Wave Turbulence: many questions remain open!
Wave turbulence is the study of the long time behavior of solutions of
nonlinear field equations, usually conservative and Hamiltonian,
describing weakly interacting waves in the presence of sources and
sinks. Think of a sea of ocean waves stirred by a storm, with waves of
all lengths and directions running about on the surface, interacting,
causing occasional shoots of white spray. What kind of statistically
stationary states can we expect?
Wave turbulence is an excellent paradigm for nonisolated statistical
systems for which the usual rules of equilibrium thermodynamics do
not apply. The reason is that wave turbulence has a natural asymptotic
closure captured by a kinetic equation for the energy density and a
frequency modulation which keeps
the asymptotic expansions for all cumulants higher than the second
uniformly valid in time.Moreover, the kinetic equation exhibits a
class of stationary solutions, the Kolmogorov-Zakharov (KZ) solutions,
which describe the transport of energy or other conserved densities of
the unforced,
undamped equations, from their sources to their sinks. The process by
which these conserved densities are transported from one length scale
to another is resonance, either between three or four (in very rare
cases, five) waves.
There tends, therefore, to be a general misconception that the wave
turbulence problem is solved and trivial. Nothing could be further
from the truth. We have shown that unless the KZ solutions carry the
same symmetry properties as the original governing equations, wave
turbulence always fails at either very small or very large scales and that,
in these regions, the dynamics is dominated by fully nonlinear events.
Are these regions compatible with the regions in which the KZ
solutions obtain? Maybe yes and maybe no. There have been several
examples of one dimensional systems where the fully nonlinear
solutions dominate at all wavenumbers and the kind of wave turbulence
behavior associated with resonances is not seen at all.
Our current research focusses on two aspects of the challenge. First,
with Benno Rumpf, I am looking at the mechanisms which play the key
role in the so called MMT turbulence.
Second, Volodja Zakharov and myself are examining a scenario we had
suggested about ten years ago for combining a description of
coexisting weak and strong turbulence. The model is the nonlinear
Schrodinger equation in two and three
dimensions. It is known that the inverse cascade of power (or wave
action in other contexts) builds condensates. These may survive if the
nonlinearity is defocussing (as is the case in Bose Einstein
condenstations) and require one to account for new kinds of
fluctuations which can ride on the condensate or they may break up
into collapsing filaments when the system is focussing. It is the
latter case in which we are interested. If the dissipation sinks are
only available at short scales, then in order to reach a statistically
steady state, the system must use its nonlinearity to carry the power
transported to large scales by the inverse cascade back to short
scales. The collapses do this. Moreover, because of incomplete burnout
in two dimensions, the partially destroyed collapse at short scales
becomes an additional source for energy and power for the wave
turbulence field. The cascade rate increases and continues to do so
until it reaches the value of Q/f where f is the burnout fraction and
Q the original flux rate of power by which the system is driven at
some intermediate scale. Only then can the system reach a
statistically steady state. The reason there may be a chance to solve
the composite problem is that, because the collapses are so fast, the
local interactions between waves and collapses are small and each
feels the other only through bulk properties such as Q and the
universal nature of the failed collapse.
A third challenge is to revisit and understand better the problem of
gravity waves. It is known that wave turbulence fails at scales larger
than those at which capillary effects can regularize the breakdown of
the weak interaction theory if the energy flux to small scales is
sufficiently large. The sea surface in such situations is pockmarked
with whitecaps which introduce a totally different dissipation
mechanism and, as in the case with collapses,
modify the spectrum. The challenge is to marry the KZ spectrum for
energy or waveaction flux (waveaction cascades to large scales) with
the kind of behavior associated with a surface spotted with whitecaps.
A fourth area of interest is to understand the manner by which the
stationary spectrum is attained. Work, initiated with Galtier and
Nazarenko, and carried on with Connaughton and Jakobsen, is trying to
understand the reason for the anomolous way in which KZ spectra are
realized. In particular, we are exploring the idea that a functional
one can associate with entropy production plays a key role in this
asymptotic behavior.
