Open core of the self-regulated coronal heating paradigm: reconnection regime and quantitative evaporation–collisionality feedback in solar coronal loops

Uzdensky (2007) proposed that solar coronal heating is a self-regulating process that keeps coronal plasma marginally collisionless: density controls the transition between slow collisional Sweet–Parker reconnection and fast collisionless reconnection, while the released energy drives chromospheric evaporation that raises the density and temporarily shuts reconnection off. The proposal was order-of-magnitude only, and its two load-bearing ingredients remain unresolved: (i) which fast-reconnection mechanism or regime (Hall-mediated, localized anomalous resistivity, or — as later work requires — plasmoid-mediated collisionless reconnection) is actually realized in coronal current sheets as a function of coronal parameters, and (ii) a first-principles quantification of the feedback loop — the fraction of released magnetic energy conducted to the chromosphere and converted into evaporated mass, the per-event density increment, and the resulting statistical distribution of coronal density relative to the critical (marginal-collisionality) value.

Background

The solar corona is several hundred times hotter than the solar surface, and identifying the mechanism that maintains this temperature is one of the classic unsolved problems of solar physics. In the direct-current (DC) channel of the leading proposal, photospheric footpoint motions braid coronal magnetic field lines, building up magnetic stress that is released intermittently by magnetic reconnection — the reconfiguration of magnetic field lines that converts stored magnetic energy to heat and particle kinetic energy — in events called nanoflares (Parker 1988).

How fast reconnection proceeds is controlled by the structure of the diffusion region, the thin layer where field lines change connectivity. In the classical Sweet–Parker picture, this layer has a large aspect ratio and reconnection is slow, with an inflow speed suppressed by the small ratio of layer thickness to system size. Petschek proposed a configuration with standing shock waves attached to a small diffusion region that allows much faster reconnection, but numerical and laboratory studies showed that Petschek's mechanism does not operate in resistive magnetohydrodynamics (MHD) with classical (e.g. Spitzer) collisional resistivity — there, the slow Sweet–Parker scaling holds. Fast reconnection becomes possible when the layer thins to collisionless scales, where kinetic physics takes over: the ion skin depth $d_i = c/\omega_{pi}$ (the scale at which ions decouple from magnetic fields) or, when a strong guide field (a magnetic component along, rather than reconnecting across, the layer) is present, the ion-acoustic gyroradius $\rho_s$ (the gyroradius computed with the ion sound speed, which replaces $d_i$ as the relevant kinetic scale when the reconnecting field is weak compared with the guide field). Two mechanisms were identified by 2007 as candidates for fast collisionless reconnection: the Hall effect (decoupled ion and electron motions inside a layer thinner than $d_i$) and spatially localized anomalous resistivity (resistivity enhanced above the collisional Spitzer value in a small region). Which of them operates under which conditions — and how the transition is triggered — was already an open question, formalized as an onset (catastrophe) model by Cassak, Shay and Drake (2005, 2007): fast reconnection switches on when the Sweet–Parker thickness $\delta_{SP}$ falls below $d_i$, or below $\rho_s$ with a strong guide field.

