A methodology is developed to describe time-dependent phenomena associated with nonlocal transport in complex, two-dimensional geometries. It is an extension of the ''iterative method" introduced previously to solve steady-state transport problems [Maggs and Morales, Phys. Rev. E 99, 013307 (2019)10.1103/PhysRevE.99.013307], and it is based on the ''jumping particle" concepts associated with the continuous-time random walk (CTRW) model. The method presented explicitly evaluates the time integral contained in the CTRW master equation. A modified version of the Mittag-Leffler function is used for the waiting-time probability distributions to incorporate memory effects. Calculations of the propagation of ''anomalous transport waves" in various systems, with and without memory, illustrate the technique.

A recently developed model of nonlocal transport in two dimensions (2D), the “iterative method” [J. E. Maggs and G. J. Morales, Phys. Rev. E 99, 013307 (2019)], is applied to the published results of ECRH experiments [Takahashi et al. AIP Conf. Proc. 1580, 145 (2014)] performed in the Large Helical Device. It is found from the temperature profiles obtained from the 2D model that the sharply peaked electron temperature profiles observed in such experiments, and previously associated with local transport and “internal transport barriers” or Core Electron-Root Confinement, are also consistent with nonlocal transport. It is shown that the iterative method is suited to handling both scenarios.

The concept of transport mediated through the dynamics of "jumping" particles is used to develop an iterative method for obtaining steady-state solutions to the nonlocal transport equation in two dimensions. The technique is self-adjoint and capable of correctly treating spatially nonuniform, asymmetric systems. An appropriate reduced version of the iteration method is used to compare with results obtained with a self-adjoint one-dimensional transport matrix approach [Maggs and Morales, Phys. Rev. E 94, 053302 (2016)10.1103/PhysRevE.94.053302]. The transport "jump" probability distribution functions are based on Lévy α-stable distributions. The technique can handle the entire Lévy α-parameter range from one (Lorentz distributions) to two (Gaussian distributions). Cases with α=2 (standard diffusion) are used to establish the validity of the iterative method. The capabilities of the iterative method are demonstrated by presenting examples from systems with various source configurations, boundary shapes, boundary conditions, and spatial variations in parameters.

Summary form only given. Results are presented from basic heat transport experiments and gyrokinetic simulations of multiple magnetized electron temperature filaments in close proximity. This arrangement samples cross-field transport from nonlinear drift-Alfven waves and large scale convective cells. Experiments are performed in the Large Plasma Device (LAPD) at UCLA. A biased LaB 6 cathode injects low energy electrons (below ionization energy) along a strong magnetic field into a pre-existing large and cold plasma forming an electron temperature filament embedded in a colder plasma, and far from the machine walls. A carbon masking plate with several holes (each 1cm diameter, 1.5 cm apart) is used to create 3 electron temperature filaments. By covering two holes in the mask drift-Alfven and thermal waves from a single filament have been characterized and compared to previous studies with a different electron beam source 1 . The observed eigenmode structures also compares favor ably with recent 3D gyrokinetic simulations 2 . The 3-filament case exhibits a complex wave pattern and enhanced cross-field transport. Detailed mode analysis and comparison with non linear simulations is reported.

An integral operator is developed to describe nonlocal transport in a one-dimensional system bounded on both ends by material walls. The "jump" distributions associated with nonlocal transport are taken to be Lévy α-stable distributions, which become naturally truncated by the bounding walls. The truncation process results in the operator containing a self-consistent, convective inward transport term (pinch). The properties of the integral operator as functions of the Lévy distribution parameter set [α,γ] and the wall conductivity are presented. The integral operator continuously recovers the features of local transport when α=2. The self-adjoint formulation allows for an accurate description of spatial variation in the Lévy parameters in the nonlocal system. Spatial variation in the Lévy parameters is shown to result in internally generated flows. Examples of cold-pulse propagation in nonlocal systems illustrate the capabilities of the methodology.

