All Figures

$\begin{figure} \par\begin{tabular}{cc} \includegraphics[width=3.8cm,clip]{1511f... ...ps} & \includegraphics[width=3.7cm,clip]{1511f4.eps} \end{tabular} \end{figure}$ Figure 1: Density wave perturbation in the disk caused by the free wave propagation for the azimuthal mode m=2. Some examples are shown for different eigenvalues and for Newtonian geometry, on the left, as well as for Schwarzschild geometry, on the right. The vertical bar indicates the location of the inner Lindblad resonance. The normalization of the eigenfunctions is arbitrary.
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In the text

$\begin{figure} \par\includegraphics[width=6.7cm,clip]{1511f5.eps} \end{figure}$ Figure 2: Variation of the highest eigenvalue, corresponding to the eigenfunction having no node, as a function of the location of the inner edge of the disk $R_{\rm in}$ . There is a monotonic increase as the disk approaches the ISCO at $R_{\rm in}=6R_{\rm g}$ . Results are shown for the m=2,5 modes in the Newtonian (N) and Schwarzschild (S) spacetime, red and blue curves respectively. The gravitational radius is defined by $R_{\rm g}=G~M_*/c^2$ .
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In the text

$\begin{figure} \par\includegraphics[width=6.6cm,clip]{1511f6.eps} \end{figure}$ Figure 3: Non-wavelike disturbance in the inner part of the accretion disk due to the asymmetric component of the magnetic field. The potential is Newtonian and m=2. The numerical values used are described in Sect. 5.
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In the text

$\begin{figure} \par\includegraphics[width=6.6cm,clip]{1511f7.eps} \end{figure}$ Figure 4: Non-wavelike disturbance in the inner part of the accretion disk due to the asymmetric component of the magnetic field. The potential is Newtonian, m=2 and the disk is counterrotating.
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In the text

$\begin{figure} \par\begin{tabular}{cc} \includegraphics[width=3.6cm,clip]{1511f8.eps} & \includegraphics[width=3.6cm,clip]{1511f9.eps} \end{tabular} \end{figure}$ Figure 5: Thin disk approximation in which the ratio H/R never exceeds 0.1, on the left for the Newtonian potential and on the right for the Schwarzschild geometry.
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In the text

$\begin{figure} \par\begin{tabular}{cc} \includegraphics[width=3.6cm,clip]{1511f... ...s} & \includegraphics[width=3.6cm,clip]{1511f11.eps} \end{tabular} \end{figure}$ Figure 6: Plasma $\beta$ parameter for the weakly magnetized disk plotted on a logarithmic scale. The magnetic field dominates only very close to the inner edge, on the left for the Newtonian potential and on the right for the Schwarzschild geometry.
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In the text

$\begin{figure} \par\includegraphics[width=7.4cm,clip]{1511f12.eps} \end{figure}$ Figure 7: Final snapshot of the density perturbation $\delta \rho /\rho _0$ in the accretion disk evolving in a quadrupolar perturbed Newtonian potential. The disk extends from R₁=6.0 to R₂=60.0. The rotation rate of the star is $\Omega _*=0.0043311$ . The time is normalized to the spin period T_*=1450.7. The m=2 structure emerges in relation to the m=2 quadrupolar magnetic perturbation.
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In the text

$\begin{figure} \par\includegraphics[width=7cm,clip]{1511f13.eps} \end{figure}$ Figure 8: Cross section of the density perturbation $\delta \rho /\rho _0$ in the Newtonian disk at the final time of the simulation. The inner and outer Lindblad resonances are shown by vertical lines at $r_{\rm L}^{\rm in/out}=23.7/49.3$ . Nevertheless the disk feels only the inner resonance, the outer one being much too weak because of the very low perturbation amplitude in this region.
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In the text

