Astron. Astrophys. 336, 1029-1038 (1998)

2. Method

2.1. Magnetohydrodynamic equations

We consider a thin, vertically oriented magnetic flux tube in the solar atmosphere which is embedded in a non-magnetic external medium. Following Defouw (1976), Roberts & Webb (1979) as well as Herbold et al. (1985) the magnetohydrodynamic equations in the thin flux tube approximation can be written

In our present adiabatic calculations we set , while in subsequent work we will consider radiation damping . In the thin tube approximation the tube is seen as a slender string of mass elements where the variations of the physical quantities across the tube are assumed to be small. Hence, the state of each mass element can be given by the single values , p, S, u and B at the tube axis. Here is the density, p the gas pressure, S the entropy, u the gas velocity, and B the magnetic field strength. All these variables are functions of height z and time t. Any wave disturbance traveling along the tube is assumed not to perturb the outside atmosphere. Therefore, the gas pressure outside the tube is assumed to be a function of z only. At this point we must stress that the thin tube approximation assumption must be further tested by detailed 2-D and 3-D MHD computations. Theoretical work by Huang (1996) on 2-D slabs akin to solar flux tubes indicates that for longitudinal tube waves the energy leakage is relatively small and thus the thin tube approximation is acceptable.

The four Eqs. (1) to (4) are sufficient to compute the five quantities , p, S, u and B, because only two of the three thermodynamic variables are independent. We now eliminate B from the system by using Eqs. (1) and (3). From (1) we get

while from (3) we have

where

is the Alfvén speed. With (6) and (7), the B derivatives in (5) can be replaced and one obtains

The system (1) to (4) is now reduced to the three differential equations (2), (4) and (9) for the unknowns , p, S and u. We also must compute the thermodynamic variables temperature T and adiabatic sound speed . Only two of the thermodynamic variables are independent. We thus express all thermodynamic variables by the variables and S. The relationship between the thermodynamic variables is given by the ideal gas law

and

where is the ratio of specific heats, µ the molecular weight, and the gas constant. Combining the first and second law of thermodynamics yields

Using the above relations, Eqs. (2), (4) and (9) can now be written as a system of three partial differential equations for the three unknowns , S and u. To employ the method of characteristics we transform this system of three partial differential equations into a system of six ordinary differential equations defined along the characteristic lines. We obtain the two compatibility relations

along the and characteristics given by

where the upper (lower) sign corresponds to the () characteristic. The tube speed is given by

Eqs. (14) and (15) are four ordinary differential equations. The two remaining ordinary differential equations are obtained from (4) written in characteristic form

along the (fluid path) C⁰ characteristic

2.2. Shocks

During propagation longitudinal tube waves form shocks due to the density gradient. At the shocks we have the variables , , , , , , in front of the shock and corresponding variables with index 2 behind the shock. Here is the enthalpy and the tube cross-section. The Rankine-Hugoniot relations for longitudinal tube waves can be written (Herbold et al. 1985)

where , are the gas velocities in the shock frame. The transformation of the gas velocity from the laboratory (Euler) frame into the shock frame is accomplished by

where is the shock velocity in the laboratory frame. The Rankine-Hugoniot relations are solved by assuming that the front shock state is known and by specifying a post-shock velocity . The front shock state is updated through the iteration process for the respective hydrodynamic time-step. With and known quantities in front of the shock the cubic equation

with

is solved for by using a Newton-Raphson iteration scheme. Here is the magnetic flux. The other post-shock quantities are then computed using

and

2.3. Flux tube models

In our investigation we consider three different magnetic flux tube models. We start by constructing a non-grey radiative and hydrostatic equilibrium solar atmosphere model with an effective temperature K and gravity cm s^-2 using the temperature and flux correction procedure of Cuntz et al. (1994). In this model, NLTE H^- continuum and Mg II k line cooling have been assumed. The atmosphere has a temperature which continuously decreases in outward direction and there is no mechanical heating. Magnetic flux tubes are subsequently embedded in this atmosphere. At height , where in the external atmosphere the optical depth is , the tubes have a specified radius km and a magnetic field strength . The tubes are allowed to spread with height in pressure balance with the surrounding gas pressure (cf. Eq. 3) up to a height . The spreading rate from height to the maximum height km is chosen differently for the different tube models.

The case with is called exponential tube . For the so-called wine-glass tube model we have . Here for the heights we fit a radius function , which ensures that the slope at is continuous and that the tube approaches a specified radius at great height. is the radius at and L is determined by the slope at . To ensure that the tube has this specified radius, an additional time-independent external pressure is assumed, which is supposed to originate from magnetic interaction with neighboring tubes. The gas pressure in the tube is computed in hydrostatic equilibrium and the temperature is determined by radiative equilibrium using the above-mentioned temperature correction procedures. Choosing and leads to our third tube model, the constant cross-section tube . For the wine-glass tube, we take km and km. The three types of tubes obtained are displayed in Fig. 1.

Fig. 1. Shape of the exponential (E), constant cross-section (C), and wine-glass (W) magnetic flux tubes as used in the present work. The radii of the tubes are shown as function of height.

The exponential tube with its unobstructed spreading may be typical for tubes in the interior of supergranulation cells or in areas just outside the network region, while the constant cross-section tube may be representative for the inner part of a very crowded network region. The wine-glass tube with a filling factor thus corresponds to the situation at the outer parts of a normal network region (Solanki 1997, private communication).

2.4. Wave computations

Using a time-dependent magnetohydrodynamic wave code (Herbold et al. 1985; Zhugzhda et al. 1995) various monochromatic waves with periods P and wave energy fluxes were introduced at height at the bottom of the tube by means of a piston. The values of P and have been varied to present a set of case studies. The piston velocity u is prescribed by

where is the velocity amplitude. Here and are the density in the tube and the tube speed at height . During the propagation, shocks form, which are treated as discontinuities and permitted to grow to arbitrary strength and to merge. The positions of shock formation are found by monitoring intersection points of adjacent characteristics.

© European Southern Observatory (ESO) 1998

Online publication: July 27, 1998
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