Non-linear torsional Alfv\'en waves evolving in stratified viscous plasmas: Coronal hole plumes

We model solar atmospheric structures characterised by parallel structuring. We focus on Alfv\'en waves in the weakly non-linear regime to highlight the efficiency of non-linear wave steepening when dissipative effects are prominent. We also consider the local and equilibrium conditions involved in shock formation and the shock's contributions to coronal seismology. Coronal plumes were modelled analytically by implementing the magnetohydrodynamic (MHD) theory in cylindrical geometry. Here, the stratification and viscosity are present internal to the plume, whilst effects of the external medium, together with equilibrium conditions, are implied where the magnetic fields are parallel to the plume axis. We implemented a second-order thin flux tube approximation to obtain a wave equation that points to effects tied to non-linear, dissipative, and stratification terms, as well as terms representing atmospheric conditions. The impact of shear viscosity on non-linear Alfv\'en waves extracted by the Cohen-Kulsrud-Burgers-type equation proves more efficient when propagated to higher altitudes. The dissipative effects linked to the dimensionless viscosity indicate that the dissipative effects are not linear. Meanwhile, the delay in shock formation enables energy conversions at higher altitudes, thereby maintaining coronal heating at higher levels. The efficiency of parallel structuring and viscous damping is enhanced by such transverse structuring, as it is directly proportional to the external plasma-beta . It is observed that Alfv\'en pulses may undergo a backward shock, either in the lower levels of coronal plasma or as they propagate toward higher regions, implying a conversion of energy occurring at various altitudes. A peak was observed, indicating that the interplay reverses at heights around $1.5$ solar radii. Such effects are shown to play a key role in the context of coronal seismology.