the surface height over the entire drop surface, and hence more accurate than the individual surface height values.

The spectrum in $m$ (but not in $\ell$) depends on the orientation. We can rotate a pattern by localizing the coordinates $(\theta,\phi)$ of one of its features, such as a maximum, rotating in longitude $\phi$ to place it in the $(x,z)$ plane and then rotating in colatitude $\theta$ to place this feature at the north pole. This allows us to interpret our spectra by using the explicit representations given by [@Busse1975] of various patterns in terms of $Y_\ell^m$. Conversely, if the shape of the pattern is known or constant, the spectrum in $m$ can be used to track changes in its orientation.

Validation
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![Time evolution of normalized deviation of $\Delta V (t)/V$ for a simulation with $\ell=6$. The value of $|\Delta V|/V$ remains less than about $3\times 10^{-6}$ throughout the simulation.[]{data-label="fig:volume_validation"}](fig1.eps){width="80.00000%"}

The front-tracking approach inherently conserves mass to high accuracy compared to other numerical interface methods [@ShJu09]. We confirm mass conservation by showing the fractional deviation of the volume. Figure \[fig:volume\_validation\] shows the time evolution of the bounding envelope of $\Delta V/V$ for a simulation with resolution $128^3$. The fluid parameters 