The Reynolds number is defined as 1Re=VmeanDυ, where υ is the dynamic viscosity of the fluid, Vmean is the mean vertical velocity of the particle, and D is the maximum dimension of the particle.

identified that Re 100–200 is the critical point for the onset of unstable motions for circular discs, after which periodic behaviour begins. The value of the critical Reynolds number varies depending on particle shape. report an experimental study of how metal circular discs fall through water and glycerol mixtures and how paper discs fall through air. Different falling regimes were observed depending upon the experimental parameters; discs could fall steadily, exhibit oscillating periodic motions, or tumble.

Periodic behaviour includes zigzag and spiralling sub-types of behaviour, and more recently an in-between behaviour was identified as transitional through experimental investigations by . find that for circular discs, at low Re and I*, the most common behaviour is spiralling, whereas at high Re and I*, zigzagging behaviour is most common, with transitional behaviour occurring at intermediate I* and Re (Fig. ).

At Reynolds numbers below the critical Reynolds number, flow around crystals is stable. For planar crystals in this regime, laboratory and field measurements have shown that the largest dimension becomes normal to the axis of gravity, and the plate crystals achieve a horizontal orientation, corresponding to a constant inclination angle of zero ( (see their Sect. 10), )

At a critical Reynolds number, the flow around the crystal becomes unstable, forming vortices as part of the boundary layer of fluid at the surface of the particle. When shedding of these vortices in the wake of crystals occurs, the distribution of pressure on the crystal changes, exerting forces that cause it to rotate . These unstable motions are observed as oscillations in orientation, as well as in the vertical and horizontal velocities, such that they are non-zero, fluctuating, and have a distribution. There is a current lack of understanding about the orientation of ice crystals in unstable regimes, and one of the aims of this paper is to explore this.

The dimensionless moment of inertia for a circular disc is defined as the ratio of the moment of inertia of a circular disc about its diameter and a quantity proportional to the moment of inertia of a rigid sphere of fluid of the same diameter, such that 2Idisc*=π64ρpρftD, where ρp is the density of the particle, ρf is the density of the fluid, t is the thickness of the disc, and D is its diameter . For more complex shapes, the more general non-dimensional moment of inertia is 3I*=IaρfD5, where D is the maximum dimension of the particle. The moment of inertia for rotation around the three principal axes of the crystals is calculated, where Ia is the smallest of these three moments, aligned in the a-axis direction of the crystal (Fig. ) . For reference, further information regarding the calculation of the moments of inertia can be found in p. 570.

Figure  presents a summary of the coverage of the data presented in , also used in this study, on the Re and I* phase space and the key prior experiments on ice crystal shapes and circular discs that are discussed in Sects. 1.3 and 1.4. Using a mass–diameter relationship from for planar dendritic crystals and methods for estimating I* from , we find that a 10, 1, and 0.1 mm planar dendritic crystal, where the density of ice is 917 kg m-3 and the density of air is 1.2 kg m-3, has an I* of 0.02, 0.2, and 2.0 respectively. The hatched region of Fig.  displays this expected range of I* for planar dendritic ice crystals – this matches well with the range of previous observations of ice crystals.

Our study focuses on planar crystals. These range from hexagonal plates (which present a solid obstacle to the flow at all Reynolds numbers) to stellar crystals and dendrites, which have much more open projections. One way to characterise this shape variability is by area ratio: the ratio of the maximum cross-sectional area of the particle and the area of its circumscribing circle, which helps compare with circular discs. Fluid experiments on planar shapes report that the amplitude of oscillations in the descent velocity was maximum for circular discs and decreased with the area ratio, suggesting that unstable motions are inhibited by more complex shapes .

The first of the periodic motion types seen is the zigzag case study: the particle swings back and forth in one plane, and as the particle swings away from its centre of fall, its inclination angle increases, akin to a planar pendulum motion. The zigzag–transitional case introduces an element of rotation around the vertical axis, such that the plane of swing slowly moves anticlockwise, and had the experimental run been longer, it may have rotated back to its original position. The transitional–spiral case is similar, but the rate of rotation around the vertical is faster, producing wider loops. Spiralling, the final sub-type of periodic motion remains at a near-constant inclination angle and does not swing back and forth and instead precesses around its central point without touching its mean centre of fall.

