A multi-split rainbow from south-east China, August 12th, 2014
Twinned rainbows are rare sightings, in the sense that one may see on average only one per year in Central Europe even when paying close attention. Much rarer still, and maybe restricted to regions closer to the equator, are multi-split rainbows. Only few cases have been documented so far [1, 2, 3], though more snapshots can be found on image sharing platforms labeled as “triple rainbow” etc. It is always a very favorable situation if an archivist and analyst like myself can establish direct communication with a skilful observer, who recorded details of a rainbow display that provide some insight beyond the pretty pictures.
In April 2019 I emailed Mr. Ji Yun, who manages a Facebook group dedicated to atmospheric optical phenomena in China, asking about a spectacular photograph of a multi-split rainbow which had been shared there. He kindly relayed my request to Mr. Liu Hai-Cheng, the original observer. Mr. Liu agreed to answer a long list of questions and I also received two sets of photographs from August 12th, 2014, one from his Sony NEX-5C camera (equipped with a Nikon AF 28mm f/2.8 lens) and the other from his cell phone (Coolpad 8720L). The camera clock’s time stamps were calibrated with respect to the actual local time by comparing camera and cell phone pictures, and assuming the cell phone clock to be synchronized over the network. All time data are given here in Chinese standard time (UTC+8h).
Mr. Liu observed this rainbow rarity in the beautiful landscape of the karst mountains near the Yulong bridge (Yangshuo County, Guilin City, Guangxi province, about 400 km northwest of Hong Kong, 24.8° N, 110.4° E) during a boat trip on the Yulong river. He remembers that it was very hot that afternoon. It began to rain before he passed through the tunnel of the bridge (at about 16:50), with some heavier rain lasting for about 25 minutes. There was no lightning, thunder or strong wind.
Judging from the photos, the rainbow appeared at about 17:10 within 30 s or less. Already on the early photographs there are hints of the unusual splitting of the primary:
However, Mr. Liu’s visual impression was that the splitting became prominent only later, after the (seemingly ordinary) primary and secondary bow had appeared successively. He also noted that the visibility of the split branches changed over time, while the main primary could always be seen clearly.
Towards the end of the shower, the display reached its peak quality. The following pictures cover the full right-hand side of the rainbow and some of the left. They are presented without additional filtering to allow for a better assessment of the natural contrast conditions.
For a deeper analysis, I chose the title picture, recorded at 17:18. In the contrast-enhanced version, three primary branches are directly visible, with the most intense one in the center. The secondary rainbow, as far as it is included in the frame, does not exhibit any anomalies. This is a typical feature in (almost) all split rainbow observations known so far. My goal was now to transform the photograph into the scattering angle vs. clock angle coordinate system (in equirectangular projection), as I did on previous occasions [1, 4]. The scattering angle is the angular distance from the sun, and the clock angle the azimuth around the rainbow’s circumference, with the 0° position corresponding to its top.
The sun’s position is easily obtained from standard astronomy software (giving an elevation of 25.4°, and azimuth of 275.4°). Additionally, the precise focal length of the lens (in pixel units) and distortion characteristics need to be known, as well as the camera pointing direction in elevation and azimuth, and the angle describing the rotation of the sensor’s pixel grid with respect to the vertical.
To precisely determine these quantities, a rather extensive calibration must be carried out. Here I had to try some reasonable guessing: There is a nominal focal length in mm, the sensor data (pixel pitch) can be looked up, as well as some distortion information for this specific lens. From aerial pictures showing the river and individual mountains, the viewing direction can be estimated. The appearance of the water surface gives some clues about the camera rotation. In combination, all these estimations allow for a plausible transformation:
Assuming this reconstruction to be not too far off, it is immediately obvious that the bright central branch does indeed fit to the conventional primary rainbow locus at a constant scattering angle of about 138°. As expected, the secondary ends up at about 129°, also as a straight line. The lower branch (i.e. at higher scattering angles) can in principle be explained by aerodynamically flattened raindrops, following a long tradition in rainbow physics [5, 6, 7, 8, 4]. However, the upper branch penetrating into Alexander’s dark band requires elongated raindrops, whose existence cannot be accounted for by aerodynamics alone. Electrostatic fields  can elongate raindrops, but in the absence of any lightning activity it is speculative if any higher fields were present. Elongated shapes do also occur as transitory states during oscillations of larger drops in the appropriate (axisymmetric) modes .
The problematic element in this explanation is, however, that in the case of the rainbow we deal with a large number of contributing raindrops and a temporal average due to the finite exposure time. So we need an argument why contributions from transitory states are not simply wiped out. The resonance frequencies of the individual drops depend on their size, so no singular event such as an acoustic shock wave from thunder (if there had been any at all) can synchronize the oscillations. The only plausible idea for a formation of stable rainbow branches by drop oscillations in a stochastic ensemble might be that the two extremal states of the oscillation (flattened and elongated) are encountered with a higher probability than intermediate ones, as the momentary velocity decreases to zero at the turning points of any classical oscillation. Admittedly, this requires a rather narrow distribution of amplitudes throughout the ensemble (at least in the dominant drop size range), as otherwise the branches will be wiped out again due to the spread in extremal axis ratios. To my knowledge, there is not enough data on the statistical properties of oscillations in large ensembles of natural raindrops published yet to draw a definitive conclusion here.
Some further details of this observation are worth to be noted: The three branches of the primary bow appear each in a distinct fashion: The lowest is broad and rather diffuse, the middle one is bright and shows the features of a typical primary rainbow, the top one is narrow with a sharp uppermost outer rim. Moreover, it gives the impression of having developed a downward sub-branch in the –10°…+5° clock angle interval, resulting in a four-fold split bow there.
Rainbows certainly go on fascinating people all over the world, and rightfully so: Even in the 21st century, some outstanding displays occur from time to time that still challenge our understanding. Maybe those in hotter climates with intense rain showers have better chances of catching such rarities. In any case, we have to go out and take a look and a picture at the right time.