Plant patterns and plant phyllotaxis.
Plant patterns, namely the way in which plant surfaces are tiled, and
plant phyllotaxis, the arrangements of leaves, flowers and stickers on
the surface, have fascinated and intrigued natural scientists for over
four hundred years.
A particular challenge has been to explain the reasons for the
appearance of the Fibonacci sequence in the families of spirals on
which the plant stickers lie. Attempts to provide rational theories
for the observations fall into four categories.
First, there are the rules of Hofmeister written over a century and a
half ago, which essentially say that primordia ( bumps which are the
forerunners to the more mature phylla) are generated in an annular
region near the shoot apical mersitem (SAM) of the plant, move
outwards relative to the plant until they enter a nonactive region as
far as pattern creation is concerned (although their flowers still
continue to mature). New primordia are initiated at the inner edge of
the generative region in the "most open space available" at regular
intervals. These rules were modified a century later by Snow and Snow.
Second, informed by the Hofmeister rules, Douady and Couder (DC), in a
series of pioneering papers in the 1990's, created an ingenious magneto-
mechanical experiment which mimiced the rules and found that
the primordia configurations of mutually repelling oil droplets
simulating primordia closely ressembeled much of what was
observed. It also helped reproduce the transitions to higher and
higher members of the Fibonacci sequence as the "plant size"
parameter increased. But the DC
original theory did not account for whorls and decussates.
To remedy this, they added more contrived rules which gave a
particular shape to the primordium. From these, they obtained whorls
but no ridge like surface shapes such as one sees on pumkins or
certain kinds of cacti. Nevertheless their theory provided a most valuable
paradigm. The phyllotactic configurations seen on plants can be
understood by looking for the minima on some "energy" landscape.
Third, at about the same time, Green and colleagues Steele and Dumais
at Stanford suggested a physical mechanism for primordia formation.
they argued that differential growth between the plant's corpus and
tunica (skin) would lead to compressive stresses in the generative
region. The tunica would buckle as a result and then the bumps caused
by the buckling would develop into phylla. A graduate student, Patrick
Shipman and I recently developed a general nonlinear theory based on
this idea and the pioneering work with respect to the central roles
played by quadratic interactions and geometric bias introduced by
Koiter in the mid fifties. In that work we were able to show that many
of the observations, including the plant surface shapes such as ridges
and parallelograms (as one sees on pinecones, for example), could be
explained. The role of biochemistry, and in particular the role of
growth hormones such as auxin, was a passive one. Nonuniform stresses
created by the buckled surface would generate auxin so that the
bucking bumps would be auxin sinks which would enable the primordium
to grow fully mature phylla at that location.
Fourth, recent work by Reinhardt and colleagues, and by Meyerowitz,
Traas and colleagues suggests however that auxin plays much more than
a passive role. They have shown that not only do phylla grow and
flourish in the presence of auxin but that there are mechanisms akin
to osmosis whereby a uniform auxin concentration can destabilize. The
driving influence for this instability is the action of PIN 1
molecules in the cell walls which can orient so as to drive auxin
against its local concentration gradient. The patterns seen from the
biochemical models are reminiscent of what one observes but there are
many open questions not the least of which is how to explain the
anisotropy of many of the surface deformations.
Shipman and a graduate student, SunZhiying, are currently exploring a
combination of both mechanisms. It is known that growth affects the
stress strain relationship so growth is an additional variable in the
surface deformation. It provides the in surface compressive stress
which buckling. Even as the buckled state develops it continues to
affect surface deformation because of the modification of the stress
strain relationship. On the other hand, nonuniform stress induces
growth enhancement or inhibition. Combining these two ideas gives us a
model from which we see cooperation in that the development of
primordia is enhanced greatly by having both effects present.
Several predictions emerge. First, we see in what circumstances the
surface deformation follows or is greatly different from the auxin
concentration distribution. Second, we see that as the plant grows and
as transitions move the number of primordium connecting spirals up the
Fibonacci sequence, the shapes stay self similar.