$\begin{figure} \par\includegraphics[width=7cm,clip]{1511f14.eps} \end{figure}$ Figure 9: Amplitude of the Fourier components of the density perturbation. The odd modes are numerically zero. Due to small nonlinearities, the even modes are apparent but with weak amplitude. The components $m\ge 9$ are set to zero because of the desaliasing process.
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In the text

$\begin{figure} \par\includegraphics[width=7.4cm,clip]{1511f15.eps} \end{figure}$ Figure 10: Same as in Fig. 7 but the Newtonian potential is replaced by a pseudo-Schwarzschild one.
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In the text

$\begin{figure} \par\includegraphics[width=7cm,clip]{1511f16.eps} \end{figure}$ Figure 11: Same as in Fig. 8 but for the pseudo-Schwarzschild potential. The Lindblad resonances are located at $r_{\rm L}^{\rm in/out}=21.6/45.5$ .
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In the text

$\begin{figure} \par\includegraphics[width=7cm,clip]{1511f17.eps} \end{figure}$ Figure 12: Same as Fig. 9 but for the pseudo-Schwarzschild potential. The Fourier coefficients form a decaying geometric series.
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In the text

$\begin{figure} \par\includegraphics[width=7.4cm,clip]{1511f18.eps} \end{figure}$ Figure 13: Same as in Fig. 10 but with an azimuthal mode m=1.
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In the text

$\begin{figure} \par\includegraphics[width=7cm,clip]{1511f19.eps} \end{figure}$ Figure 14: Same as in Fig. 11 but with an azimuthal mode m=1. The Lindblad resonances are located at $r_{\rm L}^{\rm in/out}=12.9/56.0$ .
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In the text

$\begin{figure} \par\includegraphics[width=7cm,clip]{1511f20.eps} \end{figure}$ Figure 15: Same as Fig. 12 but with an azimuthal mode m=1. The Fourier coefficients form a decaying geometric series.
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In the text

$\begin{figure} \par\includegraphics[width=7.4cm,clip]{1511f21.eps} \end{figure}$ Figure 16: Final snapshot of the density perturbation in the accretion disk evolving in a perturbed pseudo-Kerr potential with a=0.5. The disk extends from R₁=4.24 to R₂=42.4. The outer Lindblad resonance is not on the grid.
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In the text

$\begin{figure} \par\includegraphics[width=6.8cm,clip]{1511f22.eps} \end{figure}$ Figure 17: Same as in Fig. 8 but for the pseudo-Kerr potential. The inner Lindblad resonance appears clearly at $r_{\rm L}^{\rm in}=19.2$ while the outer one at $r_{\rm L}^{\rm out}=45.8$ lies outside the computational grid.
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In the text

$\begin{figure} \par\includegraphics[width=7cm,clip]{1511f23.eps} \end{figure}$ Figure 18: Same as Fig. 9 but for the pseudo-Kerr potential.
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In the text

$\begin{figure} \par\includegraphics[width=7.4cm,clip]{1511f24.eps} \end{figure}$ Figure 19: Final snapshot of the density perturbation in the counterrotating accretion disk evolving in a perturbed Newtonian potential. Same values as in Fig. 7 apply except for the sign of $\Omega _*$ . A trailing spiral density wave of m=2 propagates in the entire disk.
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In the text

$\begin{figure} \par\includegraphics[width=7cm,clip]{1511f25.eps} \end{figure}$ Figure 20: Same as Fig. 8 but for the retrograde disk. The Lindblad resonances are no longer present.
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In the text

$\begin{figure} \par\includegraphics[width=7cm,clip]{1511f26.eps} \end{figure}$ Figure 21: Same as Fig. 9 but with a retrograde disk.
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In the text

$\begin{figure} \par\includegraphics[width=7.4cm,clip]{1511f27.eps} \end{figure}$ Figure 22: Final snapshot of the density perturbation in the counterrotating accretion disk evolving in a perturbed Newtonian potential. Same values as in the Fig. 7 caption excepted for the sign of $\Omega _*$ . A trailing spiral density wave of m=1 is propagating in the whole disk.
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In the text