Uzdensky (2007) applied this transition condition to coronal heating. Because the Sweet–Parker thickness depends on the Coulomb-collision mean free path, and hence on plasma density, the transition between slow and fast reconnection is controlled by the ambient electron density $n_e$: reconnection is fast only below a critical density $n_c$ (of order $10^{10}\,\mathrm{cm^{-3}}$ for typical active-region parameters, comparable to observed coronal densities). The released energy is partly conducted along field lines to the chromosphere, where it evaporates dense plasma up into the loop, raising $n_e$ above $n_c$ and shutting fast reconnection off again; radiative cooling and draining then lower the density until the cycle restarts. In this picture coronal heating self-regulates so that the plasma stays marginally collisionless, and heating consists of repeating cycles of nanoflares, evaporation episodes, and long cooling phases. Uzdensky's estimates were deliberately approximate: he used Spitzer resistivity, an energy-balance argument that ohmic heating raises the layer electron temperature to an equipartition value $T_e \sim B_0^2/(16\pi k_B n_e)$ without much density increase, neglect of radiative and conductive losses on the Alfvén crossing time, an extension to strong guide fields, and a toy single-flux-tube model $B_0(t) = \gamma t B_z$ predicting secular growth laws — $n_e \sim t^2$ and emission measure $\mathrm{EM} \sim t^4$ in the PRL, whereas the ApJ companion's baseline, in which the loop temperature evolves with the density, instead gives $n_e \sim t$ and $\mathrm{EM} \sim t^2$, with the $t^2$ and $t^4$ laws recovered when that temperature evolution is neglected (emission measure, the line-of-sight integral of $n_e^2$, sets the extreme-ultraviolet (EUV) and soft-X-ray brightness). Cassak, Mullan and Shay (2008) independently proposed self-organization to marginal collisionality through a related but distinct argument.

Two later developments reshape (but do not settle) this framework. First, the plasmoid instability — the tearing fragmentation of Sweet–Parker layers above a Lundquist number (the ratio of the resistive diffusion time to the Alfvén transit time across the layer) of order $10^4$ — discovered by Loureiro, Schekochihin and Cowley (2007) and elaborated by Huang and Bhattacharjee (2010) and Uzdensky, Loureiro and Schekochihin (2010), means that at coronal Lundquist numbers ($\sim 10^{12}$) the laminar Sweet–Parker layer entering the original onset criterion is always unstable; how the transition to collisionless reconnection proceeds in the resulting plasmoid-unstable regime — which of the candidate regimes organized in the reconnection phase diagram of Ji and Daughton (2011) and reviewed for coronal conditions by Cassak and Shay (2012) is actually realized — is therefore itself part of the open problem. Second, Imada and Zweibel (2012) performed the only dedicated quantitative test of the self-regulation idea: one-dimensional hydrodynamic loop simulations with a prescribed density-dependent heating rate (only current sheets thinner than $d_i$ reconnect) found limit-cycle behavior (evaporation quenching reconnection, followed by cooling and draining) and a time-averaged density near the marginal-collisionality value — supporting the paradigm within a reduced model with non-self-consistent heating. Modern observations of instability-driven nanojets (Sukarmadji, Antolin and McLaughlin 2022) and of loop intermittency unexplained by the footpoint Poynting flux (the upward electromagnetic energy flux through the loop footpoints; Judge and Kuin 2024) touch the eruption-threshold and loading aspects but do not quantify them.

Problem Statement

This question asks for the quantitative physical closure of the self-regulated coronal-heating paradigm of Uzdensky (2007): turn the order-of-magnitude proposal into a validated theory by determining, from first principles and against solar observations,

(1) which fast-reconnection mechanism or regime — Hall-mediated reconnection, spatially localized anomalous resistivity, their interplay, or plasmoid-mediated collisionless reconnection — is actually realized in coronal current sheets as a function of the ambient loop parameters (electron density $n_e$, reconnecting field $B_0$, guide field $B_z$, global length $L$). This includes establishing the status of the onset criterion, originally expressed as the Sweet–Parker layer becoming thinner than the ion skin depth ($\delta_{SP} < d_i$, or $\delta_{SP} < \rho_s$ with a strong guide field), equivalently $L < L_c \sim 6 \times 10^9\,\mathrm{cm}\,(n_e/10^{10}\,\mathrm{cm^{-3}})^{-3}(B_0/30\,\mathrm{G})^4$ with an extra factor $(B_0/B_z)^2 \ll 1$ in the strong-guide-field case, in the plasmoid-unstable regime that prevails at coronal Lundquist numbers — whether the criterion remains valid as stated, requires reformulation, or is superseded by a different transition condition;

