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

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4. Diverging magnetic field and no gravitational stratification: non-zero dissipation

In this section dissipation is included, i.e. [FORMULA], but gravitational stratification is neglected and numerical results to Eqs. (5) and (7) are presented. We use the same numerical code as De Moortel et al. (1999). We note here that, altough all the results presented in this paper are numerical, the WKB solutions obtained in the Appendix give very good agreement in all cases considered and are useful in understanding the behaviour of the computational results.

Only considering resistivity, the WKB solution (see Appendix) for the perturbed magnetic field in spherical coordinates, including the dominant second order damping terms is given by

[EQUATION]

with [FORMULA], [FORMULA] and [FORMULA] as defined in Eq. (11). Considering viscosity, the solution for the perturbed velocity becomes

[EQUATION]

with [FORMULA].

Again only including the dominant damping terms, corresponding non-diverging WKB solutions in Cartesian coordinates are given by

[EQUATION]

with [FORMULA] and

[EQUATION]

with [FORMULA].

From Fig. 5, we see that including dissipation does not change the effect of an initially diverging magnetic field on the perturbed magnetic field. When the background field is radially diverging, wavelengths get shorter as the waves travel outward from the solar surface. This enhances the overall damping of the wave amplitudes and therefore we expect heat to be deposited into the plasma at lower heights in a diverging atmosphere. From comparing Figs. 5(a) and 5(b) we see again that the perturbed magnetic field and the perturbed velocity behave similarly when the resistive damping coefficient [FORMULA] and the viscous damping coefficient [FORMULA] have the same value. However, we have to remark here that this is only the case at [FORMULA] (or [FORMULA] in the Cartesian case). At other values of [FORMULA] there is a slight difference between the damping rate of the perturbed magnetic field and the perturbed velocity as Eqs. (22) and (23) show that for the magnetic field the damping term is proportional to [FORMULA] while for the velocity the damping term is proportional to [FORMULA]. Fig. 5 (c) confirms that at low heights, i.e. [FORMULA], and for small initial wavelengths [FORMULA] there is a very good agreement between the spherical and the Cartesian case.

[FIGURE] Fig. 5. 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].

We now again look at the current density [FORMULA] and the vorticity [FORMULA] to find out how the heat is deposited into the plasma through ohmic or viscous dissipation. From what we know about the no dissipation case and from the behaviour of the perturbed magnetic field, we expect strong current densities to build up when we consider a background field with radially diverging field lines.

Fig. 6 (a) and (b) are contour plots of the current density [FORMULA] with a diverging and non-diverging background magnetic field. This figure shows clearly that the current density is concentrated at lower heights in the radially diverg ing geometry. However, although the maximum of the current density occurs at lower height in the spherical case, its value is less than the corresponding Cartesian case. Due to the combination of strong phase mixing and the shortening of the length scales caused by the divergence of the background magnetic field, the perturbed magnetic field is damped more quickly.

[FIGURE] Fig. 6. A contour plot of the current density [FORMULA] with [FORMULA], [FORMULA], [FORMULA] and [FORMULA] for a a diverging background magnetic field and b a non-diverging background magnetic field.

Fig. 7 (a) shows the behaviour of the current density [FORMULA] for different values of the phase mixing parameter [FORMULA] in both the spherical and Cartesian geometry. We see that when phase mixing is weak, the results are the same as in the non-dissipative case. The current density builds up to a higher maximum, at a lower height when the background magnetic field is radially diverging. However, when phase mixing is stronger, i.e. [FORMULA], we see again that the maximum of the current density is situated at a lower height in the spherical case but reaches a higher maximum in the Cartesian case. The vorticity [FORMULA] behaves in a similar manner to the current density so that the deposition of heat into the plasma will occur at similar heights, whether we consider ohmic or viscous dissipation. Therefore we concentrate on the current density [FORMULA] to see the effect of changing the plasma parameters.

