disc, while $\kappa^2$ dominates for larger radii. \[Results:fig:dispersion\_relation\_1\]](f5.eps){width="84mm"}

General evolution of the surface density {#Results:subsec:General evolution}
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Axisymmetric overdensities (rings) are forming inside–out (Fig. \[Results:fig:disk\_sim\_A\]), as expected from our considerations in Section \[Results:subsec:Perturbation theory\]. They grow discretely at a certain radius, and their surface density increases, while they accrete mass from the inter-ring regions. At a certain point they begin to collapse to a thin circular line and finally break up into several bound clumps. A highly irregular and clumpy disc is developing within 500 Myr.\
The first structures become visible in the surface density after $50$ Myr (Figs \[Results:fig:disk\_sim\_A\] a and b) and evolve to distinct rings (Figs \[Results:fig:disk\_sim\_A\]c-h). At later times, outer rings form with larger radial widths and on longer time-scale (Figs \[Results:fig:disk\_sim\_A\]i-p). The last visible ring cannot fully evolve, since it is getting disturbed by the inner clumpy structure (Figs \[Results:fig:disk\_sim\_A\]k-p).

![image](f6q.eps){width=".5\linewidth"}

Perturbation growth {#Results:subsec:perturbation growth}
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The appearing rings are a result of growing perturbations of the dominant modes, from a very small density perturbation seed, caused by a superimposed wave spectrum, due to the initial AMR grid. To follow this process, we determine the relative changes of the surface density compared to the initial 