The photospheric surface integral 14π∫S1Bz(xBx+yBy) dx dy\frac{1}{4\pi}\int_{S_1}B_z(xB_x+yB_y)\,dx\,dy, the standard virial-theorem energy estimator for force-free fields, was shown by Zhu, Wiegelmann & Inhester (2020) to typically underestimate the magnetic energy of magnetohydrostatic (MHS) equilibria of solar active regions — but only through axisymmetric monopole-spot and two-spot model arguments. The same work introduced preprocessing of vector magnetograms for MHS extrapolation in which the weighting parameters μn\mu_n are selected empirically. Two quantitative gaps remain open. (A) A comprehensive characterization, for general non-axisymmetric active-region geometries, of when and by how much the surface integral underestimates the true MHS magnetic energy. (B) A quantification of how the preprocessing choices — the weights μn\mu_n, the smoothing operator, and the preservation of the integrals a0,a1,a2a_0,a_1,a_2 — propagate into the accuracy and determinacy of the resulting MHS magnetic-field and plasma extrapolations. No work in the literature to date resolves either part; both belong to one open quantitative-reliability question about the MHS reconstruction problem from photospheric data.

Background

Solar active-region magnetic fields above the photosphere (the Sun's visible surface layer; an active region is a localized, strongly magnetized area on the Sun) are commonly reconstructed by extrapolating the measured photospheric vector magnetogram — a map of all three components of the magnetic field on the photosphere. In the corona the field is often modeled as nonlinear force-free (electric current parallel to the magnetic field B\mathbf B), but in the photosphere and chromosphere (the atmospheric layer just above the photosphere) the plasma beta (the ratio of gas pressure to magnetic pressure) is not small, so a magnetohydrostatic (MHS) model is more appropriate: the Lorentz force of the field balances the gas-pressure gradient and gravity, (∇×B)×B/(4π)−∇p−ρz^=0(\nabla\times\mathbf B)\times\mathbf B/(4\pi)-\nabla p-\rho\hat{\mathbf z}=0 with ∇⋅B=0\nabla\cdot\mathbf B=0 (Gaussian cgs units, gravity along −z^-\hat{\mathbf z}), where pp is gas pressure and ρ\rho mass density. Measured magnetograms, however, are generally inconsistent with any such equilibrium above the surface. For force-free fields the classical consistency conditions require the net Lorentz force and net torque over the boundary to vanish, and Wiegelmann, Inhester & Sakurai (2006) introduced a preprocessing procedure that drives measured data toward those conditions while deviating from the observation by no more than the measurement noise and enforcing smoothness.

Zhu, Wiegelmann & Inhester (2020) derived the analogous conditions for an MHS equilibrium by applying the MHD virial theorem: multiplying the force-balance equation by the position vector r\mathbf r and integrating over a semi-infinite volume VV above the photosphere S1S_1 relates the magnetic energy and plasma forces in VV to surface integrals over S1S_1. They found that an isolated active region in MHS equilibrium requires only the two horizontal components of the net Lorentz force and the vertical component of the net torque to vanish; the vertical net force and the horizontal torques may be nonzero, being balanced by plasma pressure and weight. On this basis they constructed a preprocessing functional L=μ1L1+μ2L2+μ3L3+μ4L4L=\mu_1L_1+\mu_2L_2+\mu_3L_3+\mu_4L_4, where L1L_1 and L2L_2 measure the two force and one torque consistency integrals, L3L_3 the deviation from the measured field, and L4L_4 smoothness, subject to preserving three measured surface integrals a0a_0, a1a_1, a2a_2 — the integral of Bz2−Bx2−By2B_z^2-B_x^2-B_y^2 over the photosphere and its two horizontal first moments, which encode the vertical net force and horizontal torques.

