Hello readers! After the success of my lava calculations last month, I have been inspired to devote an entire blog post to pulling apart the physical inaccuracies of fantasy worlds. It turns out that the water hydrants in Tears of the Kingdom are the least of its lava-based problems (if you missed it, see: Is lava wet? And other questions).
SPOILER WARNING: I will be showing pictures from the Death Mountain region in Tears of the Kingdom (TotK) and its predecessor, Breath of the Wild (BotW). I won’t mention anything in terms of story, but there may well be characters and shrines in the background of photos.
Introduction to the problem
For those that don’t know, TotK is set a few years after BotW. The time between games is never quantified, but Link doesn’t look that much older – which is helpful for us players, because it wouldn’t be much fun if Link had aged fifty years and succumbed to crippling arthritis. Still, in the few years between the games there have been some significant changes in Hyrule, one of which is that Death Mountain has stopped erupting. In BotW, it was spewing lava down into a huge lava lake, but in TotK, most of these flows have solidified.
You might see where this is going. If Death Mountain was swimming with lava in BotW, should this have solidified before the start of TotK? There are two ways of looking at this problem. We can calculate the time required for the Death Mountain lava lake to solidify (which gives us a measure of the time that passed between the games), or we can assume a relatively short time between games and establish the properties of the lava.
Setting up the problem
We’re going to make some big assumptions before tackling this. Firstly, let’s only think about conduction: we don’t want to overcomplicate things with convection in the lava or the air above it. Let’s also ignore the latent heat of fusion, which is the heat given out by the crystals that grow as the lava starts to solidify. We can also assume the lava lake is big enough that the heat loss around its edges is insignificant. As such, the time taken to cool a lava lake boils down to a simple 1D conduction problem (see the “Appendix” to this post for more info).
To solve this conduction problem, we’ll assume that the surface of the lava is held at air temperature, so the wind is carrying all the heat away. We’ll also assume that the rocks deep beneath the lava lake remain at a constant temperature.
We now need to estimate some key parameters. How deep is the lava lake? What is the composition (and therefore the conductivity) of the lava? What temperature is the lava to begin with? How hot are the rocks beneath the lava lake? To answer these, we need to do some research.
Death Mountain fieldwork
As with any video game volcano, Death Mountain laughs in the face of Earth Science. However, its lava has a very low viscosity (almost as runny as water) but I’m going to be forgiving and classify it as a basalt, giving it a thermal diffusivity of the order 10-6 m2 s-1. We’ll assume its starting temperature is around 1200 °C, similar to basaltic eruptions on Earth.
Unfortunately, the depth of the lava lake in BotW is unknown. I can’t hit the bottom by dropping items using ‘magnesis’, so the lake is at least 20 m deep – probably more. We can get a better idea of its depth from TotK, where we can drop into caves that have formed beneath its solidified surface. In fact, we find that the lava lake is still present underground, having dropped in height by 50 m (I’ll assume the in-game elevations are in metres, rather than some obscure Hyrule system).
One explanation for this drop in lava level is that the lake drained before it fully solidified. The lake must have been very deep initially, and judging by the depth of the caves, it developed a crust over 40 m thick. Some of the lava must have drained away, possibly in the ‘upheaval’ at the start of TotK (maybe down a chasm into the depths?) and so the lake level dropped beneath this crust. As such, we can assume that the lava lake is at least 50 m deep, probably more. We’ll use this value as a conservative lower bound.
Finally, we need to think about the temperature of the rocks beneath the lava lake. Our in-game thermometer breaks when we go underground, but the air is hot enough to ignite our wooden tools, so 500 °C seems a reasonable estimate. Don’t ask how Link is surviving in these conditions. The guy is just built different.
Running the numbers…
I have solved the conduction problem for two lava lake scenarios: one 50 m deep, and one 100 m deep. The 50-m-deep lake takes a little over five years to solidify entirely – which is reasonably similar to the time which has passed between the games. In fact, if we wanted to create the caves seen in TotK, we don’t want the lake to have entirely solidified; we want about 10 m of molten material to have remained, which can then drain away and leave a gap. It only takes 4.6 years to leave a molten region 10 m thick, which feels even more reasonable for the game timeline.
