A high-order asymptotic analysis of the Benjamin–Feir instability spectrum in arbitrary depth

Abstract

We investigate the Benjamin–Feir (or modulational) instability of Stokes waves, i.e. small-amplitude, one-dimensional periodic gravity waves of permanent form and constant velocity, in water of finite and infinite depth. We develop a perturbation method to describe to high-order accuracy the unstable spectral elements associated with this instability, obtained by linearizing Euler's equations about the small-amplitude Stokes waves. These unstable elements form a figure-eight curve centred at the origin of the complex spectral plane, which is parametrized by a Floquet exponent. Our asymptotic expansions of this figure-eight are in excellent agreement with numerical computations as well as recent rigorous results by Berti et al. (Full description of Benjamin–Feir instability of Stokes waves in deep water, 2021, arXiv:2109.11852) and Berti et al. (Benjamin–Feir instability of Stokes waves in finite depth, 2022, arXiv:2204.00809). From our expansions, we derive high-order estimates for the growth rates of the Benjamin–Feir instability and for the parametrization of the Benjamin–Feir figure-eight curve with respect to the Floquet exponent. We are also able to compare the Benjamin–Feir and high-frequency instability spectra analytically for the first time, revealing three different regimes of the Stokes waves, depending on the predominant instability.

Canonical conformal variables based method for stability of Stokes waves

“…Both waves are well below the threshold of $$ka = 0.429153$$ for the superharmonic instability 14,15,17 . The asymptotic results in the work 11 provide theoretical curves for BF instability and work particularly well in the limit $$ka\rightarrow 0$$. They provide an excellent agreement for the wave with $$ka = 0.1$$ (blue circles), where the red dashed line corresponds to the theoretical curve with $$O(\varepsilon ^2)$$, and the red solid line corresponds to the curve with $$O(\varepsilon ^3)$$.…”

“…The theoretical results from Ref. 11 with $$O(\varepsilon ^2)$$ (dashed line) and $$O(\varepsilon ^3)$$ (solid line) are drawn with red and gold lines for $$ka = 0.1$$ and $$ka = 0.25$$, respectively. We observe that $$O(\varepsilon ^2)$$ overestimates the growth rate and $$O(\varepsilon ^3)$$ underestimates it.…”

“…An accurate description of the instability spectrum for small amplitude Stokes waves was established in Refs. 10 and 11. New high‐frequency subharmonic instabilities for finite‐amplitude Stokes waves have been discovered in Ref.…”

“…Equations governing the motion of a fluid surface in conformal variables were derived in [29,30,31,32,33], and wave breaking was numerically explored in [34,35]. Instabilities of Stokes wave have been studied in many works [36,37,38,39,28,40,41,42,43] including numerical studies [44,45,46,47]. Stokes waves are unstable to long wave perturbations, a.k.a the subharmonic perturbations, such as the Benjamin-Feir (BF) or modulational instability, first reported in [36,48,39] and high-frequency instability [44,49,50,42].…”

“…Instabilities of Stokes wave have been studied in many works [36,37,38,39,28,40,41,42,43] including numerical studies [44,45,46,47]. Stokes waves are unstable to long wave perturbations, a.k.a the subharmonic perturbations, such as the Benjamin-Feir (BF) or modulational instability, first reported in [36,48,39] and high-frequency instability [44,49,50,42]. Recent theoretical studies of Benjamin-Feir instabillity for small amplitude waves are described in [49,41,43].…”