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

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5. Gravitational stratification in a diverging atmosphere

Gravitational stratification was shown by De Moortel et al. (1999) to inhibit phase mixing but the results of Sect. 4 of a diverging magnetic field indicate an enhancement of energy dissipation. In this section we investigate the effect of both gravitational stratification of the density and a radially diverging background magnetic field. Therefore we solve Eqs. (5) and (7) with a finite scale height H. Again considering either resistivity or viscosity and including the dominant second order derivatives in the damping term, the WKB solutions (see Appendix) for the perturbed magnetic field and velocity are given by

[EQUATION]

with [FORMULA] and [FORMULA] and,

[EQUATION]

with [FORMULA]. In the limit [FORMULA], these solutions agree with solutions (22) and (23) for a radially diverging background magnetic field without stratification. By setting [FORMULA] and [FORMULA], we recover the Cartesian solutions

[EQUATION]

with [FORMULA] and [FORMULA] and

[EQUATION]

with [FORMULA]. Compared with the results in De Moortel et al. (1999), these solutions have an extra damping term. The extra term arises from the need to include both second order derivatives in the damping term when considering spherical geometry rather than just the transverse derivatives. However, the contribution from this extra term is almost negligible in the Cartesian limit.

5.1. No dissipation

From Fig. 10 we see that the gravitational stratification of the density has a very strong influence on the behaviour of both the perturbed magnetic field and velocity. Rather than staying constant, as in the radially diverging atmosphere, the amplitude of the magnetic field decreases with height due to the stratification. The amplitude of the velocity on the other hand, increases with height. Indeed, as mentioned earlier, [FORMULA] and [FORMULA] (Wright & Garman 1998) and as the density decreases with height in a stratified atmosphere, the amplitude of b will decrease with height and v will increase.

[FIGURE] Fig. 10. A cross-section of the perturbed magnetic field and velocity for a radially diverging background magnetic field at [FORMULA] with [FORMULA], [FORMULA] and [FORMULA] for [FORMULA]. The dotted lines represent the corresponding solution for [FORMULA].

The Alfvén speed [FORMULA] and the wavelength [FORMULA] now behave as [FORMULA] in the diverging case and as [FORMULA] in the Cartesian case. This means that the wavelength will increase everywhere in the Cartesian case, as we see from Fig. 11 (b) but that will only increase everywhere for [FORMULA] in the diverging case. This is clearly seen in Fig. 11 (a). The wavelength only increases for all values of r for [FORMULA]. When [FORMULA], the wavelengths will decrease everywhere, which in Fig. 11 (a) happens for [FORMULA] and [FORMULA]. For [FORMULA], the wavelengths will increase till they reach the turning point [FORMULA] and then decrease. Indeed, for [FORMULA], [FORMULA] initially increases till [FORMULA] and then decreases.

[FIGURE] Fig. 11. a Behaviour of the wavelength [FORMULA] for a radially diverging background magnetic field at [FORMULA] with [FORMULA] for different values of the scale-height (solid line: [FORMULA], dot-dashed line: [FORMULA], dashed line: [FORMULA], dotted line: [FORMULA]). b Behaviour of the wavelength [FORMULA] in the corresponding Cartesian case.

From Fig. 12, it is clear that gravitational stratification introduces dramatic changes. When we look at the results for the ohmic heating [FORMULA] we retrieve the effects found when studying either a purely stratified or a diverging atmosphere. Despite the fact that the amplitude of b decreases due to the density stratification, [FORMULA] still builds up for most values of the scale height H, due to the shorter wavelengths caused by the area divergence of the background magnetic field. It is only when stratification is very strong that [FORMULA] decreases. We also see that [FORMULA] builds up stronger in the diverging atmosphere that in the Cartesian case, a result already noted in the case without gravitational stratification of the density. When the value of the initial wavelength [FORMULA] is decreased, we find that the current density reaches higher values, as found in the purely diverging atmosphere.

[FIGURE] Fig. 12. A cross-section of the (left) current density and (right) vorticity for a (top) radially diverging and (bottom) uniform background magnetic field at [FORMULA] with [FORMULA] for different values of the scale-height (solid line: [FORMULA], dot-dashed line: [FORMULA], dashed line: [FORMULA], dotted line: [FORMULA]).

The results are quite different when considering the viscous heating [FORMULA]. We see that the vorticity builds up higher than the corresponding current density and that the effect of changing the scale height is reduced. This different behaviour is due to the increase of the velocity amplitude, a result which we also found in the purely stratified atmosphere. The effect of the changing the value of the initial wavelength is the same for the current density and the vorticity. If we analyse the results for different geometries, we see that, unlike the current density, the vorticity builds up higher in the Cartesian case. We also notice that while the vorticity decreases as the scale height is increased in the diverging atmosphere, the opposite happens in the Cartesian case. We see that, as expected, the (Cartesian) vorticity initially builds up less high as the stratification increases. This is due to the lengthening of the wavelengths caused by the stratification and is in agreement with previous results. However, we see that very quickly the vorticity reaches higher values for stronger stratification due to the extremely rapid increase of the velocity amplitude caused by the radially decreasing density. The effect of changing the initial wavelength [FORMULA] is nevertheless maintained. Decreasing the initial wavelength causes the vorticity to start of with a higher initial value and to reach higher values as the waves propagate up.

