Download Spectra of nonlinear unidirectional sea waves

Spectra of nonlinear unidirectional sea waves
A. Slunyaev1,2
1
Institute of Applied Physics, Nizhny Novgorod, Russia
2
Nizhny Novgorod State Technical University, Russia
The spatial and temporal scales of linear waves are related by the dispersion law.
If the nonlinearity is in force, then the dispersion relation is disturbed (i.e, wave
frequencies are different for a given wavelength), combination harmonics may
appear, and wave harmonics may effectively exchange energy due to resonances.
The consideration of exact resonance conditions (based on the linear theory) is
a fruitful approach to elaboration of effective and beautiful models [1]. At
the same time near-resonant interactions are not less important for the wave
evolution [2, 3]. Due to nonlinearity the graphs of spatial and frequency spectra
become hard to read. The spectral images of wave phenomena are obtained in
a number of recent works by virtue of direct measurements (e.g., [4]). In this
paper the spectral data are retrieved from numerical stochastic simulations of
collinear surface gravity water waves, similar to [5].
From approximate envelope equations (nonlinear Schrödinger, NLS, and Dysthe
equations) to fully nonlinear Euler equations are simulated. The initial condition
for simulations of the Euler equations and the Dysthe equation corresponds to
the JONSWAP spectrum. Relatively small waves (with no breaking), and also
steep waves with occasional breaking are considered. In the latter case the
artificial damping at short scales is introduced to regularize the breaking effect.
The NLS equation is used to simulate waves with equivalent Gaussian spectrum.
The situations from deep (kp h ≫ 1) to intermediate depths (kh ≈ 0.8) are
considered (kp is the spectrum peak wavenumber and h is the water depth).
One example of the wavenumber-frequency spectrum S(k, ω) is shown with
colours in Fig. 1 in logarithmic coordinates. Several energetic areas in the form
of strips may be clearly seen; they are well-separated on the (k, ω) plane. The
main spectral strip, which corresponds
√ to free waves, is spread along the linear
dispersion curve ω = Ω(k), Ω(k) = kg tanh kh, g is the gravity acceleration.
Other spectral strips correspond to bound waves. Multiple harmonics (marked
with white broken lines) may be easily distinguished in Fig. 1; there is also a
low-frequency strip due to interactions between free wave harmonics.
The wavenumber spectrum or frequency spectrum may be obtained by corresponding integration of S(k, ω). The spectral amplitudes of opposing waves
(k < 0) are at least 103 times smaller than the maximum spectral amplitude.
The frequency spectrum in Fig. 1 for rightward propagating waves (k > 0)
agrees well with the JONSWAP function.
The investigation of spectral pictures for various situations allows to estimate
1
Figure 1: Normalized wavenumber-frequency spectrum log10 S(ω, k) for waves
with JONSWAP spectrum (Tp = 10 s, kp = 0.04 rad/m, γ = 3) in a relatively
deep basin kp h ≈ 2, with ηrms ≈ 0.8 m (significant wave steepness 2kp ηrms ≈
0.06). Simulations of the Euler equations. The white broken lines show curves
nω = Ω(nk) for n = 1, 2, 3, 4.
the most productive nonlinear combinations of waves; to see intersections of
spectral areas occupied by free and bound waves; to observe the occurrence of
a new system of long waves; to visualize restrictions of the envelope equations
due to occurrence of non-physical effects.
References
[1] E. Kartashova. Nonlinear resonance analysis: theory, computation, applications. Cambridge University Press (2010).
[2] M. Stiassnie, L. Shemer. On the interaction of four water-waves. Wave
Motion 41, 307 (2005).
[3] S.Y. Annenkov, V.I. Shrira. Role of non-resonant interactions in the evolution of nonlinear random water wave fields. J. Fluid Mech. 561, 181
(2006).
[4] T.M.A. Taklo, K. Trulsen, O. Gramstad, H.E. Krogstad, A. Jensen. Measurement of the dispersion relation for random surface gravity waves. J.
Fluid Mech. 766, 326 (2015).
[5] A. Sergeeva, A. Slunyaev. Rogue waves, rogue events and extreme wave
kinematics in spatio-temporal fields of simulated sea states. Nat. Hazards
Earth Syst. Sci. 13, 1759 (2013).
2