Non-linear damped standing slow waves in cooling coronal magnetic loops

We study standing non-linear sausage waves in coronal loops with the plasma temperature varying with time. In our analysis we use the simplest model of a coronal loop in the form of a straight magnetic tube with a circular cross-section. We also assume that the plasma-beta is low. This enables us to neglect the magnetic field variation and consider standing waves occurring in a tube with rigid boundaries. Then the plasma motion is described by pure gas-dynamic equations. The background plasma temperature can vary with time, however we assume that its density remains constant. We consider perturbations with small amplitudes and use the Reductive Perturbation Method to derive the governing equations for standing waves. We show that a standing non-linear wave is a superposition of two identical non-linear waves propagating in the opposite directions in a complete analogy with the linear theory. Each of the two propagating non-linear waves are described by a modified Burgers equation that reduces to the standard Burgers equation when the plasma temperature does not change. The modified Burgers equation contains only one dimensionless parameter R determining the relative strength of non-linearity and dissipation related to viscosity and thermal conduction. It also contains one arbitrary function related to the background plasma temperature variation. We then assume that the temperature either increases or decreases exponentially. We study the standing waves in three cases: When dissipation strongly dominates non-linearity, when non-linearity strongly dominates dissipation, and when they are of the same order. The main conclusion that we make on basis of our analysis is that plasma cooling weakens the wave damping.