(2) the quantitative feedback loop: the fraction $\epsilon$ of released magnetic energy that is conducted to the chromosphere and converted into evaporated mass (the source's evaporation efficiency, which sets the relative per-event density increment), the density increment $\delta n_e$ added per reconnection event and its dependence on loop geometry and energy partition, and the resulting statistical distribution of coronal density relative to the critical density $n_c$ — over the time evolution within loops and across the loop population — i.e. whether and on what time-scales loops self-organize to marginal collisionality, as opposed to remaining persistently above $n_c$ (fast reconnection never triggered) or below $n_c$ (never shut off by evaporation);

(3) verification or correction of the analytic approximations underlying the estimates: the use of Spitzer resistivity, the equipartition electron-temperature estimate, the neglect of radiative and conductive losses on the Alfvén crossing time of the collisional layer, and the extension of the relations to the strong-guide-field regime; and

(4) relatedly, the threshold separating continued self-regulated nanoflare cycling (with the toy twisting-flux-tube growth laws — $n_e \sim t^2$, emission measure $\sim t^4$ in the PRL version; $n_e \sim t$, emission measure $\sim t^2$ in the ApJ companion's baseline) from global loop instability — kink instability, sigmoid formation, and eruptive transition with catastrophic energy release — a bifurcation the source raises as one of two speculative termination channels of the toy model's secular growth.

Items (1)–(3) are facets of one coherent objective — establishing whether the DC reconnection channel of coronal heating self-regulates the coronal density, and with what quantitative consequences. Item (4) is a related open aspect of the same problem: the source raises the cycling-versus-eruption bifurcation only as a speculative termination channel of an admittedly idealized toy model, and its quantitative threshold couples to the separate active research area of 3D loop stability.

