63

Rosenthal

64

wave propagation vary strongly in three dimensions. Furthermore, the wave

modes are intrinsically more complicated as they include the anisotropic

e¬ect of the magnetic ¬eld. Further complications arise from non-linearity

(e.g. in K-grains and umbral ¬‚ashes) and radiative e¬ects.

There exists a large body of literature dealing with radiative and/or mag-

netic e¬ects in one-dimensional atmospheres. In the magnetic case this

allows the study of vertical or slanted uniform ¬elds, or height-dependent

horizontal ¬elds. Radiative calculations exist at many levels of complex-

ity, culminating in non-LTE radiation hydrodynamical calculations such as

those of Carlsson & Stein (1997). Radiative calculations of waves in the so-

lar atmosphere have also been carried out in three dimensions by Skartlien

(2000) using a multigroup technique and including the e¬ects of scattering.

However, there is relatively little literature dealing with the magnetohy-

drodynamics of waves in the solar atmosphere in more than one dimen-

sion. One such numerical calculation was carried out as long ago as Shibata

(1983), but there have been no further attempts to solve the full wave equa-

tions for a two-dimensional ¬eld in a strati¬ed atmosphere until the recent

work of Cargill et al. (1997).

In this paper I will report on some calculations of the propagation of waves

in two-dimensional magnetised atmospheres. We (for a list of collaborators

see the author list of Rosenthal et al. 2002) thereby remove the debilitating

restriction of considering only one-dimensional atmospheric structures, and

we also allow for the possible development of shocks by solving the full (not

linearised) equations. At this stage, however, we have not included the

e¬ects of radiative transfer in the calculations. I will describe the models in

more detail in Section 5.2. In Section 5.3 I report on a calculation of wave

propagation in the vicinity of a network element and in Section 5.4 I report

on a surprising result on the e¬ect of weak magnetic ¬elds on the e¬ciency

of acoustic sources. Section 5.5 is a summary of the results.

5.2 Description of the models

The models to be described here are strictly two-dimensional in the sense

that all the dynamical variables are functions of only two co-ordinates, and

the vector ¬elds (velocity and magnetic ¬eld) have components only in those

two directions. In each case the initial state of the magnetic ¬eld is a poten-

tial ¬eld. This allows us to choose a hydrostatic strati¬cation for the initial

pressure and density. We here consider only an ideal-gas equation of state.

The most subtle issues, both physically and numerically, arise in regard

to the boundary conditions. At the vertical boundaries we take the straight-

Waves in the magnetised solar atmosphere 65

forward approach of assuming horizontal periodicity. We drive the system

by shaking at the lower boundary. For atmospheric models, this is intended

to simulate the forcing of the atmosphere by photospheric turbulence. For

the models to be presented in this paper, the forcing consists of a vertically-

oscillating piston operating at a single frequency, although any combination

of time-dependent vertical and horizontal motions could be implemented

relatively easily. At the upper boundary, we would like to allow all wave

motions to escape without re¬‚ection. In practice there exists no perfect way

to implement transmitting upper boundary conditions, but we have found

that the method of characteristics, in the formalism developed by Korevaar

(1989) for a hydrodynamic problem, can be extended to this magnetohydro-

dynamic case with considerable success.

Our numerical scheme, based on a code by Nordlund and Galsgaard, is

described in more detail in Rosenthal et al. (2002). It uses a high order ¬nite-

di¬erence scheme for spatial derivatives and a predictor-corrector scheme for

time-stepping. A staggered mesh di¬erencing scheme is used to ensure high-

accuracy ¬‚ux-conservation.

5.3 Network and internetwork oscillations

In the quiet Sun, the internetwork chromosphere manifests a ¬ligree pattern

of wave like motions. These motions vary greatly in amplitude and in some

internetwork regions are entirely absent.

