Radiation-magnetohydrodynamic (rMHD) simulations show that the small-scale dynamo (SSD) - the turbulent dynamo thought to generate the weak internetwork fields covering most of the quiet solar surface - produces fields strong enough to dominate the energy balance of the mid-to-upper chromosphere and to carry an energy flux exceeding canonical quiet-Sun heating requirements. Yet the purely-SSD simulation of Przybylski et al. (2025) forms only transient pockets of million-degree plasma and no steady corona, omits the non-ideal terms of a generalized Ohm's law (ambipolar diffusion and the Hall effect) as well as ion-neutral physics, and leaves the transport and thermalization pathways of the SSD energy flux undetermined. The open problem is to close the SSD energy budget of the quiet-Sun atmosphere under realistic partially ionized chromospheric physics and network/canopy fields: determine how non-ideal effects modify Poynting-flux dissipation and the chromospheric energy balance, whether SSD action in a domain containing supergranular network fields can sustain a steady million-degree corona, and how chromospheric heating is partitioned spatially and temporally between Alfvénic and compressive transport and their dissipation channels.

Background

The quiet Sun is covered mostly by the internetwork - the regions between the magnetic network, the strong-field concentrations that collect at the boundaries of supergranular convection cells (horizontal scale of order 20-30 Mm). Observations indicate that internetwork magnetic flux is ubiquitous and is generated locally by a small-scale dynamo (SSD): a turbulent dynamo operating on scales comparable to or below the energy-carrying scales of convection, which amplifies a seed field without any imposed large-scale flux and with zero net magnetic flux through the surface (reviewed by Rempel et al. 2023).

The chromosphere - the partially ionized, roughly 1-2 Mm thick layer between the photosphere and the transition region - loses energy by radiation at a rate that must be balanced by mechanical and magnetic heating. In magnetohydrodynamics the flux of magnetic energy is the Poynting flux, carried upward by motions that displace and braid the field; plasma beta is the ratio of gas to magnetic pressure. Amari et al. (2015) argued from SSD calculations that the resulting Poynting flux (about 4×1064\times10^{6} erg cm−2^{-2} s−1^{-1}) suffices to heat the quiet chromosphere, and Tilipman et al. (2023) quantified quiet-Sun photospheric Poynting fluxes observationally and numerically.

Przybylski et al. (2025) performed the first radiation-magnetohydrodynamic (rMHD) simulation of the solar atmosphere - photosphere to 10 Mm above, in a 12 x 12 Mm zero-net-flux domain, with non-equilibrium (time-dependent) hydrogen ionization - in which all magnetic field is generated by the SSD, with no imposed field. They found that the SSD field falls off with height more slowly than a potential (current-free) field extrapolation of the photospheric field, that magnetic energy exceeds kinetic (and, higher up, thermal) energy densities from the mid-chromosphere upward, and that the energy flux into the chromosphere (8.16×1068.16\times10^{6} erg cm−2^{-2} s−1^{-1}, dominated by the shear component of the Poynting flux and by enthalpy) exceeds canonical quiet-Sun requirements. Nevertheless only small, fluctuating pockets of million-degree gas form - no steady corona - which the authors attribute to the limited domain, in which no supergranular network and no large magnetic loops reaching high into the atmosphere can develop.

The chromosphere is only weakly ionized, so the standard single-fluid induction equation is approximate there. A generalized Ohm's law adds terms that become important in this layer: ambipolar diffusion (also called ion-neutral or Cowling resistivity - the drift of neutrals through the magnetized ion-electron fluid, which dissipates currents perpendicular to the magnetic field and converts magnetic energy locally into heat) and the Hall effect (a dispersive term arising from the differential motion of ions and electrons). Fully multi-fluid (ion-neutral) treatments go further by tracking the neutral and charged species separately. Khomenko et al. (2018) performed the first 3D magnetoconvection simulations including ambipolar diffusion and battery terms (the Biermann battery, which generates magnetic field wherever the electron and ion pressure gradients misalign), but their domain ends at 1.4 Mm, in the low chromosphere, and they report a drastic reduction of Poynting flux below that boundary. Martinez-Sykora et al. (2019) simulated the quiet chromosphere-to-corona with the Bifrost code (a radiation-MHD code widely used for solar-atmosphere simulations) starting from a weak imposed guide field, finding photospheric net flux insufficient and shocks able to amplify the chromospheric field; Martinez-Sykora et al. (2023) then showed that adding ambipolar diffusion and non-equilibrium ionization to a model with locally driven chromospheric field growth heats the upper chromosphere and improves agreement with IRIS (Interface Region Imaging Spectrograph) Mg II observations. Neither is a convection-driven, zero-net-flux SSD model spanning the atmosphere.