References

1. Dmitri A. Uzdensky, "Self-Regulation of Solar Coronal Heating Process via the Collisionless Reconnection Condition", Physical Review Letters 99, 261101 (2007), doi:10.1103/PhysRevLett.99.261101, https://doi.org/10.1103/PhysRevLett.99.261101; preprint arXiv:0712.3941, https://arxiv.org/abs/0712.3941
2. Dmitri A. Uzdensky, "The Fast Collisionless Reconnection Condition and the Self-Organization of Solar Coronal Heating", The Astrophysical Journal 671, 2139–2153 (2007), doi:10.1086/522915, https://doi.org/10.1086/522915
3. P. A. Cassak, M. A. Shay, J. F. Drake, "Catastrophe Model for Fast Magnetic Reconnection Onset", Physical Review Letters 95, 235002 (2005), doi:10.1103/PhysRevLett.95.235002, https://doi.org/10.1103/PhysRevLett.95.235002
4. P. A. Cassak, J. F. Drake, M. A. Shay, "Catastrophic onset of fast magnetic reconnection with a guide field", Physics of Plasmas 14, 054502 (2007), doi:10.1063/1.2734948, https://doi.org/10.1063/1.2734948
5. P. A. Cassak, D. J. Mullan, M. A. Shay, "From Solar and Stellar Flares to Coronal Heating: Theory and Observations of How Magnetic Reconnection Regulates Coronal Conditions", The Astrophysical Journal 676, L69 (2008), doi:10.1086/587055, https://doi.org/10.1086/587055
6. N. F. Loureiro, A. A. Schekochihin, S. C. Cowley, "Instability of current sheets and formation of plasmoid chains", Physics of Plasmas 14, 100703 (2007), doi:10.1063/1.2783986, https://doi.org/10.1063/1.2783986
7. W. Daughton, V. Roytershteyn, B. J. Albright, H. Karimabadi, L. Yin, Kevin J. Bowers, "Transition from collisional to kinetic regimes in large-scale reconnection layers", Physical Review Letters 103, 065004 (2009), doi:10.1103/PhysRevLett.103.065004, https://doi.org/10.1103/PhysRevLett.103.065004
8. Ellen G. Zweibel, Masaaki Yamada, "Magnetic Reconnection in Astrophysical and Laboratory Plasmas", Annual Review of Astronomy and Astrophysics 47, 291 (2009), doi:10.1146/annurev-astro-082708-101726, https://doi.org/10.1146/annurev-astro-082708-101726
9. Yi-Min Huang, A. Bhattacharjee, "Scaling laws of resistive magnetohydrodynamic reconnection in the high-Lundquist-number, plasmoid-unstable regime", Physics of Plasmas 17, 062104 (2010), doi:10.1063/1.3420208, https://doi.org/10.1063/1.3420208
10. D. A. Uzdensky, N. F. Loureiro, A. A. Schekochihin, "Fast Magnetic Reconnection in the Plasmoid-Dominated Regime", Physical Review Letters 105, 235002 (2010), doi:10.1103/PhysRevLett.105.235002, https://doi.org/10.1103/PhysRevLett.105.235002
11. Hantao Ji, William Daughton, "Phase diagram for magnetic reconnection in heliophysical, astrophysical, and laboratory plasmas", Physics of Plasmas 18, 111207 (2011), doi:10.1063/1.3647505, https://doi.org/10.1063/1.3647505
12. P. A. Cassak, M. A. Shay, "Magnetic Reconnection for Coronal Conditions: Reconnection Rates, Secondary Islands and Onset", Space Science Reviews 172, 283–302 (2012), doi:10.1007/s11214-011-9755-2, https://doi.org/10.1007/s11214-011-9755-2
13. J. Goodman, D. Uzdensky, "Reconnection in Marginally Collisionless Accretion Disk Coronae", The Astrophysical Journal 688, 555–558 (2008), doi:10.1086/592345, https://doi.org/10.1086/592345
14. Jonathan C. McKinney, Dmitri A. Uzdensky, "A reconnection switch to trigger gamma-ray burst jet dissipation", Monthly Notices of the Royal Astronomical Society 419, 573–607 (2012), doi:10.1111/j.1365-2966.2011.19721.x, https://doi.org/10.1111/j.1365-2966.2011.19721.x
15. S. Imada, E. G. Zweibel, "Self-Organization of Reconnecting Plasmas to Marginal Collisionality in the Solar Corona", The Astrophysical Journal 755, 93 (2012), doi:10.1088/0004-637X/755/2/93, https://doi.org/10.1088/0004-637X/755/2/93; preprint arXiv:1206.2706, https://arxiv.org/abs/1206.2706
16. Yi-Min Huang, A. Bhattacharjee, "Plasmoid instability in high-Lundquist-number magnetic reconnection", Physics of Plasmas 20, 055702 (2013), doi:10.1063/1.4802941, https://doi.org/10.1063/1.4802941
17. P. A. Cassak, Yi-Hsin Liu, M. A. Shay, "A review of the 0.1 reconnection rate problem", Journal of Plasma Physics 83, 715830501 (2017), doi:10.1017/S0022377817000666, https://doi.org/10.1017/S0022377817000666
18. Leonid Ledentsov, "Thermal Trigger for Solar Flares I: Fragmentation of the Preflare Current Layer", Solar Physics 296, 74 (2021), doi:10.1007/s11207-021-01817-1, https://doi.org/10.1007/s11207-021-01817-1
19. A. Ramada C. Sukarmadji, Patrick Antolin, James A. McLaughlin, "Observations of Instability-driven Nanojets in Coronal Loops", The Astrophysical Journal 934, 190 (2022), doi:10.3847/1538-4357/ac7870, https://doi.org/10.3847/1538-4357/ac7870
20. Philip G. Judge, N. Paul M. Kuin, "On the Intermittency of Hot Plasma Loops in the Solar Corona", The Astrophysical Journal 970, 130 (2024), doi:10.3847/1538-4357/ad5202, https://doi.org/10.3847/1538-4357/ad5202
21. Eugene N. Parker, "Nanoflares and the solar X-ray corona", The Astrophysical Journal 330, 474 (1988), doi:10.1086/166485, https://doi.org/10.1086/166485