K and the disc has an exponential surface density profile with scalelength $h = 5.26$ kpc and truncation at $R_{\mathrm{d}} = 16$ kpc (Fig. \[fig:surface\_density\]). The central density is $\rho_{\mathrm{c}} = 3 \; \mathrm{M_{\sun} \; pc^{-3}}$. The parameters were chosen to resemble an initially unstable massive high-redshift disc galaxy [@2011ApJ...733..101G], with a relatively large scalelength, and a stable outer part with a relatively flat rotation curve.\
The disc setup is initially in vertical hydrostatic equilibrium (see Appendix \[App:Hydrostatic equilibrium\]), which naturally leads, for an isothermal disc, to an increasing scaleheight with radius (Fig. \[fig:scale\_height\]), see also @Wang:2010wr. Hydrodynamical equilibrium is achieved by following the steps described in @Wang:2010wr, with the total rotation curve as shown in Fig. \[fig:rotation\_curve\], which consists of the contributions of the gas disc and the dark matter halo. For our ${\, \mathrm{sech} }^2$-profile, the disc is unstable for axisymmetric perturbations between $R_{\mathrm{u}}=0-10.517$ kpc (solid line in Fig. \[Q\]), becoming more stable for larger radii and totally stable for $R \geq 10.517$ kpc. The exponential profile approximation assumes a slightly more stable disc with a smaller unstable regime $R_{\mathrm{exp}} = 0-9.58$ kpc (dashed curve in Fig. \[Q\]). Using the classical Toomre $Q_0$, that is ignoring $z_0$, leads 