Discrete wave turbulence of rotational capillary water waves Adrian Constantin $^{†}$, Elena Kartashova $^{♯}$, Erik Wahlén ‡ † Faculty of Mathematics, University of Vienna, 1090 Vienna, Austria ♯ RISC, J. Kepler University, Linz 4040, Austria We study the discrete wave turbulent regime of capillary water waves with constant non-zero vorticity. The explicit Hamiltonian formulation and the corresponding coupling coefficient are ob- tained. We also present the construction and investigation of resonance clustering. Some physical implications of the obtained results are discussed. ‡ Centre for Mathematical Sciences, Lund University, PO Box 118, 221 00 Lund, Sweden; FR 6.1 - Mathematik, Universität des Saarlandes, Postfach 151150, 66041 Saarbrücken, Germany ∗ big frequency broadening do not change their energy at the appropriate time scales (Fig.3, [25]). A similar effect for irrotational capillary waves has been discovered by Pushkarev and Zakharov [38, 39] who investigated how the discreteness of the spectrum of ir- rotational capillary water waves influences the statistical properties of a weakly nonlinear wave field described by the kinetic equation. It was shown in a series of numer- ical simulations with the dynamical equations of motion (with energy pumping into small wavenumbers) that the wave spectrum consists of ∼ 10 2 excited harmonics which do not generate an energy cascade toward high wavenum- bers — frozen or fluxless modes. "There is virtually no energy absorption associated with high-wavenumbers damping in this case" ([39], p.107). The quasi-resonance channels of the energy flux have also been observed and it was shown that the size of (quasi)-resonance clusters grows with increasing of the resonance broadening ([39], p.113). Two mechanisms for increasing the number of flux-generating modes have been identified: 1) growing the modes‘ amplitudes, and 2) enlarging the number of modes. The pioneering result of Zakharov et al. [61] was the discovery of the mesoscopic wave turbulence regime which is characterized by the coexistence of discrete and kinetic regimes in numerical simulations modeling tur- bulence of gravity waves on the surface of a deep ideal incompressible fluid. The mathematical justification of this co-existence was given in [27] in the frame of the model of laminated wave turbulence. Wave kinetic equations are deduced for the continu- ous k -spectrum in Fourier space while in bounded or periodical systems the wave spectrum is discrete and described by resonance clustering [29]. An important fact first shown by Kartashova in [25] is that indepen- dent resonance clusters might exist in the infinite spectral space and are nonlocal in k -space [24, 25]. Moreover, it was shown numerically (for atmospheric planetary waves) that increasing the wave amplitudes till some limiting magnitude yields no energy spreading from a cluster to all the modes of the spectral domain but rather goes along some "channels" formed by quasi-resonances, that is, resonances with small enough frequency broadening (Fig.2, [25]). Non-resonant modes with comparatively The difference between the results of [25] and [38, 39] is as follows. Irrotational capillary waves do not pos- sess exact resonances [26] which yields the appearance of only fluxless modes; on the other hand, resonances do exist among atmospheric planetary waves. Accordingly, in [25] two types of modes had been singled out — frozen and flux-generating. The specifics of the discrete wave turbulent regime is due to the fact that its time evolu- tion is governed by the dynamical equations, not by the kinetic equation [29]; accordingly, phases of single modes are coherent and random phase approximation (necessary for deducing the wave kinetic equation) does not hold. The role of capillary water waves in the general theory of wave turbulence is invaluable. One can say that most new groundbreaking results and effects in (weak) wave turbulent systems have been first discovered for capillary water waves. Seminal work of Zakharov and Filonenko [58] laid in 1967 the very foundation for what later be- came known as statistical (or kinetic) wave turbulence theory: the first power energy spectrum has been ob- tained for capillary water waves, as an exact solution of the wave kinetic equation. The wave kinetic equation describes the wave field evolution at the finite inertial in- terval 0 < ( k$_{1}$, k$_{2}$ ) < ∞ in k -space. At this interval the energy of a wavesystem is supposed to be constant while forcing (usually in the small scales 0 < k$_{for}$ < k$_{1}$ ) and dissipation (in the scales k$_{2}$ < k$_{dis}$ ) are balanced [59]. Capillary water waves are free surface water waves, the range of wavelengths being λ < 17 mm . In spite of their small amplitude, capillary waves play an important role in the dynamics of the ocean surface. Wind gen- erated capillary waves provide roughness on the ocean surface which then yields subsequently the appearance of capillary-gravity and gravity waves [34, 37, 43]. On the other hand, capillary waves can be pumped by gravity waves [36, 54] and therefore they also transport energy from bigger to smaller scales. arXiv:1001.1497v2 [math-ph] 12 May 2010 I. INTRODUCTION