On disc protoplanet interactions in a non-barotropic disc with thermal   diffusion
  We study the disc planet interactions of low-mass protoplanets embedded in a circumstellar disc. We extend the standard theory of planet migration from the usual locally isothermal assumption to include non-barotropic effects, focusing on the validity of linear theory. We compared solutions of the linear equations with results from non-linear hydrodynamic simulations, where in both cases we adopted a background entropy gradient and solved the energy equation. We show that the migration behavior of embedded planets depends critically on the background radial entropy gradient in the disc. The presence of such a gradient not only changes the corotation torque on the planet, but also always guarantees a departure from linear behavior, which gives a singular density response at corotation, in the absence of thermal or viscous diffusion. A negative entropy gradient tends to give rise to positive, non-linear corotation torques apparently produced as material executes horseshoe turns at approximately constant entropy. These torques have no counterpart in linear theory, but can be strong enough to push the planet outwards until saturation starts to occur after a horseshoe libration period. Increased thermal diffusion acts to reduce these non-linear torques, but, at the same time, it can help to prevent their saturation. In combination with a small kinematic viscosity that is able to maintain a smooth density profile the positive torque could be sustained. 
S.-J. Paardekooper, J. C. B. Papaloizou