form $$t_{\mathrm{sech^2}} = \left(  \frac{4 \upi^2 G \Sigma}{\lambda_{f_{\mathrm{sech^2}}}} F_{\mathrm{sech^2}} - \frac{4 \upi^2 c_{\mathrm{s}}^{2}}{\lambda_{f_{\mathrm{sech^2}}}^{2}} - \kappa^{2} \right) ^{-1/2},$$ and by substituting $\lambda_{f_{\mathrm{sech^2}}}$ with equation (\[eq:fastest\_growing\_wavelength\_sech2\]) and inserting $Q_{0}$ it leads to $$\label{eq:t_sech^2}
t_{\mathrm{sech^2}} =\kappa^{-1} \left(  \frac{ 1}{ Q_{0}^{2} \; C_{\mathrm{sech^2}}^2}  - 1 \right) ^{-1/2},$$ which goes to infinity for a marginally stable disc.

Numerical modelling {#sec:Numerical modelling}
===================

In order to test the predictions of the linear stability analysis, we employ simulations of an idealized gas disc. In the following, we describe the code and the disc model.

The simulation code {#subsec:The simulation code}
-------------------

We use the hydrodynamical AMR (adaptive mesh refinement) code <span style="font-variant:small-caps;">ramses</span> to perform simulations of a self-gravitating isolated gas disc with an isothermal equation of state (EoS), embedded in a dark matter halo. Since we are interested in the early phases without any strong discontinuities, here the Euler equations are being solved with the local Lax–Friedrichs scheme. The dark matter is handled as a static external density field added to the source term in the Poisson solver.\
The mesh in the 48-kpc simulation box is structured by nested AMR grids from the coarsest level of 187.5 pc outside the disc to a maximum resolution of $\Delta 