Fractional diffusion models of nonlocal transport

Journal Article · Tue Aug 15 00:00:00 EDT 2006 · Physics of Plasmas

DOI:https://doi.org/10.1063/1.2336114· OSTI ID:20860184

Castillo-Negrete, D del ^[1]

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1. Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-8071 (United States)

A class of nonlocal models based on the use of fractional derivatives (FDs) is proposed to describe nondiffusive transport in magnetically confined plasmas. FDs are integro-differential operators that incorporate in a unified framework asymmetric non-Fickian transport, non-Markovian ('memory') effects, and nondiffusive scaling. To overcome the limitations of fractional models in unbounded domains, we use regularized FDs that allow the incorporation of finite-size domain effects, boundary conditions, and variable diffusivities. We present an {alpha}-weighted explicit/implicit numerical integration scheme based on the Grunwald-Letnikov representation of the regularized fractional diffusion operator in flux conserving form. In sharp contrast with the standard diffusive model, the strong nonlocality of fractional diffusion leads to a linear in time response for a decaying pulse at short times. In addition, an anomalous fractional pinch is observed, accompanied by the development of an uphill transport region where the 'effective' diffusivity becomes negative. The fractional flux is in general asymmetric and, for steady states, it has a negative (toward the core) component that enhances confinement and a positive component that increases toward the edge and leads to poor confinement. The model exhibits the characteristic anomalous scaling of the confinement time, {tau}, with the system's size, L, {tau}{approx}L{sup {alpha}}, of low-confinement mode plasma where 1<{alpha}<2 is the order of the FD operator. Numerical solutions of the model with an off-axis source show that the fractional inward transport gives rise to profile peaking reminiscent of what is observed in tokamak discharges with auxiliary off-axis heating. Also, cold-pulse perturbations to steady sates in the model exhibit fast, nondiffusive propagation phenomena that resemble perturbative experiments.

• OSTI ID: 20860184
• Journal Information: Physics of Plasmas, Vol. 13, Issue 8; Other Information: DOI: 10.1063/1.2336114; (c) 2006 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA); ISSN 1070-664X
• Country of Publication: United States
• Language: English

Related Subjects

70 PLASMA PHYSICS AND FUSION TECHNOLOGY
ASYMMETRY
BOUNDARY CONDITIONS
BOUNDARY LAYERS
CONFINEMENT TIME
DIFFUSION
DISTURBANCES
INTEGRO-DIFFERENTIAL EQUATIONS
MARKOV PROCESS
NUMERICAL SOLUTION
PINCH EFFECT
PLASMA
PLASMA CONFINEMENT
PLASMA HEATING
PULSES
RADIATION TRANSPORT
STEADY-STATE CONDITIONS
TOKAMAK DEVICES