ENERGY SPECTRUM OF THE ENSEMBLE OF WEAKLY NONLINEAR GRAVITY-CAPILLARY WAVES ON A FLUID SURFACE
E. Tobisch*
Institute for Analysis, Johannes Kepler University 4040, Linz, Austria
Received March 27, 2014
We consider nonlinear gravity-capillary waves with the nonlinearity parameter s ~ 0.1-0.25. For this nonli-nearity, time scale separation does not occur and the kinetic wave equation does not hold. An energy cascade in this case is built at the dynamic time scale (D-cascade) and is computed by the increment chain equation method first introduced in [15]. We for the first time compute an analytic expression for the energy spectrum of nonlinear gravity-capillary waves as an explicit function of the ratio of surface tension to the gravity acceleration. We show that its two limits — pure capillary and pure gravity waves on a fluid surface — coincide with the previously obtained results. We also discuss relations of the D-cascade model with a few known models used in the theory of nonlinear waves such as Zakharov's equation, resonance of modes with nonlinear Stokes-corrected frequencies, and the Benjamin-Feir index. These connections are crucial in understanding and forecasting specifics of the energy transport in a variety of multicomponent wave dynamics, from oceanography to optics, from plasma physics to acoustics.
DOI: 10.7868/S0044451014080173
1. INTRODUCTION
Until recently, the notion of an "energy cascade" in a weakly nonlinear wave system was traditionally associated with kinetic wave turbulence theory (WTT), where the energy spectrum is a stationary solution of the wave kinetic equation. The wave kinetic equation was first introduced in 19C2 by Hasselmann [1], and its first stationary solution (for capillary waves) was found in 19C7 by Zakharov and Filonenko [2]. Subsequently, their method of finding stationary solutions was generalized to various weakly nonlinear wave systems with dispersion [3].
The kinetic equation can be solved numerically for any nontrivial dispersion function ui with a given dispersion relation u) = u>(k), where k is the wave vector. In the particular case of the dispersion function of the form
ui ~ ka, k = |k|, a: > 1,
kinetic WTT gives an analytic prediction for the energy spectrum in the power-law form ~ k^^, 3 > 0, where 3 is different for wave systems with different disper-
* E-mail: Elena.Tobisch'ffljku.at
sion functions but does not depend 011 the excitation parameters.
The prediction holds in the so-called inertial interval, where forcing and dissipation are balanced such that energy is conserved within this interval (it is assumed that pumping and dissipation are spaced far apart in Fourier space). The basic physical mechanism leading to the formation of a kinetic energy cascade (Iv-cascade) is the ,s-wave resonance interactions of linear Fourier modes
A(t/es^'2) exp i[Am: - ujt] (1)
with slowly changing amplitudes. The ,s-wave resonances occur independently at different time scales t/es~2, .1 > 3, where 0 < e -C 1 is a small parameter, e.g., e ~ 10-2 for water waves.
Rapid technological progress in the field of measurement methods and measuring techniques allowed a systematic study of the spectrum in various fluid systems in the past two decades. The experimental data turned out to be rather contradictory, including chains such as: the energy spectrum is not formed, and energy exchange within a small set of Fourier modes occurs instead; the energy spectrum and a power law are observed, but the exponent differs from the one predicted by kinetic WTT; the exponent depends 011 the param-
otors of the initial excitation; the inertial interval does not exist, and so on. Without claiming to be exhaustive, we give a few references to the most thorough and credible recent experiments [48]. A very respectable list of references can be found in a recent review by Newell and Rumpf [9].
Some of these effects have found their explanation in the framework of the discrete WTT [10, 11]; for instance, the absence of the inertial interval is duo to the nonlocality of resonance interaction, for some types of dispersion functions. The locality of interaction in kinetic WTT is understood as follows: only the interaction of waves with wavelengths of the same order is allowed. However, it has been known for more than 20 years [12, 13] that, say, capillary waves with wavelengths of the orders k and k?' can interact directly, i.e., build a joint resonance triad; more examples can be found in [14].
The model of the energy spectra formation in wave systems with weak and moderate nonlinearity allowing the observed experimental shape of the energy spectrum to be reconciled with the predictions made for the Iv-cascade was first proposed in 2012 by Ivar-tashova [15]. In this model, the triggering physical mechanism for an energy cascade formation is the modulation instability (MI), and the corresponding energy-cascade is called a dynamical cascade (D-cascade); a D-cascade is a sequence of distinct modes in Fourier space. The use of the specially developed increment chain equation method (ICEM) allows computing the energy spectrum of a D-cascade.