The sub-types of periodic motion can be approximated by the sub-types of spherical pendulums: zigzagging is similar to a planar pendulum, spiralling is comparable to a conical pendulum, and transitional motion captures the range of pendulum motion between the two extremes, with the horizontal displacement approximating a rhodonea curve . A conical pendulum characteristically traces out a circle in the horizontal plane, akin to spiralling cases, which also trace out a circle in the horizontal plane. Similarly, a planar pendulum serves as an analogy for the zigzagging motion, as they are both constrained to movement in a single plane, tracing out a line in the horizontal plane.

Series of inclination angles, θ, for the periodic motion types can be approximated as sinusoidal (Fig. ). To distinguish between the regimes, rectified sine waves are fit to these inclination angle time series, using a fast Fourier transform as a first guess of the frequency of the sine wave and the SciPy curve fit function , such that 4θ=|θampsin⁡(ωt+ω0)+θtilt|, where θ is the inclination angle, t is the time in seconds, θamp is the amplitude of the wave, and θtilt is the angular displacement of the sine wave; the period of the sine wave is 2π/ω (in seconds) and ω0 is the phase shift.

All unstable motion presented in this study was observed to be periodic and is approximated through Eq. (). We observed no complex tumbling, chaotic fluttering, or behaviour with significantly non-sinusoidal motion to it. However, we note that there are some experiments in the study in which additional (weaker) modes of oscillation, in addition to the primary frequency, seem to be present. For example, the transitional case in Fig. c has an amplitude that fluctuates slightly in time at a lower frequency than the primary mode of oscillation captured by the simple single-frequency fit. We did not attempt to capture these finer details in our fitting procedure.

The root mean square error of the rectified sine wave fits to the four case studies is 1.1, 1.3, 2.7, and 0.9° for the zigzag, transitional–zigzag, transitional, and spiral cases respectively. The root mean square error of all fits to the data is 1° and is provided in the Supplement.

θamp and θtilt are found to summarise the motion types well, as they represent the variability and tilt of the particle respectively and constrain the pendulum model. They also allow the periodic sub-types to be distinguished quantitatively: a spiralling particle has a low θamp and a high θtilt, as it is consistently inclined and does not flutter.

Zigzagging behaviour is the opposite: a potentially high θamp and a near-zero θtilt, as it swings around a horizontal orientation, but flutters much more than a spiralling particle (Fig. ). For instance, the zigzag example case (Fig. a) has a fitted θamp of 34° and a θtilt of 0°.

The transitional–zigzag case behaves similarly but never samples (close to) θ=0, and it is just starting to transition to having a slow rotational component. It should be noted that although the transitional–zigzag case has a higher θamp (42°) than the zigzag case (where θamp is 34°), this does not negate categorisation of behaviour for either case, as the trajectory of the transitional–zigzag case is close to zigzag behaviour but never samples exactly θ=0, and it is just starting to transition to having a slow rotational component.

The transitional case (Fig. c) has a smaller amplitude than both zigzag and zigzag–transitional cases; θamp is 9°, but θtilt is much higher, at 24°. The spiralling case (Fig. d) has an even smaller amplitude still, with θamp at 1°but θtilt at 20°. Spiralling behaviour can have non-zero θamp, although θamp is small. This small variation in θ is referred to as “wobble” for spiralling cases. It may be worth considering how much of this wobble is a physical phenomenon versus an artefact or experimental uncertainty; a wobble of 2.5° could easily originate from experimental uncertainty. Given the wobble in Fig. d appears to have a uniform period, we believe the wobble in this case is partly a physical phenomenon, but you can see the impact of experimental uncertainty in the traces within Fig. a–c at the limits of the inclination angle (e.g. for zigzag the angle often does not reach 0). This will partly be due to the finite time resolution of the measurements and partly due to the accuracy of the orientation reconstruction.