However, the 100-m-deep lake takes much longer to solidify. For full solidification, we require 21 years, and for a crust 40 m thick, we require 12-15 years. This is a little on the long side, considering the lack of ageing on the game characters, and the fact that Lady Impa is still alive and kicking.
Revisiting our assumptions
A 50-m-deep lake of basalt can solidify in around 5 years, which fits nicely with the game timeline. However, this was only a lower bound: the lake is likely to be much deeper, and so it likely took longer to cool. Most of our assumptions also act to reduce the cooling time. For example, we can see in BotW that the lava is flowing, and if the lake is constantly being replenished with hot, fresh, lava, it will take much longer to cool down. We also ignored the latent heat of fusion released when the rock solidifies, which greatly increases cooling times (see the reference list for more sophisticated cooling models). As such, I’m afraid we have to conclude that Death Mountain’s lava lake should not have cooled to the extent that is depicted in TotK.
That being said, we know that Death Mountain lava isn’t normal basalt. It is far too runny, for a start, as it can splash like water (see image below). If we assume that the lava is actually molten iron at 1600 °C, with a solidification point of around 1500 °C and a thermal diffusivity of 2.3×10-5 m2 s-1, a lake 100 m deep takes only 173 days to solidify! It might just be that the ‘lava’ coming out of Death Mountain isn’t ‘lava’ at all.
In summary…
Calculating the cooling time of the Death Mountain lava lake is difficult, as the problem is poorly constrained. We don’t know the depth of the lake, except that it is more than 50 m, and we don’t know the composition of the lava. If we assume that it is basalt, then the lake is unlikely to have solidified in the time between BotW and TotK – not that I’m going to be sending Nintendo any angry emails about this. However, if we take the time between games as our starting point (i.e., about five years), then we must conclude that the Death Mountain lava isn’t actually lava at all.
Hope you enjoyed this more technical post! If you spotted any glaring errors in my work, or have any requests for further game- or film-based calculations, feel free to let me know.
Appendix: thinking about conduction
Conduction is quite an intuitive process. Heat moves from hot regions to cold regions, and the bigger the temperature difference, the faster it flows. This is summed up in Fourier’s law, Q = k.dT, where the heat flux Q is the product of the conductivity k and the temperature difference dT. The conductivity is a property of the material; in fact, rocks and lava are very poor conductors, so a lava lake will take a long time to cool down.
Fourier’s law can be broken down into the heat equation: dT/dt = a.d2T/dx2 (and it is at this point that I wish I knew how to present equations in WordPress). This equation is still fairly intuitive when we examine each of its parts. The rate of temperature change dT/dt is partly a product of the thermal diffusivity a of the material (which is another way of expressing its conductivity), and on the rate of change in thermal gradient d2T/dx2. If dT/dx is just the thermal gradient (i.e., the difference in temperature with distance), then d2T/dx2 is the change in thermal gradient as we move through our material. Things will change temperature faster where the thermal gradient varies rapidly over shorter distances – for example, at the boundary between hot and cold regions.
The only way to solve the heat equation is using some sort of iterative numerical method. I solved the problem in MATLAB, and all my assumptions and parameters are outlined in the blog post. Safe to say, this was a major oversimplification. The omission of latent heat release on solidification will increase cooling times dramatically, as will the fact that the lava is flowing. The reference list contains some proper models, which show how complicated the problem can be!
Reference list:
Worster, M.G., Huppert, H.E. and Sparks, R.S.J. (1993) The crystallization of lava lakes. Journal of Geophysical Research: Solid Earth, 98(B9), 15891-15901.
Delaney, P.T. and Pollard, D.D. (1982) Solidification of basaltic magma during flow in a dike. American Journal of Science, 282(6), 856-885.
Carracedo, J.C., Troll, V.R., Day, J.M., Geiger, H., Aulinas, M., Soler, V., Deegan, F.M., Perez‐Torrado, F.J., Gisbert, G., Gazel, E. and Rodriguez‐Gonzalez, A. (2022) The 2021 eruption of the Cumbre Vieja volcanic ridge on La Palma, canary islands. Geology Today, 38(3), 94-107.
C. W. Clayton 