At right - another view of the flare topped pillars seen by Aigar Truhin at Sigulda, Latvia. This image was taken ~10m above ground level. Part 1 and more images here.
Jon Seamans of Redmond WA did a systematic series of ray tracings with HaloSim to produce the simulation at middle right. The flared shape in the simulation is an upper tangent arc produced by horizontal column crystals refracting rays slanting upwards from a light source that (to the crystals) is 10° below the horizon. The vertical spike in the simulation was from plate crystals but that is not discussed here as the vertical components of Latvia pillars could be formed in other ways.
The apparent match of the simulation is only indicative because artificial light halos come from divergent light whereas HaloSim uses only near parallel rays.
Nonetheless, Jon's simulation gives interesting clues to the flares as a form of upper tangent arcs (UTAs) from upward shining rays. Additional support is provided by the Deventer arcs which were accompanied by a lunar circumscribed halo also formed by column crystals.
There is a problem! If the crystals were everywhere in the air we would expect the individual tangent arcs from crystals in any one locality to appear in a different angular position in the sky to those formed by crystals elsewhere. The set of individual halos might then overlap to form a blur or a relatively featureless glow. To avoid that by assuming a special spatial position for the halo forming crystals is unreasonable. Are there other ways in which distinct UTA, flared, shapes might form?
The answer lies in a rigorous solution of a 3D geometry problem but we can glean some clues of the halo formation and behaviour from an alternative quite simple approach.
Let's look (Fig 3) at only the very lowest tip of the UTA that is formed by light deviated through column crystals by 22 degrees.
Crystals that form the tip lie on a curve (locus) in a vertical plane intersecting the eye and the light. At each position along the locus a crystal deviates light from the lamp by 22 degrees. For simplicity the light is shown level with the eye but it could be otherwise.
The precise* locus is the green line in Fig 4. The second upper dotted locus is for crystals that deviate light by 27° to form a higher region in the UTA. For clarity the vertical scale in Fig4 is stretched.
The halo source is now apparent. Rays towards the eye along line B (say) come from severely overlapping UTA rays from many points above the lower tip. No distinct halo is seen except perhaps a contribution to the vertical pillar sometimes below the flare.
Rays to the eye near the line A are different. They see the tips of UTAs along a gentle curve and they reinforce each other. The corresponding 27° curve is initially 5° above that for the lower tip showing that the UTA shapes overlap little and are relatively undistorted. Crystals in the shaded red region produce distinct UTAs, the flare shapes seen in Latvia and Netherlands.
Halos in the shaded red region have their lowest tips 15 - 22° above the source lamp.
To check this prediction, measurements were made on an original Latvia image kindly supplied by Aigar Truhin. There was some uncertainty in the lamp positions but the heights of the lower flare tips on the major halos were estimated as 15.1 and 14.8°. Rays forming the 14.8° lower UTA tip would have come as from 7° below the horizon in reasonable agreement with Jon's simulation.
The Deventer halos were estimated from the lens focal length and also the known diameter of the circumscribed halo. The UTA tip heights are less certain than the Latvia ones because of the variable image scale and less well defined halos. However, they measure 14, 17 and 20° - within the predicted bounds.
To sum up:
The flare shapes are 3D variants of upper tangent arcs. They are formed by column crystals within 20-25% of the distance to the lamp compared with plate pillars where the halo forming crystals are half way. No special crystal location is needed but a thin crystal layer helps. Flares have their lower tips 15-22° above the light source - the closer the crystals the higher is the tip. For the halo tips this corresponds to rays from the light slanting upwards 7 to 0°.
Thanks to Jon Seamans for his systematic work on simulating the halos and subsequent interest and comments. He provided the key.