Re: How does the thickness of the earth's mantel effect the weather?

Your question is a good one, but there are several incorrect assumptions inherent in it.

1. The Earth's core is not constantly cooling. The famous physicist Lord Kelvin fell into this trap when he assumed that the heat in the Earth's interior was the remnant of the formation of the planet, and thus concluded that the Earth was no more than 20 million years old. We now know this number is too small by a factor of 300. He went wrong because he didn't know about the heat constantly generated by radioactive elements in the core, which maintains the core temperature over billions of years. So you're in very good company in making this error.
2. The mantle does not act as a traditional insulator, in which the insulation strength (R-value) is proportional to its thickness. This is because the mantle is constantly convecting (i.e., flowing in an overturning motion as hot rock rises and cold rock sinks), and the movement of this rock carries heat efficiently from the core to the crust. You should think of the mantle as being like a pot of molasses on a stove, rather than a sheet of styrofoam.
3. In any case, the thickness of the mantle is not significantly different under the continents versus the oceans. What does differ is the thickness of the crust. Unlike the mantle, the crust does not flow, so it does act like a traditional insulator.
4. Thus, there are significant differences in the amount of heat flowing out through the surface of the Earth, but they're due to differences in the thickness of the crust, not the mantle. However, this heat amounts to no more than one or two hundred milliwatts per square meter. This is a miniscule fraction of the heat provided by the sun, which averages several hundred watts per square meter.

On the other hand, it is possible that geothermal heat could have a significant impact on the circulation of the cold darkness of the deepest parts of the ocean. Here, the influence of the Sun is much, much weaker. Two scientists here at MIT (Jeff Scott and Alistair Adcroft) are looking into this. I'm not sure if they've published their work yet.

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