## 4. The analytical solution around the resonanceIn ideal MHD, the solutions diverge at the resonance points as
defined by (8). To remove the singularity from the ideal MHD
Eqs. (7) we include, for regions close to the resonances, the
dissipative terms into the equations. The overall solutions for
and where , and the coefficients are evaluated at . Also, is assumed but this has to be checked a posteriori when the solutions are obtained. In the numerical code the resistivity is assumed to be important only inside the narrow dissipative layer whose half-thickness is evaluated in our calculations as where defines the length scale of the resonance layer as can be estimated from the first equation in (9). In the Eqs. (9) the highest derivative term is multiplied with
the electrical resistivity. Hence for very high Reynolds numbers, the
Eqs. (9) represents a singular perturbation problem. Now it is
convenient to introduce a new scaled variable ,
which is of the order 1 in the dissipative layer. With this new
variable the Eqs. (9) for and where . Hence, in the zeroth order, the total
pressure perturbation where and are constants of integration. In the case of the cusp resonance, an analogous procedure yields the following set of equations (see e.g. Erdélyi 1997) where now , , and with all coefficients evaluated at .
According to (12), the total pressure perturbation where and are constants of integration. © European Southern Observatory (ESO) 1998 Online publication: March 23, 1998 |