The energy spectrum in the D-model is a solution of the so-called chain equation. It connects frequencies and amplitudes of two adjacent modes in D-cascade. Energy spectra for capillary and surface water waves (with the respective dispersion functions ui2 = ak?' and
= 9k) are computed in [15] for different values of a small parameter e ~ 0.1 0.4 chosen as the ratio of the wave amplitude to the wave length.
Here, we sketch the ICEM and compute the energy spectrum of an ensemble of weakly nonlinear gravity-capillary waves with the dispersion function
ui2 = gk + ak3.
We also demonstrate intrinsic mathematical connections between the D-model and other models describing nonlinear wave interaction at the same temporal and spatial scales: Zakharov's equation, resonances of nonlinear Stokes waves, and the Benjamin Foir index.
2. INCREMENT CHAIN EQUATION METHOD (ICEM)
The physical mechanism underlying the formation of a D-cascade is modulation instability, which can be described as the decay of a carrier wave u>o into two side bands uii and ui-2-
+ uj~2 = 2u?o, h + k-2 = 2k0 + ©, (2)
uii = uo + Au>, u>2 = u;o — Au;, 0 < Aui -C 1. (3)
A wave train with the initial real amplitude ,4, wa-venumber k = and frequency ui is modulationally unstable if
0 < Aui/Akui < s/2. (4)
Equation (4) describes an instability interval for the wave systems with a small nonlinearity of the order of e ~ 0.1 to 0.2, first obtained in [16]. It is also established for gravity surface waves that the most unstable modes in this interval satisfy the condition
Auj/Akuj = 1. (5)
The essence of the ICEM is the use of (5) for computing the frequencies of the cascading modes. At the first step of the D-cascade, a carrier mode has a frequency tJo and the distance to the next cascading mode
(Aui)l = |u>0 — ijJ\ I
with the frequency u>o chosen such that condition (5) is satisfied, i.e.,
|u;0 - u;i| = A0k0^0.
At the second step of the D-cascade, a carrier mode has the frequency uii, the distance to the next cascading mode
(Au?) 2 = |u?l — 0u?2 | is chosen such that
|u>i — UJ2I = Aikiuii,
and so 011.
In this way, a recursive relation for the cascading modes can easily be obtained:
s/pnAn = A(uj„, ± ujnAnkn). (6)
Here, we let pn denote the fraction of energy transported from the cascading mode
An = .4(u;„)
to the cascading mode
3) Chain equations:
= .4(u;„.+i), ,4„. This fraction p is called the cas-
i. e., A„+i = y/jxi] cade intensity [15].
Equation (6) describes two chain equations: one chain equation with the plus sign for a direct D-cascade with uin < ui„+i and another chain equation for the inverse D-cascade with uin > u>„+i.
Speaking generally, the cascade intensity
Pn />„(.•1,).^,).//)
might be a function of the excitation parameters .4o,u;o and the step n. But because numerous experiments have established that p„, depends only on the excitation parameters and does not depend on the step n, all the formulas below are given for a constant cascade intensity. Accordingly, the notation p is used instead of pn. This means in particular that
-4„+i —
— P ' -*lo
and because the energy behaves as En~ A*
it follows that
E„ ~ p" -4g
i.e., the energy spectrum of the D-cascade amplitudes has an exponential form.
Taking a Taylor expansion of the right-hand side of the chain equation and retaining only the first two terms of the resulting series, we can derive an ordinary differential equation describing stationary amplitudes of the cascading modes. The consequent steps of the ICEM are given below.
1) Relation between neighboring amplitudes:
-4„+i — s/pAn.
2) Condition for the maximal increment:
f (^nAnkn
= 1.
(7)
(8)
where f(uinAnkn) is a known function of the product u)nAnkn. For instance,
,f(^nAnkn) = u)nAnkn
for gravity surface waves with the small parameter of the order of 0.1 0.2. Examples for bigger nonlinearities and also for other wave types can be found in [17, 18].
-4„+i = .4(u;„.+i ) =
= A(uj„, ± f(ujnAnkn)) =
0)
where the plus sign should be taken for the direct cascade and the minus for the inverse cascade.
4) Approximate ODE(s) for the amplitude An:
\fpAn ~ A„ ± Anf(u)nAnkn
5) Discrete energy spectrum En ~ ,4 C) Spectral density
5 M =
dEn
lim
n-s-oo UUln
(10)
(ID
In particular, the formula below gives an explicit expression for wave amplitudes (for the direct cascade) in the case of a small initial nonlinearity e ~ 0.1 0.25:
-4(ui„) =
1)
du,,
(12)
Let us remark, that it is known that if the autocorrelation function involves temporal measurements at a single point, then the power spectrum has the units m2/'Hz. It is easy to verify that the spectral density S(ui) has the correct units. Indeed, as we compute the amplitudes at single points, the amplitudes .4(u;„) have the units m, then their squares ,4(u;„)2 have the unit
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