In larger-domain simulations with the MURaM code (a solar radiation-MHD code; 50 x 50 Mm, extending to the corona) seeded with a 10−310^{-3} G rms zero-net-flux random field amplified by SSD action, Chen et al. (2021, 2022) showed that a supergranular network of field and a self-consistently heated hot corona do form - but those runs use ideal induction with tabulated radiative losses assuming local thermodynamic equilibrium (LTE), no generalized Ohm's law, and no energy-budget attribution. Noraz et al. (2026) have recently quantified shock-versus-current-sheet heating partition in weakly magnetized (non-SSD) Bifrost quiet-Sun models. What no existing study does is combine SSD-only fields, a network/canopy-containing domain reaching the corona, and the non-ideal partially ionized physics of the chromosphere in one self-consistent model - the combination of missing ingredients Przybylski et al. (2025) identify before the SSD's role in the quiet-Sun energy budget can be settled.

Problem Statement

As posed by Przybylski et al. (2025) in the discussion and conclusions of their purely-SSD chromosphere simulation, the problem is to establish whether small-scale-dynamo action alone can heat and sustain the quiet-Sun atmosphere once the missing ingredients - non-ideal partially ionized physics and a domain containing network/canopy fields - are included, and in doing so to close the SSD energy budget. Concretely, in a self-consistent 3D radiation-MHD model of the quiet Sun in which all magnetic field is SSD-generated (zero imposed field, zero net flux), spanning photosphere, chromosphere and corona, and with a horizontal extent large enough to contain supergranular network fields:

  1. Determine how the inclusion of non-ideal, partially ionized plasma physics - for example a generalized Ohm's law with ambipolar diffusion and the Hall term, or a full ion-neutral (multi-fluid) treatment - modifies the transport and dissipation of Poynting flux and the chromospheric energy balance found in the ideal-induction case: does the magnetic-energy dominance of the mid-to-upper chromosphere and the sufficiency of the SSD energy flux (8.16×1068.16\times10^{6} erg cm−2^{-2} s−1^{-1} into the chromosphere) survive these effects, how much magnetic energy do they convert to heat, and where?

  2. Establish whether SSD action in such a network-containing domain can sustain a steady (persistent, not merely transient) million-degree corona when non-ideal effects are included - the 12 x 12 Mm zero-net-flux domain of Przybylski et al. (2025) produced only small, fluctuating pockets of million-degree gas - and whether longer time series and different upper-boundary placements change the slow evolution of the chromospheric field and the Poynting-flux gradients.

  3. Quantify the spatial and temporal partitioning of chromospheric heating: the relative contributions of Alfvénic (transverse, largely incompressible magnetic) and compressive (magnetoacoustic) energy transport, and the locus and channel of thermalization of magnetic and kinetic energy.

These three components are the terms of a single energy equation for the same SSD-driven system and are jointly required: the non-ideal dissipation rates feed the heating partition, the partition determines whether the supplied flux can maintain coronal temperatures, and the corona's steadiness cannot be interpreted without both.

Because the horizontal extent and the duration of the model are outcome-determining - a 12 x 12 Mm zero-net-flux domain produces only transient coronal pockets, whereas a 50 x 50 Mm domain containing supergranular network forms a self-consistently heated corona - items 1-2 are completed only by a model whose horizontal extent spans at least several supergranular cells (of order 50 x 50 Mm) and whose coronal steadiness is defined and assessed operationally: a corona counted as steady only if maintained over a span long compared with both the roughly 30-minute temporal sampling of prior work and the multi-hour evolution of the chromospheric magnetic field, and not merely as transient, fluctuating million-degree pockets. Any physically justified extent and runtime resolve the items, provided the choice is stated and its influence on the conclusions of items 1-2 is reported.

References

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