are those of section \[sec:phys\_par\] and table \[tab:forcing\_response\] for the $\ell=6$ case. (See figure \[fig:S\_ell2\_L\] for an illustration of the extraction of the bounding envelope.) We see that $|\Delta V(t)|/V$ remains less than about $3\times 10^{-6}$ throughout the simulation.

![Calculation of thresholds. a) Timeseries of interface height oscillate and grow or decay exponentially, depending on the imposed acceleration amplitude, as shown by the linear time dependence of the logarithms of the peaks. b) The slopes of the logarithms of the timeseries maxima constitute the growth or decay rates, which are interpolated to determine the acceleration amplitude threshold for $\ell=6$ shown as the rightmost cross in figure \[fig:validation\].[]{data-label="fig:valid_slopes_rates"}](fig2.eps){width="80.00000%"}

![Calculation of thresholds. a) Timeseries of interface height oscillate and grow or decay exponentially, depending on the imposed acceleration amplitude, as shown by the linear time dependence of the logarithms of the peaks. b) The slopes of the logarithms of the timeseries maxima constitute the growth or decay rates, which are interpolated to determine the acceleration amplitude threshold for $\ell=6$ shown as the rightmost cross in figure \[fig:validation\].[]{data-label="fig:valid_slopes_rates"}](fig3.eps){width="\textwidth"}

In our previous investigation [@Ali1], we extended the method of [@KT1994] for computing the Faraday threshold via Floquet theory to a spherical geometry. In figure 