Figure  also displays the distributions of θ for each case study. The zigzag case has a distribution that is consistent with that expected for simple harmonic motion, where the most likely inclination is at the end of each swing where dθ/dt is smallest and least likely is an angle of zero. Spiral has an almost constant inclination angle, and hence a very narrow distribution of θ, centred on an angle significantly higher than zero. Transitional has a distribution that is in between the other two cases, and in common with spiral cases θ is always above zero. This is in significant disagreement with the common assumption that orientation is a Gaussian distribution where the most common orientation is horizontal (θ=0).

An azimuth angle, ϕ, represents where the c axis of the crystal when projected into plane view is pointing relative to the x axis (in the laboratory reference frame) (Fig. c). A spiralling particle has a linear increase (or decrease, in cases not shown) of azimuth angle, and dϕ/dt is constant (as seen in Fig. d). The saw-tooth shape is produced by the angle being limited to ±180°.

Purely zigzag cases swing around one axis: when the particle goes from pointing one way to pointing another, ϕ changes by Δ180° seen by the square-wave shape (Fig. a). For pure zigzag cases, dϕ/dt is constant except for close to the instant where the particle becomes horizontal (θ=0) and ϕ becomes highly uncertain, as the c axis is momentarily pointing towards the vertical (and can be ignored). The axis that zigzagging cases pivot around tends to be the branches of the crystal, in the plane of the a and a′ axes. For shapes that are non-hexagonally symmetric (S, F, D, DP, and PB), the shortest branches are the axis the crystal pivots around.

Transitional cases have a non-constant dϕ/dt, a combination of the saw-tooth and square waves seen in the spiral and zigzag cases. For the zigzag–transitional case (Fig.  b), ϕ increases during its time along each loop of the rhodonea curve and then jumps by a value close to 180° when the orientation of the c axis is close to vertical. The transitional case (Fig. c) has no visible jumps in ϕ, except for the aliasing at ±180°, as θ never becomes close to zero.

The amplitudes of the u and w components of velocity (in the x and z directions respectively) changed in the presence of any component of rotation around the vertical. Therefore, a combined horizontal component of velocity, U, was calculated as follows: 5U=u2+w2. Sine waves can be fit to the vertical component of velocity, V, and the horizontal component of velocity, U, such that 6V=Vampsin⁡(ωVt+ωV0)+Vmean,7U=|Uampsin⁡(ωUt+ωU0)+Uoffset|.

These velocity components of the particles were found to follow a sinusoidal pattern consistent with the pendulum model (Fig. ). U is fit with a rectified sine wave, as horizontal speed can become zero in zigzag cases but cannot be negative (Eq. ).

For all experiments with periodic motion, θ, U, and V are all found to have sinusoidal patterns with the same period (i.e. ωV=ωU) but different offsets and amplitudes. Uoffset is not always zero: it is non-zero for particles that drift as they descend or particles with a spiralling component.

A summary of the sine wave fit components can be found in Table . Horizontal velocity, U, was found to peak when the tilt was lowest (i.e. when the particle was flat or the inclination angle of the particle was closest to zero) just after vertical velocity, V, peaks.

The amplitude of V fluctuations relative to the mean vertical fall speed (Vamp/Vmean) is 0.37, 0.47, 0.42, and 0.03 for the zigzag, zigzag–transitional, transitional, and spiral cases respectively. The amplitude of horizontal velocity fluctuations relative to the mean fall speed (Uamp/Vmean) is similar to their vertical components (0.41, 0.55, 0.49, and 0.03 for each case respectively). In the case of the spiralling particle, U and V are held relatively constant compared to the other cases, effectively making the spiralling cases a quasi-steady mode with a non-zero near-constant inclination.

For the non-spiralling cases, large fluctuations suggest that the mean vertical velocity, Vmean, does not sufficiently characterise the velocity of a particle. The distributions shown in Fig.  for the non-spiralling cases are broad, suggesting that a broad spectrum of instantaneous velocities for a single type of oscillating particle should be considered when interpreting Doppler spectra.