$\begin{figure} \par\includegraphics[width=7cm,clip]{1511f28.eps} \end{figure}$ Figure 23: Same as Fig. 8 but with a retrograde disk. The Lindblad resonances have disappeared.
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In the text

$\begin{figure} \par\includegraphics[width=7cm,clip]{1511f29.eps} \end{figure}$ Figure 24: Same as Fig. 9 but with a retrograde disk.
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In the text

$\begin{figure} \par\begin{tabular}{cc} \includegraphics[width=3.8cm,clip]{1511f... ...ps} & \includegraphics[width=3.7cm,clip]{1511f4.eps} \end{tabular} \end{figure}$	Figure 1: Density wave perturbation in the disk caused by the free wave propagation for the azimuthal mode m=2. Some examples are shown for different eigenvalues and for Newtonian geometry, on the left, as well as for Schwarzschild geometry, on the right. The vertical bar indicates the location of the inner Lindblad resonance. The normalization of the eigenfunctions is arbitrary.
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$\begin{figure} \par\includegraphics[width=6.6cm,clip]{1511f6.eps} \end{figure}$	Figure 3: Non-wavelike disturbance in the inner part of the accretion disk due to the asymmetric component of the magnetic field. The potential is Newtonian and m=2. The numerical values used are described in Sect. 5.
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$\begin{figure} \par\includegraphics[width=6.6cm,clip]{1511f7.eps} \end{figure}$	Figure 4: Non-wavelike disturbance in the inner part of the accretion disk due to the asymmetric component of the magnetic field. The potential is Newtonian, m=2 and the disk is counterrotating.
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$\begin{figure} \par\begin{tabular}{cc} \includegraphics[width=3.6cm,clip]{1511f8.eps} & \includegraphics[width=3.6cm,clip]{1511f9.eps} \end{tabular} \end{figure}$	Figure 5: Thin disk approximation in which the ratio H/R never exceeds 0.1, on the left for the Newtonian potential and on the right for the Schwarzschild geometry.
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$\begin{figure} \par\begin{tabular}{cc} \includegraphics[width=3.6cm,clip]{1511f... ...s} & \includegraphics[width=3.6cm,clip]{1511f11.eps} \end{tabular} \end{figure}$	Figure 6: Plasma $\beta$ parameter for the weakly magnetized disk plotted on a logarithmic scale. The magnetic field dominates only very close to the inner edge, on the left for the Newtonian potential and on the right for the Schwarzschild geometry.
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$\begin{figure} \par\includegraphics[width=7.4cm,clip]{1511f12.eps} \end{figure}$	Figure 7: Final snapshot of the density perturbation $\delta \rho /\rho _0$ in the accretion disk evolving in a quadrupolar perturbed Newtonian potential. The disk extends from R₁=6.0 to R₂=60.0. The rotation rate of the star is $\Omega _=0.0043311$ . The time is normalized to the spin period T_=1450.7. The m=2 structure emerges in relation to the m=2 quadrupolar magnetic perturbation.
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$\begin{figure} \par\includegraphics[width=7cm,clip]{1511f13.eps} \end{figure}$	Figure 8: Cross section of the density perturbation $\delta \rho /\rho _0$ in the Newtonian disk at the final time of the simulation. The inner and outer Lindblad resonances are shown by vertical lines at $r_{\rm L}^{\rm in/out}=23.7/49.3$ . Nevertheless the disk feels only the inner resonance, the outer one being much too weak because of the very low perturbation amplitude in this region.
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$\begin{figure} \par\includegraphics[width=7cm,clip]{1511f14.eps} \end{figure}$	Figure 9: Amplitude of the Fourier components of the density perturbation. The odd modes are numerically zero. Due to small nonlinearities, the even modes are apparent but with weak amplitude. The components $m\ge 9$ are set to zero because of the desaliasing process.
