# The Nonlinear Tearing Mode

- Edward W. Hones Jr.

Published Online: 19 MAR 2013

DOI: 10.1029/GM030p0292

Copyright 1984 by the American Geophysical Union.

Book Title

## Magnetic Reconnection in Space and Laboratory Plasmas

Additional Information

#### How to Cite

Van Hoven, G. and Steinolfson, R. S. (1984) The Nonlinear Tearing Mode, in Magnetic Reconnection in Space and Laboratory Plasmas (ed E. W. Hones), American Geophysical Union, Washington, D. C.. doi: 10.1029/GM030p0292

#### Publication History

- Published Online: 19 MAR 2013
- Published Print: 1 JAN 1984

#### Book Series:

#### ISBN Information

Print ISBN: 9780875900582

Online ISBN: 9781118664223

- Summary
- Chapter
- References

### Keywords:

- Dynamic reconnection;
- Magnetic reconnection;
- Nonlinear tearing mode;
- Velocity

### Summary

The nonlinear behavior of the tearing instability is investigated with numerical solutions of the resistive, incompressible, MHD equations. The initial state for the non-linear computations is provided by the linear instability, with the amplitude selected such that the nonlinear terms just equal the dominant linear term in one of the equations at some location in the spatial grid. Typical simulations are described for a magnetic Lundquist number S of 10^{4} and wavelength parameters α(= 2πa/λ, where a is the shear scale and λ the instability wavelength) from 0.05 to 0.5. In all cases, the nonlinear mode initially evolves at the linear growth rate, followed by a period of reduced growth. Another common feature is the formation of secondary flow vortices, near the tearing surface, which are opposite in direction to the initial linear vortices. At high S and low α these vortices result in the creation of a new island centered at the initial X-point. The one constant-ψ solution investigated had markedly different behavior from the remaining nonconstant-ψ solutions. Not only was its growth reduced (approximately an order of magnitude less over the same time period) but, whereas the nonconstant-ψ computations showed a reduction by about 20% of the initial magnetic energy in the shear layer, the constant-ψ simulation indicated a reduction of magnetic energy two orders of magnitude smaller. The island width of the nonconstant-ψ solutions became larger than twice the width of the shear layer.