Abstract
Context. Obtaining observational constraints on the role of turbulent effects for the solar dynamo is a difficult, yet crucial, task. Without such knowledge, the full picture of the operation mechanism of the solar dynamo cannot be formed. Aims. The magnetic helicity spectrum provides important information about the α effect. Here we demonstrate a formalism in spherical geometry to infer magnetic helicity spectra directly from observations of the magnetic field, taking into account the sign change of magnetic helicity across the Sun’s equator. Methods. Using an angular correlation function of the magnetic field, we develop a method to infer spectra for magnetic energy and helicity. The retrieval of the latter relies on a fundamental definition of helicity in terms of linkage of magnetic flux. We apply the two-scale approach, previously used in Cartesian geometry, to spherical geometry for systems where a sign reversal of helicity is expected across the equator on both small and large scales. Results. We test the method by applying it to an analytical model of a fully helical field, and to magneto-hydrodynamic simulations of a turbulent dynamo. The helicity spectra computed from the vector potential available in the models are in excellent agreement with the spectra computed solely from the magnetic field using our method. In a next test, we use our method to obtain the helicity spectrum from a synoptic magnetic field map corresponding to a Carrington rotation. We observe clear signs of a bihelical spectrum of magnetic helicity, which is in complete accordance to the previously reported spectra in literature from the same map. Conclusions. Our formalism makes it possible to infer magnetic helicity in spherical geometry, without the necessity of computing the magnetic vector potential. It has many applications in solar and stellar observations, but can also be used to analyse global magnetoconvection models of stars and to compare them with observations.
Highlights
The solenoidal nature of magnetic fields enables us to examine them in terms of the topology of closed curves (Berger & Field 1984)
This shows that spherical magnetic helicity spectrum (SMHS) (Fig. 4b) computed using the angular correlation function of magnetic field recovers a bihelical spectrum with positive sign at large scales, which is in very good agreement with the actual helicity spectrum of the simulation (Fig. 4a)
In order to investigate mechanisms responsible for largescale magnetic fields present in the Sun and other stars, having a knowledge of magnetic helicity and its distribution over scales is of importance
Summary
The solenoidal nature of magnetic fields enables us to examine them in terms of the topology of closed curves (Berger & Field 1984). Using a toroidal-poloidal decomposition (Chandrasekhar 1961) of vector fields in spherical geometry, Pipin et al (2019) compute the magnetic helicity from solar synoptic maps by reconstructing A, again making a particular gauge choice In their analysis they separate the large and small scales relying on azimuthal averages, wherein the small-scale magnetic helicity density includes contributions from all scales except the axisymmetric mean field, including the large-scale non-axisymmetric contributions. Prior et al (2020) demonstrated a wavelet based multi-resolution analysis to infer magnetic helicity which is applicable for inhomogeneous systems with non-periodic boundaries They avoid computing the vector potential and avoid making a gauge choice by relying on a topologically meaningful definition of helicity in terms of B.
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