The same virial identity exposes an energy-accounting issue. For a force-free field decaying sufficiently fast, the magnetic energy E=∫VB2/(8π) dVE=\int_V B^2/(8\pi)\,dV equals the photospheric integral

14π∫S1Bz(xBx+yBy) dx dy,\frac{1}{4\pi}\int_{S_1}B_z(xB_x+yB_y)\,dx\,dy,

with the coordinate origin taken in the photospheric plane (under the consistency criteria the value is invariant under horizontal shifts of the origin), and this integral is widely used to estimate active-region magnetic energies from magnetograms alone. In an MHS equilibrium the identity instead reads E=14π∫S1Bz(xBx+yBy) dx dy−∫Vf⋅r dVE=\frac{1}{4\pi}\int_{S_1}B_z(xB_x+yB_y)\,dx\,dy-\int_V\mathbf f\cdot\mathbf r\,dV with f=−∇p−ρz^\mathbf f=-\nabla p-\rho\hat{\mathbf z} the non-magnetic force density, so the magnetogram-based estimator differs from EE by exactly ∫Vf⋅r dV\int_V\mathbf f\cdot\mathbf r\,dV, and it underestimates the energy precisely when ∫Vf⋅r dV<0\int_V\mathbf f\cdot\mathbf r\,dV<0. (Zhu et al. 2020 write the same identities with the 4π4\pi factor absorbed into the field normalization; all expressions here are in Gaussian cgs, which changes no sign or comparison statement.) For an axisymmetric sunspot-like field, the pressure-depleted spot makes f⋅r\mathbf f\cdot\mathbf r negative near the spot, so the integral underestimates EE; Zhu et al. extended the argument heuristically to multi-spot regions and cited statistical studies of predominantly downward net Lorentz forces in observed active regions. But this reasoning rests on axisymmetry, monopole-like spot cartoons, and decay assumptions: a comprehensive, quantitative characterization of the sign and magnitude of the error for general (non-axisymmetric, arbitrary) active-region geometries was not given, and the preprocessing weights μn\mu_n were stated to be 'as yet undetermined', selected empirically for one dataset. These are the open quantitative cores examined here.

Problem Statement

For magnetohydrostatic (MHS) equilibria of realistic, general (non-axisymmetric) solar active regions reconstructed from photospheric vector magnetograms, establish a quantitative reliability characterization of the boundary-value problem, consisting of two jointly supporting parts:

(A) Energy-bound characterization. Determine, for general active-region geometries, when the photospheric virial integral 14π∫S1Bz(xBx+yBy) dx dy\frac{1}{4\pi}\int_{S_1}B_z(xB_x+yB_y)\,dx\,dy — the Gaussian-cgs surface integral of Bz(xBx+yBy)B_z(xB_x+yB_y) over the photospheric boundary S1S_1, where Bx,By,BzB_x,B_y,B_z are the Cartesian components of the magnetic field and the coordinate origin lies in the photospheric plane — underestimates the true magnetic energy E=∫VB2/(8π) dVE=\int_V B^2/(8\pi)\,dV of the MHS equilibrium (the energy in the semi-infinite volume VV above the photosphere), and by how much: either prove that the underestimate (equivalently, ∫Vf⋅r dV<0\int_V\mathbf f\cdot\mathbf r\,dV<0 with f=−∇p−ρz^\mathbf f=-\nabla p-\rho\hat{\mathbf z} the non-magnetic force density, pp gas pressure, ρ\rho mass density) holds under stated geometric and plasma conditions, or exhibit explicit equilibrium classes for which it fails, and in either case give a quantitative bound or calibrated error law for the magnitude of the underestimate as a function of active-region properties.

(B) Preprocessing-determinacy characterization. Quantify how the preprocessing choices applied to the measured magnetogram before MHS extrapolation — the weighting parameters μn\mu_n of the consistency functional, the smoothing operator, and the selection of the preserved surface integrals a0,a1,a2a_0,a_1,a_2 (the integral of Bz2−Bx2−By2B_z^2-B_x^2-B_y^2 over the photosphere and its two horizontal first moments) — propagate into the accuracy and the determinacy (sensitivity, non-uniqueness) of the extrapolated magnetic field and plasma variables (pressure, density), across representative active-region classes.

The common objective is a quantitative account of how much the photospheric boundary data of an MHS reconstruction can be trusted: part (A) fixes the error of the energy budget read off from the boundary integrals that the preprocessing preserves, and part (B) fixes the uncertainty of the reconstruction itself. The problem is completely answered when both parts are discharged with stated, expert-checkable quantitative criteria — bounds with explicit assumptions, or calibrated error laws with explicitly covered classes of active-region geometries — rather than illustrative model cases or a single empirical parameter choice.

References

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