We consider the long-standing problem of Rayleigh-Taylor instability with variable acceleration, and focus on the early-time scale-dependent dynamics of an interface separating incompressible ideal fluids of different densities subject to an acceleration being a power-law function of time for a spatially extended three-dimensional flow periodic in the plane normal to the acceleration with symmetry group p6mm. By employing group theory and scaling analysis, we discover two distinct subregimes of the early-time dynamics depending on the exponent of the acceleration power-law. The time scale and the early-time dynamics are set by the acceleration for exponents greater than (-2), and by the initial growth-rate (due to, e.g., initial conditions) for exponents smaller than (-2). At the exponent value (-2) a transition occurs from one subregime to the other with varying acceleration strength. For a broad range of the acceleration parameters, the instability growth rate is explicitly found, the dependence of the dynamics on the initial conditions is investigated, and theory benchmarks are elaborated.

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Summary 1.Studies of ecosystem engineering may use burrow volume and soil displacement rate to quantify the impact of burrowing vertebrates. Calculations of burrow volume from morphometric measurements have previously treated burrows as rectangular or elliptical prisms. 2. Here we use burrows of the Wedge-Tailed Shearwater (Puffinus pacificus Gmelin, 1789) to demonstrate a new method for collecting morphometric data, and for mathematically modelling burrow shape used to calculate burrow volume. 3. Our method improves on previous estimates by better estimating the cross-sectional burrow shape, and by accounting for some of the variation in burrow width and height. 4. Wedge-Tailed Shearwater burrows were parabolic in cross-section, averaged 1·99 ± 0·04SE m in length and had a mean volume of 0·06 ± 0·00SE m 3 . The shearwaters excavate at a rate of 7·75 m 3 ha −1 year −1 (10·51 t ha −1 year −1 ) which ranks them above many geomorphic mammals. We suggest that this warrants further investigation into the role of burrowing seabirds as ecosystem engineers.

Richtmyer-Meshkov instability (RMI) plays an important role in many areas of science and engineering, from supernovae and fusion to scramjets and nano-fabrication. Classical Richtmyer-Meshkov instability is induced by a steady shock and impulsive acceleration, whereas in realistic environments the acceleration is usually variable. We focus on RMI induced by acceleration with power-law time-dependence and apply group theory to solve the long-standing problem. For early-time dynamics, we find the dependence of the growth-rate on the initial conditions and show that it is independent of the acceleration parameters. For late-time dynamics, we find a continuous family of regular asymptotic solutions, including their curvature, velocity, Fourier amplitudes, and interfacial shear, and we study their stability. For each solution, the interface dynamics is directly linked to the interfacial shear, the non-equilibrium velocity field has intense fluid motion near the interface and effectively no motion in the bulk. The quasi-invariance of the fastest stable solution suggests that nonlinear coherent dynamics in RMI is characterized by two macroscopic length-scales -the wavelength and the amplitude, in agreement with observations. The properties of a number of special solutions are outlined, these being respectively, the Atwood, Taylor, convergence, minimum-shear, and critical bubbles, among others. We also elaborate new theory benchmarks for future experiments and simulations.

The effectiveness of a particular technique of using nonlinear approximations to control chaotic dynamical systems in the case when the systems are of dimension higher than two is investigated.