By Sergey Nazarenko (auth.)
Wave Turbulence refers back to the statistical conception of weakly nonlinear dispersive waves. there's a huge and transforming into spectrum of actual purposes, starting from sea waves, to plasma waves, to superfluid turbulence, to nonlinear optics and Bose-Einstein condensates. past the basics the e-book therefore additionally covers new advancements reminiscent of the interplay of random waves with coherent constructions (vortices, solitons, wave breaks), inverse cascades resulting in condensation and the transitions among susceptible and powerful turbulence, turbulence intermittency in addition to finite procedure measurement results, corresponding to “frozen” turbulence, discrete wave resonances and avalanche-type strength cascades. This publication is an outgrow of a number of lectures classes held via the writer and, therefore, written and dependent particularly as a graduate textual content than a monograph, with many routines and options provided alongside the way in which. the current compact description basically addresses scholars and non-specialist researchers wishing to go into and paintings during this field.
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Extra info for Wave Turbulence
Sample text
The Kraichnan dual-cascade picture presented above was recently confirmed numerically in high-resolution (163842) DNS in [12]. 5 Yet Another Argument about the Cascade Directions In Sects. 3 we already presented two versions of the arguments predicting the energy and the enstrophy cascade directions, and here we will give yet another simple argument. Let us consider a power-law spectrum nk = Ak-x (not necessarily a stationary one) and let us construct plausible plots for the energy flux e and the enstrophy flux g as functions of x, see Fig.
3. Note that further CB cascade will become more and more anisotropic, k? =kk ! 1: Thus, the energy cascade undergoes a reversal, which is especially interesting in the case of a continuous forcing of 2D turbulence: the net (integrated over kk) flux in the resulting steady state will be zero in between the forcing and the cascade reversal scales, even though locally the flux is finite in the two-dimensional (kk, k\) space. 2 Gravity Water Waves CB approach for the gravity water waves, or equating the linear and the nonlinear timescales, leads to the famous Phillips spectrum [31], ð1DÞ Ek $ gk À3 : ð3:31Þ This is easy to understand if we remember that both the linear and the nonlinear scalings in this system are completely determined by the same single dimensional parameter g.
Four-Wave? N-Wave? N is the minimal number of waves for which: • The N-wave resonant conditions for the wavevectors and the frequencies are satisfied simultaneously for a non-trivial set of wavenumbers, xðk1 Þ Æ xðk2 Þ Æ Á Á Á Æ xðkN Þ ¼ 0; ð3:1Þ k1 Æ k2 Æ Á Á Á Æ kN ¼ 0; ð3:2Þ where xðkÞ is the frequency of the wave with wavenumber k. g. for the three-wave processes 2 ? 1 two of the signs must be ‘‘+’’ and one sign ‘‘-’’, and for the four-wave processes 2 ? 2 there must be two ‘‘+’’es and two ‘‘-’’es.