Pis’ma v ZhETF, vol. 112, iss. 12, pp. 799 - 800
© 2020 December 25
Numerical simulation of collinear capillary-wave turbulence
E. Kochurin+1), G. Ricard∗, N. Zubarev+×, E. Falcon∗
+Institute of Electrophysics, Ural Division, Russian Academy of Sciences, 620016 Ekaterinburg, Russia
∗Université de Paris, Univ. Paris Diderot, MSC Laboratory, UMR 7057 CNRS, F-75 013 Paris, France
×P. N. Lebedev Physical Institute, Russian Academy of Sciences, 119991 Moscow, Russia
Submitted 28 October 2020
Resubmitted 28 October 2020
Accepted 9 November 2020
DOI: 10.31857/S1234567820240027
We report on direct numerical simulation of
of the fluid, and ω is the angular frequency. Equation
quasi-one-dimensional bidirectional capillary-wave tur-
(2) being obtained from (1) with S(k)dk = S(ω)dω.
bulence. Although nontrivial three-wave and four-wave
The expressions (1) and (2) are similar to the well
resonant interactions are absent in this peculiar geom-
known Zakharov-Filonenko spectrum of isotropic cap-
etry, we show that an energy transfer between scales
illary wave turbulence [2]. Equations (1)-(2) are ob-
still occurs concentrated around the linear dispersion
tained under the assumption of the dominant influence
relation that is broadened by nonlinearity. The wave
of three-wave interactions.
spectrum displays a clear wave number power-law
At the next order (four-wave interactions), the
scaling that is found to be in good agreement with
anisotropic capillary-wave turbulence spectrum reads
the dimensionally prediction for capillary-wave tur-
bulence involving four-wave interactions. The carried
(σ)-1/2
S(k) = C4w1DP1/3
k-7/2,
(3)
out high-order correlation analysis (bicoherence and
ρ
tricoherence) confirms quantitatively the dominant
2
(σ)1/3
role of four-wave quasi-resonant interactions. The
S(ω) =
C4w1DP1/3
ω-8/3.
(4)
3
ρ
Kolmogorov-Zakharov’s (KZ) spectrum constant is
also estimated numerically. We interpret our results as
Note that the predictions of (3)-(4) involving four-wave
the first numerical observation of anisotropic capillary-
interactions differ from the ones of (1)-(2) obtained for
wave turbulence in which four-wave interactions play a
the three-wave system. The difference in the k-power-
dominant role.
law exponent is roughly 7 %: -15/4 = -3.75 (three-
Let us now rewrite the wave-elevation spectrum pre-
wave system) vs. -7/2 = -3.5 (four-wave system). The
diction of weak turbulence obtained by dimensional
aim of this work is to perform direct numerical simula-
analysis following [1]. Introduce the wave elevation
tion of anisotropic capillary-wave turbulence with high
power spectrum as S(k) = |ηk|2, where ηk is the spa-
accuracy.
tial Fourier transform of the wave elevation η(x) at the
The numerical model used in the present work is
location x. Assuming that the leading order process is
based on the cubic nonlinear equations of the boundary
three-wave interaction, we obtain the prediction of the
motion. The numerical integration scheme for the solu-
anisotropic (quasi-1D) capillary-wave turbulence spec-
tion of the governing equations in time is based on the
trum
fourth-order explicit Runge-Kutta method. The spatial
)-3/4
derivatives and integral operators are calculated using
(σ
the pseudo-spectral methods. The numerical simulation
S(k) = C3w1DP1/2
k-15/4,
(1)
ρ
results indeed show that a capillary-wave turbulence
)1/6
2
(σ
regime is observed in the peculiar 1D bidirectional ge-
S(ω) =
C3w1DP1/2
ω-17/6,
(2)
ometry, once a high enough level of pumping is reached.
3
ρ
Figure 1
shows the time-averaged spatial-power
where C3w1D is the KZ constant, P is the energy dissipa-
spectrum S(k) of the wave height η(x, t) in the station-
tion rate, σ is the surface tension, ρ is the mass density
ary state. A clear power-law scaling is observed on more
than one decade in k. The best fit is S(k) ∼ k-3.5±0.1
1)e-mail: kochurin@iep.uran.ru
which is closer to the prediction of (3) than to the
Письма в ЖЭТФ том 112 вып. 11 - 12
2020
799
