Direct numerical experiment on measuring of dispersion relation for gravity waves in the presence of condensate

During previous numerical experiments on isotropic turbulence of surface gravity waves we observed formation of the long wave background (condensate). It was shown (Korotkevich, Phys. Rev. Lett. vol. 101 (7), 074504 (2008)), that presence of the condensate changes a spectrum of direct cascade, corresponding to the flux of energy to the small scales from pumping region (large scales). Recent experiments show that the inverse cascade spectrum is also affected by the condensate. In this case mechanism proposed as a cause for the change of direct cascade spectrum cannot work. But inverse cascade is directly influenced by the linear dispersion relation for waves, as a result direct measurement of the dispersion relation in the presence of condensate is necessary. We performed the measurement of this dispersion relation from the direct numerical experiment. The results demonstrate that in the region of inverse cascade influence of the condensate cannot be neglected.

💡 Research Summary

The paper presents a direct numerical investigation of how a long‑wave background, termed “condensate,” influences the dispersion relation of surface gravity waves and, consequently, the inverse cascade spectrum. Using the weakly nonlinear equations for surface elevation η and surface velocity potential ψ (Eq. 1), the authors simulate an ideal incompressible fluid on a doubly periodic domain (2π × 2π) with high‑resolution grids (1024 × 1024 and a longer run on 256 × 256). Gravity is normalized to g = 1, and a pseudo‑viscous damping γₖ is applied for k > k_d (k_d = 256 for the large grid, 64 for the small grid) with a quadratic dependence on (k − k_d). Energy is injected through an isotropic, narrow‑band forcing in wave‑number space (k ≈ 28–32) with random phases and a small amplitude (F₀ = 1.5 × 10⁻⁵). The initial condition is low‑level white noise across all modes.

The simulations reproduce both a direct energy cascade toward high wave numbers and an inverse cascade of wave action toward low wave numbers. Importantly, a strong condensate forms near a characteristic wave number k_c ≈ 30, where the wave‑action flux stalls because the discrete wave‑number grid prevents exact resonant interactions. The condensate’s amplitude exceeds that of neighboring modes by more than an order of magnitude, creating a quasi‑static large‑scale background.

To assess the impact of this condensate on the linear dispersion relation ω = √(gk), the authors record the complex normal variables aₖ(t) for a set of modes along the kₓ‑axis (from k = 4 up to k = 180, sampling every tenth mode at higher k). They then compute the temporal Fourier transform aₖ(ω). In a purely linear system, each aₖ(ω) would be a delta‑function at ω = √(gk). Instead, the spectra display a central peak at the linear frequency plus pronounced sidebands. The sidebands are interpreted as the result of three‑wave interactions between a mode aₖ₀ and the condensate mode a_{k_c}: the product aₖ₀ a_{k_c} generates an upper sideband at ω_up = ω(k₀ + k_c) + ω(k_c), while aₖ₀ a*_{k_c} yields a lower sideband at ω_low = ω(k₀ − k_c) ± ω(k_c). The authors plot theoretical sideband locations (various line styles in Fig. 2) and find good agreement with the measured spectra.

Crucially, in the inverse‑cascade region (k ≈ 10–30) the sideband amplitudes are comparable to, or even larger than, the central linear peak, indicating that the condensate substantially distorts the effective dispersion relation. By contrast, in the direct‑cascade region (k ≫ k_c) the sidebands are negligible and only a slight nonlinear frequency shift of the central line is observed. This asymmetry explains why the inverse‑cascade spectral slope observed in the simulations (≈ k⁻³·⁵) deviates from the weak‑turbulence prediction (k⁻²³⁄⁶ ≈ k⁻³·⁸³). The deviation can be attributed to the condensate‑induced modification of the dispersion relation, which directly enters the Zakharov–Hasselmann kinetic equation through the exponent α in ω ∝ k^α.

The authors conclude that the presence of a condensate invalidates the assumption of an unchanged linear dispersion relation in the inverse‑cascade regime. Consequently, the standard kinetic equation, which relies on that assumption, cannot fully describe the observed spectra. They suggest that future theoretical work must incorporate condensate‑wave interactions, possibly by extending the kinetic equation to include a modified dispersion relation or by treating the condensate as a coherent background field. Experimental verification would require high‑resolution wave‑tank measurements capable of resolving the sideband structure. Overall, the paper provides the first direct numerical measurement of the dispersion relation in a condensate‑laden gravity‑wave system and highlights its pivotal role in shaping inverse‑cascade dynamics.