For Hawaii, $H\propto\sqrt{A}$.

I looked at the islands on the Hawaiian-Emperor Seamount Chain with surface volcanoes, and plotted their surface area vs. their maximum elevation, using data from Wikipedia. I checked that the topography of each island is indeed dominated by the central volcano or volcanoes, although this is a bit complicated on some of the larger islands, as there are multiple active cones.

I left out a number of islands, atolls, seamounts, and guyots. The islands I used in my dataset are the main volcanic islands; the few extinct islands are extremely small. Seamounts and guyots are underwater, and therefore aren't helpful. Atolls, of course, are old and too eroded to give us any helpful data.

I then plotted the points logarithmically and fit a power-law model to the data, using this code by Aziz Alto. Power-laws are great in general, and are often the physicist's method of choice, so I figured it would be a decent starting point.

I found that $H\propto A^{0.54}$, or, essentially, $H\propto \sqrt{A}$, where $H$ is the maximum height of the volcano and $A$ is the area of the island. In particular, the best-fit equation I found was $$H=34.67\left(\frac{A}{\text{km}^2}\right)^{0.54}\text{ meters}$$ The $R^2$ statistic is rather good; $R^2=0.796$.

I doubt this relation has a physical meaning; I'm interpreting it as merely a statistical fit that will help you in your calculations and worldbuilding. Although the fact that the exponent is close to $1/2$ is nice, and maybe indicative of some physical law, I'm inclined to doubt it Maybe there's underlying physics; maybe there isn't. But I suppose it's a good enough solution, for your intents and purposes.

Here's the data again, with the best-fit model plotted:

For this chain, a good approximation is $H\propto\sqrt{A}$. Since the islands are roughly circular and $A\propto r^2$, $H\propto r$, which is something of an interesting result. It does seem to agree with a model of shield volcanoes as having straight sides of constant slope - although that is still a simplistic picture.

Other chains

I decided to test this model on other chains. In particular, you asked about the Canary Islands and the Azores. For the Canary Islands, I found a reasonably poor fit ($R^2=0.0375$) for a best-fit power-law model of $H\propto A^{0.13}$. The distribution of heights for the seven main islands appears to be somewhat random, which is unhelpful. A possible reason for the discrepancy could be that the Hawaiian-Emperor Seamount Chain is composed largely of shield volcanoes, and this trend may not apply to other types, like stratovolcanoes.

The Azores were interest, yielding $H\propto A^{0.28}$, with $R^2=0.368$. If we remove an outlier, Pico Island, we get $H\propto A^{0.20}$, with $R^2=0.541$. This is better than the Canary Islands, but still not as good as Hawaii. I do notice that all three of these exponents are lower than Hawaii's, which is interesting. I suspect it's partially because of the lack of a clear power-law trend, from what I've seen.

Notes

The Big Island of Hawaii does lower the power-law model a bit, but not by a huge amount (the exponent becomes $0.60$). It also lowers the proportionality constant somewhat.

posted over 6 years ago

HDE 226868‭

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