1. S.I.Bezrodnykh, V.I.Vlasov, B.V.Somov
An analytical model of magnetic reconnection with disrupting curren layer and MHD shock waves
S.I.Bezrodnykh, V.I.Vlasov, B.V.Somov
Sternberg Astonomical Institute; Dorodnicyn Computing Centre of RAS, Moscow

2. The Syrovatskii current layer model
Magnetic field
Magnetic potential
Equations system for , :
[1] S. I. Syrovatskii // Zh. Eksp. Teor. Fiz. Vol. 60. P (1971 [Sov. Phys. JETP. Vol. 33. P. 933 (1971)]
[2] B. V. Somov and S. I. Syrovatskii // Tr. Fiz. Inst. AN SSSR. Vol. 74. P. 14 (1974).
[3] B. V. Somov, Plasma Asrophysics. Parts I, II. New York: Springer, 2006. 2.

3. [5] D. Biskamp // Phys. Fluids. Vol. 29. P. 1520 (1986).
Presence of MHD shocks
Petschek’s model (1964)
[4] K. V. Brushlinsky, A. M. Zaborov, S. I. Syrovatskii // Fiz. Plazmy. Vol. 6. P. 297 (1980)
[Sov. J. Plasma Phys. 6. P. 165 (1980)]
[5] D. Biskamp // Phys. Fluids. Vol. 29. P (1986).
[6] D. Biskamp. Nonlinear Magnetohydrodynamics (Cambridge Univ., Cambridge, 1997).
[7] T. Yokoyama and K. Shibata // Astrophys. J L61 (1997).
[8] P. F. Chen, C. Fang, Y. H. Tang, et al. // Astrophys. J. Vol P. 516 (1999).
[9] K. Kondoh, M. Ugai, and T. Shimizu // Proceedings of the International Scientific Conference on Chromospheric and Coronal Magnetic Fields, 30 August - 2 September 2005 (ESA SP-596), 72.1 (2005)
[10] H.E.Petschek, Magnetic field annihilation // AAS-NASA Symposium on the physics of Solar flares. - NASA Spec. Publ. P (1964) 3.

4. Somov’s model of magnetic reconnection
Magnetic field in reconnection region :
Domain :
Riemann – Hilbert problem for in :
Riemann – Hilbert problem for in :
[11] S.A.Markovskii, B.V.Somov // Solar Plasma Physics. Collected vol. Moscow: Nauka, P. 45. 4.

5. Scheme of the Riemann – Hilbert Problem Solution
Mappings and
are the Schwarz – Christoffel
transformations 5.

6. Magnetic field in Somov’s model
Formula for :
where , and
were found explicitly.
Formula for : 6.

7. Bibliography for the solution of Somov’s model
The magnetic field for Somov’s was found in [12] with the use of contemporary mathematical method [13]:
[12] S. I. Bezrodnykh and V. I. Vlasov // Zh. Vychisl.Mat. Mat. Fiz. Vol. 42.
P. 277 (2002) [Comp. Math. Math. Phys. Vol. 42. P. 263 (2002)]
[13] S. I. Bezrodnykh and V. I. Vlasov // Spectral Evolut. Probl. Vol. 16. P. 112 (2006).
Physical interpretation of this field was given in [14]–[16]:
[14] B.V.Somov, S.I.Bezrodnykh, V.I.Vlasov // Izvestiya of RAS. Physics. Vol. 70. № 1. P. 16 (2006).
[15] S.I.Bezrodnykh, V.I.Vlasov, B.V.Somov // Pis'ma Astron. Zh. Vol. 33. P. 153 (2007)
[Astron. Lett. Vol. 33. P. 130 (2007)].
[16] S.I.Bezrodnykh, V.I.Vlasov, B.V.Somov // Pis'ma Astron. Zh. Vol. 37.
№ 2. P. 133 (2011) [Astron. Lett. Vol. 37. № 2. P. 113 (2011)]. 7.

8. Arguments for disruption of current layer
The disruption can emerge from a tearing instability (Furth 1963 [17]) or
when a region of higher electrical resistivity appears (Kadomtsev 1975 [18]).
[17] H.P.Furth, J.Killen, and M.N.Rosenbluth // Phys. Fluids. Vol. 6. P. 459 (1963).
[18] B.B.Kadomtsev. Collective Phenomena in Plasmas (Nauka, Moscow, Pergamon, Oxford, 1982).
Somov and Syrovatskii in 1975 suggested a simple analytical model [19], which considers disrupting current layer of infinite width. In this model the force of magnetic tensions acts on the edges of the discontinuity in the layer. This force is proportional to the magnitude of the discontinuity and tends to increase it. A strong electric field is induced inside the discontinuity (Syrovatskii 1981 [20]). The force is capable to accelerate charged particles to high energies under conditions of solar flares.
[19] B.V.Somov, S.I.Syrovatskii // Izv. AN SSSR, Ser. Fiz. Vol. 39. P. 375 (1975).
[20] S.I.Syrovatskii // Ann. Rev. Astron. Astrophys. Vol. 19. P. 163 (1981). 8.

9. New models of current layer disruption
Disrupting current layer of finite width.
Current configuration contains four MHD shocks.
Formulation of mathematical problem is the same as in above Somov’s model: 9.

10. Magnetic field in new models
The first model (current layer without shocks):
The second model (current configuration contains four shocks):
Here conformal mapping is given by the same formula as in Somov’s model:
Function is given by the formula:
where is a cubic polynomial with real coefficients.
10.

11. Magnetic field near disrupting current layer
Without shocks
With four shocks
11.

12. Angles between the magnetic field line and the normal to the shock wave
Slow
Trans-Alfvenic
Fast
1) – Trans-Alfvenic shock wave;
2) – Fast Shock Wave;
3) – Slow Shock Wave.
12.