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$\begin{figure} \par\includegraphics[width=7.4cm,clip]{1511f15.eps} \end{figure}$	Figure 10: Same as in Fig. 7 but the Newtonian potential is replaced by a pseudo-Schwarzschild one.
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$\begin{figure} \par\includegraphics[width=7cm,clip]{1511f16.eps} \end{figure}$	Figure 11: Same as in Fig. 8 but for the pseudo-Schwarzschild potential. The Lindblad resonances are located at $r_{\rm L}^{\rm in/out}=21.6/45.5$ .
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$\begin{figure} \par\includegraphics[width=7cm,clip]{1511f17.eps} \end{figure}$	Figure 12: Same as Fig. 9 but for the pseudo-Schwarzschild potential. The Fourier coefficients form a decaying geometric series.
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$\begin{figure} \par\includegraphics[width=7.4cm,clip]{1511f18.eps} \end{figure}$	Figure 13: Same as in Fig. 10 but with an azimuthal mode m=1.
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$\begin{figure} \par\includegraphics[width=7cm,clip]{1511f19.eps} \end{figure}$	Figure 14: Same as in Fig. 11 but with an azimuthal mode m=1. The Lindblad resonances are located at $r_{\rm L}^{\rm in/out}=12.9/56.0$ .
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$\begin{figure} \par\includegraphics[width=7cm,clip]{1511f20.eps} \end{figure}$	Figure 15: Same as Fig. 12 but with an azimuthal mode m=1. The Fourier coefficients form a decaying geometric series.
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$\begin{figure} \par\includegraphics[width=7.4cm,clip]{1511f21.eps} \end{figure}$	Figure 16: Final snapshot of the density perturbation in the accretion disk evolving in a perturbed pseudo-Kerr potential with a=0.5. The disk extends from R₁=4.24 to R₂=42.4. The outer Lindblad resonance is not on the grid.
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$\begin{figure} \par\includegraphics[width=6.8cm,clip]{1511f22.eps} \end{figure}$	Figure 17: Same as in Fig. 8 but for the pseudo-Kerr potential. The inner Lindblad resonance appears clearly at $r_{\rm L}^{\rm in}=19.2$ while the outer one at $r_{\rm L}^{\rm out}=45.8$ lies outside the computational grid.
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$\begin{figure} \par\includegraphics[width=7cm,clip]{1511f23.eps} \end{figure}$	Figure 18: Same as Fig. 9 but for the pseudo-Kerr potential.
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$\begin{figure} \par\includegraphics[width=7.4cm,clip]{1511f24.eps} \end{figure}$	Figure 19: Final snapshot of the density perturbation in the counterrotating accretion disk evolving in a perturbed Newtonian potential. Same values as in Fig. 7 apply except for the sign of $\Omega _*$ . A trailing spiral density wave of m=2 propagates in the entire disk.
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$\begin{figure} \par\includegraphics[width=7cm,clip]{1511f25.eps} \end{figure}$	Figure 20: Same as Fig. 8 but for the retrograde disk. The Lindblad resonances are no longer present.
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$\begin{figure} \par\includegraphics[width=7cm,clip]{1511f26.eps} \end{figure}$	Figure 21: Same as Fig. 9 but with a retrograde disk.
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$\begin{figure} \par\includegraphics[width=7.4cm,clip]{1511f27.eps} \end{figure}$	Figure 22: Final snapshot of the density perturbation in the counterrotating accretion disk evolving in a perturbed Newtonian potential. Same values as in the Fig. 7 caption excepted for the sign of $\Omega _*$ . A trailing spiral density wave of m=1 is propagating in the whole disk.
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$\begin{figure} \par\includegraphics[width=7cm,clip]{1511f28.eps} \end{figure}$	Figure 23: Same as Fig. 8 but with a retrograde disk. The Lindblad resonances have disappeared.
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$\begin{figure} \par\includegraphics[width=7cm,clip]{1511f29.eps} \end{figure}$	Figure 24: Same as Fig. 9 but with a retrograde disk.
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