All Figures

$\begin{figure} \par\includegraphics[width=7.8cm,clip]{7169f1.eps} \end{figure}$ Figure 1: Real frequency of the most unstable modes for a gap opened by a Jupiter-mass planet after 10 periods as a function of the azimuthal mode number. The solid line shows the mode frequency at the outer edge of the gap and the dashed line is the mode frequency at the inner edge divided by $m \Omega _{{\rm K}}(r_0)$ where $\Omega _{{\rm K}}(r_0)$ is the Keplerian angular frequency at the planet position.
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In the text

$\begin{figure} \par\includegraphics[width=7.8cm,clip]{7169f2.eps} \end{figure}$ Figure 2: Growth rate for a gap opened by a Jupiter-mass planet after 10 orbits against the azimuthal mode number. The solid line shows the growth rate at the outer edge of the gap and the dashed line is the growth rate at the inner edge. The growth rate of the instability peaks at mode numbers 5-6.
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In the text

$\begin{figure} \par\includegraphics[width=8.1cm,clip]{7169f3.eps} \end{figure}$ Figure 3: Radial profile of the eigenfunctions for the outer edge of the gap at t = 10 periods and mode number m = 5. From top to bottom the pressure perturbation and radial and azimuthal perturbed velocity components are shown. The dotted and dashed lines are the real and imaginary part of the eigenfunctions. The amplitude is shown by the solid line which peaks at the position of the edge for the eigenfunction of the perturbed pressure.
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In the text

$\begin{figure} \par\includegraphics[width=8.1cm,clip]{7169f4.eps} \end{figure}$ Figure 4: Radial profile of the eigenfunctions for the inner edge of the gap at t = 10 periods and mode number m = 5. From top to bottom the pressure perturbation and radial and azimuthal perturbed velocity components are shown. The dotted and dashed lines are the real and imaginary part of the eigenfunctions.
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In the text

$\begin{figure} \par\includegraphics[width=7.5cm,clip]{7169f5a.eps}\vspace*{5mm} \includegraphics[width=7.5cm,clip]{7169f5b.eps} \end{figure}$ Figure 5: Real frequency and growth rates of the unstable modes with azimuthal number m=5, as a function of time, for NIRVANA and polar FLASH simulations with resolution n_r $\times$ $n_{\phi }=256$ $\times$ 768. In the top panel, the crosses represent the mode frequencies at the outer edge of the gap and the dots are the mode frequencies at the inner edge for NIRVANA. The circles are the mode frequencies at the outer edge of the gap and the stars are the mode frequencies at the inner edge for polar FLASH. The frequencies are divided by $m \Omega _{{\rm K}}(r_0)$ , where $\Omega _{{\rm K}}$ is the Keplerian angular frequency and r₀ is the radius at the edge of the gap. The crosses in the bottom panel are the growth rates at the outer gap edge and the dots are the growth rates at the inner edge for a NIRVANA calculation. The circles are the growth rates at the outer edge of the gap and the stars are the growth rates at the inner edge for a polar FLASH calculation. Growth rates are divided by the Keplerian frequency at the edge of the gap.
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In the text

$\begin{figure} \par\includegraphics[width=12cm,clip]{7169f6CMJN.eps} \end{figure}$ Figure 6: Surface density contours for cylindrical simulations in logarithmic scale. From top to bottom, NIRVANA simulation using a damping wave region as described in Sect. 3 (see also de Val-Borro et al. 2006), NIRVANA simulation using outgoing-wave boundary conditions defined by Godon (1996) and polar FLASH. The left panels show the density after 50 orbital periods and the right panels show the density after 100 orbits with the same color scale. The resolution is $n_r \times n_{\phi } =256\times 768$ and the planet is located at $x \times y =(1,0)~a$ . There are two elongated vortices forming next to the outer gap edge at 50 orbits that move with different velocities. The vortices merge in both simulations and one large vortex is observed in the outer disk at 100 orbits.
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In the text

$\begin{figure} \par\includegraphics[width=7.7cm,clip]{7169f7.eps} \end{figure}$ Figure 7: Growth rates of the unstable modes with mode number m=5, as a function of time divided by the local Keplerian frequency. Dots represent a NIRVANA calculation with resolution $n_r \times n_{\phi } =256\times 768$ and stars are obtained from a FLASH calculation with resolution $n_r \times n_{\phi }=128\times 384$ .
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In the text

$\begin{figure} \par\includegraphics[width=7.6cm,clip]{7169f8CMJN.eps} \end{figure}$ Figure 8: Azimuthally averaged surface density profiles for the Jupiter simulations after 100 orbital periods. The solid line is the NIRVANA simulation with resolution $n_r \times n_{\phi } =256\times 768$ and damping wave boundary conditions and the short dashed line is the NIRVANA simulation with resolution $n_r \times n_{\phi } =256\times 768$ and outgoing-wave boundary conditions. The profiles are averaged over 5 orbits.
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In the text