5.2. Gravitationally stratified, diverging atmosphere, non-zero dissipation

Figs. 13 and 14 show that including dissipation gives familiar results for the behaviour of the perturbed magnetic field. We see that, in both the spherical and the Cartesian case, the magnetic field initially decays faster when we include gravitational stratification. But, overall the damping rate is reduced in a stratified atmosphere. For weak gravitational stratification the radial divergence of the background magnetic field still causes the waves to dissipate faster in the spherical case compared to the Cartesian case. We notice an initial increase in the amplitude of the velocity in the stratified plasma which is the remnant of the amplitude increase of perturbed velocity noted in the zero dissipation case. We also see that the differences between the velocity results for the atmosphere with and without gravity, are considerably smaller than the magnetic field results. When considering viscous dissipation we see that the perturbed velocity decays faster than the perturbed magnetic field damped by ohmic dissipation. However, in general, the wave amplitudes decay faster in an atmosphere without gravitational stratification. For both the perturbed magnetic field and the perturbed velocity we mainly recover the results we found when studying the effect of (only) gravitational stratification on phase mixing of Alfvén waves. The effect of a radially diverging background magnetic field on phase mixing does not seem to be strong enough to compensate for the stratification of the density when the dimensionless pressure scale height H is smaller than [FORMULA].

[FIGURE] Fig. 13. A cross-section of the perturbed magnetic field and velocity for a radially diverging background magnetic field at [FORMULA] with [FORMULA], [FORMULA] and [FORMULA] for [FORMULA]. The dotted lines represent the corresponding solution for [FORMULA].

[FIGURE] Fig. 14. A cross-section of the (Cartesian) perturbed magnetic field and velocity at [FORMULA] with [FORMULA], [FORMULA] and [FORMULA] for [FORMULA]. The dotted lines represent the corresponding solution for [FORMULA].

The cross section (Fig. 15) of the current density [FORMULA] only confirms the dominant effect of the stratified density. In both the spherical and the Cartesian case we see that the current density is spread out over a wider area when the pressure scale height is smaller. The maximum of [FORMULA] is less high and situated higher up. However, we do see that the divergence of the background magnetic field still has some effect. When comparing corresponding different geometries we see that in the spherical case the maximum of the current density is situated at a lower height but is also smaller in magnitude, a result noticed and explained when studying the effect of divergence on phase mixing of Alfvén waves. We also recover the effect of changing the initial wavelength [FORMULA]. When [FORMULA] is decreased, [FORMULA] obtaines a higher maximum at a lower height.The effect of stratification on the vorticity is a lot smaller than the effect on the current density. The vorticity is only spread out very slightly due to the lengthening of the wavelengths in the stratified atmosphere. This different behaviour is due to the initial increase in the amplitude of the perturbed velocity and the fact that the dynamic viscosity [FORMULA] is constant, rather than the kinematic viscosity [FORMULA].

[FIGURE] Fig. 15. A cross-section of the (left) current density and (right) for a (top) radially diverging and (bottom) uniform background magnetic field at [FORMULA] with [FORMULA] for different values of the scale-height (solid line: [FORMULA], dot-dashed line: [FORMULA], dashed line: [FORMULA], dotted line: [FORMULA]).

Overall, we can make two conclusions about the combined effect of a gravitational density stratification and a radially diverging background magnetic field on the phase mixing of Alfvén waves. The stratification generates longer wavelengths, therefore phase mixing is less efficient and heat is deposited into the plasma at higher heights compared to a purely diverging atmosphere without gravitational stratification. At the same time the divergence results in shorter wavelengths which enhances phase mixing and heat is deposited at lower heights compared to a non-diverging atmosphere. So, comparing the gravity results with the Heyvaerts and Priest solution, phase mixing can be more or less efficient depending on the value of the scale height H. A similar conclusion can be found in Ruderman et al. (1998) but a direct comparison cannot be made. They assumed an exponentially diverging magnetic field and an exponentially decreasing density in such a manner that the resulting Alfvén velocity was depending on the horizontal coordinate only. In this study we have a different Alfvén speed and a truly open atmosphere in the sense that no magnetic field lines connect back to the solar surface. However, the general conclusions are broadly in agreement with Ruderman et al. (1998).

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© European Southern Observatory (ESO) 2000

Online publication: January 31, 2000
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