4. Solutions close to singularities
As it was already mentioned in Sect. 1 and in Sect. 3, Eqs. (6) are valid only away from the resonant surfaces defined by Eqs. (8). Close to the ideal singularity we use the SGHR method to determine the solutions. This means that we use the ideal MHD equations everywhere except within a comparatively narrow dissipative layer around the resonances where the ideal solutions for P and tend to diverge and the dissipative terms should be included in the equations.
In this paper, we are not particularly interested in the details of the solutions within the dissipative layer and all we need to know is how to cross the layer containing the singularity during the numerical computations of and P. For this purpose, we use connection formulae (GRH1995 ) that relate the solutions for both, P and , at the end points of the dissipative layer. A brief description of how this is done for the cusp and the Alfvén resonant dissipative layers is given below. The dominant dissipation is taken to arise from the isotropic electric resistivity .
4.1. The cusp resonance
When dissipation is included, an analysis by SGH(1991) and GRH(1995) leads to the following set of simplified differential equations for resonant driven slow waves in dissipative MHD around the cusp resonant point
where the new variable is , is the electric resistivity and . In Eqs. (20) all coefficients have their values taken at .
The solutions of Eqs. (20) yield the jump conditions, which connect the solutions accross the dissipative resonant layer for both the Lagrangian displacement and the total pressure perturbation P:
where all equilibrium quantities in (21) are taken at . The second relation in (21) represents the conservation law for the Eulerian perturbation of the total pressure .
The extent of the dissipative layer can be measured by the parameter as in GRH(1995) . The value of is obtained by equating the dissipative and the non-dissipative terms in the left hand side of Eqs. (20). The asymptotic analysis in GRH(1995) shows that dissipation is important only within the interval around the resonant point . Outside this interval, the dissipative term becomes negligible in comparison to the ideal terms in (20) and the ideal MHD equations can be used. In particular, when isotropic electric resistivity is considered as the dominant dissipative process, the width of the dissipative layer is proportional to:
4.2. The Alfvén resonance
An analogous procedure can be applied to the Alfvén resonance, which yields the approximate dissipative differential equations for and P close to
where , and all equilibrium quantities have their values taken at .
Eqs. (23) imply that the conservation law for resonant Alfvén waves is the same as for resonant slow waves. The jump conditions for the Alfvén resonance are:
Jump conditions (24) are applied to connect the solutions for P and accross the interval around the Alfvén singularity. An expression is determined by
© European Southern Observatory (ESO) 1997
Online publication: April 8, 1998