The Self-Regulated Winds of Long Period Variable Stars
  Numerical models of the extended atmospheres of long period variable or Mira stars have shown that their winds have a very simple, power law structure when averaged over the pulsation cycle. This structure is stable and robust despite the pulsational wave disturbances, and appears to be strongly self-regulated. Observational studies support these conclusions. The models also show that dust-free winds are nearly adiabatic, with little heating or cooling. The classical, steady, adiabatic wind solution to the hydrodynamic equations fails to account for an extensive region of nearly constant outflow velocity. We investigate analytic solutions which include the effects of wave pressure, heating, and the resulting entropy changes. Wave pressure is represented by a term like that in the Reynolds turbulence equation for the mean velocity. Although the pressure from individual waves is modest, the waves are likely the primary agent of self-regulation of dust-free winds. In models of dusty winds, the gas variables also adopt a power law dependence on radius. Heating is required at all radii to maintain this flow, and grain heating and heat transfer to the gas are significant. Both hydrodynamic and gas/grain thermal feedbacks transform the flow towards self-regulated forms. (Abridged) 
Curtis Struck, Daniel C. Smith, Lee Anne Willson, Gary Turner and   George H. Bowen