3 research outputs found
Small BGK waves and nonlinear Landau damping
Consider 1D Vlasov-poisson system with a fixed ion background and periodic
condition on the space variable. First, we show that for general homogeneous
equilibria, within any small neighborhood in the Sobolev space W^{s,p}
(p>1,s<1+(1/p)) of the steady distribution function, there exist nontrivial
travelling wave solutions (BGK waves) with arbitrary minimal period and
traveling speed. This implies that nonlinear Landau damping is not true in
W^{s,p}(s<1+(1/p)) space for any homogeneous equilibria and any spatial period.
Indeed, in W^{s,p} (s<1+(1/p)) neighborhood of any homogeneous state, the long
time dynamics is very rich, including travelling BGK waves, unstable
homogeneous states and their possible invariant manifolds. Second, it is shown
that for homogeneous equilibria satisfying Penrose's linear stability
condition, there exist no nontrivial travelling BGK waves and unstable
homogeneous states in some W^{s,p} (p>1,s>1+(1/p)) neighborhood. Furthermore,
when p=2,we prove that there exist no nontrivial invariant structures in the
H^{s} (s>(3/2)) neighborhood of stable homogeneous states. These results
suggest the long time dynamics in the W^{s,p} (s>1+(1/p)) and particularly, in
the H^{s} (s>(3/2)) neighborhoods of a stable homogeneous state might be
relatively simple. We also demonstrate that linear damping holds for initial
perturbations in very rough spaces, for linearly stable homogeneous state. This
suggests that the contrasting dynamics in W^{s,p} spaces with the critical
power s=1+(1/p) is a trully nonlinear phenomena which can not be traced back to
the linear level
Bursting and large-scale intermittency in turbulent convection with differential rotation
The tilting mechanism, which generates differential rotation in two-dimensional turbulent convection, is shown to produce relaxation oscillations in the mean flow energy integral and bursts in the global fluctuation level, akin to Lotka-Volterra oscillations. The basic reason for such behavior is the unidirectional and conservative transfer of kinetic energy from the fluctuating motions to the mean component of the flows, and its dissipation at large scales. Results from numerical simulations further demonstrate the intimate relation between these low-frequency modulations and the large-scale intermittency of convective turbulence, as manifested by exponential tails in single-point probability distribution functions. Moreover, the spatio-temporal evolution of convective structures illustrates the mechanism triggering avalanche events in the transport process. The latter involves the overlap of delocalized mixing regions when the barrier to transport, produced by the mean component of the flow, transiently disappears