reader to Crida et al. (2007).]{}

In all simulations presented below, the 2D grid is as before (from 0.3 to 5, with 282x325 resolution). The inner 1D grid starts at $r=0.016$ and the outer 1D grid ends at $r=40$ (which corresponds to about 200 AU in our units). They have the same radial resolution as the 2D grid. The initial surface density profile of the disk is of type $\Sigma(r)= 3\times 10^{-4} \exp(-r^2/53)$, as illustrated with a dash-dotted line in Fig. \[profiles\], and is derived from the analysis of Guillot and Hueso (2006) for a disk evolving under the collapse of new matter onto the plane from the proto-stellar cloud, viscous evolution and photo-evaporation. The viscosity is assumed constant, for simplicity (we have verified, as MS01, that an $\alpha$ prescription for the viscosity would not change the results significantly, as the planets are very close to each other, although it can affect the global evolution of the disk).

Conversely to what we did in the previous section, we first let the planets evolve in the disk for 8,000 Jupiter orbits without feeling the disk’s perturbations, assuming an aspect ratio $H/r=3$% and a viscosity $\nu=10^{-5.5}$ (in our units, see above). [This 