Author:Max McMurdo at the Plasma Dynamics Group, University of Sheffield

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### What Is The Coronal Heating Problem?

Imagine approaching a radiator that, although turned on, feels inexplicably cold to the touch. Yet, as you step away, the temperature begins to rise. In essence, this is what happens at our Sun. With a surface temperature of approximately 5800 K at the base of the photosphere to over a million K in the corona, the temperature gradient is exceptionally pronounced. This unsolved problem regarding plasma heating in the solar atmosphere is one of the most enigmatic questions that still eludes a definite answer. Modern ground and space-based instruments (DKIST, DST, Hinode, SDO, SST, Solar Orbiter), have provided observational evidence of waves with sufficient energy flux to heat the solar atmosphere. Despite this, analytical theory does not predict wave damping to occur over sufficiently short length scales, preventing them from being a dominant mechanism for effective plasma heating.

Pioneering work carried out by Heyvaerts & Priest (1983) first applied the theory of phase mixing as a mechanism to enhance the efficiency of Alfvén wave damping in the corona. Inhomogeneities in local plasma parameters causes the Alfvén speed to vary spatially. This in turn causes waves on neighbouring magnetic surfaces to propagate out of phase with one another leading to the development of large transversal gradients. These large gradients enhance the efficiency of resistive mechanisms leading to more dramatic wave damping, compared with a homogeneous plasma. Although significant progress has been made in advancing this theory, researchers often find themselves having to assume dissipative coefficients many orders of magnitude larger than theoretical formulae predict. This is typically justified by invoking the influence of turbulence, which is believed to amplify these values.

### What Difference Do Neutrals Make?

In a partially ionized plasma, theory predicts the values of these dissipative coefficients can be as much as six orders of magnitude larger compared with the fully ionised corona, precisely the values required to provide effective plasma heating. Treating the plasma as partially ionised introduces a new dissipative process known as Ambipolar diffusion, closely related to the Cowling resistivity. Due to neutrals lack of interaction with magnetic forces, neutrals are decoupling from the magnetic field, and the magnetic field diffuses through the neutral gas. Figure 1 presents typical values for Cowling resistivity, shear viscosity, and magnetic resistivity in the lower solar atmosphere:

We introduce a parameter μ, which defines the ionisation degree of the plasma, determined by the relative number density of ions to the total number density of particles, and investigate the effects of varying this on the damping lengths of Alfvén waves. In what is to follow, we apply the theory of phase mixing to the partially ionised lower solar atmosphere and construct a numerical model to obtain solutions modelling the attenuation of propagating phase mixed Alfvén waves, demonstrating that the damping lengths and associated heating rates are sufficient to heat the solar chromosphere.

### Numerical modelling

We present a numerical approach combining finite difference approximations, a Runge-Kutta 4th order time stepping algorithm, and sparse matrices to obtain solutions for phase mixed Alfvén waves in partially ionised plasmas. We vary the background Alfvén speed through an imposed inhomogeneous plasma density profile and investigate the damping of Alfvén waves for four different profiles, P1 − P4, increasing in gradient. We retain a homogeneous Alfvén speed (P1) for comparisons to a homogeneous base case. Our results show that the damping length is highly dependent on the gradient in the Alfvén speed as shown by Figure 2.

For the steepest profile, the damping length is approximately 1 Mm, while for the homogeneous case, the amplitude has not reduced to half its original value after 4 Mm of propagation, highlighting the efficiency of phase mixing on damping Alfvén waves (see Figure 3).

### Heating Rates?

The heating rates are calculated for a wavelength of 400 km, for a range of ionization degrees that range from weakly ionised (large presence of neutral gas) to strongly ionised (very few neutrals). In order to estimate the efficiency of the phase-mixed Alfvén waves to heat the plasma, we use the estimated average heating rate of the quiet chromosphere. The radiative losses estimated from commonly used semi-empirical models of the quiet-Sun chromosphere is 4.3 kWm^{-2}, which is shown by the black horizontal line in Figure 3.

Our analysis reveals that the maximum heating rate produced by Alfvén waves varies by more than one order of magnitude and it attains its maximum value for an ionisation degree of μ = 0.576. The results show that waves propagating in a partially ionised plasma with ionisation degrees in the range μ = 0.518 − 0.657 provide sufficient heating rates to balance chromospheric radiative losses. In the AL c7 atmospheric model, these values correspond to a ratio of neutrals to ions, n_{n}/n_{i} = 0.0567 − 0.917, respectively. The prevailing factor contributing to this heating is associated with the significant value of the Cowling resistivity, as indicated in Figure 1, showcasing the importance of neutrals on the damping of Alfvén waves and heating of the lower solar atmosphere.

### Conclusions

- A numerical code has been developed to study the damping of propagating phase mixed Alfvén waves in the presence of an arbitrary density inhomogeneity.
- Investigations concluded that the damping lengths are heavily influenced by the levels of phase mixing and the proportional presence of neutrals.
- The heating rates obtained for ionisation degrees that fall in the range μ = 0.518 − 0.657 provide sufficient energy flux to balance chromospheric radiative losses in the quiet Sun.
- The location of maximal heating occurs at heights above the solar surface that correspond to the location of the transition region, where we find the rapid increase in atmospheric temperature.

### References

- Avrett, E. H., & Loeser, R. 2008, ApJS, 175, 229, doi: 10.1086/523671
- Heyvaerts, J., & Priest, E. R. 1983, A&A, 117, 220
- McMurdo, M., Ballai, I., Verth, G., Alharbi, A., & Fedun, V. 2023, ApJ, 958, 81, doi: 10.3847/1538-4357/ad0364
- Vernazza, J.