Effect of Thermal Conductivity, Compressive Viscosity and Radiative Cooling on the Phase Shift of Propagating Slow Waves with and Without Heating–Cooling Imbalance

Abstract

We study the phase shifts of propagating slow magnetoacoustic waves in solar coronal loops invoking the effects of thermal conductivity, compressive viscosity, radiative losses, and heating–cooling imbalance. We derive the general dispersion relation and solve it to determine the phase shifts of density and temperature perturbations relative to the velocity and their dependence on the equilibrium parameters of the plasma such as the background density [$\rho _{0}$] and temperature [$T_{0}$]. We estimate the phase difference [$\Delta \phi$] between density and temperature perturbations and its dependence on $\rho _{0}$ and $T_{0}$. The role of radiative losses, along with the heating–cooling imbalance for an assumed specific heating function [$H(\rho , T) \propto \rho ^{-0.5} T^{-3}$], in the estimation of the phase shifts is found to be significant for the high-density and low-temperature loops. Heating–cooling imbalance can significantly increase the phase difference ($\Delta \phi \approx 140^{\circ }$) for the low-temperature loops compared to the constant-heating case ($\Delta \phi \approx 30^{\circ }$). We derive a general expression for the polytropic index [$\gamma _{\rm eff}$] using the linear MHD model. We find that in the presence of thermal conduction alone, $\gamma _{\rm eff}$ remains close to its classical value $5/3$ for all the considered $\rho _{0}$ and $T_{0}$ observed in typical coronal loops. We find that the inclusion of radiative losses (with or without heating–cooling imbalance) cannot explain the observed polytropic index under the considered heating and cooling models. To make the expected $\gamma _{\rm eff}$ match the observed value of $1.1 \pm 0.02$ in typical coronal loops, the thermal conductivity needs to be enhanced by an order of magnitude compared to the classical value. However, this conclusion is based on the presented model and needs to be confirmed further by considering more realistic radiative functions. We also explore the role of different heating functions for typical coronal parameters and find that although the $\gamma _{\rm eff}$ remains close to $5/3$, but the phase difference is highly dependent on the form of the heating function.