brief outline, focusing on the quantities needed to analyze the turbulent flow in our 2D models.

The domain considered is the volume enclosed between two concentric spheres of inner and outer radii $r_{\rm in}$ and $r_{\rm out}$, respectively. These spherical surfaces can be any pairwise combination of the neutrinosphere, gain radius, or shock radius, depending on the particular region to be studied. The radial basis functions $g_\ell(k_{n\ell}r)$ are linear combinations of spherical Bessel functions $j_\ell$ and $y_\ell$, with coefficients chosen to satisfy specific boundary conditions at both interfaces (Appendix \[s:sfb\_appendix\]).

Imposing these boundary conditions generates a set of discrete radial wave numbers $k_{n\ell}$, in analogy with the modes of a membrane in cylindrical coordinates. In addition to its quantum numbers $n$ and $\ell$, these wave numbers depend on the chosen ratio of inner and outer radii $r_{\rm in}/r_{\rm out}$. Appendix \[s:sfb\_appendix\] derives the wave numbers, relative coefficients, and normalization of the radial basis functions for the cases of vanishing (Dirichlet) and zero gradient (Neumann) boundary conditions. For low $n$, $\ell$, and $r_{\rm in}/r_{\rm out}\to 1$, these wave numbers approach $$k_{n\ell}\to \frac{\pi}{(r_{\rm out}-r_{\rm in})}(n+1), \quad (n=0,1,2,...)$$ increasing in value for stronger curvature. The normalized basis functions satisfy the orthogonality relation (equations 