If I told you that to determine the size of a volcanic eruption all you needed were three important measurements, you’d most likely think I’d lost the plot. Well, there’s a little more to it – aka some amazingly simple maths, but that is exactly how it’s done.
Suppose you were given the daunting task of trying to predict the blast radius of a volcanic eruption: where would you start? Perhaps some experiments are in order. If you could measure the size of several different eruptions from volcanoes of different shapes, sizes and strengths then maybe you could spot a pattern. It’s actually a pretty good idea, and for most situations a good starting point, but volcanic eruptions are pretty rare. Like 1 per year for the past 10,000 years rare (there’s an amazing interactive tool here that I highly recommend you play with). So, let’s try something a little different…
A volcano erupts due to a build-up in pressure which is suddenly released in a large explosion. Now, for better or worse, as humans we happen to be experts in the field of explosions. We’ve studied them for centuries and have data available from thousands of experiments to tell us what the most important factors are when it comes to determining the size of a blast.
Number 1 – time. The longer the time after the explosion, the further the fireball will have travelled. Number 2 – energy. Perhaps as expected, increasing the energy of the explosion leads to an increased fireball radius. The third and final variable was a little less obvious – air density. For a higher air density, the resultant fireball is smaller. If you think of density as how ‘thick’ the air feels, then a higher air density will slow down the fireball faster and therefore cause it to stop at a shorter distance.
Now, the exact relationship between these three variables, time t, energy E, density p, and the radius r of the explosion, happens to be a highly classified military secret – or at least it used to be before British mathematician G. I. Taylor came along. His ingenious approach was to use the method of scaling analysis. For the three variables identified as having an important effect on the blast radius, we have the following units:
- Time = [T]
- Energy = [M L2 T-2]
- Density = [M L-3]
where T represents time in seconds, M represents mass in kilograms and L represents distance in metres. The quantity that we want to work out – the radius of the explosion – also has units of length L in metres. Taylor’s idea was to simply multiply the units of the three variables together in such a way that he obtained an answer with units of length L. Since there is only one way to do this using the three given variables, the answer must tell you exactly how the fireball radius depends on these parameters! It may sound like magic, but let’s give it a go and see how we get on.
To eliminate M, we must divide energy by density (this is the only way to do this):
Now to eliminate T, we must multiply by time squared (again this is the only option without changing the two variables we have already used):
And finally, taking the whole equation to the power of 1/5 we get an answer with units equal to length L:
This gives the final result that can be used to calculate the radius r of the fireball created by an explosion:
And that’s it! A simple equation for the radius of an explosion which can be applied to an erupting volcano. All that we need to know is the energy – or pressure stored within the volcano, the time after the eruption and the density of the air and we have our prediction of the eruption radius.
Of course, this is only an estimate and much more complicated models are required to accurately monitor and control eruption zones, but I wanted to share it with you to demonstrate the awesome power of the technique of scaling analysis. It’s something you most likely are yet to see at school, and yet it is an incredibly simple and powerful tool in higher level study of maths and science at university. For many problems the equations will be too difficult for you to solve exactly and so instead you have to rely on estimation techniques such as this to be able to gain some insight into the correct solution.
If you’re yet to be convinced just how amazing scaling analysis is, check out an article here explaining the use of scaling analysis in my PhD thesis on river outflows into the ocean.
And if that doesn’t do it, then I wish you the best of luck waiting for that next volcanic eruption…
Competition 2.4: Exploring scaling analysis
Dr Tom Crawford (article author) specialises in Applied Maths and completed his PhD in Fluid Dynamics at the University of Cambridge under the supervision of Prof. Paul Linden. He obtained his undergraduate degree in Maths from Oxford in 2012 where he studied at St John’s College. Alongside his teaching commitments, Tom works closely with the outreach team at St Edmund Hall and regularly gives talks in schools and universities across the UK. understandable to all.
Professor Stuart White (competition author) is a Tutorial Fellow in Mathematics at St John’s College. He teaches tutorials for a range of predominately pure mathematics courses in first and second year, and supervises DPhil students on topics related to his research interests. His research focuses on operator algebras, a branch of functional analysis with connections to many other branches of pure mathematics.