fluids Article Turbulence of Capillary Waves on Shallow Water Natalia Vladimirova 1 , Ivan Vointsev 2 , Alena Skoba 2 and Gregory Falkovich 2,3, *   Citation: Vladimirova, N.; Vointsev, I.; Skoba, A.; Falkovich, G. Turbulence of Capillary Waves on Shallow Water. Fluids 2021, 6, 185. https://doi.org/10.3390/fluids6050185 Academic Editor: Alexander I. Dyachenko Received: 11 February 2021 Accepted: 11 May 2021 Published: 13 May 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1 Department of Physics, Brown University, Providence, RI 02912, USA; nvladimirova@gmail.com 2 Landau Institute for Theoretical Physics, 142432 Moscow, Russia; vointsevivan@gmail.com (I.V.); aoskoba@mail.ru (A.S.) 3 Weizmann Institute of Science, Rehovot 76100, Israel * Correspondence: gregory.falkovich@weizmann.ac.il; Tel.: +972-8934-2830 Abstract: We consider the developed turbulence of capillary waves on shallow water. Analytic theory shows that an isotropic cascade spectrum is unstable with respect to small angular perturbations, in particular, to spontaneous breakdown of the reflection symmetry and generation of nonzero momentum. By computer modeling we show that indeed a random pumping, generating on average zero momentum, produces turbulence with a nonzero total momentum. A strongly anisotropic large-scale pumping produces turbulence whose degree of anisotropy decreases along a cascade. It tends to saturation in the inertial interval and then further decreases in the dissipation interval. Surprisingly, neither the direction of the total momentum nor the direction of the compensated spectrum anisotropy is locked by our square box preferred directions (side or diagonal) but fluctuate. Keywords: turbulence; capillary wave; spectrum; anisotropy 1. Introduction Most of the ripples one observes on the puddles are capillary waves with wavelengths exceeding fluid depth, so they are one of the most ubiquitous waves in nature. Extensive literature on the turbulence of waves is summarized in two monographs [1,2]. As far as waves on the water surface are concerned, one finds a vast body of research on gravity waves and a substantial research on the capillary waves on a deep water. Yet surprisingly little is known about the turbulence of the capillary waves on a shallow water, except isotropic weak-turbulence spectrum and its small perturbations [1]. Needless to say that neither nature nor human activity generally provides us with nearly isotropic turbulence. From a fundamental viewpoint it is of much interest to understand how anisotropy of forcing or geometry of the container impacts an anisotropy of developed turbulence at small scales. That problem has not been solved completely for capillary waves on a deep water either: Theory predicts that weak anisotropy imposed by forcing at large scales leads to more and more anisotropic turbulence at smaller and smaller scales, as the turbulence cascade develops [1,3]. On the other hand, initially anisotropic force-free turbulence was found numerically to undergo isotropization during turbulence decay [4], and also isotropization along the cascade was found for steady turbulence generated by a strongly anisotropic pumping [5]. This work is devoted to anisotropic turbulence of capillary waves on shallow water. The main research question that we pose is as follows: how anisotropy of the environment (forcing and/or container shape) influences the anisotropy of small-scale turbulence. Un- derstanding far-from-equilibrium states of capillary waves on thin fluid layers in different geometries can be important, among other things, for an emerging field of liquid meta- materials—wave-driven matrices of vortices, akin to optical lattices [6]. Such wave-driven flows can give one an ability to control and separate active particles and chemicals in fluid layers in biological and engineering contexts, as well in controlled self-assembly [6,7]. The Fluids 2021, 6, 185. https://doi.org/10.3390/fluids6050185 https://www.mdpi.com/journal/fluids