disc for several timesteps. Due to axisymmetry, we only consider one quadrant of the disc and average the surface density azimuthally within $\Delta R =20$ pc bins. At each time step ($\sim$1.49 Myr), we subtract the initial surface density profile from the current one $(\delta \Sigma = \Sigma_t - \Sigma_0)$. This method is only possible, because of the, initially, relatively well-balanced disc setup, and gives the quantities, which arise from the perturbation growth with a relatively clear measurable signal (see Fig. \[U\_Zoom\_Rings\_2\]). We measure the initial seed overdensities with $\delta \Sigma (t=0)  \sim 10^{-3} \; \mathrm{M_{\sun} \;pc^{-2}}$. They grow up to maximum values $\delta \Sigma (t) \sim 10^5 \; \mathrm{M_{\sun} \; pc^{-2}}$ over time within the clumps. In Fig. \[Results:fig:bubble\_plot\], we limit the range to $\delta \Sigma =10^{-2} - 8 \times 10^2 \; \mathrm{M_{\sun} \;pc^{-2}}$ to illustrate the amplitude of the ring overdensities. Most of the low-density disturbances disappear very early in the evolution. Only a few survive and lead to very high densities, form rings and finally fragment into a clumpy, disordered structure. The innermost ring, $R1$, is moving slightly outwards and comes very near to the second one, $R2$, while it is growing and finally begins to absorb 