Nonlocal Transport Model Research Articles

Non-diffusive, non-local transport in fluids and plasmas

Abstract. A review of non-diffusive transport in fluids and plasmas is presented. In the fluid context, non-diffusive chaotic transport by Rossby waves in zonal flows is studied following a Lagrangian approach. In the plasma physics context the problem of interest is test particle transport in pressure-gradient-driven plasma turbulence. In both systems the probability density function (PDF) of particle displacements is strongly non-Gaussian and the statistical moments exhibit super-diffusive anomalous scaling. Fractional diffusion models are proposed and tested in the quantitative description of the non-diffusive Lagrangian statistics of the fluid and plasma problems. Also, fractional diffusion operators are used to construct non-local transport models exhibiting up-hill transport, multivalued flux-gradient relations, fast pulse propagation phenomena, and "tunneling" of perturbations across transport barriers.

Simple collision operators for direct Vlasov simulations of laser plasma interaction and transport

Non-local electron transport effects have a direct influence on the compression of cryogenic targets in laser driven ICF and target heating in high energy density experiments. There is a growing need for self-consistent models of laser plasma interactions coupled to nonlocal transport. We present a direct Vlasov solver that includes multiple species and a simple collision operator. This BGK model operator – which conserves particle density, energy and momentum – is fully implicit. For collisionless plasmas it has been shown that a double layer may be formed in which an accelerated, kinetic ion population satisfies the zero current condition. Here we extend this result to collisionalities of interest to laser driven ignition to assess the validity of nonlocal electron transport models based on fluid ions.

A nonlocal theory of sediment transport on hillslopes

Hillslopes are typically shaped by varied processes which have a wide range of event‐based downslope transport distances, some of the order of the hillslope length itself. We hypothesize that this can lead to a heavy‐tailed distribution of displacement lengths for sediment particles. Here, we propose that such a behavior calls for a nonlocal computation of the sediment flux, where the sediment flux at a point is not strictly a function (linear or nonlinear) of the gradient at that point only but is an integral flux taking into account the upslope topography (convolution Fickian flux). We encapsulate this nonlocal behavior in a simple fractional diffusive model which involves fractional derivatives, with the order of differentiation (1 < α ≤ 2) dictating the degree of nonlocality (α = 2 corresponds to linear diffusion and strictly local dependence on slope). The model predicts an equilibrium hillslope profile which is parabolic close to the ridgetop and transits, at a short downslope distance, to a power law with an exponent equal to the parameter α of the fractional transport model. Hillslope profiles reported in previously studied sites support this prediction. Furthermore, we show that the nonlocal transport model gives rise to a nonlinear dependency on local slope and that variable upslope topography leads to widely varying rates of sediment flux for a given local hillslope gradient. Both of these results are consistent with available field data and suggest that nonlinearity in hillslope flux relationships may arise in part from nonlocal transport effects in which displacement lengths increase with hillslope gradient. The proposed hypothesis of nonlocal transport implies that field studies and models of sediment fluxes should consider the size and displacement lengths of disturbance events that mobilize hillslope colluvium.

Nonlocal transport model in equilibrium two-component plasmas

The full set of linearized Fokker–Planck kinetic equations with Landau collision terms have been solved as an initial-value problem for equilibrium electron-ion plasmas. This work is a generalization of the nonlocal transport theory by Bychenkov et al. [Phys. Rev. Lett. 75, 4405 (1995)] to both electron and ion components with frequency-dependent responses. Transport closure relations have been obtained for the complete set of electron and ion fluid equations which are valid over the full range of particle collisionality and for an arbitrary ionic charge. The well-known limits of collisionless and strongly collisional plasma transport theory have been recovered. Practical fits based on the numerical calculations have been introduced for nonlocal and frequency-dependent ion transport coefficients.

