Abstract
To illustrate his theory of coronal heating, Parker initially considers the problem of disturbing a homogeneous vertical magnetic field that is line-tied across two infinite horizontal surfaces. It is argued that, in the absence of resistive effects, any perturbed equilibrium must be independent of z. As a result random footpoint perturbations give rise to magnetic singularities, which generate strong Ohmic heating in the case of resistive plasmas. More recently these ideas have been formalized in terms of a magneto-static theorem but no formal proof has been provided. In this paper we investigate the Parker hypothesis by formulating the problem in terms of the fluid displacement. We find that, contrary to Parker's assertion, well-defined solutions for arbitrary compressibility can be constructed which possess non-trivial z-dependence. In particular, an analytic treatment shows that small-amplitude Fourier disturbances violate the symmetry ∂ z = 0 for both compact and non-compact regions of the (x, y) plane. Magnetic relaxation experiments at various levels of gas pressure confirm the existence and stability of the Fourier mode solutions. More general footpoint displacements that include appreciable shear and twist are also shown to relax to well-defined non-singular equilibria. The implications for Parker's theory of coronal heating are discussed.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Ali, F. and Sneyd, A.: 2001, Geophys. Astrophys. Fluid Dyn. 94(3–4), 221.
Ali, F. and Sneyd, A.: 2002, Solar Phys. 205, 279.
Aly, J. and Amari, T.: 1989, Astron. Astrophys. 221, 287.
Antiochos, S.: 1990, Soc. Astron. Italiana 61, 369.
Bernstein, I., Frieman, E., Kruskal, M., and Kulsrud, R.: 1958, Proc. R. Soc. Lond. 244, 17.
Bobrova, N. and Syrovatskii, S.: 1979, Solar Phys. 61, 379.
Bogoyavelenskij, O.: 2000a, Phys. Rev. Lett. 84, 1914.
Bogoyavelenskij, O.: 2000b, Phys. Rev. Lett. 85, 4406.
Craig, I. and Litvinenenco, Y.: 2005. Phys. Plasmas 12, 032301.
Craig, I. and McClymont, A.: 1993, Astrophys. J. 405, 405.
Craig, I. and Sneyd, A.: 1990, Astrophys. J. 357, 653.
Fisher, G., Longcope, D., Metcalf, T., and Petsov, A.: 1998, Astrophys. J. 508, 885.
Longbottom, A., Rickard, G., Craig, I., and Sneyd, A.: 1998, Astrophys. J. 500, 471.
Low, B.: 1987, Astrophys. J. 323, 358.
Mikic, Z., Schnack, D., and van Hoven, G.: 1989, Astrophys. J. 338, 1148.
Moffatt, H.: 1985, J. Fluid Mech. 159, 359.
Parker, E.: 1972, Astrophys. J. 174, 499.
Parker, E.: 1979, Cosmical Magnetic Fields, Their Origin and Activity, The International Series of Monographs on Physics, Clarendon, Oxford.
Parker, E.: 1994, Spontaneous Current Sheets in Magnetic Field, Oxford University Press, New York, Oxford.
Parker, E.: 2000, Phys. Rev. Lett. 85, 4405.
Rosner, R. and Knobloch, E.: 1982, Astrophys. J. 262, 349.
Ruzmaikin, A. and Berger, M.: 1998, Astron. Astrophys. 337, L9.
Syrovatskii, S.: 1971, Sov. phys. JETP 33, 933.
Syrovatskii, S.: 1981, Ann. Rev. Astron. Astrophys. 19, 163.
van Ballegooijen, A.: 1985, Astrophys. J. 298, 421.
Zweibel, E. and Li, H.-S.: 1987, Astrophys. J. 312, 423.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Craig, I.J.D., Sneyd, A.D. The Parker Problem and the Theory of Coronal Heating. Sol Phys 232, 41–62 (2005). https://doi.org/10.1007/s11207-005-1582-8
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s11207-005-1582-8