Direct Numerical Investigation of Turbulence of Capillary Waves

Abstract

We consider the inertial range spectrum of capillary wave turbulence. Under the assumptions of weak turbulence, the theoretical surface elevation spectrum scales with wave number $k$ as ${I}_{\ensuremath{\eta}}\ensuremath{\sim}{k}^{\ensuremath{\alpha}}$, where $\ensuremath{\alpha}={\ensuremath{\alpha}}_{0}=\ensuremath{-}19/4$, energy (density) flux $P$ as ${P}^{1/2}$. The proportional factor $C$, known as the Kolmogorov constant, has a theoretical value of $C={C}_{0}=9.85$ (we show that this value holds only after a formulation in the original derivation is corrected). The ${k}^{\ensuremath{-}19/4}$ scaling has been extensively, but not conclusively, tested; the ${P}^{1/2}$ scaling has been investigated experimentally, but until recently remains controversial, while direct confirmation of the value of ${C}_{0}$ remains elusive. We conduct a direct numerical investigation implementing the primitive Euler equations. For sufficiently high nonlinearity, the theoretical ${k}^{\ensuremath{-}19/4}$ and ${P}^{1/2}$ scalings as well as value of ${C}_{0}$ are well recovered by our numerical results. For a given number of numerical modes $N$, as nonlinearity decreases, the long-time spectra deviate from theoretical predictions with respect to scaling with $P$, with calculated values of $\ensuremath{\alpha}<{\ensuremath{\alpha}}_{0}$ and $C>{C}_{0}$, all due to finite box effect.