Inwardly Rotating Spirals in a Nonoscillatory Medium Harunori N. Yoshikawa * and Christian Mathis Universit´ e Côte d’Azur, CNRS, UMR 7351, Laboratoire J.-A. Dieudonn´ e, 06108 Nice Cedex 02, France Shu Satoh and Yuji Tasaka Faculty of Engineering, Hokkaido University, N13W8, Sapporo 060-8628, Japan (Received 24 August 2018; published 10 January 2019) We report the spontaneous formation of spiral patterns observed at a downward-facing free surface of a horizontal liquid film. The surface is unstable to the Rayleigh-Taylor instability and the resulting liquid discharge from the film can occur in the form of propagating liquid curtains. They are born at the film circular periphery and exhibit patterns of inwardly rotating spiral arms. With the help of a phenomeno- logically constructed cellular automaton, we show that the patterns arise from the phase locking leading to periodic liquid discharge at constant flow rate over the whole film surface. DOI: 10.1103/PhysRevLett.122.014502 The formation of patterns in nonequilibrium systems has attracted much interest in a variety of scientific fields, as understanding self-organization processes provides rich insights into their complex behavior [1]. Spiral patterns emerging through chiral symmetry breaking are of particular interest. The formation has been investigated intensively for waves in excitable media, e.g., in Belousov-Zhabotinsky chemical reaction systems [2]. The behavior of these reaction-diffusion systems is modeled either by coupled differential equations or by cellular automata for excitation and recovery processes [3,4]. Spiral patterns in forced dissipative systems have also been studied. Bodenschatz et al. [5] observed patterns of stable spirals and of chaotic spiral defects in a Rayleigh-B´ enard convection of a non- Boussinesq fluid. Numerical simulations of the generalized Swift-Hohenberg model reproduce the observed patterns [6]. A theoretical analysis [7] by a phase diffusion equation, which is applicable to a wide variety of nonequilibrium systems [1], shows that the frustration of the local wave number vector drives the motion of convection rolls to produce spiral patterns. The patterns observed in these systems consist of single or multiple curved arms rotating around a core. The rotation occurs with their arms trailing the direction of rotation so that the advancing front of each arm moves outward from the core. Nevertheless, the phase diffusion equation allows spirals propagating both inward and outward [7]. Vanag and Epstein, indeed, discovered inwardly rotating spiral arms in the Belousov-Zhabotinsky reaction in a water-in-oil microemulsion [8]. The observed pattern is called antispiral. The formation of antispirals is reproduced by numerical simulations in excitable and oscil- latory media [9]. In the present Letter, we report our observation of antispirals at the downward-facing free surface of a liquid film, to which liquid is supplied continuously. The destabi- lization of the film by the Rayleigh-Taylor (RT) instability leads to liquid discharge from the film in different modes, drops, columns, and curtains, depending on the rate of liquid supply. The RTinstability of a film under continuous liquid supply and the dynamics of resulting patterns have been investigated, in particular, for the column mode of discharge [10–15]. We explore the discharge in curtain mode, where spiral patterns are observed, with varying the film lateral extension, the liquid viscosity, and the rate of liquid supply. The experimental setup is the same as in previous work [13]. A liquid film forms under a grid plate. A given amount 2R Vacuum pump Rotary pump LED LED Depressurized reservoir Tank Annular disk (a) (c) (b) FIG. 1. Experimental setup. (a) Schematic illustration of the whole setup. Holes of the grid plate have a diameter a ¼ 1 mm and distributed on a hexagonal lattice with an interhole distance d ¼ 2 mm. Uniform liquid supply is assured by a static liquid layer maintained in a depressurized reservoir above the plate. An annular light-emitting diode (LED) lighting system allows us to detect sharp deflections of the free surface by optical observation in top view. (b) A snapshot of a pattern of two spiral curtains in top view. (c) A snapshot of a pattern of single spiral curtain in side view (see Movie 1 in Supplemental Material [16]). PHYSICAL REVIEW LETTERS 122, 014502 (2019) Editors' Suggestion Featured in Physics 0031-9007=19=122(1)=014502(5) 014502-1 © 2019 American Physical Society