\[eq:normalization\_condition\_dirichlet\] and \[eq:normalization\_condition\_neumann\]) $$\int_{r_{\rm in}}^{r_{\rm out}}r^2\totd r\, g_{\ell}(k_{n\ell}r)g_{\ell}(k_{m\ell}r) = \delta_{nm}.$$ The coefficients for the spherical Fourier-Bessel expansion in equation (\[eq:sfb\_2D\]) are thus $$f_{n\ell}(t) = \frac{2\ell+1}{2}\int f(r,\theta,t) g_\ell(k_{n\ell}r)\,P_\ell(\cos\theta)\,
r^2dr\,\sin\theta\totd\theta.$$

From Parseval’s identity, $$\label{eq:parseval_2d}
\int |f|^2\, \totd^2 x = \sum_{n,\ell}  \frac{2}{2\ell+1}\,|f_{n\ell}|^2,$$ one can define a discrete power spectral density in 2D space $$P_{n\ell} = \frac{2}{2\ell+1}\,|f_{n\ell}|^2.$$ The coefficients $f_{n\ell}(t)$ can also be Fourier analyzed in time, yielding an individual power spectrum for each $(n,\ell)$ mode. Using a Discrete Fourier Transform (DFT) in time, the normalization can be taken to be the time-average of the volume integral of the variable in question (e.g., @NR), $$\begin{aligned}
\frac{1}{N_q}\sum_q \int |f|\, \totd^2 x & = & \frac{1}{N_q^2}\sum_{n\ell q} \frac{2}{2\ell+1}\,|\widehat{f}_{n\ell q}|^2\\
\label{eq:3d_spectrum}
&\equiv & \sum_{n\ell q} \mathcal{P}_{n\ell q},\end{aligned}$$ where $N_q$ is the number of time samples, and $\widehat{f}_{n\ell q}$ is the DFT of $f_{n\ell}(t)$ at frequency $q$.

In practical applications, the series in equation (\[eq:parseval\_2d\]) must be truncated at a finite value of the indices. In our analysis we set these maximum indices to be at most half the number of grid points in the corresponding direction, in analogy with the Nyquist limit in cartesian coordinates.

General Relativistic, Radiation-Hydrodynamic Simulations
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We use the set of two-dimensional, general-relativistic, 