During solar flares, the photospheric magnetic field near polarity inversion lines changes abruptly and permanently: the horizontal field component strengthens, the field tilt relaxes toward its potential-field value, and the Lorentz force on the solar surface changes downward — yet photospheric magnetic shear is not systematically relieved. Petrie (2012), who documented this pattern in six major neutral-line flares observed with the Helioseismic and Magnetic Imager (HMI), posed four follow-up questions: statistical representativeness, sunspot-internal responses, the altitude dependence of shear relief, and the line-tying explanation for the absence of photospheric shear relief. Later literature (2016–2024) has settled the statistical question with multi-event studies and has documented sunspot responses with a nuanced (partly negative) answer. The open core, addressed here, is the physically linked pair: (A) determine, as a function of height in the solar atmosphere, where flare-related shear and free magnetic energy are actually relieved — photospheric measurements alone cannot answer this, and force-free extrapolations suggest shear may decrease above a certain height even when photospheric shear does not — and (B) quantitatively test the proposal that photospheric line-tying (anchoring of field-line footpoints in the dense photosphere) inhibits the lateral footpoint motions needed to relieve shear while still permitting tilt relaxation, by characterizing effective anchoring strength and predicting observable signatures that distinguish line-tying-dominated from footpoint-mobile photospheric responses.

Background

A solar flare is a sudden release of magnetic free energy — the energy a coronal magnetic field stores above that of the lowest-energy potential (current-free) configuration — through magnetic reconnection, the process in which magnetic field lines break and reconnect, converting free energy into radiation, plasma heating, and eruption. The flare's back-reaction on the solar atmosphere was predicted by Hudson (2000): if the pre-flare coronal field is inflated and sheared, energy release must make it contract, an "implosion". Fisher et al. (2011) showed that such a contraction transfers a net Lorentz-force change — the magnetic body force J×B\mathbf{J}\times\mathbf{B} per unit volume, where J\mathbf{J} is the current density and B\mathbf{B} the magnetic field — downward onto the photosphere (the visible solar surface) and the solar interior.

The relevant observations come from vector magnetographs, instruments that measure all three components of the photospheric magnetic field; the key modern instrument is the Helioseismic and Magnetic Imager (HMI) on board NASA's Solar Dynamics Observatory (SDO), which provides full-disk vector magnetograms at high cadence. A polarity inversion line (PIL, also called a neutral line) separates opposite vertical magnetic polarities in an active region, and strong flares occur along PILs where the horizontal field is strong and highly sheared. Magnetic shear is measured by the angle between the observed horizontal field vector and the horizontal field of a potential field extrapolated from the same vertical-flux distribution; a large shear angle signals strong nonpotentiality, i.e., stored free energy. Flare-related "magnetic imprints" are the abrupt, step-like, and permanent (hours-long) changes of the photospheric field observed at flares.

Petrie (2012) analyzed six major neutral-line flares observed with SDO/HMI and found a consistent pattern: a dominant increase of the horizontal field in a band centered on the PIL; relaxation of the field tilt angle toward its potential-field value; a downward Lorentz-force change consistent with the implosion picture; but no uniform relief of photospheric shear — the shear angle did not systematically decrease, even though flare energy release should reduce nonpotentiality. This poses a puzzle: the flare demonstrably releases free energy, yet the photospheric boundary shows no systematic shear relief. Nonlinear force-free field (NLFFF) extrapolation — the standard technique that reconstructs the three-dimensional coronal field from photospheric boundary data under the assumption that the Lorentz force vanishes in the corona — suggested in the studies Petrie cited that shear may decrease above a certain height even when photospheric shear does not. Petrie proposed that line-tying, the standard boundary condition of coronal modeling in which the footpoints of coronal field lines are treated as anchored in the high-density photosphere and below, is the reason: it is far harder to move footpoints laterally (which relieving shear requires) than to change their tilt. He noted this explanation was plausible but not demonstrated.

Petrie (2012) posed four follow-up questions: (1) whether the documented pattern is representative of flares generally, requiring large statistical samples from high-cadence vector measurements; (2) whether an analogous pattern of flare-related magnetic and Lorentz-force changes exists within sunspots; (3) the altitude-dependent redistribution of shear and free energy — where in the atmosphere shear is actually relieved, requiring coordinated coronal-field modeling and multi-height observations; and (4) a quantitative test of the line-tying explanation, including the anchoring-strength role and observable signatures distinguishing line-tying-dominated from footpoint-mobile responses. Questions (1) and (2) have since been substantially addressed in the literature, while (3) and (4) — the two faces of one physical objective, an account of how the flare back-reaction is distributed with height and why the photosphere responds as observed — remain open.

Problem Statement

Determine where in the solar atmosphere flare-related magnetic shear and free magnetic energy are actually relieved, as a function of height. Establish the altitude-dependent redistribution of shear and free energy during solar flares through coordinated coronal magnetic-field modeling and multi-height observations — resolving the discrepancy between the absence of systematic photospheric shear relief (Petrie 2012; confirmed statistically in Petrie 2019, whose 15 X-class flares show strengthening of the horizontal shear in the central structure) and nonlinear force-free extrapolations indicating that shear may decrease above a certain height. In the same framework, determine quantitatively whether photospheric line-tying explains why the photospheric tilt relaxes toward potential values while photospheric shear does not: characterize the effective anchoring strength of photospheric footpoints, and predict observable signatures that distinguish line-tying-dominated photospheric responses (tilt relaxation without lateral footpoint motion or shear relief) from footpoint-mobile responses. These two questions form one coherent objective — a physical account of how the flare back-reaction on the atmospheric magnetic field is distributed with height, and of why the photospheric signature takes the observed form. No restriction is imposed on event class, sample size, height range, model class, or answer form.

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