Dynamics of falling snow

Motions of falling objects are generally complex  even regularly-shaped objects such as coins and cards flutter (oscillate from side to side), tumble, and drift sideways

Demonstation: business card  change of initial orientation from 0°to 90°is enough to cause big change in fall behavior Business card held horizontally and vertically

Origin of complex motions  Qualitative description dates back to Maxwell (1853)  Cause: torque created by combination of gravity and lift forces  Forces acting on different points on a body produce several moment arms  Result is complex descending motions

Observations: falling of circular disks through a viscous fluid medium  Stereoscopic cameras used to record the trajectories of disks falling through fluid  disk material: steel and lead  diameter d = 5.1 to 18.0 mm  thickness t = to 1.63 mm  fluid: water-glycerol mixture

There are 5 parameters  Disk diameter d, thickness t, density ρ, fluid density ρ f, and kinematic viscosity ν.  reduce to 3 dimensionless ratios  moment of inertia I*  Reynolds number Re  aspect ratio t/d  Disks are thin: omit effect of aspect ratio

Definitions of I* and Re

Four distinct motions observed and mapped on “phase diagram”  1) Steady falling  2) Periodic oscillation  3) Chaotic  4) Tumbling

Steady falling Periodic oscillation Chaotic Tumbling with drift

Field et al. Letters refer to details in Field et al. I* Re “(The disks’) ultimate location of the bottom of a large container could never be predicted.” Today we recognize this as a hallmark of chaos. Transition to chaotic behavior with increasing Re and I*

Deterministic chaos  Extreme sensitivity to initial conditions  Example: ski hill with moguls

AHS 260 Day 38 Spring Final positions of skis are very sensitive to their initial positions! Since there is uncertainty in the exact initial position, the final position is unpredictable. Ski positions at bottom of hill are much farther apart than at top of hill Paths of seven skis released from slightly different positions Ski tracks become widely spaced! All ski tracks are closely bunched at first

Zhong et al. found a spiral motion as well planar zigzag transitional spiral

Zhong et al.

interactions between disk and induced vortices play a significant role Vortices shed from zigzagging disk Wake forms helicoidal shape Vorticity is produced and fed into wake, scrolling into a roll

Can fall behavior be modeled numerically?  In principle, we could solve Navier-Stokes equations in 3D  both daunting and unrealistic at higher Re  Computational approaches limited to simple shapes and 2D so far

What about snow crystals or snowflakes (aggregates)?  Casual observation in light winds shows motion is not just downward  snowflake trajectories vary in direction, and vary relative to each other  sideways motion easy to see, but not full swing or spiral  spin around vertical axis occasionally seen  tumbling very occasionally seen  Some due to eddy motions of air, but some due to unstable mode of descent of particle itself

Why study snowflake fall motions?  Computation of aggregation of snow crystals requires knowledge of  fall velocities (well-studied)  fall trajectories (few studies)  Work of Kajikawa on unrimed and rimed crystals, and “early” snowflakes

Importance of fall behavior for cloud physics  Collection efficiency depends on motion of both collector and collected particle horizontal motions of collector would increase sweep volume allow collisions between particles with same fallspeed

Other applications  Flutter of cloud particles influences  radiation transfer in clouds  differential reflectivity signals

Research of M. Kajikawa time interval = 0.01 s Melted crystal

Results for unrimed plate-like crystals  Unstable motion first appears as flutter (like (a))  Next swinging (b) appears, followed by spiral  No mention made of chaotic regime  Tumbling did not appear  Latter two points call for further observations From Field et al.

Fig. 5 Kajikawa 1992

Dendritic crystals  Fall with a stable pattern over larger range of Re than simple plates  may be due to internal ventilation of crystals, seen in model experiments

Motion of dendritic crystals  SD of vertical velocity < 3% of fall velocity, but SD of horizontal velocity up to 20% of fall velocity  variation in horizontal velocity likely has significant effect on aggregation Standard deviation of V H (cm s -1 ) V H (cm s -1 )

Rimed crystals  Similar to diagram for unrimed crystals Fig. 2 Kajikawa 1997

Tracking software  I experimented with Tracker, a video analysis tool for physics lab experiments  Allows tracking of objects and analysis and modeling of their motion  Position, velocity and acceleration overlays can be made

autotracking

Problem with snowflakes  Snowflakes fall at ~150 cm s -1  At 30 fps, snowflake falls 5 cm between frames  This is 5x its size, so it’s blurred  Need high-speed camera (problem: cost, for now)  Also need distance scale  relative horizontal vs. vertical motion may be enough to start with

Eddy motions also important  we can assume that mean wind motion does not affect collision process  however fluctuations in wind velocity because of turbulence may enhance collisions  How can turbulence increase collision rate?  by changing the  relative velocities  spatial distribution  collision and coalescence efficiencies between crystals

Even with good measurements….  We cannot characterize motion of each snowflake  can measure average horizontal displacement and velocity, and oscillation frequency  Thus we still need parameterization in numerical cloud or mesoscale models