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Abstract: Wave turbulence is about understanding the long time statistical behavior of a random sea of weakly interacting, dispersive wavetrains in the presence of sources and sinks. The topic will be discussed in three lectures.

Lecture one sets the menu. It will discuss the big picture and give an overview of how the challenge of wave turbulence relates to, but is arguably richer than, 3D, high Reynolds number hydrodynamics. I will indicate why one achieves natural asymptotic closure and schematically derive kinetic equations for the spectral densities of conserved quantities (such as energy and particles/wave action). The kinetic equations have a nontrivial family of solutions of both thermodynamic (Rayleigh-Jeans, Bose-Einstein and Fermi-Dirac) and finite flux (Kolmogorov-Zakharov) types. The KZ spectra describe how energy and particles/wave action cascade from sources to sinks. We will introduce the notions of finite/infinite capacity, explain how the KZ spectra are set up in finite/infinite time and discuss how the wave turbulence closure can break down at very small or large scales and lead to fully nonlinear and intermittent behavior. I will illustrate the ideas by discussing applications to water waves and whitecapping and to optical waves of diffraction and collapses. This self-contained lecture may be of interest to a broader audience than the participants of the summer school.

Lecture two will address systems such as surface tension, MHD, Rossby and interval gravity waves whose long time statistics are dominated by three wave interactions. We will learn how to set up the problem and derive the first asymptotic closure (the kinetic equation for the spectral energy density and frequency renormalization). Using a dimensional argument, we will derive the KZ finite energy flux spectrum, discuss the criteria for breakdown of the wave turbulence closure and illustrate the ideas in the context of surface tension dominated surface waves.

Lecture three will address systems such as deep water gravity waves, optical waves of diffraction in nonlinear media and semiconductor lasers which are dominated by four wave interactions. Starting from the kinetic equation, we will discuss its properties (conservation laws and equilibrium solutions) and give a careful derivation of the Kolmogorov-Zakharov spectra using the Zakharov transformation. Using a local (Fokker-Planck) approximation, we discuss how the KZ spectra are realized. Finally, we discuss breakdown and the onset of intermittency in the context of deep water gravity waves (whitecapping) and optical waves (collapses).

Abstract: Natural patterns of an almost periodic structure turns up all over the place. One sees them on wavewashed long sandy beaches, on saguaro cacti, as tiger and zebra stripes and on the markings of fish skins, as epidermal ridges on fingers, palms and soles, as cloud formations, in geologic and galactic structures and in megalithic art. In the laboratory, one sees them in experiments on convecting fluids, on buckled shells, on laser beams and on flame fronts. The striking similarities between pattern textures, in both planform and defect structure, arising from widely different microscopic contexts suggests that patterns are macroscopic objects whose behaviors are governed by shared symmetry properties rather than individual details. The goal of theory is to find macroscopic descriptions which serve to unify and simplify our understanding of the patterns arising in equivalence (symmetry sharing) classes of microscopic systems. Such a theory has the difficult task of capturing both the smooth (e.g. patches of roll or hexagon planforms) and singular (defect lines and points between the smooth patches) features. The aim of these three lectures is to introduce the student to the means of building these descriptions.

Lecture one, which will be self contained and may therefore be of interest to a broader audience, will survey the methods and results in near onset and far from onset situations. In particular, we will illustrate (and give results from experiment, simulation and theory) the challenges in the context of convection in a large aspect ratio (ratio of diameter to depth) elliptical container whose sidewalls are heated.

Lecture two will discuss, using simple models, the derivation of the various order parameter equations (Landau-Stuart-Watson, Newell-Whitehead-Segel, complex Ginzburg-Landau, complex Swift-Hohenberg) applicable near onset. We will learn how the competition between different planforms is resolved and about some of the various instabilities (zig-zag, Eckhaus-skew varicose, modulational) which can destroy the exact periodicity of patterns. Several examples from optics and fluids will be used as illustrations and as homework exercises.

Lecture three will discuss the methods of handling patterns far from onset. We will discuss the results of Busse and coworkers who examined the linear stability of rolls and the nature of the instabilities and their boundaries in the Rayleigh number-wavenumber plane (the Busse balloon) and the modulational theories of Cross-Newell. We will see in particular how the latter connects with other fields (e.g. the bending of their elastic sheets) and with minimization problems belonging to the family of harmonic maps.