An electromagnetic, 3D gyrokinetic particle code is used to study the relaxation of a magnetized electron temperature filament embedded in a large, uniform plasma of lower temperature. The study provides insight into the role played by unstable drift-Alfvén waves observed in a basic electron heat transport experiment [D. C. Pace et al., Phys. Plasmas 15, 122304 (2008)] in which anomalous cross-field transport has been documented. The simulation exhibits the early growth of temperature-gradient-driven, drift-Alfvén fluctuations that closely match the eigenmodes predicted by linear theory. At the onset of saturation, the unstable fluctuations display a spiral spatial pattern, similar to that observed in the laboratory, which causes the rearrangement of the temperature profile. After saturation of the linear instability, the system exhibits a markedly different behavior depending on the inclusion in the computation of modes without variation along the magnetic field, i.e., kz = 0. In their absence, the initial filament evolves into a broadened temperature profile, self-consistent with undamped, finite amplitude drift-Alfvén waves. But the inclusion of kz = 0 modes causes the destruction of the filament and damping of the drift-Alfvén modes leading to a final state consisting of undamped convective cells and multiple, smaller-scale filaments.

Analysis of the time series obtained with the Doppler backscattering system (Hillsheim et al 2009 Rev. Sci. Instrum. 80 0835070) in the DIII-D tokamak (Luxon 2005 Fusion Sci. Technol. 48 828) shows that intermediate wave number plasma density fluctuations in low confinement (L-mode) tokamak plasmas are chaotic. The supporting evidence is based on the shape of the power spectrum; the location of the signal in the complexity-entropy plane (C-H plane) (Rosso et al 2007 Phys. Rev. Lett. 99 154102); and the population of the corresponding Bandt–Pompe (Bandt and Pompe 2002 Phys. Rev. Lett. 88 174102) probability distributions.

It is demonstrated that a novel heating configuration applied to a large and cold magnetized plasma allows the study of avalanche phenomena under controlled conditions. Intermittent collapses of the plasma pressure profile, associated with unstable drift-Alfvén waves, exhibit a two-slope power-law spectrum with exponents near -1 at lower frequencies and in the range of -2 to -4 at higher frequencies. A detailed mapping of the spatiotemporal evolution of a single avalanche event is presented.

A radial fractional transport model [Kullberg et al., Phys. Rev. E 87, 052115 (2013)], that correctly incorporates the geometric effects of the domain near the origin and removes the singular behavior at the outer boundary, is compared to results of off-axis heating experiments performed in the Rijnhuizen Tokamak Project (RTP), ASDEX Upgrade, JET, and DIII-D tokamak devices. This comparative study provides an initial assessment of the presence of fractional transport phenomena in magnetic confinement experiments. It is found that the nonlocal radial model is robust in describing the steady-state temperature profiles from RTP, but for the propagation of heat waves in ASDEX Upgrade, JET, and DIII-D the model is not clearly superior to predictions based on Fick's law. However, this comparative study does indicate that the order of the fractional derivative, α, is likely a function of radial position in the devices surveyed.

The project expanded and developed mathematical descriptions, and corresponding numerical modeling, of non-diffusive transport to incorporate new perspectives derived from basic transport experiments performed in the LAPD device at UCLA, and at fusion devices throughout the world. By non-diffusive it is meant that the transport of fundamental macroscopic parameters of a system, such as temperature and density, does not follow the standard diffusive behavior predicted by a classical Fokker-Planck equation. The appearance of non-diffusive behavior is often related to underlying microscopic processes that cause the value of a system parameter, at one spatial position, to be linked to distant events, i.e., non-locality. In the LAPD experiments the underlying process was traced to large amplitude, coherent drift-waves that give rise to chaotic trajectories. Significant advances were made in this project. The results have lead to a new perspective about the fundamentals of edge transport in magnetically confined plasmas; the insight has important consequences for worldwide studies in fusion devices. Progress was also made in advancing the mathematical techniques used to describe fractional diffusion.

Continuous control over azimuthal flow and shear in the edge of the Large Plasma Device (LAPD) [W. Gekelman et al., Rev. Sci. Instr. 62, 2875 (1991)] has been achieved using a biasable limiter. This flow control has allowed a careful study of the effect of flow shear on pressure-gradient-driven turbulence and particle transport in LAPD. The combination of externally controllable shear in a turbulent plasma along with the detailed spatial diagnostic capabilities on LAPD makes the experiment a useful testbed for validation of shear suppression models. Motivated by these models, power-law fits are made to the density and radial velocity fluctuation amplitudes, particle flux, density-potential crossphase, and radial correlation length. The data show a break in the trend of these quantities when the shearing rate (γs=∂Vθ/∂r) is comparable to the turbulent decorrelation rate (1/τac). No one model captures the trends in the all turbulent quantities for all values of the shearing rate, but some models successfully match the trend in either the weak (γsτac<1) or strong (γsτac>1) shear limits.