$\begin{figure} \par\includegraphics[width=12cm,clip]{7169f9CMJN.eps} \end{figure}$ Figure 9: Surface density distribution after 100 orbital periods for NIRVANA on the left hand side and FLASH on the right hand side using the same logarithmic color scale. Both models use the same wave damping condition in the outer disk between 2.1-2.5 a. while FLASH does not have a damping condition in the inner boundary. NIRVANA has density enhancements close to the gap opened by the protoplanet. FLASH has a smooth density distribution and a larger density peak at the planet position which is saturated in the image.
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In the text

$\begin{figure} \par\includegraphics[width=12cm,clip]{7169f10CMJN.eps} \end{figure}$ Figure 10: Vortensity in polar coordinates is shown at times t = 10, 20, 50 and 100 orbital periods from left to right and top to bottom. The simulation has resolution $n_{\rm r} \times n_\phi = 512 \times 1536$ .
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In the text

$\begin{figure} \par\includegraphics[width=7.65cm,clip]{7169f11.eps} \end{figure}$ Figure 11: Vortensity profiles averaged over azimuth at different times t = 10, 20, 50 and 100 orbital periods for the NIRVANA simulation at resolution $n_{\rm r} \times n_\phi = 256 \times 768$ .
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In the text

$\begin{figure} \par\includegraphics[width=7.65cm,clip]{7169f12.eps} \end{figure}$ Figure 12: Vortensity profiles averaged over azimuth at different times t = 10, 20, 50 and 100 orbital periods for the FLASH simulation in Cartesian coordinates.
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In the text

$\begin{figure} \par\includegraphics[width=14.5cm,clip]{7169f13CMJN.eps} \end{figure}$ Figure A.1: Velocity vectors in the corotating frame after 100 periods for the NIRVANA simulation at resolution $n_{\rm r} \times n_\phi = 512 \times 1536$ . The left panel shows the vortex at the inner gap edge and the right panel shows the vortex at the outer gap edge.

In the text

$\begin{figure} \par\includegraphics[width=12cm,clip]{7169f14CMJN.eps} \end{figure}$ Figure A.2: Streamlines in the corotating frame of the inner and outer vortices plotted in Fig. 13. The radial extent of the vortices is about 0.15a. Spiral arms created by the planet and weaker shocks associated with the vortices are observed.

In the text

$\begin{figure} \par\includegraphics[width=8.65cm,clip]{7169f15.eps} \end{figure}$ Figure A.3: Dependence on time of growth rates of modes with m=5 for NIRVANA simulations with resolution $n_{\rm r} \times n_\phi = 256 \times 768$ (dots) and FLASH with resolution $n_{\rm r} \times n_\phi = 128 \times 384$ (stars). The Navier-Stokes viscosity is $\nu =10^{-6}$ .

In the text

$\begin{figure} \par\includegraphics[width=7.8cm,clip]{7169f2.eps} \end{figure}$	Figure 2: Growth rate for a gap opened by a Jupiter-mass planet after 10 orbits against the azimuthal mode number. The solid line shows the growth rate at the outer edge of the gap and the dashed line is the growth rate at the inner edge. The growth rate of the instability peaks at mode numbers 5-6.
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$\begin{figure} \par\includegraphics[width=8.1cm,clip]{7169f4.eps} \end{figure}$	Figure 4: Radial profile of the eigenfunctions for the inner edge of the gap at t = 10 periods and mode number m = 5. From top to bottom the pressure perturbation and radial and azimuthal perturbed velocity components are shown. The dotted and dashed lines are the real and imaginary part of the eigenfunctions.
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$\begin{figure} \par\includegraphics[width=7.7cm,clip]{7169f7.eps} \end{figure}$	Figure 7: Growth rates of the unstable modes with mode number m=5, as a function of time divided by the local Keplerian frequency. Dots represent a NIRVANA calculation with resolution $n_r \times n_{\phi } =256\times 768$ and stars are obtained from a FLASH calculation with resolution $n_r \times n_{\phi }=128\times 384$ .
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$\begin{figure} \par\includegraphics[width=12cm,clip]{7169f10CMJN.eps} \end{figure}$	Figure 10: Vortensity in polar coordinates is shown at times t = 10, 20, 50 and 100 orbital periods from left to right and top to bottom. The simulation has resolution $n_{\rm r} \times n_\phi = 512 \times 1536$ .
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$\begin{figure} \par\includegraphics[width=7.65cm,clip]{7169f11.eps} \end{figure}$	Figure 11: Vortensity profiles averaged over azimuth at different times t = 10, 20, 50 and 100 orbital periods for the NIRVANA simulation at resolution $n_{\rm r} \times n_\phi = 256 \times 768$ .
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$\begin{figure} \par\includegraphics[width=7.65cm,clip]{7169f12.eps} \end{figure}$	Figure 12: Vortensity profiles averaged over azimuth at different times t = 10, 20, 50 and 100 orbital periods for the FLASH simulation in Cartesian coordinates.
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