Behind the curve

Savants have sought to explain the secret of Mt Fuji's subtly concave slopes for more than a century. Now they're getting close

It was the curve that captivated him – the line that falls, steeply at first, from Mt Fuji’s shoulders, then at an ever-shallower angle until, at Omiya, it blends softly into the horizontal. So elegant was this line, he speculated, that it might have inspired the elegant cusping of Japan’s house roofs and castle walls. Merely to admire this mysterious curve wouldn't do, though. As a Victorian scientist, he had to explain it.

John Milne had joined the staff of the Imperial College of Engineering in Tokyo in 1876. He’d already packed several geological expeditions into his 26 years, including forays to Sinai and Newfoundland. Even his journey to Japan was an adventure; to avoid a long sea voyage, he travelled from England via Russia, riding part of the way on a dog-sled across the Siberian snows.

As one of 3,000 or so ‘o-yatoi’ or foreign technical experts hired by the Meiji government, the young professor’s duties were to teach mining, architecture, metallurgy and chemistry. Today, Milne is remembered as a founder of seismology. During his two decades in Japan, he invented an improved seismograph and later helped set up the first global network for monitoring earthquakes.

He also toured the archipelago, inspecting its volcanoes. Their regular forms contrasted with the “very rough” volcanoes he’d seen in Iceland. Just two years after landing in Japan, he’d acquainted himself with “Fusiyama, 12,365 feet near Yokohama; Ganjosan, 7,000 feet near Morioka, Chokaisan, 6,000 feet, between Niigata and Akita, Iwakisan, 5,000 feet, near Awomori; and Kumagatake, 2,700 feet, near Hakodate”.

Now Milne was ready to start his analysis, which he published in the 1878 edition of the Geological Magazine. All the above mountains, he wrote, appear conical at first sight. Looking closer, though, “we see this upper cone expanding and sweeping outward, forming a graceful curve”.

This curve, he reasoned, could arise in two ways. Either it formed at the start of the volcano's life, from the way its ash and lava were thrown out. Or it developed later, through "denudation" by wind and rain. But which of these factors created the elegant profiles of Japanese volcanoes?

Milne doesn’t waste many pages of the Geological Magazine before delivering a crisp judgment: “This being the case, I think we are justified in regarding mountains, similar to those about which I am now writing, as having a form mainly due to the simple piling up of material and not as cones which have been subsequently modified by a number of secondary causes, such as are advocated in treatises on Physical Geology and Volcanos.”

Two pieces of evidence lead him to this conclusion. First, his volcanic profiles – they form logarithmic curves, he believes – are so similar that they must have a common cause. Secondly, the slope angles are close to what theory would predict “for the stability of a self-supporting mass of loose materials”.

A lesser scientist might have regarded the case as proven. But Milne continued to think about the curve. And he went to look at more volcanoes, in Yezo and the Kuriles. More data, however, led to less certainty. A year later, he sent the Geological Magazine some “Further notes upon the Form of Volcanos”. The tone is now circumspect: the slopes Milne saw on his recent tour, while “generally logarithmic”, are not absolutely so.

Then he puts his theories about slope stability to the test. This he does by pouring onto the floor first sand, next gravel, and finally a mixture of the two. Sadly, the resulting heaps remain resolutely cone-shaped, betraying scarcely a hint of Fuji-like curvaciousness. "Taking these experiments as a whole," Milne admits, "it will be observed that I did not obtain much evidence in favour of my views."


He revisits the ways in which a volcano might get its curve. A self-supporting heap tends to spread out at its base; large particles roll further down the slope than small ones, creating a shallower slope angle lower on the hill; and there’s always "denudation" to fall back on. Incidentally, he's tried sprinkling water on his piles of experimental sand and gravel, but that too fails to produce the elusive line.

Has he missed something? Another savant, a Mr Mallet, has proposed that a volcano's weight might crush the strata under it, causing the mountain to settle into the ground. That might result in curved slopes. Or – and surely a note of desperation creeps in here – the volcano's core might shrink as it cools, pulling its slopes inwards like a child sucking in its cheeks.

