## 5. Gravitational stratification in a diverging atmosphereGravitational 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
with . In the limit , these solutions agree with solutions (22) and (23) for a radially diverging background magnetic field without stratification. By setting and , we recover the Cartesian solutions with and and with . 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 dissipationFrom 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, and
(Wright & Garman 1998) and as
the density decreases with height in a stratified atmosphere, the
amplitude of
The Alfvén speed and the
wavelength now behave as
in the diverging case and as
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 in the diverging case. This is
clearly seen in Fig. 11 (a). The wavelength only increases for
all values of
From Fig. 12, it is clear that gravitational stratification
introduces dramatic changes. When we look at the results for the ohmic
heating we retrieve the effects
found when studying either a purely stratified or a diverging
atmosphere. Despite the fact that the amplitude of
The results are quite different when considering the viscous heating . 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 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 dissipationFigs. 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
The cross section (Fig. 15) of the current density 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 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 . When is decreased, 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 is constant, rather than the kinematic viscosity .
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 © European Southern Observatory (ESO) 2000 Online publication: January 31, 2000 |