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.