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Astron. Astrophys. 354, 334-348 (2000)

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3. Diverging magnetic field and no gravitational stratification: no dissipation

To understand each physical effect clearly, we first examine the effect of a diverging field on the phase mixing of Alfvén waves. Hence, we eliminate gravitational effects by setting the pressure scale height to infinity, i.e. [FORMULA]. In the rest of this paper, we assume that the plasma is structured in the [FORMULA]-direction. When referring to stratification, we mean radial stratification due to gravity while a diverging atmosphere refers to the area change due to spherical geometry.

If we neglect dissipation, i.e. [FORMULA], the solution for the magnetic field and velocity perturbations are given by

[EQUATION]

and

[EQUATION]

where [FORMULA] and J is a Bessel function of order either [FORMULA] or [FORMULA].

The ideal MHD solutions in spherical coordinates may be approximated by a simple WKB solution (see Appendix) of the form

[EQUATION]

where [FORMULA]. Although the exact analytical solutions for the perturbed magnetic field and velocity differ, the approximate WKB solutions are both the same since the leading asymptotic terms of the Bessel functions agree on using the large argument approximation [FORMULA] and [FORMULA] (Abramowitz & Stegun). The solutions (15) and (16) show that area divergence decreases the wavelength in agreement with the numerical solutions shown in Figs. 1 and 2. Figs. 2 (a) and 2(b) confirm that the solutions for the perturbed magnetic field and the perturbed velocity are the same as predicted by Eq. (17).

When [FORMULA], Eq. (14) becomes the standard wave equation for phase mixing in a Cartesian, non-dissipative system, i.e.

[EQUATION]

[FIGURE] Fig. 1. (left) A contour plot of the perturbed magnetic field in a diverging atmosphere, with [FORMULA]. (right) The behaviour of the wavelength with height (at [FORMULA]) with [FORMULA]. The solid line is the solution for a diverging atmosphere, the dashed line corresponds to the Cartesian case.

[FIGURE] Fig. 2. A cross-section of the perturbed a magnetic field and b velocity for a radially diverging background magnetic field at [FORMULA] with [FORMULA], [FORMULA] and [FORMULA]. c A cross-section of the perturbed magnetic field at [FORMULA] with [FORMULA], [FORMULA] and [FORMULA].The dotted lines represent the corresponding solutions in Cartesian coordinates at [FORMULA].

Therefore, the solutions for the perturbed velocity and magnetic field in Cartesian coordinates are given by

[EQUATION]

where [FORMULA].

From Fig. 2 (c) we see that low down, i.e. near [FORMULA], and for small initial wavelengths [FORMULA] (see Eq. (6)), the spherical and the Cartesian case indeed agree extremely well. The results in Fig. 1 show clearly that, unlike gravitational stratification which lengthens the wavelengths, area divergence shortens the wavelengths while the wavelengths remain constant in the Cartesian case. Indeed, in this case the Alfvén speed and the wavelength [FORMULA] behave like [FORMULA]. The amplitude of both the perturbed magnetic field and the perturbed velocity are constant in height as we expected. Wright & Garman (1998) and Torkelsson & Boynton (1998) showed that in the large wavenumber limit the amplitudes of the Alfvén waves behave as [FORMULA] and [FORMULA] and as [FORMULA] is constant with height in this case, the result follows. This suggests that phase mixing will be more efficient in a diverging medium than in a non diverging medium as the short length scales, necessary for efficient dissipation, will be created much faster. Therefore, heat could now be deposited at lower heights. Similar results were obtained by Ruderman et al. (1998). These results also show that, unlike the results due to stratification, the effect of the divergence of the background field is the same whether resistive or viscous heating is considered.

To obtain an estimate of where this heat would be deposited if dissipation were included, we now consider the current density, [FORMULA]. In spherical coordinates, only including the dominant terms, [FORMULA] is given by

[EQUATION]

while in Cartesian coordinates, [FORMULA] is given by

[EQUATION]

where [FORMULA] and [FORMULA]. The numerical results obtained for both the magnetic field and the velocity indicate that the behaviour of the current density and the vorticity will be similar. Therefore we concentrate on the current density alone.

