A&A 415, 751-754 (2004)
Kiepenheuer-Institut für Sonnenphysik, Schöneckstr. 6, 79104 Freiburg, Germany
Received 8 December 2003 / Accepted 15 December 2003
This note deals with magnetohydrodynamic body waves in a magnetic cylinder. It is shown that the solution obtained by the thin-tube expansion is, term by term, identical to the Taylor expansion of the exact solution. Each level of approximation adds a pair of modes, a slow and a fast one, and corrects the frequencies and eigenfunctions of the previous approximation. All eigenfrequencies, approximate and exact, can be read off from a single graph. All slow modes have phase velocities between the tube speed and , all fast modes have phase velocities above .
Key words: magnetohydrodynamics (MHD) - waves - methods: analytical - Sun: magnetic fields
Concentrations of magnetic flux abound on the solar surface, as first documented in the seventies by the work of Howard & Stenflo (1972) and Frazier & Stenflo (1972). Stenflo (1973) and Frazier & Stenflo (1978) found that the field strength in those concentrations often reaches 150 mT, which indicates an equilibrium between the magnetic pressure and the gas pressure.
The magnetic flux concentrations, or flux tubes, support a variety of magnetohydrodynamic waves. Theoretically, such waves have been treated in the thin-tube limit (Defouw 1976; Spruit 1981; Hasan 1984), which is equivalent to the leading order of an expansion in terms of the distance from the tube axis (Roberts & Webb 1978). Other applications of the thin-tube limit include the convective collapse of magnetic flux concentrations in a super-adiabatic environment (Spruit & Zweibel 1979), the Evershed flow in a sunspot penumbra (Schlichenmaier et al. 1998), and the heating of coronal loops that arrises from the non-linearity of the tube waves (Zhugzhda & Nakariakov 1997).
Second-order truncations of the expansion have been used by Browning & Priest (1983) and by Pneuman et al. (1986) to calculate equilibrium configurations of flux tubes. Including time-dependence, Ferriz Mas & Schüssler (1989) have considered the general case, in particular in the context of the exact solution of the magnetohydrodynamic equations which is known in a special case: waves of small amplitude in a straight circular magnetic cylinder; these waves are represented by Bessel functions (Roberts & Webb 1978). The latter authors drew confidence to the expansion procedure from the vicinity of the exact phase speed to the tube speed that characterizes the leading order. Ferriz Mas et al. (1989), using a second-order approximation, concluded that body waves are less well represented than surface waves. Zhugzhda (2002) realized that the dispersion relation for body waves is the same at all levels of approximation, and suggests a recurrence procedure for calculating the eigenfunctions of the diverse orders. Applying such a procedure to the case of body waves in a magnetic cylinder, I show in this note that the expansion is identical to the Taylor expansion of the exact solution and that, therefore, the body waves can be represented with any accuracy. In addition, I shall give a scheme that allows to evaluate the wave frequencies at any level of truncation from a single graph.
In this note I use the terms magnetic cylinder and magnetic flux tube as synonyms. I call the thin-tube limit, or leading order what is commonly known as the thin-tube approximation, while any truncated expansion is called a thin-tube approximation. Any wave that is supported by the magnetic tube is called a tube wave. As I do not consider waves that ow their existence to the existence of boundaries, the tube waves of this note are all body waves, not surface waves. The latter have been discussed in a number of contributions, including Ferriz Mas et al. (1989) and Zhugzhda & Goossens (2001).
The thin-tube approximations make use of an expansion of all dependent variables in terms of powers of s, the distance from the axis of the tube. In order to clarify the convergence problem of that expansion, I shall consider the simplest possible case, a circular magnetic cylinder. Gravity is absent, the equilibrium as well as the perturbations are assumed to be axisymmetric and poloidal (no azimuthal vector components), and the medium external to the cylinder does not participate in the perturbation.
The equilibrium shall consist of a homogeneous magnetic field
inside the cylinder, of radius
and constant values
of pressure and density. This is a special solution of the
equations describing the static case. Small perturbations of this
equilibrium will be marked with a tilde. Thus
The expansions (1)-(5) are substituted into the
equations of ideal magnetohydrodynamics, under the assumption of
adiabatic perturbations. This procedure has been described often
(e.g., Ferriz Mas & Schüssler 1989; Stix 1989), hence I can rely
on such earlier work. Comparing equal powers of s one obtains the
Eqs. (3.7)-(3.10), (3.13), and (3.15) of Ferriz Mas & Schüssler
(1989), which in the present case yield the following linearized
In addition to these equations, there is the condition of pressure
continuity at the cylinder boundary,
which in the
linearized form is
The dispersion relation for the two body waves is Eq. (53) of
Ferriz Mas et al. (1989), or Eq. (31) of Zhugzhda (2002). It appears
to me that it has not been realized that, once the phase velocity
is related to the tube speed ,
dispersion relation for
can be written in terms of a
|Figure 1: Dimensionless squared phase velocity as function of the parameter . The lower and upper branches mark the slow and fast body waves, respectively.|
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For N=4 the system of perturbation equations defines the
"four-mode approximation''. As will be clear from the fourth-order
dispersion formula (27) below, the dimensionless
phase velocities of the four modes again obey (14), with
only a single parameter. Hence these phase velocities can again be
read off Fig. 1. However, the definition of the parameter is now
Zhugzhda (2002) has pointed out that the truncation of the perturbation
equations at level N can be used to derive recurrence relations
for the solutions, and that the eigenfunctions converge to Bessel
functions of orders 0 and 1 (although he suggests the constant
in place of the frequency-dependent wave number). Here I
eliminate the functions
in order to derive a recurrence relation for
This straight-forward algebra yields
The appearance of the functions
in the combination (13) does not mean that these two unknowns cannot be determined
separately. For the N-mode approximation the process of elimination
yields, for ,
Using (22) we may evaluate the boundary condition (12) at any level of truncation. For N=2 we recover the
dispersion relation (14), for N=4 we obtain
At each level of truncation a pair of modes is being added, and the
frequencies of the modes found in the previous approximation are
corrected. Since the approximate dispersion relations converge to
the exact dispersion relation, their zeros (and the eigenfrequencies)
must also converge to the exact zeros (eigenfrequencies).
It is easy to see that all approximate frequencies as
well as all frequencies of the converged series can be obtained
by solving (14). The recipe is: take the th root
of the N/2 roots of the N-mode dispersion
relation (written as a N/2-grade polynomial in
), or take the th zero
of J0; evaluate
It seems to me that the value of the thin-tube expansion has not always been fully appreciated. In this note it has been demonstrated for a special case - a straight, circular, and homogeneous magnetic cylinder - that the expansion indeed converges to the exact solution. Moreover, at each level of truncation the solutions are precisely the truncated Taylor series of the Bessel function that represent the exact solution. Therefore each level of truncation can be considered as an improvement. This is true for the eigenfrequencies as well as for the eigenfunctions. Hence, in order to improve an approximation, it seems not advisable to restrict the result to a very thin tube, or to introduce a fictitious wave number or effective tube radius, depending on the level of approximation (Ferriz Mas et al. 1989; Zhugzhda 1996); it is better simply to go to an approximation of higher order. I concede, however, that the method of obtaining eigenfrequencies via the parameter is related to introducing those effective parameters.
The strength of the expansion method lies in its applicability to situations that are more involved than the present example. Such may include stratification, bending, twist, nonlinearity, and the interaction with the external medium. I suppose that the convergence prove of the present simple case adds confidence to the approximations used in those more complicated cases, and that similar proves might be possible there.
I thank O. Steiner and Y. D. Zhugzhda for useful comments.