$\ell=0$ growth rate bifurcates to a non-oscillatory mode lies at a much higher heating rate than the point where $t_{\rm adv-g} = t_{\rm heat-g}$ in both sequences. Nevertheless, it is shown in Appendix \[s:L0\] that non-oscillatory instability still sets in at the heating rate for which these timescales are equal in the initial condition, indicating that the expansion is a non-linear effect[^1].

Spherical Fourier-Bessel Spectral Decomposition {#s:sfb_outline}
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To analyze the properties of the flow accounting for its intrinsic spherical geometry, we employ a spherical Fourier-Bessel expansion to perform various spectral decompositions. This set of functions forms an orthogonal basis of two- or three-dimensional space in spherical coordinates, allowing the expansion of an arbitrary scalar function $f(r,\theta,t)$ in a series of the form $$\label{eq:sfb_2D}
f(r,\theta,t) = \sum_{n,\ell} f_{n\ell}(t) g_\ell(k_{n\ell}r) P_\ell(\cos\theta),$$ where $g_\ell(k_{n\ell}r)$ are the radial basis functions, $k_{n\ell}$ is the radial wave number of order $n$, $P_\ell(\cos\theta)$ are the Legendre polynomials of index $\ell$, and $f_{n\ell}(t)$ are (time-dependent) scalar coefficients. Expansions of this form have previously been used in the context of galaxy redshift surveys (e.g., @fisher1995). Appendix \[s:sfb\_appendix\] contains a detailed description of the expansion method, including the straightforward extension to three-dimensional space. In what follows we provide a 