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Comparing Values of the Relative Magnetic Helicity in Finite Volumes


Relative magnetic helicity, as a conserved quantity of ideal magnetohydrodynamics, has been highlighted as an important quantity to study in plasma physics. Due to its nonlocal nature, its estimation is not straightforward in both observational and numerical data. In this study we derive expressions for the practical computation of the gauge-independent relative magnetic helicity in three-dimensional finite domains. The derived expressions are easy to implement and rapid to compute. They are derived in Cartesian coordinates, but can be easily written in other coordinate systems. We apply our method to a numerical model of a force-free equilibrium containing a flux rope, and compare the results with those obtained employing known half-space equations. We find that our method requires a much smaller volume than half-space expressions to derive the full helicity content. We also prove that values of relative magnetic helicity of different magnetic fields can be compared with each other in the same sense as free-energy values can. Therefore, relative magnetic helicity can be meaningfully and directly compared between different datasets, such as those from different active regions, but also within the same dataset at different times. Typical applications of our formulae include the helicity computation in three-dimensional models of the solar atmosphere, e.g., coronal-field reconstructions by force-free extrapolation and discretized magnetic fields of numerical simulations.

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The authors thank the referee for helpful comments which improved the clarity of the paper. The authors thank Bernhard Kliem and Tibor Török for making the numerical solution of the TD equilibrium available. The research leading to these results has received funding from the European Commission’s Seventh Framework Programme under the grant agreement n 284461 (eHEROES project).

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Correspondence to G. Valori.

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Valori, G., Démoulin, P. & Pariat, E. Comparing Values of the Relative Magnetic Helicity in Finite Volumes. Sol Phys 278, 347–366 (2012).

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  • Active regions, magnetic fields
  • Magnetic field, photosphere, corona