Counterintuitive Results in Conduction Cooling Model

[PinkGeologist]

I am modeling the injection of hot magma sills into the Earth's crust at varying rates (and varying sill heights for each rate).

For instance, for a total of 16 km of magma ...
1a-n. 40, 400m high sills emplaced at rate of 5e-3 ma-1, 5e-4 ma-1, 1e-2 ma-1, , 2e-2 ma-1, 3e-2 ma-1, and 4e-2 ma-1
1b-n. 160, 100m high sills emplaced at the same set of rates
1c-n. 320 sills, 50 m high at each of the rates and ...
1d-n. 640 sills (25 m high) at each rate.

What you might expect is that at some rate, as sill size decreases, the hot zone will get larger (ostensibly because the magma retains more heat and cools more slowly when the time between sills is shorter). That is what I generally see.

What is strange is that for a few sets I got a results suggesting a much larger hot zone would results from say, 320, 50m sills than for 640, 25m sills (this happens at an injection rate of 1e-2 ma-1) or in another set that ... a larger hot zone is created with 160 100m sills than for 320 50m sills or for 640 25m sills emplaced at the same overall rate (2e-2 ma-1).

This is easier to understand when looking at the figures. You can see them here:
https://www.dropbox.com/s/3atk0p47sfvjb5h/Screen Shot 2015-04-20 at 2.54.59 PM.png?dl=0
https://www.dropbox.com/s/v7r6t41swis1ksd/Screen Shot 2015-04-20 at 2.54.37 PM.png?dl=0
https://www.dropbox.com/s/lwwxps2jznptmcq/Screen Shot 2015-04-20 at 2.54.24 PM.png?dl=0

Does anyone have a way to explain why this occurs?

 

[PinkGeologist]

I know there must be some kind of "sweet spot" where the repose time between sill-injections and the heat retention of the magma intercept to create a more robust hot zone, but I don't have a pinpoint on exactly why and how to articulate it.

 

[DrDu]

I have some doubts that this is the right forum. I think thermodynamics is covered by the classical physics forums and geology by the Earth forum. If you want it to be moved, contact the forum staff via the "report" button.
Maybe you could explain what your numbers mean, e.g., 40, 400m high sills, what does the 40 stand for?
If the graphics are for publication, maybe you should ascertain that the units are roman, not italic.

 

[PinkGeologist]

I apologize - when I say "40, 400m-high sills" that means there are 40 total sills, each of them 400 meters thick and emplaced at some rate (from 0.0005 to 0.04 meters per year).

I'll also look into moving the post.

 

[DrDu]

I think I more or less understand what you are talking about. So the graphics show the temperature distribution after magma intruded in the same time span but either in blocks of n sills or continuously. There are two competing effects. If the rate is very low and n is small, the temperature can equilibrate between the events and especially between the last event and the end of the time span. In contrast, even at arbitrary low rate, there will always be some hot magma in the last silt at the end of the intrusion period if the process is continuous. That seems to explain fig. 4a and e.
The other effect is that the bigger the sill, the longer it takes to cool down as its heat capacity increases but heat can only flow over the surface which changes little with volume.
I guess this two effects taken together might explain your findings.

 

[DrDu]

Did you consider some simple analytical models? I thought of replacing the sills by delta functions. For either slabs or point like sills, the heat equation has closed form solutions from which you could derive some intuition.

 

[PinkGeologist]

Well, the only analytical comparisons I've made is for cooling of a single sill ... I'm not sure how I'd implement a simple analytical solution to capture the dynamic situation of a deceleration of the cooling rate with the increased surrounding temperature as well as the different dull heights and cooling times ... I'm not well-versed in heat equation solutions to be honest; it took me longer than it should have to put the simple conduction + latent heat version into a Matlab script to get the cooling time for a body of X thickness. I'd love if someone had the patience to step me through a version that would assist me with that, but that's a pretty big request to make :-)