## Saturday, February 12, 2011

### Fun with numbers!

I was having some fun over at The Land the other week with one of the denizens claiming undersea volcanoes are responsible for the melting of the Arctic ice cap. While he wanted to rant about conspiracies and all that he does raise an interesting point which has made me think, just how much heat (and of course energy) do volcanoes need to add to the oceans to actually make any difference? So I figured I might do some calculations, there are a number of assumptions I will make and I’ll be quite open about them, I will however be generally erring on the conservative side.

It’s going to get a little complex but none of the calculations are all that difficult, grab a bit of paper and a pen.

We’ll assume a rise in average ocean temperature of 1 degree over 100years, assuming that this is constant throughout the water column (if it were due to volcanic activity the difference would be higher the lower you go as that’s where the heat source is but to make it easy I’ll stick with a one degree warming throughout the column), so how much energy does this take?

From basic physics we know that it takes 4.2 Joules of energy to raise the temperature of one gram of water by one degree Celsius (I’m assuming fresh water, I do know it’s salt water but the difference in the end will only be minor), from a quick Google search (http://hypertextbook.com/facts/2001/SyedQadri.shtml) we find that the worlds ocean volume is around 1.3-1.4 billion cubic kilometres.

Assuming one gram of water equals one millilitre (yes, I do know the density will be a little different with salt water but in the end it really won’t make much difference), one cubic metre will equal one million millilitres (1 x 106 ml) , one cubic kilometre will equal one million billion millilitres (1x 1015 ml) and will require 4.2 x 1015 Joules to warm it by one degree, so the whole ocean will require 4.2 x 1015 x 1.3 x 109 or 5.46 x 1024 Joules of energy to raise it by one degree (this does make a very, very big assumption that there’s no energy lost from the system, counting in energy loss this figure would be much higher).

That’s a lot of joules, so what does this equate to? Another Google search on “world energy consumption” tells us the average annual energy consumption is roughly 5 x 1020 Joules (my search said 4.74 x 1020 but I rounded it up just for ease of calculation) so dividing or 5.46 x 1024 by 5 x 1020 gives us 10920 years of the worlds energy use would raise our oceans by 1 degree Celsius.
So what? I hear you quite rightly say, “the energy comes from cooling magma not from burning coal” so what we need to know are the melting point of magma and the specific heat capacity (specific heat capacity is how much energy it takes to raise one gram of a substance by one degree Celsius).

Another Google search on “magma melting temperature” gives an answer of between 700 and 1300 degrees to melt it depending on the makeup of the magma, in this case for ease of calculation we’ll pick a mid point of 1000 degrees. As for specific heat capacity, granite has a specific heat capacity of 790J/kg/degree or 0.79J/g/degree (we’ll round it up to 0.8J/g/degree for easier calculations, don’t worry this works out as more heat released for less weight which helps out the conservative nature of the calculations).

Time for a breather? Just to reiterate, we’ve figured out how much energy it would take to warm the world’s oceans by one degree Celsius, what we’re doing now is figuring out how much volcanic magma we would need to heat it by that much, as I’ve said, I’ve made a number of assumptions but they should result in a fairly conservative answer.

Ok, the amount of energy going into melting something should be the same as that released when it solidifies and cools so to raise 1 gram of granite to melting point of around 1000 degrees from a starting temperature of 50 degrees (it’s a lot hotter at the bottom of the oceanic crust than it is at the top so I figured 50 degrees would be a reasonable average), would require 950 (the temperature difference) multiplied by 0.8 (the specific heat capacity) or 760 Joules, that would also be released as it cools and solidifies.

So, what does that mean? Fair question, how many tonnes of magma would need to cool from 1000 degrees to 50 degrees to release that sort of energy?

Taking our 5.46 x 1024 Joules of energy needed divided by 760 gives us 7.2 x 1021 grams or 7.2 x 1015 Tonnes. Granite has a density of around 2.7T/m3 and basalt around 3T/m3, for ease of calculations I’ll go with the basalt density which gives 2.4 x 1015 cubic metres or 2.4 million cubic kilometres of magma needing to be produced by the worlds mid ocean ridges, volcanoes etc over 100 years to raise the oceans temperatures by 1 degree celcius (remember I’m assuming no loss, the actual number would be much higher).

As this mostly happens in the oceans and the oceanic crust is between 7 and 10km thick, we’ll assume a thickness of 10km giving 240 000 square kilometres of oceanic crust formed in 100 years.
Actually, it isn’t, that assumes all the action happens at the mid ocean ridges where the crust is formed and nothing at the subduction zones so I’ll halve it assuming (quite reasonably I think) that there’s as much energy released at the subduction zone as at the mid ocean ridges, so we’re looking for 120 000 square kilometres of new ocean crust formed in 100 years (remember, this is a conservative estimate, the actual number would be much more). Again using Google there’s around 80 000km of mid ocean ridge which means they’d have to widen by 1.5km in a century, or again being nice 750 metres either side meaning the continents would have to move at least 7.5 metres a year this is around 150 times faster than they are currently moving and to move 150 times faster you’d need 22 500 times the energy (remembering basic physics F=MV2).

These figures are very much in the lower end of the ballpark so if there is an influence from oceanic volcanoes on sea temperature and climate change we can see that it is very, very little.