An analytical model of magnetic reconnection with disrupting curren layer and MHD shock waves...

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

The Syrovatskii current layer model

Magnetic field

Magnetic potential

Equations system for , :

[1] S. I. Syrovatskii // Zh. Eksp. Teor. Fiz. Vol. 60. P. 1726 (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.

[10] H.E.Petschek, Magnetic field annihilation // AAS-NASA Symposium on the physics of Solar flares. - NASA Spec. Publ. P. 425-439. (1964)

3.

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. 1520 (1986).

[6] D. Biskamp. Nonlinear Magnetohydrodynamics (Cambridge Univ., Cambridge, 1997).

[7] T. Yokoyama and K. Shibata // Astrophys. J. 474. L61 (1997).

[8] P. F. Chen, C. Fang, Y. H. Tang, et al. // Astrophys. J. Vol. 513. 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)

4.

Somov’s model of magnetic reconnectionMagnetic 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, 1989. P. 45.

5.

Scheme of the Riemann – Hilbert Problem Solution

Mappings and are the Schwarz – Christoffel

transformations

Formula for :

6.

Magnetic field in Somov’s model

Magnetic field

Formula for :

where , andwere found explicitly.

[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).

[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)].

The magnetic field for Somov’s was found in [12] with the use of contemporary mathematical method [13]:

7.

Physical interpretation of this field was given in [14]–[16]:

Bibliography for the solution of Somov’s model

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.

8.

Arguments for disruption of current layer

The disruption can emerge from a tearing instability (Furth 1963 [17]) orwhen 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, 1975. Pergamon, Oxford, 1982).

[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).

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:

where is a cubic polynomial with real coefficients.

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:

10.

Magnetic field near disrupting current layer

Without shocks With four shocks

11.

3) – Slow Shock Wave.

Angles between the magnetic field line and the normal to the shock wave

Trans-AlfvenicTrans-Alfvenic

FastFast

SlowSlow

1) – Trans-Alfvenic shock wave;

2) – Fast Shock Wave;

12.