Turbulence in a system of nonlinearly interacting waves is referred to as wave turbulence. It has been known since seminal work by Kolmogorov, that turbulent dynamics is controlled by a directional energy flux through the wavelength scales. We demons
trate that an energy cascade in wave turbulence can be bi-directional, that is, can simultaneously flow towards large and small wavelength scales from the pumping scales at which it is injected. This observation is in sharp contrast to existing experiments and wave turbulence theory where the energy flux only flows in one direction. We demonstrate that the bi-directional energy cascade changes the energy budget in the system and leads to formation of large-scale, large-amplitude waves similar to oceanic rogue waves. To study surface wave turbulence, we took advantage of capillary waves on a free, weakly charged surface of superfluid helium He-II at temperature 1.7K. Although He-II demonstrates non-classical thermomechanical effects and quantized vorticity, waves on its surface are identical to those on a classical Newtonian fluid with extremely low viscosity. The possibility of directly driving a charged surface by an oscillating electric field and the low viscosity of He-II have allowed us to isolate the surface dynamics and study nonlinear surface waves in a range of frequencies much wider than in experiments with classical fluids.
We study the original $alpha$-Fermi-Pasta-Ulam (FPU) system with $N=16,32$ and $64$ masses connected by a nonlinear quadratic spring. Our approach is based on resonant wave-wave interaction theory, i.e. we assume that, in the weakly nonlinear regime
(the one in which Fermi was originally interested), the large time dynamics is ruled by exact resonances. After a detailed analysis of the $alpha$-FPU equation of motion, we find that the first non trivial resonances correspond to six-wave interactions. Those are precisely the interactions responsible for the thermalization of the energy in the spectrum. We predict that for small amplitude random waves the time scale of such interactions is extremely large and it is of the order of $1/epsilon^8$, where $epsilon$ is the small parameter in the system. The wave-wave interaction theory is not based on any threshold: equipartition is predicted for arbitrary small nonlinearity. Our results are supported by extensive numerical simulations. A key role in our finding is played by the {it Umklapp} (flip over) resonant interactions, typical of discrete systems. The thermodynamic limit is also briefly discussed.
We report evaluations of a resonant kinetic equation that suggest the slow time evolution of the Garrett and Munk spectrum is {em not}, in fact, slow. Instead nonlinear transfers lead to evolution time scales that are smaller than one wave period at
high vertical wavenumber. Such values of the transfer rates are inconsistent with conventional wisdom that regards the Garrett and Munk spectrum as an approximate stationary state and puts the self-consistency of a resonant kinetic equation at a serious risk. We explore possible reasons for and resolutions of this paradox. Inclusion of near-resonant interactions decreases the rate at which the spectrum evolves. This leads to improved self-consistency of the kinetic equation.
Many major oceanographic internal wave observational programs of the last 4 decades are reanalyzed in order to characterize variability of the deep ocean internal wavefield. The observations are discussed in the context of the universal spectral mode
l proposed by Garrett and Munk. The Garrett and Munk model is a good description of wintertime conditions at Site-D on the continental rise north of the Gulf Stream. Elsewhere and at other times, significant deviations in terms of amplitude, separability of the 2-D vertical wavenumber - frequency spectrum, and departure from the models functional form are noted. Subtle geographic patterns are apparent in deviations from the high frequency and high vertical wavenumber power laws of the Garrett and Munk spectrum. Moreover, such deviations tend to co-vary: whiter frequency spectra are partnered with redder vertical wavenumber spectra. Attempts are made to interpret the variability in terms of the interplay between generation, propagation and nonlinearity using a statistical radiative balance equation. This process frames major questions for future research with the insight that such integrative studies could constrain both observationally and theoretically based interpretations.
We obtain a canonical form of a quadratic Hamiltonian for linear waves in a weakly inhomogeneous medium. This is achieved by using the WKB representation of wave packets. The canonical form of the Hamiltonian is obtained via the series of canonical B
ogolyubov-type and near-identical transformations. Various examples of the application illustrating the main features of our approach are presented. The knowledge of the Hamiltonian structure for linear wave systems provides a basis for developing a theory of weakly nonlinear random waves in inhomogeneous media generalizing the theory of homogeneous wave turbulence.
To investigate the formation mechanism of energy spectra of internal waves in the oceans, direct numerical simulations are performed. The simulations are based on the reduced dynamical equations of rotating stratified turbulence. In the reduced dynam
ical equations only wave modes are retained, and vortices and horizontally uniform vertical shears are excluded. Despite the simplifications, our simulations reproduce some key features of oceanic internal-wave spectra: accumulation of energy at near-inertial waves and realistic frequency and horizontal wavenumber dependencies. Furthermore, we provide evidence that formation of the energy spectra in the inertial subrange is dominated by scale-separated interactions with the near-inertial waves. These findings support oceanographers intuition that spectral energy density of internal waves is the result of predominantly wave-wave interactions.
