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The evolution of trans-Alfvénic shocks in gases of finite electrical conductivity

Published online by Cambridge University Press:  28 March 2006

L. Todd
L. Todd
Affiliation:
Department of Applied Mathematics and Theoretical Physics,
Now at the Department of Applied Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts.
University of Cambridge
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Abstract

The stability of steady, plane, one-dimensional, trans-Alfvénic shocks to small normal disturbances (i.e. those in which the perturbed quantities are functions only of time and the distance from the plane of the shock wave) is discussed. The magnetic diffusivity of the ambient gas is taken to be very much greater than each of the viscous diffusivities and the thermal diffusivity.

It is confirmed in detail that all plane-polarized trans-Alfvénic shocks, except the 2–3 type (which is the only type that has no steady-state structure), are unstable to disturbances in those components of the magnetic field and velocity which are transverse to the plane of polarization. An incident Alfvénic wave, consisting of a weak, diffusing current-sheet would initially cause the shock profiles of these transverse quantities to grow linearly with time, while outside this shock region steady, uniform states would be reached. An integral condition is obtained which, together with the relevant boundary conditions; determines the asymptotic shock profiles of the transverse quantities whenever the disturbance is such that a steady state is reached. This removes the puzzling arbitrariness of these profiles.

It is also shown that the ‘1–4’ trans-Alfvénic shock is unstable to magnetoacoustic waves and contact fronts. A qualitative description of how it may be broken up is given. If the disturbance is of finite extent, a steady state is reached. An integral equation is obtained which, together with the relevant boundary conditions, determines the asymptotic steady-state shock-profiles for this case. This removes the apparent arbitrariness of these profiles.

The behaviour of ‘2–3’ trans-Alfvénic shocks and of switch-on and switch-off shocks is discussed.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 21 , Issue 2 , February 1965 , pp. 193 - 209
Copyright
© 1965 Cambridge University Press

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References

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