Slow nonlinear sausage waves in cooling coronal magnetic loops

We study standing nonlinear 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 nonlinear wave is a superposition of two identical nonlinear waves propagating in the opposite direction in a complete analogy with the linear theory. Each of the two propagating nonlinear 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 nonlinearity and dissipation related to viscosity and thermal conduction. It also contains one arbitrary function related to the background plasma temperature variation. We then assumed that the temperature either increases of decreases exponentially. We studied the standing waves in three cases: When dissipation strongly dominates nonlinearity, when nonlinearity 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.