This time, Milne’s conclusion is not even tentative; it is an abdication. “Taken as a whole, these causes are very varied ..,” he writes. “If we examine them singly, we can but barely form an idea as to the nature of their actions, and when we remember that … they act irregularly in their relations to each other, we see that the task of unravelling their complications becomes quite hopeless.”


As if taking this warning to heart, few scholars since Milne’s day have troubled themselves about volcanic curves. Current textbooks, if they raise the question at all, often take the opposite line to Milne’s: "Initially, a pristine volcano will form a pure conical form,” intone Jon Davidson and Shan de Silva in the Encyclopaedia of Volcanoes, “[until] Mass wasting results in a transfer of mass from the upper parts of the edifice to the lower flanks, building out a talus apron. The edifice evolves to a steady-state profile with concave-upward slopes..."

Yet denudation doesn’t do it for everybody. A trio of highly numerate geologists from Oxford and Cornell proposed – this was in 1981 – that "hydraulic resistance to the flow of magma determines the geometrical form of volcanoes". Feeding their equations for the stickiness of lava into a model, they came up with a slope profile that nicely mimics that of Mt Fuji. However, they note, “the theory is not expected to be valid near the summit, where the solution is singular”.


And there the matter rested – an unresolved tussle, if you like, between the Lords of Lava Dynamics and the Maharajahs of Mass Wasting – until the Space Shuttle blew onto the scene. On one of its flights, the vehicle carried a radar that produced accurate three-dimensional profiles of all the landforms a scientist could possibly want to ogle.

Using this data, a group of researchers led by a Hungarian volcanologist named Dávid Karátson took another look at volcanic slope profiles. They started by picking out 19 highly symmetrical stratovolcanoes from around the Pacific rim. All these mountains turned out to have lower slopes that curve in a logarithmic manner, just as Milne had surmised. This the researchers ascribe to “a combination of different volcano-related mass transport processes on the lower flanks”. In other words, denudation and stuff.

But the upper slopes told a different story. Eight of the mountains looked like true cones, with ruler-straight upper skylines – the researchers called these C-type, for conical – while the rest had concave upper slopes. These were termed P-type, for parabolic. Yotei-zan in Hokkaido is a classic C-type volcano, while Mayon in the Philippines is a P-type.

Erosion cannot explain the upper-slope profiles, Karátson & Co believe. Instead, they suspect that the differing "eruption styles" of P- and C-type volcanoes may be responsible. Parabolic slopes form when volcanoes put out more lava than ash. (By what exact mechanism is left unexplained.) Conversely, straight-sided cones result when eruptions are explosive and ash-rich. Just in time for Milne's centenary, it looks as if the mystery of Mt Fuji's curve may be on the way to being solved.

But what happened to Mt Fuji in all this? Regrettably, the top Hyakumeizan didn't make it into Karátson's elite group of 11 "parabolic" volcanoes. Marred by an eighteenth-century flank eruption, its form was deemed too irregular for study. To its admirers, of course, it may be this very irregularity that transforms Mt Fuji into an image of perfection. But that raises a different kind of question altogether…

References

Milne, J (1878): On the Form of Volcanos, Geological Magazine (Decade II), 5: 337-345

Milne, J (1879): Further Notes upon the Form of Volcanos, Geological Magazine (Decade II) (1879), 6: 506-514

Lacey, A, Ockendon, J and Turcotte, D (1981): On the geometrical form of volcanoes, Earth and Planetary Science Letters, Volume 54, Issue 1, June

Davidson, J and De Silva, S (1999): Composite Volcanoes: Chapter 43 in The Encyclopedia of Volcanoes, Elsevier

Karátson, D, Favalli, M, Tarquini, S, Fornaciai, A, and Wörner, G (2010): The regular shape of stratovolcanoes: a DEM-based morphometrical approach, Journal of Volcanology and Geothermal Research, 193, 171–181. (And many thanks to Dávid Karátson for explaining some key points about this paper to Project Hyakumeizan.)

Photos: Portrait of John Milne from Wikipedia; photos of Fuji from The Volcanoes of Japan, Part 1, Fujisan by John Milne & W.K. Burton; Collotype Plates by K. Ogawa, ca 1892