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Astron. Astrophys. 327, 1-7 (1997)

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4. Large scale flux tubes

In our basic Eq. (2) the last two terms have orders of magnitude of [FORMULA] and [FORMULA], respectively. They can therefore be ignored when [FORMULA] is very large. This fact is important for two reasons. From the integration point of view, if the laplacian term is negligible, the equation is no longer elliptic. It can be treated analytically and what is most noticeable is that it can be integrated as an initial-value problem, thus restoring the prediction ability of the equation. We do not need to assume the shape and magnitude of the inhomogeneity before the acoustic regime period. The other reason is that [FORMULA] probably represents a more interesting case as dissipative effects may wipe out small scale inhomogeneities after the time epoch considered in this paper.

The analytical solution now becomes

[EQUATION]

where [FORMULA] and [FORMULA] are integration constants, which may be determined with the boundary conditions. We have several possibilities which should be discussed.

Suppose first that we are considering the time interval [0,1]. For [FORMULA], we have [FORMULA] unless [FORMULA]. But then [FORMULA]. Then only isocurvature would be a valid initial condition, and [FORMULA]. However, we must avoid [FORMULA] (Big-Bang) and begin with [FORMULA], very small but non-vanishing. Also, in our numerical outputs we have carried out the integration since [FORMULA]. We have four options to determine [FORMULA] and [FORMULA]:

1) Boundary value and homogeneity. We assume [FORMULA] and [FORMULA]. The solution is then

[EQUATION]

As [FORMULA] is low, this basically represents a linear growth.

2) Boundary value and isocurvature. We assume [FORMULA], [FORMULA]

[EQUATION]

where the last term is negligible. We again obtain a quasi-linear growth. As it was also numerically obtained, the conditions of homogeneity and isocurvature provide the same results, except in the beginning.

3) Initial value and homogeneity. We assume [FORMULA] and obtain

[EQUATION]

which also represents a nearly linear growth, but now the rate of growth [FORMULA] is larger than the [FORMULA] in the two previous cases. In the end, we obtain [FORMULA], i.e. the inhomogeneity grows [FORMULA] -fold, where [FORMULA] could be considered the magnetogenesis time, even if we adopted in the numerical integration in the previous section [FORMULA]. This third possibility has the best prediction ability.

4) Initial value and isocurvature. We assume [FORMULA] and [FORMULA]. In this case, we obtain [FORMULA] and therefore

[EQUATION]

is a constant. No evolution is to be expected.

With the exception of the fourth, these possibilities all, basically, lead to the result already obtained for small scale flux tubes: the growth is more or less linear. The growth is plotted in Fig. 7 for the four possibilities mentioned above.

[FIGURE] Fig. 7. Time evolution of the value of [FORMULA] at the centre of the filament, for [FORMULA] calculated with [FORMULA], [FORMULA] and [FORMULA]. Curve a: Boundary value and homogeneity; Curve b: Boundary value and isocurvature; Curve c: Initial value and homogeneity; Curve d: Initial value and isocurvature. Curves b and c coincide except for small values of t, where curve b gives slightly higher values

There is another analytical integration in another approach. Suppose that [FORMULA] and that [FORMULA], so that only the laplacian term is negligible. The analytical solution becomes

[EQUATION]

but this solution does not represent a realistic case: [FORMULA] has a maximum at [FORMULA], vanishes at [FORMULA], and what is more important, [FORMULA] increases until it reaches values comparable to a, in contradiction with the initial assumption.

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© European Southern Observatory (ESO) 1997

Online publication: April 8, 1998
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