From Fig. 3 (a) we see that even when there is no phase mixing, i.e. [FORMULA], the current density builds up very rapidly in a diverging magnetic field. In the non-diverging atmosphere, i.e. the Cartesian case, this current density remains constant as gradients in the vertical or horizontal direction do not build up. As there is no phase mixing, the growth in [FORMULA] in the diverging atmosphere is entirely due to the flaring out of the background field lines, i.e. the radial derivatives are building up. When we include phase mixing, i.e. [FORMULA], we see from Fig. 3 (b) that the build up of the current density, [FORMULA], increases when [FORMULA] increases. However, when the magnetic field is diverging, the effect of increasing [FORMULA] is larger than in the Cartesian case. Indeed, from comparing Eqs. (17) and (19), we see that a change in [FORMULA] causes a bigger change in the spherical [FORMULA]-derivatives than in the Cartesian x-derivatives. At [FORMULA] both the spherical r-derivatives and the Cartesian z-derivatives stay the same when the phase mixing parameter [FORMULA] changes.

[FIGURE] Fig. 3. A cross-section of the current density [FORMULA] for a diverging background magnetic field with [FORMULA] and [FORMULA] at [FORMULA] for a [FORMULA] b different values of [FORMULA] (solid line: [FORMULA], dashed line: [FORMULA]) The thin lines represent the corresponding solutions in Cartesian coordinates at [FORMULA].

Fig. 4 (a) shows the change in the current density [FORMULA] when we change the initial wavelength [FORMULA], through changes to either the frequency [FORMULA] or the background Alfvén velocity [FORMULA]. Since [FORMULA] (Eq. (6)), doubling [FORMULA] has the same effect as doubling [FORMULA] or halving [FORMULA] and so on. Therefore we concentrate on the effect of varying just the one parameter [FORMULA]. Changing [FORMULA] also causes [FORMULA] to change as [FORMULA]. We notice that [FORMULA] starts off with a higher initial value when the initial wavelength [FORMULA] is smaller, which is clear from Eqs. (17) and (19). We still see that the current density [FORMULA] builds up faster in the spherical case and even that the difference between the spherical and the Cartesian case is more pronounced as the initial wavelength [FORMULA] gets smaller. From Eqs. (17) and (19) we see that all derivatives, apart from the Cartesian z-derivative, contain a [FORMULA] factor. From Eq. (21), we see that, in the Cartesian case, the dimesionless variables introduce a factor [FORMULA] in front of the square of the z-derivative. These means that in both the spherical and Cartesian case, [FORMULA] which implies that the difference between the spherical and Cartesian results gets 4 times bigger when we halve the initial wavelength [FORMULA]. From Fig. 4 (b) we see that the current density builds up higher and that the difference between the spherical and the Cartesian case gets larger as we decrease the value of the parameter [FORMULA]. However, from Eqs. (17) and (19) it is clear that all the derivatives remain unchanged when we vary [FORMULA]. The parameter [FORMULA] only appears in expression (20) and (21) for the current density [FORMULA]. We see that the spherical [FORMULA]-derivatives and Cartesian x-derivatives are multiplied with a factor [FORMULA] or that these derivatives will become 4 times larger when we halve [FORMULA]. Again this implies that the difference between the spherical and Cartesian [FORMULA] will get larger as [FORMULA] gets smaller.

[FIGURE] Fig. 4. A cross-section of the current density [FORMULA] for a diverging background magnetic field with [FORMULA] at [FORMULA] for a [FORMULA] and different values of [FORMULA] (solid line: [FORMULA], dot-dashed line: [FORMULA], dashed line: [FORMULA]) and b [FORMULA] and different values of [FORMULA] (solid line: [FORMULA], dot-dashed line: [FORMULA], dashed line: [FORMULA]). The thin lines represent the corresponding solutions in Cartesian coordinates at [FORMULA].

From studying the non-dissipative case, we expect phase mixing to be enhanced when we increase the phase mixing parameter [FORMULA] or decrease either the intial wavelength [FORMULA] or the parameter [FORMULA]. We also expect more heat to be deposited into the plasma when we consider a diverging magnetic field compared to the Heyvaerts and Priest model in Cartesian coordinates. We now want to examine if these effects remain the same when we include dissipation in the system.

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© European Southern Observatory (ESO) 2000

Online publication: January 31, 2000
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