[FIGURE] Fig. 7. A cross-section of the current density [FORMULA] with [FORMULA] at [FORMULA] for different values of delta (solid line: [FORMULA], dashed line: [FORMULA]). The thin lines represent the corresponding solutions in Cartesian coordinates at [FORMULA].

Fig. 8 (a) shows the variation in the current density [FORMULA] when we change the initial wavelength [FORMULA]. Changing [FORMULA] causes the damping coefficient [FORMULA] to change as [FORMULA]. The resistivity [FORMULA] is kept the same for all cases. From Fig. 8 (a) we see in both the spherical and the Cartesian case that the current density [FORMULA] builds up higher and quicker. Indeed, by starting off with a smaller initial wavelength, the small length scales needed for dissipation to be effective are created faster, i.e. the current density will build up at lower heights. However, although initially [FORMULA] builds up faster and the maxima occur at lower heights in the spherical case, the maxima are smaller than in the corresponding Cartesian case. The current density [FORMULA] is dominated by the transverse derivatives. In the spherical case, we see that [FORMULA] while in the Cartesian case, [FORMULA] (see Eqs. (22) and (24)). So although the spherical [FORMULA]-derivatives initially build up as [FORMULA], the damping term [FORMULA] is significantly stronger than [FORMULA] explaining why [FORMULA] initially builds up faster and why the maxima are situated at lower height and are less high in the spherical case. Fig. 8 (b) shows the change in the current density [FORMULA] when we change the parameter [FORMULA]. Again this causes other parameters to change as well. In this case the damping coefficient [FORMULA] and [FORMULA] will change as [FORMULA] and [FORMULA]. The effect of changing [FORMULA] is largely the same as in the zero dissipation case. The current density builds up stronger at lower heights and again the maxima are higher in the Cartesian case.

[FIGURE] Fig. 8. A cross-section at [FORMULA] of the current density [FORMULA] for a different values of [FORMULA] (solid line: [FORMULA], dot-dashed line: [FORMULA], dashed line: [FORMULA]) and b 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].

Fig. 9 describes the height (in solar radii) at which the maximum of the current density would occur for different initial wavelengths [FORMULA]. This figure confirms that the maxima are situated lower down in the spherical case but also shows that the difference gets smaller as the initial wavelength [FORMULA] gets smaller. If we look back at expression (12), we see that for small [FORMULA], the spherical and the Cartesian geometry give the same results. As the maximum of [FORMULA] is situated lower down as [FORMULA] gets smaller we can conclude that higher initial frequencies or a lower background Alfvén speed will cause the deposition of heat into the plasma to occur at lower heights.

[FIGURE] Fig. 9. The height in solar radii at which the maximum of the current density is situated for different initial wavelengths [FORMULA]. The dotted line represents the corresponding solution in Cartesian coordinates.

The total ohmic heating in the spherical ([FORMULA]) and the Cartesian ([FORMULA]) case, for certain values of [FORMULA] and x respectively, agree very well. This suggests that similar amounts of heat will be deposited through ohmic dissipation into the plasma in both cases and that the total amount of heat deposited does not depend on the geometry of the background magnetic field but only on the Poynting flux of magnetic energy through the photospheric base in response to footpoint motions. As expected, we also notice that the total ohmic heating [FORMULA] does not depend on [FORMULA] or [FORMULA] (or [FORMULA] and [FORMULA] in the Cartesian case).

Overall, we can conclude that a diverging background magnetic field enhances phase mixing of Alfvén waves in the sense that wavelengths get shorter as the waves propagate up making the process of phase mixing more efficient as the small lengthscales needed for dissipation will build up lower down compared to a non-diverging atmosphere. A similar conclusion can be found in the Ruderman et al. (1998) solution for a uniform density in the vertical direction and an exponentially diverging magnetic field. We cannot make a direct comparison as we considered a truly open magnetic field, meaning that at no point do the magnetic field lines connect back to the solar surface. However, we do reach the same conclusion that wave damping due to phase mixing will be faster in a diverging atmosphere.

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© European Southern Observatory (ESO) 2000

Online publication: January 31, 2000
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