Analysis of anomalous transport with temporal fractional transport equations in a bounded domain

doi:10.1088/1674-1056/acedf3

Abstract Anomalous transport in magnetically confined plasmas is investigated using temporal fractional transport equations. The use of temporal fractional transport equations means that the order of the partial derivative with respect to time is a fraction. In this case, the Caputo fractional derivative relative to time is utilized, because it preserves the form of the initial conditions. A numerical calculation reveals that the fractional order of the temporal derivative α (α ∈ (0,1), sub-diffusive regime) controls the diffusion rate. The temporal fractional derivative is related to the fact that the evolution of a physical quantity is affected by its past history, depending on what are termed memory effects. The magnitude of α is a measure of such memory effects. When α decreases, so does the rate of particle diffusion due to memory effects. As a result, if a system initially has a density profile without a source, then the smaller the α is, the more slowly the density profile approaches zero. When a source is added, due to the balance of the diffusion and fueling processes, the system reaches a steady state and the density profile does not evolve. As α decreases, the time required for the system to reach a steady state increases. In magnetically confined plasmas, the temporal fractional transport model can be applied to off-axis heating processes. Moreover, it is found that the memory effects reduce the rate of energy conduction and hollow temperature profiles can be sustained for a longer time in sub-diffusion processes than in ordinary diffusion processes.

Keywords: anomalous transport temporal fractional transport equation Caputo fractional derivatives memory effects hollow temperature profiles

Received: 16 May 2023
Revised: 12 July 2023
Accepted manuscript online: 08 August 2023

PACS:	05.60.-k	(Transport processes)
	05.70.Ln	(Nonequilibrium and irreversible thermodynamics)
	47.27.T-	(Turbulent transport processes)
	45.10.Hj	(Perturbation and fractional calculus methods)

Fund: This work was supported by the National Key R&D Program of China (Grant No. 2022YFE03090000), the National Natural Science Foundation of China (Grant No. 11925501), and the Fundamental Research Fund for the Central Universities (Grant No. DUT22ZD215).

Corresponding Authors: Zheng-Xiong Wang
E-mail: zxwang@dlut.edu.cn

Cite this article:

Kaibang Wu(吴凯邦), Jiayan Liu(刘嘉言), Shijie Liu(刘仕洁), Feng Wang(王丰), Lai Wei(魏来), Qibin Luan(栾其斌), and Zheng-Xiong Wang(王正汹) Analysis of anomalous transport with temporal fractional transport equations in a bounded domain 2023 Chin. Phys. B 32 110502

[1] Vlad M, Reuss J, Spineanu F and Misguich J 1998 J. Plasma Phys. 59 707
[2] Pommois P, Zimbardo G and Veltri P 2007 Phys. Plasmas 14 012311
[3] Gustafson K, Ricci P, Furno I and Fasoli A 2012 Phys. Rev. Lett. 108 035006
[4] Gustafson K and Ricci P 2012 Phys. Plasmas 19 032304
[5] Bovet A, Fasoli A and Furno I 2014 Phys. Rev. Lett. 113 225001
[6] Zimbardo G, Amato E, Bovet A, Effenberger F, Fasoli A, Fichtner H, Furno I, Gustafson K, Ricci P and Perri S 2015 J. Plasma Phys. 81 495810601
[7] Zarzoso D and del-Castillo-Negrete D 2020 J. Plasma Phys. 86 795860201
[8] Boyd T J M and Sanderson J J 2003 The Physics of Plasmas (New York:Cambridge University Press) p. 301
[9] Guthrie A and Wakerling R 1949 The Characteristics of Electrical Discharges in Magnetic Fields (New York:McGraw-Hill) p. 58
[10] Taylor J B and McNamara B 1971 Phys. Fluids 14 1492
[11] Hsu J Y, Wu K B, Agarwal S K and Ryu C M 2013 Phys. Plasmas 20 062302
[12] Balescu R 2005 Aspects of Anomalous Transport in Plasmas (Boca Raton:CRC Press)
[13] Vlad M, Spineanu F, Misguich J H and Balescu R 1998 Phys. Rev. E 58 7359
[14] Vlad M, Spineanu F, Misguich J H and Balescu R 2002 Nucl. Fusion 42 157
[15] Negrea M 2020 Plasma Sci. Technol. 22 015101
[16] Crotinger J A and Dupree T H 1992 Phys. Fluids B 4 2854
[17] Balescu R 2000 Plasma Phys. Control. Fusion 42 B1
[18] Samko S G, Kilbas A A and Marichev O I 1993 Fractional Integrals and Derivatives:Theory and Applications (Amsterdam:Gordon and Breach Science Publishers)
[19] Podlubny I 1999 Fractional Differential Equations (San Diego:Academic Press)
[20] Kilbas A A, Srivastava H M and Trujillo J J 2006 Theory and Applications of Fractional Differential Equations (Netherlands:Elsevier)
[21] Caputo M 1967 Geophys. J. R. Astronom. Soc. 13 529
[22] Oldham K B and Spanier J 1974 The Fractional Calculus:Theory and Applications of Differentiation and Integration to Arbitrary Order (New York:Academic Press) p. 140
[23] Li C, Deng W H and Wu Y J 2011 Comput. Math. Appl. 62 1024
[24] Gao G H, Sun Z Z and Zhang H W 2014 J. Comput. Phys. 259 33
[25] He S B, Wang H H and Sun K H 2022 Chin. Phys. B 31 060501
[26] Povstenko Y Z 2008 J. Mol. Liq. 137 46
[27] Povstenko Y Z 2009 Quart. Appl. Math. 67 113
[28] Povstenko Y Z 2012 Arch. Appl. Mech. 82 345
[29] Qi H T and Liu J G 2010 Meccanica 45 577
[30] Povstenko Y 2013 Eur. Phys. J. Spec. Top. 222 1767
[31] Chung W S, Gungor A, Kříž J, Lütfüoǧlu B C and Hassanabadi H 2023 Chin. Phys. B 32 040202
[32] Montroll E W and Weiss G H 1965 J. Math. Phys. 6 167
[33] Sánchez R and Newman D 2018 A Primer on Complex Systems:With Applications to Astrophysical and Laboratory Plasmas (Netherlands:Springer) p. 243
[34] Mantica P, Gorini G, Hogeweij G M D, Lopes Cardozo N J and Schilham A M R 2000 Phys. Rev. Lett. 85 4534
[35] Kullberg A, del-Castillo-Negrete D, Morales G J and Maggs J E 2013 Phys. Rev. E 87 052115
[36] Kullberg A B 2014 Non-local fractional diffusion and transport in magnetized plasmas (PhD Dissertation) (Los Angeles:University of California)
[37] Kullberg A, Morales G J and Maggs J E 2014 Phys. Plasmas 21 032310
[38] Wu K B, Wei L and Wang Z X 2022 Plasma Sci. Technol. 24 045101

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