One major application of this research is for dwelling radars and lidars, whether ground-based (usually close to zenith) or spaceborne (usually close to nadir). Variation in the projected area affects assumptions in backscatter cross-section and hence the retrieval of particle size and number. Fluctuations will also affect polarimetric measurements and retrievals using these, particularly when the particles are viewed from the side. Figure  presents the time series of reconstructed projected areas as seen from below for each case study, normalised by the planar cross-sectional area of each analogue. The projected area as seen from below anti-correlates with θ in each time series, displaying an out-of-phase relationship: as θ increases, the aspect ratio of the analogues increases (becoming closer to 1), while the area ratio and projected area decrease.

For all four cases, vertical velocity and projected area exhibit a 180° out-of-phase relationship, where vertical velocity peaks just after the projected area reaches its minimum point. The projected area is more in phase with horizontal velocity, peaking just after U reaches its maxima. This supports the idea that vertical velocity increases when the projected area is minimised, as drag is minimised in the vertical direction, allowing the particle to accelerate.

In the case of an infinitely thin particle, the projected area as seen from below is equal to the cross-sectional area of the particle multiplied by cos⁡θ (and hence correlated with cos⁡θ, as evident from Fig. ). In the presence of particle thickness, the normalised projected area is expected to be greater than or equal to cos⁡θ and therefore should never fall below 0.7, as θ never exceeds 45° for periodically oscillating cases. In fact, the observed projected area occasionally exceeds that of the projected area of the particle when horizontal (i.e. the ratio in Fig. b slightly exceeds 1): this can only be achieved through the influence of the finite thickness of the particle.

Since these four cases are not discrete classes of behaviour, we propose a parameter to characterise where each experiment lies on the continuum of motions between zigzag and spiral. To quantify the spectrum of periodic behaviour, a motion-type parameter, ϵ, is defined as 8ϵ=θtiltθtilt+θamp, such that particles with ϵ=1 correspond to spiralling, as θtilt≫0 and θamp≈0 for spiralling cases. Zigzagging behaviour corresponds to ϵ=0, as zigzagging behaviour has high amplitudes, θamp≫0, and θtilt≈0. Transitional cases can have non-zero θtilt and θamp, and ϵ can therefore range between 0 and 1. The four case studies presented have ϵ= 0.01, 0.00, 0.73, and 0.94 for zigzagging, zigzag–transitional, transitional, and spiral respectively. The parameter ϵ characterises the distribution in θ and does not account for variation in ϕ. This explains why the transitional–zigzag motion case (Fig. b) has near-zero ϵ – the distribution shape of θ in that example is very close to that of a pure zigzag motion, even though there is also a weak azimuthal rotation superimposed which distinguishes it from the zigzag case. Although ϵ does not capture azimuthal variations, these details are often not practically significant when considering the statistics of a crystal population in a cloud; instead, it is the distribution of inclination angle θ which is of primary interest, and this motivates our definition of ϵ.

For bulk approximations (retrievals, microphysics schemes), it is useful to characterise the frequency of oscillatory behaviour. To non-dimensionalise the frequency of oscillation of the experiments, the Strouhal number is calculated as 9Stθ=fDVmean, where f is the frequency of oscillation found by the sine wave fit to θ . Stθ, the Strouhal number for θ, is therefore representative of the number of oscillations in θ of the particle in the time it takes for the particle to fall the vertical distance equal to its diameter.

Stθ (frequencies) of the four cases is 0.45, 0.34, 0.91, and 0.90 for zigzag, zigzag–transitional, transitional, and spiral respectively (Table ).

A secondary Strouhal number can also be calculated using the rate of change of ϕ: 10Stϕ=1360°|dϕdt‾|DVmean, where |dϕdt‾| is the mean absolute rate of precession of the particle. Stϕ is therefore the number of full turns around a vertical axis that a particle makes during the time it takes for the particle to fall the vertical distance equal to its diameter. Stϕ for the zigzag and zigzag–transitional cases is 0.028 and 0.079 respectively. The zigzag case is effectively non-rotational around the vertical axis and therefore has the lowest Stϕ. The transitional and spiral cases have substantial rotation around the vertical axis and therefore have Stϕ of 0.57 and 0.46 respectively. The transitional case spirals faster and wobbles faster than the spiral case, but otherwise both Stθ and Stϕ appear to increase with ϵ.