The permutation entropy concept of Bandt and Pompe (2002 Phys. Rev. Lett. 88 174102) is used to analyze the fluctuations in ion saturation current that spontaneously arise in a basic experimental study (Pace et al 2008 Phys. Plasmas 15 122304) of electron heat transport in a magnetized plasma. From the behavior of the Shannon entropy and the Jensen-Shannon complexity it is found that the underlying dynamics are chaotic rather than stochastic. A partitioning and scrambling technique is used to demonstrate that the exponential character of the associated power spectrum arises from individual Lorentzian pulses observed in the time series.

A two-dimensional fractional Laplacian operator is derived and used to model nonlocal, nondiffusive transport. This integro-differential operator appears in the long-wavelength, fluid description of quantities undergoing non-Brownian random walks without characteristic length scale. To study bounded domains, a mask function is introduced that modifies the kernel in the fractional Laplacian and removes singularities at the boundary. Green's function solutions to the fractional diffusion equation are presented for the unbounded domain and compared to the one-dimensional Cartesian approximations. A time-implicit numerical integration scheme is presented to study fractional diffusion in a circular disk with azimuthal symmetry. Numerical studies of steady-state reveal temperature profiles in which the heat flux and temperature gradient are in the same direction, i.e., uphill transport. The response to off-axis heating, scaling of confinement time with system size, and propagation of cold pulses are investigated.

Exponential spectra have been observed in the edges of tokamaks, stellarators, helical devices and linear machines. The observation of exponential power spectra is significant because such a spectral character has been closely associated with the phenomenon of deterministic chaos by the nonlinear dynamics community. The proximate cause of exponential power spectra in both magnetized plasma edges and nonlinear dynamics models is the occurrence of Lorentzian pulses in the time signals of fluctuations. Lorentzian pulses are produced by chaotic behavior in the separatrix regions of plasma E × B flow fields or the limit cycle regions of nonlinear models. Chaotic advection, driven by the potential fields of drift waves in plasmas, results in transport. The observation of exponential power spectra and Lorentzian pulses suggests that fluctuations and transport at the edge of magnetized plasmas arise from deterministic, rather than stochastic, dynamics.

Continuous control over azimuthal flow and shear in the edge of the Large Plasma Device (LAPD) has been achieved using a biasable limiter which has allowed a careful study of the effect of flow shear on pressure-gradient-driven turbulence and transport in the LAPD. The LAPD rotates spontaneously in the ion diamagnetic direction; positive limiter bias first reduces, then minimizes (producing a near-zero shear state), and finally reverses the flow into the electron diamagnetic direction. Degradation of particle confinement is observed in the minimum shearing state and a reduction in the turbulent particle flux is observed with increasing shearing in both flow directions. Near-complete suppression of the turbulent particle flux is observed for shearing rates comparable to the turbulent autocorrelation rate measured in the minimum shear state. Turbulent flux suppression is dominated by amplitude reduction in low-frequency (<10 kHz) density fluctuations. An increase in fluctuations for the highest shearing states is observed with the emergence of a coherent mode which does not lead to net particle transport. The variations of density fluctuations are fit well with power laws and compare favorably to simple models of shear suppression of transport.

Pulses having a temporal Lorentzian shape arise naturally from topological changes in flow trajectories or phase-space orbits associated with deterministic chaos. The pulses can appear as random intermittent events in the time series of observable quantities, and they are the cause of exponential frequency power spectra previously observed in magnetically confined plasmas and various nonlinear systems.

An experimental investigation of the low-frequency density fluctuations in the plasma edge region of the TJ-K stellarator [N. Krause et al., Rev. Sci. Inst. 73, 3474 (2002)] finds that the ensemble-averaged frequency spectra exhibit a near exponential frequency dependence whose origin can be traced to individual pulses having a Lorentzian temporal shape. Similar features have been previously observed [D. C. Pace et al., Phys. Plasmas 15, 122304 (2008)] in a linear magnetized device under conditions in which cross-field pressure gradients are present. The reported observation of such features within the turbulent environment of a toroidal confinement device provides support for the conjecture that the underlying processes are a general feature of pressure gradients. Also presented is the magnetic field strength dependence of the pulse widths and the waiting time distribution between pulses.