Rayleigh-Taylor instability of crystallization waves at the superfluid-solid $^{4}$He interface S. N. Burmistrov, ∗ L. B. Dubovskii, and V. L. Tsymbalenko Institute of Superconductivity and Solid State Physics, Kurchatov Institute, 123182 Moscow, Russia At the superfluid-solid $^{4}$He interface there exist crystallization waves having much in common with gravitational-capillary waves at the interface between two normal fluids. The Rayleigh-Taylor instability is an instability of the interface which can be realized when the lighter fluid is pro- pelling the heavier one. We investigate here the analogues of the Rayleigh-Taylor instability for the superfluid-solid $^{4}$He interface. In the case of a uniformly accelerated interface the instability occurs only for a growing solid phase when the magnitude of the acceleration exceeds some critical value independent of the surface stiffness. For the Richtmyer-Meshkov limiting case of an impulsively ac- celerated interface, the onset of instability does not depend on the sign of the interface acceleration. In both cases the effect of crystallization wave damping is to reduce the perturbation growth-rate of the Taylor unstable interface. PACS numbers: 68.03.Kn, 67.80.-s, 52.35.Py The Rayleigh-Taylor instability is a fingering instabil- ity of an interface between two fluids of different den- sities. It takes place when the heavier fluid is deceler- ated by the lighter fluid or, in other words, density and pressure gradients have opposite directions [1, 2]. The Rayleigh-Taylor instability [3] occurs in numerous phys- ical and technological situations, e.g., gravity-driven in- stability of a heavier fluid atop a lighter one and inertial confinement fusion. In essence, the Rayleigh-Taylor in- stability is the first step in a fluid-mixing mechanism, the step eventually leading via formation of bubbles, spikes, and curtains to the turbulent regime of fluid mixing. Along with the Kelvin-Helmholtz criterion for tangen- tial discontinuities at the interface between two normal fluids, the Rayleigh-Taylor criterion is among the most generic principles in the complicated subject of interface instability. For the interfaces of superfluid $^{4}$He, some of the hydro- dynamic instabilities have been found as well. First, we mention the Faraday instability which denotes the phe- nomenon of the parametric excitation of standing waves on the free surface of a fluid. The flat shape of the sur- face becomes unstable with a periodic modulation of the acceleration of gravity. Recently [4], generation of Fara- day standing waves on the free surface of $^{4}$He has been realized in the experimental cell subjected to sinusoidal vibration in the vertical direction. According to [5, 6], it is also possible to generate a dense fog of helium droplets by driving the capillary waves on a superfluid $^{4}$He surface unstable with an intense ultrasonic beam from a piezo- electric transducer under the surface. There have been observed electrohydrodynamic interface instabilities due to charges trapped at the surfaces and interfaces of vari- ous condensed helium phases [7]. The shear flow between the superfluid A and B phases of $^{3}$He can result in the Kelvin-Helmholtz interface instability [8]. Dynamics of the superfluid-solid $^{4}$He interface due to To date, the Rayleigh-Taylor phenomena have exten- sively been studied in normal fluids, but not much study has been made in superfluids or quantum solids. The classical Rayleigh-Taylor instability of the superfluid- solid $^{4}$He interface in the field of gravity is observed by Demaria, Lewellen, and Dahm [14]. In these experiments a cell in which the solid and liquid phases occupy initially the lower and upper halves, respectively, is inverted me- chanically by 180 $^{◦}$. After inversion a single finger of the liquid phase ascends at the centre of a cell, and the solid phase descends along the walls. On the other hand, in experiment [15] the flat shape of the interface remained stable for a $^{4}$He crystal grown at the needle point with its lower facet under favorable condition for developing the gravity-driven instability. A difference in the observa- tions can be associated with the following reasons. The A series of mechanical and hydrodynamical instabili- ties has been predicted and observed for the superfluid- solid $^{4}$He interface. We mention the Grinfeld instability under uniaxial stress of a solid. Warping of the flat in- terface occurs at some threshold stress when the release of elastic energy exceeds the loss of the surface energy [9, 10]. Like normal fluids, the steady flow of a superfluid in the direction tangential to the interface can result in the Kelvin-Helmholtz instability. As the flow exceeds a threshold magnitude, crystallization waves appear at the superfluid-solid $^{4}$He interface [11, 12]. The phenomenon has qualitatively been observed as a distortion of the crystal surface in the fluid jet [13]. An analogy with generating sea waves by wind is fully appropriate here. sufficiently fast processes of crystallization and melting resembles much that of the free surface of a fluid. In particular, as was predicted by Andreev and Parshin, the crystal in contact with its liquid phase can support wave-like processes of crystallization and melting, see re- view [9]. From the dynamical point of view such weakly damping crystallization waves are an immediate coun- terpart of the well-known gravitational-capillary waves at the vapor-liquid interfaces. arXiv:1001.1887v1 [cond-mat.other] 12 Jan 2010 INTRODUCTION