On Thermal Conduction in the Solar Atmosphere: An Analytical Solution for Nonlinear Diffusivity without Compact Support

TL;DR

This paper derives an analytical solution for nonlinear thermal diffusivity in the solar atmosphere, enabling validation of numerical models and studying energy transport during solar nanoflares.

Contribution

It presents the first analytical first-order perturbation solution for nonlinear diffusivity with a nonzero background, applicable in 1D, 2D, and 3D, and demonstrates its use in benchmarking solar energy transport simulations.

Findings

The analytical solution models nonlinear diffusivity accurately within its validity range.

Numerical schemes in Ebysus correctly model Spitzer conductivity.

A nanoflare's energy diffuses 9 Mm in 1 second due to Spitzer conductivity.

Abstract

The scientific community employs complicated multiphysics simulations to understand the physics in Solar, Stellar, and Interstellar media. These must be tested against known solutions to ensure their validity. Several well-known tests exist, such as the Sod shock tube test. However, a test for nonlinear diffusivity is missing. This problem is highly relevant in the Solar atmosphere, where various events release energy that subsequently diffuses by Spitzer thermal conductivity. The aim is to derive an analytical solution for nonlinear diffusivity in 1D, 2D, and 3D, which allows for a nonzero background value. The solution will be used to design a test for numerical solvers and study Spitzer conductivity in the Solar atmosphere. There existed an ideal solution assuming zero background value. We perform an analytical first-order perturbation of this solution. The first-order solution is first tested against a dedicated nonlinear diffusion solver, whereupon it is used to benchmark the single- and multifluid radiative magnetohydrodynamics code Ebysus, used to study the Sun. The theory and numerical modeling are used to investigate the role of Spitzer conductivity in the transport of energy released in a nanoflare. The derived analytical solution models nonlinear diffusivity accurately within its region of validity and approximately beyond. Various numerical schemes in the Ebysus code have been found to model Spitzer conductivity correctly. The energy from a representative nanoflare has been found to diffuse 9 Mm within the first second of its lifetime due to Spitzer conductivity alone, strongly dependent on the electron density. The analytical first-order solution is a step forward in ensuring the physical validity of intricate simulations of the Sun. Additionally, since the derivation and argumentation are general, they can easily be followed to treat other nonlinear diffusion problems.