Time and space nonlocalities underlying fractional-derivative models: Distinction and literature review of field applications

We investigate the spatiotemporal nonlocality underlying fractional-derivative models as a possible explanation for regional-scale anomalous dispersion with heavy tails. Properties of four fractional-order advection–dispersion equation (fADE) models were analyzed and compared systematically, including the space fADEs with either maximally positive or negative skewness, the time fADE with a temporal fractional-derivative 0 < γ < 1 , and the extension of the time fADE with 1 < γ < 2 . Space fADEs describe the dependence of local concentration change on a wide range of spatial zones (i.e., the space nonlocality), while time fADEs describe dynamic mass exchange between mobile and multiple immobile phases and therefore record the temporal history of concentration “loading” (i.e., the time-nonlocality). We then applied the fADEs as models of anomalous dispersion to four extensively-studied, regional-scale, natural systems, including a hillslope composed of fractured soils, a river with simultaneous active flow zones and various dead-zones, a relatively homogeneous glaciofluvial aquifer dominated by stratified sand and gravel, and a highly heterogeneous alluvial aquifer containing both preferential flowpaths and abundant aquitards. We find that the anomalous dispersion observed at each site might not be characterized reasonably or sufficiently by previous studies. In particular, the use of the space fADE with less than maximally positive skewness implies a spatial dependence on downstream concentrations that may not be physically realistic for solute transport in watershed catchments and rivers (where the influence of dead-zones on solute transport can be described by a temporal, not spatial, fractional model). Field-scale transport studies show that large ranges of solute displacement can be described by a space nonlocal, fractional-derivative model, and long waiting times can be described efficiently by a time-nonlocal, fractional model. The unknown quantitative relationship between the nonlocal parameters and the heterogeneity, and the similarity in concentration profiles that are solutions to the different nonlocal transport models, all demonstrate the importance of distinguishing the representative nonlocality (time and/or space) for any given regional-scale anomalous dispersion process.

A Fokker-Planck code for laser-produced plasmas

A Fokker-Planck code is developed based upon Epperlein's scheme [Laser Part. Beams 12, 257 (1994)] for the investigation of laser-produced plasmas in relevance to inertial confinement fusion. The equations are integrated implicitly by the time-splitting method. The test problems are simulated to show the versatility of the code. The comparisons among our computational heat flux and the classical Spitzer-Härm (SH) transport model and non-local transport models have been presented. The result shows that the non-local model of heat flux is in reasonable agreement with the FP simulation in overdense region and invalid in the hot underdense plasmas.

Numerical investigation of non-local electron transport in laser-produced plasmas

Non-local electron transport in laser-produced plasmas under inertial confinement fusion (ICF) conditions is studied based on Fokker–Planck (FP) and hydrodynamic simulations. A comparison between the classical Spitzer–Härm (SH) transport model and non-local transport models has been made. The result shows that among those non-local models the Epperlein and Short (ES) model of heat flux is in reasonable agreement with the FP simulation in overdense region. However, the non-local models are invalid in the hot underdense plasmas. Hydrodynamic simulation is performed with the flux limiting model and the non-local model, separately. The simulation results show that in the underdense region of the laser-produced plasmas the temperature given by the flux limiting model is significantly higher than that given with the non-local model.

Péclet-dependent memory kernels for transport in heterogeneous media

Transport in heterogeneous media can be described by partial differential equations, which exhibit convolutions with time and/or space memory kernels. In this work, we characterize the full time spectrum of time-memory kernels by applying a nonparametric inversion algorithm to macroscopic synthetic data for heterogeneous porous media, Our findings put into evidence the inherent nonuniqueness of the transport parameters for nonlocal transport models. Notably, we find that the Péclet number can be interpreted as an ancillary parameter of a family of probability distribution functions that characterizes the memory kernels of transport.

Non-local diffusion and the chemical structure of molecular clouds

We present an application of a non-local turbulent transport model (currently being used to model transport in magnetically confined laboratory plasmas) to the study of the chemical structure of a molecular cloud. We consider a 'toy model' chemistry with a single molecular species which is adsorbed/desorbed from grain surfaces. With this idealized chemistry, we are able to find analytic solutions to both the 'classical' turbulent diffusion model as well as to the non-local transport model. For the turbulent diffusion model, we find that for the turbulent transport to be important one needs a mixing length comparable to the size of the cloud. On the other hand, with the non-local transport model we find that the chemistry is already strongly affected by the turbulent transport for mixing lengths two orders of magnitude smaller than the cloud size. This model then has the desirable property of being able to mix material over long distances (compared with the size of a molecular cloud) without requiring an inordinately large characteristic size for the turbulent eddies.

