218 - Österreichische Mathematische Gesellschaft

independent (to the first order) of the topography of the Earth. This model allows
us to reproduce to the correct order the periods of oscillations and also to interpret
the main features which are observed in measured data [11].
II. Energy cascades of surface water waves. Based on the structure of resonance
clustering in this specific wave system with resonance conditions (25), it is
possible to demonstrate the existence of two generic types of energy transport, via
angle- and scale-resonances. Angle-resonances do not transfer energy over scales,
they just redistribute the initial energy over angles; no new k-scales appear due to
this type of energy transport. Angle-resonances are formed by two pairs of wave
vectors with pairwise equal lengths k 1 = k 2 and k 3 = k 4 with k j = (m 2 j +n2 j )1/2 ; the
overwhelming part of resonances among surface water waves consists of angleresonances.
Scale-resonances are formed by quartets in which at least three wave
vectors have different lengths; they realize energy transport over scales. Scaleresonances
are very rare (in spectral domain k j ≤ 1000 there are about 6 hundred
million resonance quartets, among them only 3945 scale-resonances). Mixed cascades
are also possible and are described by clusters consisting of angle- and
scale-resonances simultaneously [9].
III. General form of energy cascade in wave turbulent systems. The classical
theory of wave turbulence (WT) [21] is based on the statistical description of wave
systems. This yields stationary energy spectra which do not depend on the initial
energy of a system. The assumptions used are in general similar to those for the
case of strong wave turbulence, only the players in the game are different: waves
instead of vortexes. The list of the most important assumptions includes (but is
not restricted to) the following: 1) smallness of nonlinearity; 2) infinite-box limit
(L/λ → ∞, where L is the size of the box and λ is the characteristic wavelength);
3) existence of inertial interval (k 1 ,k 2 ); 4) locality of interactions in k-space; etc.
However the results of various attempts to experimentally validate this theory are
rather controversial. In some cases, no cascade is generated; instead, regular patterns
are observed. Most experiments in which a cascade has been generated
show that it consists of two distinct parts – discrete and continuous; discrete and
sometimes also the continuous part are not described by classical spectra. An interesting
scenario has been observed in experiments with surface water waves in
a laboratory flume where discrete energy cascade developed a strongly nonlinear
regime, while no continuous spectrum has been observed. Moreover, it has been
reported more often than not that the form of the energy spectra depends on the
parameters of the initial excitation, etc. etc.
A great deal of effort has been spent on improving the classical WT theory –
mostly concentrated on explaining the form of observed energy cascades based on
the statistical approach. As a result, a number of new types of WT has introduced
in last decade, e.g. frozen WT, sandpile WT, mesoscopic WT, finite-dimensional
WT, etc. However, neither finite-size effects nor the effect of initial excitation can
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