Astron. Astrophys. 324, 329-343 (1997)

Mean magnetic field and energy balance of Parker's surface-wave dynamo

A.J.H. Ossendrijver and P. Hoyng

SRON Laboratorium voor Ruimteonderzoek, Sorbonnelaan 2, 3584 CA Utrecht, The Netherlands

Received 22 April 1996 / Accepted 24 January 1997

Abstract

We study the surface-wave dynamo proposed by Parker (1993) as a model for the solar dynamo, by solving equations for the mean magnetic field , as well as for the mean 'magnetic energy tensor' . This tensor provides information about the energy balance, rms field strengths and correlation coefficients between field components. The main goal of this paper is to check whether the equations for and are compatible, i.e. whether both have "reasonable" solutions for a set of "reasonable" parameters. We apply the following constraints: has a period of 22 years and, taking into account the effect of period variations, a decay time of 10 dynamo periods, and is marginally stable. We find that under these constraints, the equations for and are compatible only if, apart from turbulent transport out of the dynamo region, an additional energy sink is introduced. If this extra term is omitted, then marginal stability of requires a turbulent diffusion in the convection zone of the order  cm² s^-1, whereas the conditions on require  cm² s^-1. Furthermore, the rms surface field strength, the maximum rms field strength and the magnetic energy flux through the upper surface of the convection zone cannot simultaneously assume solar values. We explore the possibility that the extra energy sink is provided by resistive dissipation, hitherto not accounted for in the equation for , by considering various cases. We demonstrate that with a heuristically modified equation for , the inconsistencies can be removed. Our results suggest that resistive dissipation is the dominant sink of magnetic energy, and that resistive heating may amount to several percent of the solar luminosity.

Key words: Sun: magnetic fields MHD turbulence

Contents

• 1. Introduction
• 2. Mean-field dynamo theory and resistive dissipation
• 3. The mean magnetic field
  • 3.1. Equations
  • 3.2. Boundary conditions
  • 3.3. Solutions
  • 3.4. Parameters
• 4. The mean magnetic energy
  • 4.1. Equations
  • 4.2. Boundary conditions
  • 4.3. Constraints on the parameters
  • 4.4. Solution of Eqs. (43- 44)
• 5. Case A: no resistive dissipation ()
  • 5.1. Parameters
  • 5.2. Implications for the mean magnetic field
  • 5.3. Root mean square magnetic field strength
  • 5.4. Summary
• 6. Case B: uniform resistive dissipation ()
  • 6.1. Parameters
  • 6.2. Solutions
  • 6.3. Root mean square magnetic field strength
• 7. Case C: reduced dissipation in the overshoot layer ()
• 8. Summary and discussion
• Acknowledgements
• Appendix
• Appendix A: dispersion relation for the mean magnetic field
• Appendix B: expressions for the mean magnetic energy tensor
• Appendix C: boundary conditions for the mean magnetic energy tensor
• References

© European Southern Observatory (ESO) 1997

Online publication: May 26, 1998

helpdesk.link@springer.de