Now showing 1 - 2 of 2
  • Publication
    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 characterise 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 nontrivial 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.
    Scopus© Citations 8  287
  • Publication
    Externally forced triads of resonantly interacting waves: Boundedness and integrability properties
    We revisit the problem of a triad of resonantly interacting nonlinear waves driven by an external force applied to the unstable mode of the triad. The equations are Hamiltonian, and can be reduced to a dynamical system for 5 real variables with 2 conservation laws. If the Hamiltonian, H , is zero we reduce this dynamical system to the motion of a particle in a one-dimensional time-independent potential and prove that the system is integrable. Explicit solutions are obtained for some particular initial conditions. When explicit solution is not possible we present a novel numerical/analytical method for approximating the dynamics. Furthermore we show analytically that when H=0 the motion is generically bounded. That is to say the waves in the forced triad are bounded in amplitude for all times for any initial condition with the single exception of one special choice of initial condition for which the forcing is in phase with the nonlinear oscillation of the triad. This means that the energy in the forced triad generically remains finite for all time despite the fact that there is no dissipation in the system. We provide a detailed characterisation of the dependence of the period and maximum energy of the system on the conserved quantities and forcing intensity. When View the MathML source we reduce the problem to the motion of a particle in a one-dimensional time-periodic potential. Poincaré sections of this system provide strong evidence that the motion remains bounded when View the MathML source and is typically quasi-periodic although periodic orbits can certainly be found. Throughout our analyses, the phases of the modes in the triad play a crucial role in understanding the dynamics.
      250Scopus© Citations 14