Early stage of implosion in inertial confinement fusion: Shock timing and perturbation evolution

Excessive increase in the shell entropy and degradation from spherical symmetry in inertial confinement fusion implosions limit shell compression and could impede ignition. The entropy is controlled by accurately timing shock waves launched into the shell at an early stage of an implosion. The seeding of the Rayleigh-Taylor instability, the main source of the asymmetry growth, is also set at early times during the shock transit across the shell. In this paper we model the shock timing and early perturbation growth of directly driven targets measured on the OMEGA laser system [T. R. Boehly et al., Opt. Commun. 133, 495 (1997)]. By analyzing the distortion evolution, it is shown that one of the main parameters characterizing the growth is the size of the conduction zone Dc, defined as a distance between the ablation front and the laser deposition region. Modes with kDc&amp;gt;1 are stable and experience oscillatory behavior [V. N. Goncharov, Phys. Rev. Lett. 82, 2091 (1999)]. The model shows that the main stabilizing mechanism is the dynamic overpressure due to modulations in the blow-off velocity inside the conduction zone. The long wavelengths with kDc&amp;lt;1 experience growth because of coupled Richtmyer-Meshkov-like and Landau-Darrieus instabilities [L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Pergamon, New York, 1982)]. To match the simulation results with both the shock timing and perturbation growth measurements a new nonlocal thermal transport model is developed and used in hydrocodes.

A deep sub-micrometer NMOSFET non-local transport model for ESD effect

Non-local transport characteristics of the deep sub-micrometer NMOS device are studied under electro-static discharge (ESD) stress. The result obtained shows that velocity overshoot may increase the drain current and has a great impact on the device characteristics, and that the energy relaxation time is correlated closely with the electric field of some point in the device as well as the velocity and the energy of the carriers, thus constant value is not appropriate for this parameter. The energy relaxation time and the high field mobility as functions of the carrier energy are gained by Monte Carlo calculation, and then device simulation is performed with these parameter models. Comparing ESD simulation with experimental results shows that accurate result about the I-V characteristics can be gained by using the models of the energy relaxation time and the high field mobility.

Heuristic theory of nonlocally broken gyro-Bohm scaling

Global gyrokinetic simulations of ion temperature gradient turbulent transport with piecewise-flat profiles are given to illustrate the breaking of gyro-Bohm scaling by a nonlocal mechanism. The nonlocal drainage of the turbulence from unstable regions spreading into stable (or less unstable) regions breaks the gyro-Bohm scaling toward Bohm in unstable regions and toward super-gyro-Bohm in stable (or less unstable) regions. A heuristic model for this nonlocal process is formulated in terms of a nonlocal growth rate resulting from a locally weighted radial average of the local linear ballooning mode growth rate. A nonlocality length L measured in ion gyroradii provides the exponential scale for the local weighting. The nonlocal growth rate can be incorporated into a local gyro-Bohm-scaled transport model in place of the local growth rate. The resulting nonlocal transport model will provide some transport in stable regions. A heuristic theory of this nonlocal transport mechanism based on the partial formation of global modes in toroidal geometry is detailed. The theory argues that the nonlocality length L increases with relative gyroradius and decreases with the linear growth rate.

