  Small non-spherical particles settling in a quiescent fluid tend to orient so
that their broad side faces down, because this is a stable fixed point of their
angular dynamics at small particle Reynolds number. Turbulence randomises the
orientations to some extent, and this affects the reflection patterns of
polarised light from turbulent clouds containing ice crystals. An overdamped
theory predicts that turbulence-induced fluctuations of the orientation are
very small when the settling number Sv (a dimensionless measure of the settling
speed) is large. At small Sv, by contrast, the overdamped theory predicts that
turbulence randomises the orientations. This overdamped theory neglects the
effect of particle inertia. Therefore we consider here how particle inertia
affects the orientation of small crystals settling in turbulent air. We find
that it can significantly increase the orientation variance, even when the
Stokes number St (a dimensionless measure of particle inertia) is quite small.
We identify different asymptotic parameter regimes where the tilt-angle
variance is proportional to different inverse powers of Sv. We estimate
parameter values for ice crystals in turbulent clouds and show that they cover
several of the identified regimes. The theory predicts how the degree of
alignment depends on particle size, shape and turbulence intensity, and that
the strong horizontal alignment of small crystals is only possible when the
turbulent energy dissipation is weak, of the order of $1\,$cm$^2$/s$^3$ or
less.
