Feynman diagrams and interaction rules of wave‐wave scattering processes

Abstract

The energy transfer due to weak nonlinear interactions in random wave fields is reinterpreted in terms of a hypothetical ensemble of interacting particles, antiparticles, and virtual particles. In the particle picture, the interactions can be conveniently described by Feynman diagrams, which may be regarded either as branch diagrams of the perturbation expansion or as collision diagrams. The derivation of the transfer expressions can then be reduced to a few general rules for the construction of the diagrams and the associated collision cross sections. The representation follows closely the standard treatment of nonlinear lattice vibrations, but the particle picture differs from the usual phonon interpretation of lattice waves. It has the unreMistie property that the energies and number densities of antiparticles are negative. This is offset by simpler interaction rules and a closer correspondence between the perturbation graphs and collision diagrams. The method is illustrated for scattering processes in the oceanic wave guide involving surface and internal gravity waves, horizontal currents (turbulence), seismic waves, and bottom irregularities. Nonlinear interactions between surface gravity waves have been studied byPhillips [1960], Hasselmann [1962; 1963a, b], Longuet-Higgins [1062], Benney [1962], Bretherton [1064], among others. Computations of the energy transfer for a random wave field were found to be in qualitative agreement with the observed decay of ocean swell [Snodgrass et al., 1965]. 2 K. HASSELMANN Scattering from gravity waves into elastic waves is one of the principal sources of microseisms. The conversion can occur either through direct nonlinear interactions between gravity waves [Longuet-Higgins, 1950] or through the interaction of gravity waves with an inhomogeneous ocean bottom [Wiechert, 1904]. Both processes have been quantitatively confirmed by measurements byHaubrich et al. [1963] [Hasselmann, 1963c]. The generation of long gravity waves with periods in •he minute range ('surf bea•' [Munk, 1949; Tucker, 1950]) may be a•tributed to a nonlinear gravity wave interaction plus a bottom interaction [Longuet-Higgins and Stewart, 1962; Hasselmann et al., 1963; Gallagher, 1965]. Sca•ering between surface and in•ernal gravity waves has been investigated by Ball [1964], Thorpe [1966], and Kenyon [1966]. Kenyon has computed •ransfer ra•es for both gravity wave and Rossby wave interactions. Nonlinear interactions between surface gravity waves and quasi-s•eady curren•s or •urbulence have been s•udied by Longuet-Higgins and Stewart [1961] and Phillips [1959]. Sca•ering processes very similar •o •hese geophysical examples have also been considered in o•her fields. In a fundamental paper on •he hea• conduction in solids, Peierls [1929] investigated •he •ransfer effects due •o secular interactions between random lattice vibrations. Nonlinear interactions between ligh• waves and lattice vibrations, firs• investigated in •he classic papers of Brillouin [1922] and Raman [1928], have la•ely received renewed ...

Resonant and Near-Resonant Internal Wave Interactions

“…Kenyon (1968) states (without detail) that Kenyon (1966) and Hasselmann (1966) give numerically similar results. We have found that Kenyon (1966) differs from the four approaches examined below on one of the resonant manifolds but have not pursued the question further.…”

“…The first kinetic equations for wave-wave interactions in a continuously stratified ocean appear in Kenyon (1966), Hasselmann (1966), and Kenyon (1968). Kenyon (1968) states (without detail) that Kenyon (1966) and Hasselmann (1966) give numerically similar results.…”

Noise generation in the solid Earth, oceans and atmosphere, from nonlinear interacting surface gravity waves in finite depth

“…Longuet-Higgins (1950) showed how seismic waves can be generated by the same process, with noise radiating along the Earth's crust in the form of Rayleigh waves. That theory was extended to random waves by Hasselmann (1963), and later cast in the more general framework of wave-wave interactions (Hasselmann 1966). Work on compressible flows has also been extended to the study of tsunamis.…”

Spectral energy transfer of atmospheric gravity waves through sum and difference nonlinear interactions

“…The studies on nonlinear interaction of gravity waves have made great advances by the theoretical, observational and modelling efforts. On the basis of the weak interaction approximation, many features and effects of resonant interaction of oceanic internal gravity waves have been acquired (Phillips, 1960;Bretherton, 1964;Hasselmann, 1966;Olbers, 1976;McComas and Bretherton, 1977;McComas and Müller, 1981;Müller et al, 1986). Subsequently, the properties of resonant and nonresonant interactions among atmospheric gravity waves have been extensively investigated (Dysthe et al, 1974;Liu, 1981, 1985;Klostermeyer, 1982Klostermeyer, , 1991Inhester, 1987;Dong and Yeh, 1988;Yeh and Dong, 1989), even in a sheared, dissipative and rotating atmosphere Axelsson et al, 1996;Yi and Xiao, 1997), and in the uniform and nonuniform plasmas (Stenflo, 1994;Stenflo and Shukla, 2009).…”