Spherical Harmonics Ambiguity Indicator (ISH)

• ISH is a helicity proxy that decomposes the transverse magnetic field into parity-even (E) and parity-odd (B) modes using spin-2 spherical harmonics.
• It constructs a normalized two-scale EB cross-spectrum to quantify magnetic helicity while robustly addressing the inherent 180° ambiguity in polarimetric measurements.
• Empirical validations on synthetic and solar datasets demonstrate ISH's effectiveness for inferring magnetic helicity across solar, stellar, and galactic scales.

The Spherical Harmonics Ambiguity Indicator (ISH) is a global proxy for magnetic helicity that is defined on the sphere and is invariant under the inherent 180° (“π”) ambiguity of transverse magnetic field measurements. ISH employs spin-2 spherical harmonics to decompose the transverse field into parity-even (E) and parity-odd (B) modes, constructing a two-scale cross-spectrum that functions as a helicity indicator. This methodology enables robust helicity inference from global datasets such as solar synoptic vector magnetograms or stellar/Galactic polarization maps, circumventing ambiguities that hamper conventional helicity diagnostics in weak-field regions (Brandenburg, 2019).

1. Spin-2 Spherical Harmonic Decomposition

The ISH pipeline begins with the construction of a complex linear polarization field, $p(\theta,\phi)$, defined via the components $Q(\theta,\phi)$ and $U(\theta,\phi)$ of the transverse vector field:

$$p(\theta, \phi) \equiv Q(\theta, \phi) + U(\theta, \phi)$$

This field is then expanded onto the basis of spin-weighted spherical harmonics with spin $+2$: $$\tilde{R}_{\ell m} = \int_{4\pi} p(\theta,\phi)\; {}_2Y_{\ell m}^*(\theta,\phi)\; \sin\theta\, d\theta\, d\phi,\quad (\ell \ge 2,\ -\ell \le m \le \ell)$$ Parity-even and parity-odd coefficients are constructed by combining $\tilde{R}_{\ell m}$ and its $\pm m$ conjugate: $$\tilde{E}_{\ell m} = \frac{1}{2} \left( \tilde{R}_{\ell m} + \tilde{R}_{\ell, -m}^* \right), \qquad \tilde{B}_{\ell m} = \frac{1}{2} \left( \tilde{R}_{\ell m} - \tilde{R}_{\ell, -m}^* \right)$$ By construction, $E_{\ell m}$ is even, and $Q(\theta,\phi)$0 is odd under parity.

2. Definition of the Two-Scale EB Helicity Proxy

The ISH is defined as a cross-spectrum of the E and B coefficients at nearby spherical harmonic degrees. Empirically, the sharpest and most informative helicity proxy is given by $Q(\theta,\phi)$1: $Q(\theta,\phi)$2 A normalization by the E and B mode powers at these degrees is often adopted: $Q(\theta,\phi)$3 This normalized indicator quantifies the global helicity at different angular scales on the sphere.

3. Invariance under the π-Ambiguity

The ISH inherits a crucial property of invariance under the 180° ambiguity in azimuthal orientation of the transverse field. Specifically, for a flip $Q(\theta,\phi)$4 corresponding to a rotation by π, the polarization field transforms as $Q(\theta,\phi)$5: $Q(\theta,\phi)$6 This invariance propagates to the harmonic coefficients and the two-scale proxy, rendering ISH immune to the ambiguity that typifies linear polarimetry in weak-field regions.

4. Empirical Validation on Model Fields

ISH has been numerically validated on both axisymmetric (1D) and nonaxisymmetric (2D) toy models:

• 1D (axisymmetric) models: For purely toroidal potential and field (e.g., $Q(\theta,\phi)$7 and $Q(\theta,\phi)$8), the helicity density $Q(\theta,\phi)$9 is antisymmetric about the equator. The constructed EB cross-spectrum $U(\theta,\phi)$0 shows a single prominent peak at a specific degree, with a sign opposite to the local helicity in the northern hemisphere.
• 2D (nonaxisymmetric) models: Using combinations of poloidal and toroidal superpotentials ($U(\theta,\phi)$1, $U(\theta,\phi)$2), the EB cross-spectrum reflects regions of positive and negative helicity with expected sign patterns.

These tests verify that $U(\theta,\phi)$3 robustly traces the sign and scale of net hemispheric helicity.

5. Application to Solar Synoptic Magnetograms

For empirical data, such as full-Sun Carrington-rotation synoptic maps, the implementation produces the pseudo-polarization: $U(\theta,\phi)$4 After mapping to a full-sphere pixelization (e.g., HEALPix), one computes the harmonic coefficients and evaluates $U(\theta,\phi)$5. Application to solar data yields a robust negative dip in $U(\theta,\phi)$6 at $U(\theta,\phi)$7 (corresponding to angular scales $U(\theta,\phi)$8), consistently across multiple solar rotations.

6. Physical Interpretation of the Helicity Proxy

The sign of $U(\theta,\phi)$9 is empirically found to be opposite to the sign of the true magnetic helicity in the northern hemisphere (with the converse holding by symmetry for the southern hemisphere). Accordingly, a negative $$p(\theta, \phi) \equiv Q(\theta, \phi) + U(\theta, \phi)$$0 at intermediate $$p(\theta, \phi) \equiv Q(\theta, \phi) + U(\theta, \phi)$$1 (such as $$p(\theta, \phi) \equiv Q(\theta, \phi) + U(\theta, \phi)$$2 on the Sun) is interpreted as indicating positive large-scale magnetic helicity in the northern hemisphere. At smaller scales (larger $$p(\theta, \phi) \equiv Q(\theta, \phi) + U(\theta, \phi)$$3), the proxy is significantly noisier and exhibits no systematic net sign, plausibly reflecting cancellation in weak-field regions or depth-dependent reversals of helicity above the photosphere.

7. Generalization to Stellar and Galactic Applications

The ISH formalism generalizes to any dataset providing global Stokes $$p(\theta, \phi) \equiv Q(\theta, \phi) + U(\theta, \phi)$$4 (or a transverse field modulo $$p(\theta, \phi) \equiv Q(\theta, \phi) + U(\theta, \phi)$$5) on a spherical domain. Zeeman–Doppler imaging of stellar surfaces and all-sky dust-polarization surveys (e.g., Planck) can be fed into the same spin-2 → $$p(\theta, \phi) \equiv Q(\theta, \phi) + U(\theta, \phi)$$6 → two-scale EB proxy pipeline. The indicator thus facilitates direct, global helicity spectra that are directly comparable between the Sun, stars, and the Galaxy, exploiting its invariance to $$p(\theta, \phi) \equiv Q(\theta, \phi) + U(\theta, \phi)$$7-ambiguity and insensitivity to incomplete azimuthal disambiguation (Brandenburg, 2019).