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

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3. Small and intermediate flux tube thickness

We have two basic parameters: [FORMULA], which determines the length-scale of the inhomogeneity, and a, which determines the magnetic field strength. After some initial trial calculations we decided to adopt a in the range [FORMULA] to [FORMULA]. A value much lower than this makes the problem a classical one in the absence of magnetic fields. A higher value produced unrealistic profiles with a large maximum at intermediate times: terms containing a were then of a larger order of magnitude and produced a large growth only restricted by our boundary conditions at [FORMULA]. Indeed, this leads us to an important conclusion: if [FORMULA], we have equipartition of magnetic field and radiative energy densities. This corresponds to an equivalent-to-present field strength of 3 [FORMULA]. Thus [FORMULA] would correspond to a present field strength of [FORMULA]. Fields as low as equivalent-to-present [FORMULA] are able to affect inhomogeneities in the time interval considered. If they are now measured to be higher than this, some amplification or dynamo mechanism must have taken place after recombination. Magnetic fields which are able to affect the small and intermediate scale inhomogeneities are also of the order of [FORMULA] (equivalent present values).

In Figs. 2 and 3, we plot our results for small scale flux tubes, with [FORMULA]. Fig. 2 shows the time evolution of the maximum perturbation at the flux tube axis for [FORMULA], taking two different initial boundary conditions: isocurvature and inhomogeneity. We see that both initial conditions give the same results except for very early times. Fig. 3 shows the time evolution of the inhomogeneity profile. It remains essentially gaussian throughout the whole time period, increasing more rapidly in the recent half time period.

[FIGURE] Fig. 2. Time evolution of the value of [FORMULA] at the centre of the filament, for [FORMULA] and [FORMULA]. Curve 0 for [FORMULA]. Curve -X for [FORMULA]
[FIGURE] Fig. 3. Time evolution of the filamentary inhomogeneity profile for [FORMULA] and [FORMULA] for the boundary condition [FORMULA]. The parameter characterizing the different curves is a time parameter

In Figs. 4, 5, 6, we plot the results obtained for intermediate scale flux tubes, with [FORMULA]. Fig. 4 shows the time evolution of the maximum perturbation for [FORMULA], [FORMULA] and for both isocurvature [FORMULA] and homogeneity [FORMULA] initial conditions. The first one provides curves without a short initial decrease, which does not seem to be very realistic. For small magnetic fields we see again that the growth is faster in the last part of the time considered. For large magnetic fields the situation is more or less reversed. Fig. 5 shows the time evolution of the inhomogeneity profile, for moderate magnetic field strengths, [FORMULA], and Fig. 6 for higher strengths. The latter shows significant departures from the gaussian profiles.

[FIGURE] Fig. 4. Time evolution of the value of [FORMULA] at the centre of the filament, for [FORMULA]. Curve a: [FORMULA], [FORMULA]; curve b: [FORMULA], [FORMULA]; curve c: [FORMULA], [FORMULA]; curve d: [FORMULA], [FORMULA]
[FIGURE] Fig. 5. Time evolution of the filamentary inhomogeneity profile for [FORMULA] and [FORMULA] for the boundary condition [FORMULA]. The parameter characterizing the different curves is a time parameter
[FIGURE] Fig. 6. Time evolution of the filamentary inhomogeneity profile for [FORMULA] and [FORMULA] for the boundary condition [FORMULA]. The parameter characterizing the different curves is a time parameter

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© European Southern Observatory (ESO) 1997

Online publication: April 8, 1998
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