## 4. Solutions close to singularitiesAs 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 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 ## 4.1. The cusp resonanceWhen 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
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 resonanceAn analogous procedure can be applied to the Alfvén
resonance, which yields the approximate dissipative differential
equations for and 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 © European Southern Observatory (ESO) 1997 Online publication: April 8, 1998 |