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Astron. Astrophys. 332, 786-794 (1998)

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4. The analytical solution around the resonance

In 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 [FORMULA] and P are now obtained by solving the ideal Eqs. (7) outside the dissipative layers and the dissipative Eqs. (6) inside the layers with the requirement for continuous solutions at the boundaries of the layers. In the close vicinity of the resonant points, we can simplify the initial set of Eqs. (6) by reducing them first to a form analogous to (7) and then express the coefficients as linear functions of the distance s from the resonant point (see e.g. Erdélyi, Goossens & Ruderman 1995). Due to the resonant coupling in the presence of a background flow, the waves could loose or gain energy as pointed out in the previous section and therefore their eigenfrequencies are complex: [FORMULA].
In the case of the Alfvén resonance at [FORMULA] the Eqs. (6) can be approximated in some interval [FORMULA] around the resonance, by the following set

[EQUATION]

where [FORMULA], [FORMULA] and the coefficients are evaluated at [FORMULA]. Also, [FORMULA] is assumed but this has to be checked a posteriori when the solutions are obtained. In the numerical code the resistivity [FORMULA] is assumed to be important only inside the narrow dissipative layer [FORMULA] whose half-thickness [FORMULA] is evaluated in our calculations as [FORMULA] where

[EQUATION]

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 [FORMULA], which is of the order 1 in the dissipative layer. With this new variable the Eqs. (9) for [FORMULA] and P take the form:

[EQUATION]

where [FORMULA]. Hence, in the zeroth order, the total pressure perturbation P is a conserved quantity across the resonance layer. Hence the analytical solutions for [FORMULA] and P inside the dissipative layer around the Alfvén resonance can be found in the following form (Tirry & Goossens 1996):

[EQUATION]

where

[EQUATION]

[FORMULA] and [FORMULA] 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)

[EQUATION]

where now [FORMULA], [FORMULA], [FORMULA] and

[EQUATION]

with all coefficients evaluated at [FORMULA]. According to (12), the total pressure perturbation P is a conserved quantity across the cusp resonance layer in the zeroth order. Hence the analytical solutions for [FORMULA] and P inside the dissipative layer embracing the cusp resonance are given by :

[EQUATION]

where

[EQUATION]

[FORMULA] and [FORMULA] are constants of integration.

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© European Southern Observatory (ESO) 1998

Online publication: March 23, 1998
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