Figure  displays the distributions of inclination angle across Re, excluding stable experiments, and separates the cases by particle shape and ϵ greater than or less than 0.5 to demonstrate the impact of ϵ on the distributions. The onset of unsteadiness is seen at Re ≈ 200 (Re=212 for non-circular disc shapes), the lowest onset of spiralling in particular is seen at Re =461 (for non-circular disc shapes), and the onset of spiralling for circular discs is seen at values as low as Re ≈ 300.

Different shapes show different distributions of inclination angles as well as exhibiting different θtilt and θamp (Fig. ). For a particular shape, the distribution remains similar between adjacent Reynolds number bins; once a specific motion regime is reached for a particular shape, the same θtilt and θamp are maintained. For instance, Wang-BBP shapes spiral and have a mean θ of around 21° (along with a narrow distribution), while circular discs tend to have a much wider distribution, corresponding to high-amplitude zigzag behaviour.

Many of the shapes that present spiralling behaviour at high Re first present zigzagging behaviour at intermediate Re (see also Figs. , , and ).

Sine waves were fit using Eq.  to all velocity components and to θ for all periodic experiments (Eq. ), and the motion-type parameter ϵ was subsequently calculated (Fig. ) (Eq. ).

There is discrepancy between the literature on circular discs and our observations of ice crystal shapes when taking I* into account. The motion parameter, ϵ, increases for increasing Reynolds number and dimensionless moment of inertia (Fig. ), which is the opposite of the result reported by , who found that, for circular discs exhibiting periodic motions, spiralling occurred at lower Re and I* and zigzagging occurred at higher Re and I*. Our result instead agrees with , who found that hexagonal plates exhibit a zigzag motion at low Re, while larger plates at higher Re exhibited spiralling. also finds that for falling spheres, zigzagging occurs at lower Re than spiralling, which occurs at very high Re (Re>105). Our results also deviate from the stable–periodic division line provided by , as the ice crystal shapes do not become unsteady until higher Reynolds numbers than circular discs, since shape has a strong impact on the conditions that the onset of unsteady motions occurs.

Whilst I* is used successfully for studies of circular discs, shape must also be taken into account when considering the broad range of shapes that ice crystals exhibit. Differences in shape can be quantified by area ratio (the ratio of the maximum cross-sectional area of the particle and the area of its circumscribing circle).

Across the experiments, ϵ increases for increasing Reynolds number and area ratio (Fig. . In agreement with and , stable fall behaviour is found to be much more likely for particles with lower area ratios; i.e. ice crystals with more dendritic or complex shapes are more likely to fall steadily when under the same Reynolds number.

Using linear support vector classification , we identify a line of best fit that maximises the distance between stable and periodic behaviour, such that periodic behaviour occurs when 11log⁡10(Re)>2.82-Arearatio0.87. The critical point of this expression is displayed in Fig. .

Mean inclination angle was not found to distinguish well between periodic motion types (Fig. a). θtilt is found to be near-zero for zigzagging cases (where ϵ<0.5), but θamp can reach up to 45° (Fig. b). For all experiments where ϵ>0.5, θtilt is between 7 and 28° (Fig. c).

The distribution of ϵ was found to be bimodal (Fig. d), favouring either zigzag or spiralling behaviour, with transitional motion being less likely. Out of all 131 periodic plate-like observed experiments, 65 experiments displayed ϵ < 0.2, 34 were found to have ϵ between 0.2 and 0.8, and 32 experiments had ϵ > 0.8. The potential cause of this is discussed in Sect. 3.3.4.

