Appendix A: model equation for fast waves propagating vertically in a cold coronal arcade
Let us consider the ideal magnetohydrodynamic equations for a cold plasma (Eqs. -). To make analytical progress we introduce a moving coordinate frame which follows (with the speed of the linear wave, ) vertically propagating nonlinear waves. In this frame, plasma quantities depend on the spatial coordinates x and z, which means that nonlinear waves are not strictly stationary in this frame. We can write
where is a small parameter (which measures the weakness of nonlinearity) and () corresponds to up-going (down-going) propagating waves. It is noteworthy that the case of a horizontally propagating wave can be treated analogously. The only difference would be to replace z by x in the above formula for .
All plasma variables are expanded around the unperturbed state (8), (11) as follows
This expansion means that we are concerned with weakly nonlinear waves. Our approach consists of a first order improvement over the linear theory, so it is not correct for strongly (large amplitude) nonlinear waves.
Substituting (A1) and (A2) into the system of MHD equations (with V cast in its normal and parallel components) and collecting terms at , we obtain after some algebra
A compatibility condition at leads to the equation
where the coefficient is given by
Transforming this equation back into the original z, t coordinates and using
we obtain the final form of the model equation for in the weakly nonlinear regime. This equation can be simplified by changing to the dependent variable (cf. Eq. ), already used in obtaining solutions to the linear problem,
We now restrict ourselves to the case , which corresponds to a constant Alfvén speed with height. Moreover, since the initial impulse is launched at and only vertical propagation is being considered, we also have . It is convenient to use dimensionless coordinates
and introduce the normalised velocities and
This equation is used in Sect. 7.4 to explain the basic features of the nonlinear waves observed in the numerical calculations.
Finally, it is worth mentioning that equation (A13) is less general than the nonlinear Klein-Gordon equation that was recently derived by Nakariakov & Oraevsky (1995) and Nakariakov et al. (1997) for the case of fast wave propagation along a smoothed interface. We believe a similar analysis can be done for the arcade, although it can be more difficult to obtain the most important nonlinear terms here.
© European Southern Observatory (ESO) 1998
Online publication: January 16, 1998