptions.
In this case, we can see that the Total Squared
Curvature is preserved up to the fifth decimal
place. Running a simulation with stricter error
tolerances may produce better conservation,
depending on the given initial condition.
By Kelly Epperson Under the direction of Dr.
Annalisa Calini, With assistance from Dr. Tom
Ivey and Dr. Brenton LeMesurier.