Our model of the network/internetwork consists of an isolated ¬‚ux tube

embedded in an isothermal atmosphere. (Because of the horizontally peri-

odic boundary conditions, this actually corresponds to a “picket fence” struc-

ture of ¬‚ux sheets extending to in¬nity in each direction.) The properties

of the background atmosphere (gravitational acceleration and density scale

height) are chosen to resemble those of the Sun. At the atmospheric base,

the density and pressure are 2.60 — 10’7 g cm’3 and 1.13 — 105 g cm’1 s’2 ,

respectively. A constant gravitational acceleration of 2.74 — 104 cm s’2 is

adopted, and the ratio of speci¬c heats is set at 5/3. The density scale-

height is 158 km and the adiabatic sound-speed is 8.49 km s’1 .

The vertical magnetic ¬eld imposed at the lower boundary consists of a

weak ¬‚ux tube (peak strength 250G) con¬ned to a region approximately

2 Mm across and separated by 14 Mm from the next repeat of the unit cell.

It is useful to de¬ne a plasma-β parameter as the ratio of the square of the

sound speed to the square of the Alfv´n speed.

e

Rosenthal

66

Fig. 5.1. Model of an internetwork oscillation. The solid lines are magnetic ¬eld

lines and the dashed lines are contours of constant plasma-β. The variable depicted

is the vertical velocity (in km s’1 ) after 220 seconds of simulated time. The heavy

dotted black curve at the bottom of the panel marks the distribution of displacement

in the piston. Note that the vertical scale has been greatly expanded relative to

the horizontal scale.

5.3.1 Internetwork oscillations

To model the internetwork oscillations we drive the system with a simple-

harmonic piston located in the region outside the ¬‚ux sheet. The frequency

of the piston is 42 mHz and its amplitude is 3.7% of the sound speed. Fig-

ure 5.1 shows a snapshot of the result. The wave generated by the piston

at the almost ¬eld-free region near the lower boundary is a fast mode which

travels upwards initially at the (¬xed) speed of sound. As it progresses up-

wards its amplitude grows due to the decrease in density. The fast mode

propagates at the fast speed “ the root mean square of the sound speed and

the Alfv´n speed. Hence, as the wave nears the region where the fast speed

e

and sound speed are comparable (the β = 1 layer), its speed of propaga-

tion and vertical wavelength increase. Above this region, the fast mode is

Waves in the magnetised solar atmosphere 67

increasingly magnetic in character and its propagation speed increases ex-

ponentially with height. This rapid increase in the wave propagation speed

causes the waves to be re¬‚ected back downwards in much the same way as

the increase in sound speed in the solar interior causes non-radial p-modes to

be re¬‚ected back upwards. In this model, the isosurfaces of constant phase-

velocity are not, however, horizontal. The wave packet initially propagates

vertically upwards but is therefore re¬‚ected back downwards at a slight angle

to the vertical “ an angle much exaggerated by the false aspect ratio used in

plotting Figure 5.1. Interference between the upward and downward prop-

agating components can then give rise to rapid apparent horizontal phase

propagation.

The internetwork oscillations are thus trapped in the region below β ≈ 1

and their amplitude is largest close to the re¬‚ecting layer. Thus the visibil-

ity of the waves will depend strongly on the height of this layer relative to

the height of formation of the diagnostic in which the wave are observed.

This strongly supports the idea that the observed intermittancy in the waves

(Carlsson 1999, Judge, Tarbell & Wilhelm 2001) is dependent on the mag-

netic structure (McIntosh et al. 2001, McIntosh & Judge 2001). Near the

re¬‚ection layer, the strong interference between the upward and downward

propagating wave trains gives rise to very rapid (supersonic) phase speeds,

which suggest themselves as a possible explanation for the rapid motions

seen in movies of the ¬ligree internetwork oscillation pattern.

5.3.2 Waves in a network element

We can model waves in a network element by moving the piston inside the

magnetic ¬‚ux sheet. The results, as shown in Figure 5.2, are quite di¬erent

from those of the internetwork case. While waves near the edges of the mag-

netic element continue to show re¬‚ection, waves near the centre of the ¬‚ux

sheet propagate upwards to the top of the computational domain without

undergoing any signi¬cant re¬‚ection. The velocity amplitude continues to

increase with height but, to a ¬rst approximation, the wavelength is un-

changed. The simplest description of the waves is therefore that they are

acoustic waves everywhere.