6 Conclusions

We have presented here a new formulation of the equations governing a stationary axisymmetric MHD flow in the equatorial plane. This formulation includes an exact treatment of all effects: thermal pressure, gravity and arbitrary shape of the magnetic flux tubes. The wind solution appears as the level contour of a Bernoulli-function which passes through two particular points: the slow and fast critical points. It allows a direct comparison with the classical model of Weber & Davis (1967), in particular in the formulation given by Sakurai (1985). Thus the specifically relativistic effects are easily identified.

We have used our model to extend the study of the magnetic to kinetic energy conversion made by Begelman & Li (1994). We show that the main parameter which fixes this efficiency is the shape of the magnetic flux tubes. In the case of a constant opening angle, non-relativistic flows have a good efficiency of the magnetic to kinetic energy conversion but as soon as the terminal Lorentz factor is greater than $\sim$1.5, this efficiency decreases rapidly. Such relativistic winds are not able to transfer a large fraction of their magnetic energy to the matter. On the other hand, regions where the opening angle diverges from the constant case are very efficient in converting magnetic into kinetic energy, even in the ultra-relativistic case. This is true as long as such regions are located beyond the fast critical point. Gravity and the thermal pressure play only a minor role.

In Sect. 5, we apply this model in the context of gamma-ray bursts (GRBs hereafter). In the case where the wind produced by the source of GRBs is initially Poynting flux dominated, we have shown that the efficiency of the acceleration strongly depends on the geometry of the magnetic flux tubes. We found that a large variety of situations is expected. If the magnetic tubes have the possibility to diverge strongly from a constant opening angle, it is possible that most of the energy is eventually in kinetic form. On the other hand it is very likely that the magnetic to kinetic energy conversion is incomplete and that the wind is still Poynting flux dominated when it has reached its terminal Lorentz factor. We have demonstrated on one example that such a wind can lead to very promising situations compared to the standard picture: a large amount of the magnetic energy can be dissipated at large radii by reconnection. This reconnection can start when the wind is optically thick or already transparent. So the large magnetic energy reservoir could have two effects: a supplementary acceleration phase increasing the final magnetic to kinetic energy conversion efficiency and/or a direct contribution to the emission. These two possibilities will be investigated in a future work.

Acknowledgements

The authors would like to thank Dr. H. C. Spruit for many stimulating discussions, important suggestions and reading the manuscript.