Precession and Recession of the Rock'n'roller

We study the dynamics of a spherical rigid body that rocks and rolls on a plane under the effect of gravity. The distribution of mass is non-uniform and the centre of mass does not coincide with the geometric centre. The symmetric case, with moments of inertia I_1=I_2, is integrable and the motion is completely regular. Three known conservation laws are the total energy E, Jellett's quantity Q_J and Routh's quantity Q_R. When the inertial symmetry I_1=I_2 is broken, even slightly, the character of the solutions is profoundly changed and new types of motion become possible. We derive the equations governing the general motion and present analytical and numerical evidence of the recession, or reversal of precession, that has been observed in physical experiments. We present an analysis of recession in terms of critical lines dividing the (Q_R,Q_J) plane into four dynamically disjoint zones. We prove that recession implies the lack of conservation of Jellett's and Routh's quantities, by identifying individual reversals as crossings of the orbit (Q_R(t),Q_J(t)) through the critical lines. Consequently, a method is found to produce a large number of initial conditions so that the system will exhibit recession

Percolation transition in the kinematics of nonlinear resonance broadening in Charney–Hasegawa–Mima model of Rossby wave turbulence

We study the kinematics of nonlinear resonance broadening of interacting Rossby waves as modelled by the Charney-Hasegawa-Mima equation on a biperiodic domain. We focus on the set of wave modes which can interact quasi-resonantly at a particular level of resonance broadening and aim to characterize how the structure of this set changes as the level of resonance broadening is varied. The commonly held view that resonance broadening can be thought of as a thickening of the resonant manifold is misleading. We show that in fact the set of modes corresponding to a single quasi-resonant triad has a non-trivial structure and that its area in fact diverges for a finite degree of broadening. We also study the connectivity of the network of modes which is generated when quasi-resonant triads share common modes. This network has been argued to form the backbone for energy transfer in Rossby wave turbulence. We show that this network undergoes a percolation transition when the level of resonance broadening exceeds a critical value. Below this critical value, the largest connected component of the quasi-resonant network contains a negligible fraction of the total number of modes in the system whereas above this critical value a finite fraction of the total number of modes in the system are contained in the largest connected component. We argue that this percolation transition should correspond to the transition to turbulence in the system