Effect of particle inertia on the alignment of small ice crystals in turbulent clouds

TL;DR

The study shows that particle inertia can substantially increase tilt-angle fluctuations of small ice crystals settling in turbulence, challenging overdamped theories that predict near-perfect alignment at large settling speeds. It develops a comprehensive model that couples translational and angular dynamics with convective fluid inertia, deriving a phase diagram of regimes where tilt variance scales differently with the settling number $\mathrm{Sv}$ and Stokes number $\mathrm{St}$. Analytical results, supported by statistical-model simulations and direct numerical simulations (DNS) of turbulence, reveal multiple mechanisms—translational slip fluctuations and viscous torque interactions—that drive misalignment, with inertia causing variance growth by orders of magnitude in some regimes. The findings imply that strong horizontal alignment in cirrus clouds requires very weak turbulence, and they offer a framework to predict tilt statistics from particle size, shape, and turbulence intensity, with potential implications for cloud radiative balance and remote sensing.

Abstract

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.

Figures (5)

• Figure 1: Platelet (left) and column (right) settling in a turbulent flow. The particle symmetry axis is $\hat{\boldsymbol{n}}$, and the particle velocity is denoted by $\boldsymbol{v}$. Gravity $\boldsymbol{g} = g \hat{\boldsymbol{g}}$ points downwards. The tilt angle is defined as $\cos\varphi=\pm \hat{\boldsymbol{n}}\cdot\hat{\boldsymbol{g}}$ (see text). In a quiescent fluid small columns fall with steady-state orientation $\hat{\boldsymbol{n}}\cdot\hat{\boldsymbol{g}}=0$, while platelets fall with steady-state orientation $\hat{\boldsymbol{n}}\cdot\hat{\boldsymbol{g}}=\pm 1$ (see text).
• Figure 2: Coordinate system for angular dynamics: direction of gravitational acceleration $\hat{\boldsymbol{g}} = \hat{\bf e}_1$, projection $\hat{\boldsymbol{p}}$ of $\hat{\boldsymbol{n}}$ onto the plane perpendicular to gravity, and $\hat{\boldsymbol{s}} = \hat{\bf e}_1\wedge \hat{\boldsymbol{p}}$.
• Figure 3: Phase diagram of asymptotic regimes for the tilt-angle variance $\langle\delta\varphi^2\rangle$ in the statistical model, together with results of numerical statistical-model simulations of Eqs. (\ref{['eq:eom']}) for platelets with $\beta=0.1$, $\ell=10$, and ${\rm Ku}=10$ (colour coded, see legend). The conditions separating the different regimes are discussed in the text: $\tau_{\varphi}=1$ (dotted line), $\tau_{\varphi}=\tau_{\rm d}^{({\rm tr})}$ (solid line), $\tau_{\rm d}^{({\rm tr})}=\tau_{\rm s}$ (dashed line), and $\tau_{\rm d}^{({\rm rot})}=1$ (dash-dotted line). 'MFT' stands for mean-field theory (Section \ref{['sec:ts']}.\ref{['sec:mft']}).
• Figure 4: Tilt-angle variance as a function of particle aspect ratio $\beta$ keeping ${\rm St}/A^{(g)}$ and ${\rm Sv}/A^{(g)}$ constant. Results obtained using DNS of turbulence: empty symbols are with the inertial drag correction (\ref{['eq:drag_correction']}), filled symbols without this correction. The overdamped approximation (\ref{['eq:largesvod']}) is shown as a black solid line. Also shown is the theoretical prediction (\ref{['eq:phiSqrPlateletsUnevaluated']}) for regimes ➁ to ➃ for $\ell=14.7$, coloured lines. Other parameters: ${\rm Sv}/A^{(g)}= 22$ and ${\rm St}/A^{(g)}=0.11$ (red,$\circ$), $0.45$ (green,$\Box$), and $2.2$ (blue,$\diamond$).
• Figure 5: Phase diagram, similar to Fig. \ref{['fig:PhaseDiagram']} for platelet-shaped crystals, for $\ell=10$. Symbols show the values of the dimensionless parameters corresponding to experimental and numerical studies of non-spherical crystals settling in turbulence (details in Supplemental Material). Dimensionless parameters estimated from: Bre04 ($\Box$); numerical study of collisions between disks settling in turbulence Jucha2018 ($\Diamond$); experiments by Kramel ($\blacksquare$), and esteban2020disks ($\blacklozenge$). Blue solid lines show contours of constant turbulent dissipation rate, $\mathscr{E}=0.1,1,10,100\,$cm$^2$/s$^3$, using Eq. (\ref{['eq:froude']}) with $\nu=0.1\,$cm$^2$/s, $\rho_{\rm p}/\rho_{\rm f}=1000$, and $g=980$cm/s$^2$. Red solid lines show contours of constant $a\sqrt{\beta}=10,20,40\,\mu$m (see text).