Hydrodynamics simulations of 2ω laser propagation in underdense gasbag plasmas

Recent 2ω laser propagation and stimulated Raman backscatter (SRS) experiments performed on the Helen laser have been analyzed using the radiation-hydrodynamics code HYDRA [M. M. Marinak, G. D. Kerbel, N. A. Gentile, O. Jones, D. Munro, S. Pollaine, T. R. Dittrich, and S. W. Haan, Phys. Plasmas 8, 2275 (2001)]. These experiments utilized two diagnostics sensitive to the hydrodynamics of gasbag targets: a fast x-ray framing camera (FXI) and a SRS streak spectrometer. With a newly implemented nonlocal thermal transport model, HYDRA is able to reproduce many features seen in the FXI images and the SRS streak spectra. Experimental and simulated side-on FXI images suggest that propagation can be explained by classical laser absorption and the resulting hydrodynamics. Synthetic SRS spectra generated from the HYDRA results reproduce the details of the experimental SRS streak spectra. Most features in the synthetic spectra can be explained solely by axial density and temperature gradients. The total SRS backscatter increases with initial gasbag fill density up to ≈0.08 times the critical density, then decreases. Data from a near-backscatter imaging camera show that severe beam spray is not responsible for the trend in total backscatter. Filamentation does not appear to be a significant factor in gasbag hydrodynamics. The simulation and analysis techniques established here can be used in ongoing experimental campaigns on the Omega laser facility and the National Ignition Facility.

Nonlocal Hot-Electron Transport and Capture Model for Multiple Quantum Well Structures Excited by Infrared Radiation

In this paper, a model for hot-electron transport and capture phenomena in semiconductor heterostructures with multiple uncoupled quantum wells (QW) excited by infrared radiation is presented. The model takes into account the nonlocal character of the electron transport and capture of mobile electrons propagating over the barriers. It includes the Poisson equation and balance equations for electrons and their energy. The model is used for the calculation of steady-state spatial distributions of the electric-field and the average electron energy in multiple QW structures. The conditions of the formation of periodic electric-field domains revealed recently in ensemble Monte Carlo particle simulations are found. The obtained results are compared with the previous analytical calculations based on a simplified model. The relationships between phenomenological parameters of the latter and QW structure parameters are obtained. The developed analytical model highlights the origin of periodic electric-field domains in multiple QW structures and explains some features of such domain behavior.

Bioturbation and irrigation in natural sediments, described by animal-community parameters

The key objective of this study was to compare tracer-based methods that measure bioturbation and irrigation rates with predictive models based on benthic community parameters such as animal size and abundance. The individual parameters of each model were investigated separately to assess their relative influence on model output. An important aspect of this study was its focus on benthic processses in natural marine sediments, where the well-defined model assumptions are not necessarily accomodated. Faunal parameters were incorporated into a biodiffusion model proposed to describe bioturbation, and direct comparisons were made to tracer-based depth distributions of an inert particle tracer, 51 Cr. The results showed that this biodiffusion model under-estimated bioturbation rates (Db) compared to 51 Cr tracer measurements. It was shown that the model predictions were highly sensitive to estimates of faunal weight and sediment reworking depth. The model suggests that large, deposit-feeding animals dominate biogenous mixing and that mixing rates are more influenced by animal size than density. In the box-cosm system, the large urchins Echinocardium cordatum did in fact dominate sediment-mixing processes. Irrigation was modelled by a combination of the non-local transport model and the radial-diffusion model, using animal size and density, and was compared to down-core distributions of a dissolved, conservative tracer bromide (Br - ). Modelled irrigation rates increased as a function of burrow radius and animal density, with the strongest dependency on animal density. Given the simplicity of the assumptions for the idealized radial-diffusion model, agreement between measured and predicted rates was relatively good in 50% of the box-cosms. Over-estimation in the remainder may be attributable to unrealistic assumptions that all individuals inhabit vertical burrows and irrigate them continuously.

Enhanced transport of impurity ions and electron heat in Tore Supra ohmic discharges following nickel laser blow-off injection

