In both of these approaches, disc–planet interactions causing planet migration are modelled using analytical formulae for the disc torque experienced by the planet. This gravitational torque consists of two components. The differential Lindblad torque results from the angular momentum exchange between the planet and the spiral density waves it generates inside the disc. For sufficiently low-mass planets, it scales linearly with disc mass and planet mass and as the inverse square of the disc aspect ratio (Tanaka, Takeuchi & Ward 2002). Although its sign depends on the density and temperature gradients inside the disc, the differential Lindblad torque is generally negative for typical disc models and is therefore responsible for inward migration. The corotation torque is due to the torque exerted by the material located in the coorbital region of the planet. It is composed of a barotropic part which scales with the vortensity (i.e. the ratio between the vertical component of the vorticity and the disc surface density) gradient (Goldreich & Tremaine 1979) plus an entropy-related part which scales with the entropy gradient (Baruteau & Masset 2008; Paardekooper & Papaloizou 2008). A negative vortensity (resp. entropy) gradient gives rise to a positive vortensity (resp. entropy) related corotation torque. It has been shown that for mildly positive surface density gradients or negative entropy gradients, a positive corotation torque can eventually counteract the effect of a negative differential Lindblad torque, which may stall or even reverse migration (Masset, D'Angelo & Kley 2006; Paardekooper & Papaloizou 2009). In isothermal discs, the corotation torque is a non-linear process generally referred as the horseshoe drag and whose amplitude is controlled by advection and diffusion of vortensity inside the horseshoe region. In non-isothermal discs, the corotation torque is also powered by singular production of vortensity due to an entropy discontinuity on downstream separatrices (Masset & Casoli 2009; Paardekooper et al. 2010). In the absence of any diffusion processes inside the disc, vortensity and entropy gradients across the horseshoe region tend to flatten through phase mixing, which causes the two components of the horseshoe drag to saturate. Consequently, desaturating the horseshoe drag requires that some amount of viscous and thermal diffusions are operating inside the horseshoe region. In that case, the amplitude of the horseshoe drag depends on the ratio between the diffusion time-scales and the horseshoe libration time-scale and its optimal value, also referred as the fully unsaturated horseshoe drag, is obtained when the diffusion time-scales are approximately equal to half the horseshoe libration time (e.g. Baruteau & Masset 2013). In the limit where the diffusion time-scales become shorter than the U-turn time-scale, the corotation torque decreases and approaches the value predicted by linear theory. Therefore, the corotation torque can be considered as a linear combination of the fully unsaturated horseshoe drag and the linear corotation torque with coefficients depending on the ratio between the diffusion time-scales and the horseshoe libration time-scale. Corotation torque formulae as a function of viscosity and thermal diffusivity were recently proposed by Paardekooper, Baruteau & Kley (2011) and Masset & Casoli (2010).
Citation: Paardekooper & Papaloizou 2009
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