Across all periodic experiments, the amplitude of the vertical velocity, Vamp, was found to be approximately 85 % of the amplitude of the horizontal speed, Uamp (Fig. a), using least-squares linear regression. In contrast to our findings, reported that the standard deviation of the horizontal velocity, U, was considerably larger (5 %–20 % of the fall velocity) than the standard deviation of the vertical velocity (< 3 % of the fall velocity) for dendritic-shaped particles undergoing periodic oscillation. The reason for the difference between our findings and those of Kajikawa is unknown, and it is hard to understand why fluttering particles would have large horizontal velocity fluctuations but almost constant vertical velocity. More investigation of natural particles using modern observations, such as by , may help explore this in the future.

For zigzagging particles, as θamp increases, the amplitudes of both the vertical and horizontal speed components increase exponentially (Fig. b). For spiralling particles, as θtilt increases, the amplitude of vertical velocity increases slightly (i.e. there is more wobble). θtilt has no influence on the amplitude of vertical velocity for zigzagging particles, as it is near-zero (Fig. c). Other relationships between the variables mentioned in this study were also explored; however, no clear patterns or simple relationships were evident.

Strouhal numbers were calculated for all experiments, as detailed in Eqs. () and (). For experiments where ϵ≤0.2, Stθ has a mean of 0.29 and a standard deviation of 0.10, with no significant trend with Re (Fig. ). Zigzagging particles (ϵ≤0.2) never have Stθ above 0.50. Despite having typically much smaller amplitudes than zigzagging particles, spiralling experiments (ϵ≥0.8) have a larger range of Stθ, with a potential for Stθ up to 1.50 and a mean Stθ of 0.5. Stθ for spiralling particles may be greater than for zigzagging; the wobbling of a spiralling particle is at a much smaller amplitude (θamp) than that of a zigzagging particle.

Similar to Stθ and previous literature on circular discs, there is no particular trend in Stϕ with Reynolds number (Fig. ). Stϕ is close to zero for zigzagging particles (where ϵ≤0.2), as the rate of spiralling is very low (by definition), whereas for spiralling particles (ϵ≥0.8), Stϕ can be as high as 0.7. No systematic relationship was found between either Strouhal number (Stθ or Stϕ) and area ratio.

When spiralling analogues rotate faster, they tend to also wobble more frequently. For high ϵ (and correspondingly, small θamp), Stθ is approximately half of Stϕ, corresponding to the classic observation that wobbling plates are found to wobble twice as fast as they rotate (Fig. ) . When θamp is non-zero, the spiral motion that the centre of mass of the analogue makes is not a perfect circle: the smaller the angle of wobble, the closer the traces of the analogue are to circles. When the spin is not a perfect circle, Stϕ no longer matches double the wobble rate, Stθ. For higher θamp (low ϵ) cases, Stϕ and Stθ appear to both remain low and not depend on one another.

For particles that are already spiralling, the higher θamp is, the more likely the behaviour is to be transitional (non-perfect spiralling) and the larger the swing the particle makes; therefore the wobble is less frequent and Stϕ is lower. This may also be the cause of the bimodal distribution in ϵ (Fig. ). Particles that have high Stϕ spiral quickly relative to their vertical velocity. This fast rotation around the vertical means that any torque at 90° to the vertical axis of rotation (which causes wobble in θ) will have less effect because it is small relative to the gyroscopic torque. Therefore, the rotation of spiralling particles most likely inhibits any potential zigzagging motion. Across a set of experiments for a given particle, at low Reynolds numbers, mean inclination angle, θ‾, is close to zero, as particles are stable, and at higher Reynolds numbers, particles become unstable and θ‾ increases to a steady, non-zero value. To quantify and compare the onset of unstable motions for different particle properties, a logistic curve is fit to the data using a least-squares method and the Trust Region Reflective algorithm from the SciPy curve fit function for each particle shape (Fig. a) and area ratio (Fig. b) as follows: 12θ‾=θ‾unstable1+e-k(log⁡10(Re)-log⁡10(Re)onset)+θ‾stable, where Reonset is the value of the function's midpoint, θ‾unstable+θ‾stable is the supremum of the values of the function, k is the steepness of the curve, and b is the minimum θ‾ value of the function. Particles with higher area ratio typically have bigger oscillations with larger θ‾. Dendrites with low area ratios and stellar crystals are stable at very high Re and typically have a smaller θ‾ when unstable.