Nickel has been injected into ohmic Tore Supra (TS) discharges with and without activation of the ergodic divertor (ED). The nickel ion transport is studied by means of an impurity transport code, and the radial profiles of the diffusion coefficient D and of the inward convection velocity V are obtained by simulating the soft X ray brightness and emissivity time evolutions. The simulations confirm two previously reported conclusions: the presence of a central core with reduced transport coefficients and, with the ED activated, the necessity of introducing a peripheral transport barrier (with increased inward convection and/or reduced diffusion). To make satisfactory simulations in both experimental conditions of the initial phase of the nickel ion inflow before the first inverted sawtooth, it is necessary to increase transiently the diffusion coefficient (by 1 order of magnitude or more) during the first few milliseconds after the injection. By comparison of the experimental and simulated bolometric signals, it is possible to verify quantitatively the available calculations of the global radiative losses for nickel. The observed discrepancy by a factor of 2 is discussed. The inward propagating cold wave, following the nickel injection, has been observed with a heterodyne radiometer (monitoring the electron temperature). Both with and without the ED, good simulations of the cold wave propagation are obtained using a predictive non-local transport model, previously successfully tested in JET during transient regimes. The model predicts large increases of the electron thermal diffusivity χe, by a factor of 5, immediately after the injection and describes well the electron temperature profile in the so-called heat transport dominated region outside the q = 1 surface. The experimental signals are smoothed with the `generalized singular value decomposition' technique, reducing the effect of the sawtooth modulation; the simulation can, therefore, be more clearly compared with the measurements. It is not possible, using present experimental data, to demonstrate the existence of an electron diffusivity transport barrier inside the q = 1 region, as done for the impurity transport, because the temperature profile is strongly affected by sawteeth in this region. Similarities have been found in the time evolutions of the electron temperature and of the Abel inverted soft X ray emissivities at fixed radii, implying similar time behaviour of D and χe outside the q = 1 surface.

Time-Resolved Study of Nonlocal Electron Heat Transport in High Temperature Plasmas

Exploiting the high absorption efficiency of intense, ultrashort laser pulses in gases of atomic clusters we have created plasma filaments with temperatures of $g1\mathrm{keV}$ and electron densities in excess of ${10}^{20}{\mathrm{cm}}^{\ensuremath{-}3}$. Using picosecond laser pulses, we have interferometrically measured the temporal and spatial evolution of the electron density in these plasmas on a fast $(l50\mathrm{ps})$ time scale. Our measurements indicate that nonlocal heat transport by hot electrons drives a fast ionization wave, and the data agree well with a nonlocal heat transport model.

On nonlocal electron heat conduction

An improvement of the Albritton nonlocal electron heat transport model is proposed for high-Z plasmas. The thermal decay of the temperature perturbation in a uniform plasma as calculated by this model is compared with that obtained by Fokker–Planck simulations. Complete agreement is found up to values kλe≂0.1, where k is the wave number of the perturbation and λe is the thermal electron mean-free path.

Non-local transport effects on the tokamak SOL plasma parameters

The effect of non-local transport on the SOL-plasma parameters in a tokamak with a poloidal divertor is considered. A self-consistent model of plasma and neutral gas transport has been developed. The plasma is described by a set of 1D equations for the particle, and parallel momentum and heat transport, taking into account non-local heat fluxes and viscous stresses. The non-local fluxes are determined by a set of equations for higher-order moments, obtained on the basis of the Grad method. The neutral gas behaviour is represented in the diffusion approximation. Calculations have been performed using the PNL1D code for ASDEX-like plasma parameters. A comparison between the longitudinal distribution of plasma parameters in the SOL, calculated using both local and non-local transport models will be presented.

Nonlocal transport models of the self-consistent potential distribution in a plasma sheath with charge transfer collisions

Plasma sheaths are often assumed to be collision free; however, high-voltage cathode sheaths are typically thicker than the mean free path for charge transfer collisions at pressures encountered in glow discharge processing equipment (greater than 10 mTorr). In this paper, the potential distribution in a plasma sheath is determined by solving Poisson’s equation self-consistently using a kinetic theory nonlocal ion transport model for charge transfer collisions. The relationship between the potential distribution, ion flux, and thickness of a plasma sheath is presented for arbitrary values of the sheath thickness relative to the mean free path for charge transfer. The results may be used to estimate the ion flux from measurements of the sheath thickness and potential drop across the sheath. Ion energy distribution functions and a one-parameter approximation to the numerically determined potential distribution are also presented. These results apply to rf discharges in a time-averaged sense when the ion sheath transit time is much longer than the rf cycle time, and they apply to high-voltage cathode sheaths in ‘‘abnormal’’ dc and low-frequency rf discharges. The present model is compared to earlier self-consistent sheath models, including the collision-free approximation, the local mobility model, and a nonlocal fluid approximation known as the viscous drag model.