Self-consistent model of the plasma staircase and nonlinear Schrödinger equation with subquadratic power nonlinearity.

A new basis has been found for the theory of self-organization of transport avalanches and jet zonal flows in L-mode tokamak plasma, the so-called "plasma staircase" [Dif-Pradalier et al., Phys. Rev. E 82, 025401(R) (2010)PLEEE81539-375510.1103/PhysRevE.82.025401]. The jet zonal flows are considered as a wave packet of coupled nonlinear oscillators characterized by a complex time- and wave-number-dependent wave function; in a mean-field approximation this function is argued to obey a discrete nonlinear Schrödinger equation with subquadratic power nonlinearity. It is shown that the subquadratic power leads directly to a white Lévy noise, and to a Lévy fractional Fokker-Planck equation for radial transport of test particles (via wave-particle interactions). In a self-consistent description the avalanches, which are driven by the white Lévy noise, interact with the jet zonal flows, which form a system of semipermeable barriers to radial transport. We argue that the plasma staircase saturates at a state of marginal stability, in whose vicinity the avalanches undergo an ever-pursuing localization-delocalization transition. At the transition point, the event-size distribution of the avalanches is found to be a power law w_{τ}(Δn)∼Δn^{-τ}, with the drop-off exponent τ=(sqrt[17]+1)/2≃2.56. This value is an exact result of the self-consistent model. The edge behavior bears signatures enabling to associate it with the dynamics of a self-organized critical (SOC) state. At the same time the critical exponents, pertaining to this state, are found to be inconsistent with classic models of avalanche transport based on sand piles and their generalizations, suggesting that the coupled avalanche-jet zonal flow system operates on different organizing principles. The results obtained have been validated in a numerical simulation of the plasma staircase using flux-driven gyrokinetic code for L-mode Tore-Supra plasma.

Diamond-Coated Plasma Probes for Hot and Hazardous Plasmas.

Plasma probes are simple and inexpensive diagnostic tools for fast measurements of relevant plasma parameters. While in earlier times being employed mainly in relatively cold laboratory plasmas, plasma probes are now routinely used even in toroidal magnetic fusion experiments, albeit only in the edge region, i.e., the so-called scrape-off layer (SOL), where temperature and density of the plasma are lower. To further avoid overheating and other damages, in medium-size tokamak (MST) probes are inserted only momentarily by probe manipulators, with usually no more than a 0.1 s per insertion during an average MST discharge of a few seconds. However, in such hot and high-density plasmas, their usage is limited due to the strong particle fluxes onto the probes and their casing which can damage the probes by sputtering and heating and by possible chemical reactions between plasma particles and the probe material. In an attempt to make probes more resilient against these detrimental effects, we tested two graphite probe heads (i.e., probe casings with probes inserted) coated with a layer of electrically isolating ultra-nano-crystalline diamond (UNCD) in the edge plasma region of the Experimental Advanced Superconducting Tokamak (EAST) in Hefei, People's Republic of China. The probe heads, equipped with various graphite probe pins, were inserted frequently even into the deep SOL up to a distance of 15 mm inside the last closed flux surface (LCFS) in low- and high-confinement regimes (L-mode and H-mode). Here, we concentrate on results most relevant for the ability to protect the graphite probe casings by UNCD against harmful effects from the plasma. We found that the UNCD coating also prevented almost completely the sputtering of graphite from the probe casings and thereby the subsequent risk of re-deposition on the boron nitride isolations between probe pins and probe casings by a layer of conductive graphite. After numerous insertions into the SOL, first signs of detachment of the UNCD layer were noticed.

Lévy flights on a comb and the plasma staircase.

We formulate the problem of confined Lévy flight on a comb. The comb represents a sawtoothlike potential field V(x), with the asymmetric teeth favoring net transport in a preferred direction. The shape effect is modeled as a power-law dependence V(x)∝|Δx|^{n} within the sawtooth period, followed by an abrupt drop-off to zero, after which the initial power-law dependence is reset. It is found that the Lévy flights will be confined in the sense of generalized central limit theorem if (i) the spacing between the teeth is sufficiently broad, and (ii) n>4-µ, where µ is the fractal dimension of the flights. In particular, for the Cauchy flights (µ=1), n>3. The study is motivated by recent observations of localization-delocalization of transport avalanches in banded flows in the Tore Supra tokamak and is intended to devise a theory basis to explain the observed phenomenology.

Beam stabilization in the two-dimensional nonlinear Schrödinger equation with an attractive potential by beam splitting and radiation.

The effect of attractive linear potentials on self-focusing in-waves modeled by a nonlinear Schrödinger equation is considered. It is shown that the attractive potential can prevent both singular collapse and dispersion that are generic in the cubic Schrödinger equation in the critical dimension 2 and can lead to a stable oscillating beam. This is observed to involve a splitting of the beam into an inner part that is oscillatory and of subcritical power and an outer dispersing part. An analysis is given in terms of the rate competition between the linear and nonlinear focusing effects, radiation losses, and known stable periodic behavior of certain solutions in the presence of attractive potentials.

Attraction of nonlocal dark optical solitons.

We study the formation and interaction of spatial dark optical solitons in materials with a nonlocal nonlinear response. We show that unlike in local materials, where dark solitons typically repel, the nonlocal nonlinearity leads to a long-range attraction and formation of stable bound states of dark solitons.

Collapse arrest and soliton stabilization in nonlocal nonlinear media.

We investigate the properties of localized waves in cubic nonlinear materials with a symmetric nonlocal nonlinear response of arbitrary shape and degree of nonlocality, described by a general nonlocal nonlinear Schrödinger type equation. We prove rigorously by bounding the Hamiltonian that nonlocality of the nonlinearity prevents collapse in, e.g., Bose-Einstein condensates and optical Kerr media in all physical dimensions. The nonlocal nonlinear response must be symmetric and have a positive definite Fourier spectrum, but can otherwise be of completely arbitrary shape and degree of nonlocality. We use variational techniques to find the soliton solutions and illustrate the stabilizing effect of nonlocality.

Generic features of modulational instability in nonlocal Kerr media.

The modulational instability (MI) of plane waves in nonlocal Kerr media is studied for a general response function. Several generic properties are proven mathematically, with emphasis on how new gain bands are formed through a bifurcation process when the degree of nonlocality, sigma, passes certain bifurcation values and how the bandwidth and maximum of each individual gain band depends on sigma. The generic properties of the MI gain spectrum, including the bifurcation phenomena, are then demonstrated for the exponential and rectangular response functions. For a focusing nonlinearity the nonlocality tends to suppress MI, but can never remove it completely, irrespectively of the shape of the response function. For a defocusing nonlinearity the stability properties depend sensitively on the profile of the response function. For response functions with a positive-definite spectrum, such as Gaussians and exponentials, plane waves are always stable, whereas response functions with spectra that are not positive definite (such as the rectangular) will lead to MI if sigma exceeds a certain threshold. For the square response function, in both the focusing and defocusing case, we show analytically and numerically how new gain bands that form at higher wave numbers when sigma increases will eventually dominate the existing gain bands at lower wave numbers and abruptly change the length scale of the periodic pattern that may be observed in experiments.