Computational Physics 3 JAN, 2022

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

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 experiment aimed at measuring the dispersion relation of surface gravity waves when a large‑scale background, or “condensate,” is present. In previous isotropic turbulence simulations of gravity waves, the authors observed that energy injected at large scales accumulates in a low‑wavenumber region, forming a coherent, high‑amplitude long‑wave component that they refer to as a condensate. Earlier work (Korotkevich, Phys. Rev. Lett. 101, 074504, 2008) demonstrated that this condensate modifies the direct cascade spectrum (the flux of energy toward smaller scales), changing the spectral exponent from the weak‑turbulence prediction of k⁻⁵ᐟ² to a steeper k⁻⁴. The mechanism proposed there relies on the condensate acting as an intermediate mediator in three‑wave interactions, thereby altering the efficiency of energy transfer from the forcing band to the dissipative scales.

However, the inverse cascade—characterized by a flux of wave action toward larger scales—depends critically on the linear dispersion relation ω(k)=√(gk). If the condensate perturbs the actual ω‑k relationship, the inverse cascade spectrum will also be affected, but the previously suggested mechanism (mediated energy transfer) cannot explain such changes because the inverse cascade is not driven by energy flux. Consequently, a direct measurement of the dispersion relation in the presence of the condensate is required.

To achieve this, the authors performed high‑resolution DNS of the Euler equations for an incompressible, infinitely deep fluid with periodic boundary conditions. Energy is injected through a narrow band of low‑wavenumber modes (the “pumping region”), and the system evolves under fully nonlinear interactions. The surface elevation η(x,t) is recorded over many wave periods. By applying a two‑dimensional Fourier transform in space and time, they obtain η(k,ω) spectra. For each wavenumber k, the peak of the ω‑distribution defines a measured central frequency ω_center(k). Comparing ω_center(k) with the theoretical √(gk) curve reveals systematic deviations that correlate with the presence of the condensate.

The results can be summarized as follows:

  1. Condensate‑dominated low‑k region (k ≲ k_c) – where k_c marks the condensate peak – the measured frequencies lie below the linear prediction. The shift is substantial, indicating that the condensate effectively reduces the phase speed of waves, possibly by providing an additional “mass” or “pressure” background that modifies the restoring force.

  2. Intermediate scales (k_c < k < k_d) – with k_d denoting the onset of the inverse cascade – the measured dispersion gradually approaches the linear curve but retains a small systematic offset.

  3. High‑k region (k ≫ k_d) – well within the direct cascade – the condensate influence disappears and ω_center(k) aligns with √(gk) to within numerical precision.

Crucially, in the wavenumber band where the inverse cascade is active, the slight downward shift of ω(k) reduces the group velocity and, consequently, the flux of wave action toward larger scales. This manifests as a steeper inverse‑cascade spectrum than the classical weak‑turbulence prediction of k⁻²³ᐟ²; the numerical data are closer to a k⁻³ scaling. Hence, the condensate alters the inverse cascade not through a three‑wave energy‑mediating mechanism, but by directly modifying the linear dispersion relation that underpins the cascade dynamics.

The authors conclude that the condensate should be regarded as a dynamical entity that influences both nonlinear energy redistribution and the fundamental linear wave properties. Any theoretical or operational model of oceanic gravity‑wave spectra that seeks to describe the inverse cascade must therefore incorporate the possibility of a condensate‑induced dispersion‑relation shift. The study opens a pathway for future work to embed condensate effects into kinetic equations, improve the interpretation of field measurements, and refine forecasting tools for sea‑state evolution.