Probing the solar coronal heating function with slow magnetoacoustic waves

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Nugget
Number: 394
1st Author: Dmitrii Y. Kolotkov
2nd Author: et al.
Published: 16 November 2020
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Contents

Introduction

The solar corona contains waves of many types, and their great significance follows immediately from the simple fact that the corona consists mainly of electrodynamic fields, not matter - the energy and its sometimes violent conversions come from the field directly, although we detect the dynamics only via the properties of the visible gas content.

For compressive waves, the coronal plasma that appears to be in an apparent thermal equilibrium formed by the balance of energy losses and gains acts as an active medium. In other words, the wave experiences a back-reaction of the modification of the cooling and heating rates by the density and temperature perturbations, either losing or gaining energy from the energy supply that heats the plasma. This natural mechanism of the energy exchange between a thermodynamically active plasma and its wave population is known as a wave-induced heating/cooling misbalance.

In Ref. [1] we have presented a novel seismological method allowing us to determine empirical constraints on the unknown coronal heating function by combining the theory of perturbed thermal equilibrium and the observed behaviour of slow magnetoacoustic waves.

Stability of acoustic and thermal modes in the corona for varying heating models

In the continuously heated and cooled coronal plasma with thermal conduction and plasma β tending to zero, the dynamics of coupled linear slow magnetoacoustic and thermal (entropy-related) modes, perturbing both heating and cooling processes, strongly depends on the ratio between the wave period and the characteristic timescales of the parallel thermal conduction and thermal misbalance:

393fa.png

with the wavenumber k, adiabatic index γ, specific heat capacity CV, parallel Spitzer conduction coefficient κ, and the derivatives QT = ∂Q/∂T and Qρ = ∂Q/∂ρ (evaluated at the equilibrium) of the function Q = L(ρ,T) - H(ρ,T), combining unknown coronal heating (H) and radiative losses (L). The heating function H is taken to have no dependence upon the magnetic field strength B, since in the zero-β limit (used here) the slow wave does not perturb B.

We synthesise the coronal optically thin radiation function L(ρ,T) from the CHIANTI atomic database, and pin down the functional form of the heating process as H(ρ,T) ~ ρaTb, treating the power indices a and b as free parameters to be constrained seismologically. With this, the slow magnetoacoustic and thermal modes become unstable due to the violation of the coronal thermal equilibrium if

393fb.png

Here, the overstable regime of the acoustic mode refers to an acoustic oscillation with a growing amplitude; and an unstable regime of the thermal mode may lead to rapid condensations of the coronal plasma and the phenomenon of coronal rain (Ref. [2]). Since we know from observations that the coronal condensations are not a frequent phenomenon, and acoustic modes are usually seen to rapidly decay without manifesting instabilities, we can obtain empirical constraints on the coronal heating function, for it to sustain a long-lived stable corona. We evaluate conditions (1) and (2) for a range of coronal densities 108-1011 cm-3 and temperatures 0.5 - 20 MK, typical of the vast majority of the coronal plasma structures (see Fig. 1). This allows us to find that the corona is stable to both the acoustic and thermal modes for the values of a and b outlined by a triangle with approximate vertices (-2.5, -5), (1, -1.5), and (3.5, -5).

Figure 1: The heating power indices a and b determining stability of acoustic and thermal modes. The red triangles show the Rosner-Tucker-Vaiana heating models (Ref. [3]). The green triangle shows the heating model used as an example in Figure 2.

Estimation of the thermal misbalance timescale for a broad range of coronal conditions

From the identified parametric region of a and b allowing for a stable corona, for illustration we pick a specific heating model a=0.5 and b=-3.5 and estimate the value of the misbalance timescale τM= τ1τ2/(τ12) characterising the stability and dynamics of the acoustic mode, for a broad range of coronal densities and temperatures (Figure 2). In particular, the values of τM are seen to coincide with typical slow wave periods τ detected in

393fc.png

see Ref. [4] for the most recent review of MHD waves in the corona. The detected similarity between τM and the periods of slow magnetoacoustic waves demonstrates the broad applicability of this approach for diagnostics of heating functions in various specific coronal structures. For example, for those long-lived coronal plasma structures, the thermal mode is either stable or grows slowly, and the observed acoustic-mode period (and decay time) is about τM for -1 < a < 0.5 and -1.5 < b < -0.5 (for propagating slow waves in quiescent loops and plumes), and -2 < a < 0 and -1.5 < b < -0.5 (for standing slow-mode oscillations in hot loops).

Figure 2: Characteristic thermal misbalance time τM for a fixed heating model (see Figure 1). The blue symbols illustrate combinations of density and temperature in hot and dense loops in active regions (star), warm and less dense quiescent loops (circle), and polar plumes (box), see Eq. (3) for details.

Conclusions

The coronal heating function can be constrained seismologically using observations of damped slow magnetoacoustic waves in long-lived coronal plasma structures and theory of perturbed thermal equilibrium. The applicability of this approach depends on the ratio of the misbalance time τM to the wave period. We empirically revealed the heating models allowing for a stable corona in general and obtained the heating models stable to thermal and acoustic modes in specific coronal plasma structures, such as quiescent and active-region loops, and plumes in coronal holes. We showed that for these heating models, the effect of thermal misbalance leads to the behaviour of slow waves consistent with observations.

Acknowlegements

The co-authors of this Nugget, and of Ref. [1], are Timothy J. Duckenfield and Valery M. Nakariakov.

Bibliography

[1] In press in Astron. Astrophys., doi: 10.1051/0004-6361/202039095.

[2] "Thermal instability and non-equilibrium in solar coronal loops: from coronal rain to long-period intensity pulsations"

[3] "Dynamics of the quiescent solar corona"

[4] "Magnetohydrodynamic Waves in the Solar Corona"

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