Reduced Lorenz models for anomalous transport and profile resilience

Abstract

The physical basis for the Lorenz equations for convective cells in stratified fluids, and for magnetized plasmas imbedded in curved magnetic fields, are reexamined with emphasis on anomalous transport. It is shown that the Galerkin truncation leading to the Lorenz equations for the closed boundary problem is incompatible with finite fluxes through the system in the limit of vanishing diffusion. An alternative formulation leading to the Lorenz equations is proposed, invoking open boundaries and the notion of convective streamers and their back-reaction on the profile gradient, giving rise to resilience of the profile. Particular emphasis is put on the diffusionless limit, where these equations reduce to a simple dynamical system depending only on one single forcing parameter. This model is studied numerically, stressing experimentally observable signatures, and some of the perils of dimension-reducing approximations are discussed.

Evidence of low-dimensional chaos in magnetized plasma turbulence

“…Since these frequencies are smaller than the Doppler-shifted frequency of the observed modes in the Helimak plasma, we conclude that their origin is the oscillation of the background density and potential profile. Our analysis confirms theoretical results for plasma streamers, which were described as a system of three ordinary differential equations in [2]. These results may have relevance for other magnetized plasma configurations where interchange instability, cross-field turbulent transport, and resilient plasma profiles appear as a strongly coupled dynamic entity.…”

“…Our hypothesis is that the streamer dynamics in the Helimak configuration can be described by a set of three differential equations [2] and hence the attractor dimension of its dynamics is C≤3 (and probably >2). According to Takens embedding theorem [4], in the absence of noise, the embedding dimension D is approximately equal to 2C+1, and for C≤3, D≤7.…”

“…From the cross-coherence and cross-phase analysis of the electron density fluctuations it was shown in [1] that the dominant modes are flute-like electrostatic interchange modes, which are in a monochromatic regime when the magnetic field is below 50 mT (close to threshold of the interchange instability) and in a turbulent regime when magnetic field is greater than 50 mT. It has been pointed out by several authors that the Rayleigh-Benard instability has its counterpart in flute interchange instabilities in magnetized plasmas, and a set of reduced Lorenz equations for the dynamics of plasma streamers providing cross-field plasma transport in the Helimak configuration was derived in [2]. The motivation for this paper is to investigate whether experimental data obtained from the Helimak device could reveal the existence of low dimensional, possibly chaotic, dynamics.…”

“…1) produced as a consequence of azimuthal rotation of the plasma column. In this way, we extract larger scale components which show the response of the background plasma which is predicted to exhibit chaotic behavior [2]. We have also applied a "mexican hat'' wavelet filter in order to test the validity of the moving average filter, and we plot the filtered signal on top of the raw signal in the upper panel of Fig.…”

Global and local characterization of turbulent and chaotic structures in a dipole-confined plasma

“…7 Quasi-two-dimensional turbulence appears in certain laboratory plasma devices, [8][9][10] in the scrape off layers of toroidal fusion confinement devices, 11 in the solar wind, 12 and in driven thin sheets of fluid. 18,19 First, we describe the collisionless terrella experiment [14][15][16] ͑CTX͒ along with the operational method used to create interchange turbulence.…”