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What proportion of Earth's internal heat is generated by tidal flexion as a result of the orbiting moon, and would it have been significantly greater four billion years ago when the moon was closer? Has it ever been enough to affect surface temperature to an easily measurable extent?
note: This is not a duplicate of the following because there are no clear, quantitative answers specifically about heat generated by tidal flexion. Instead those answers focus on heat produced in the oceans.
The complications of calculating the moon's contributions to earth's heat budget are touched on in Emergence of a Habitable Planet (2007) by Zahnle, et.al. in Space Science Reviews, a paper trying to determine when in earth's history the planet's surface would become cool enough to support life. The portion of the paper that relates to tidal heating suggests that the early moon contributed substantially to the interior heat of the earth system for about 1.2 million years, but as the moon's orbit widened, the earth cooled and solidified, and water condensed into oceans, the moon's contributions gradually became minimal.
Viscous damping of tidal motions generates heat. Therefore tidal heating occurs most strongly in materials that are solid but close to melting. This introduces the possibility of a governing feedback that works through the dependence of viscosity on temperature. If tidal dissipation exceeds what the atmosphere can radiate, the excess heat raises the temperature, which lowers the viscosity, which in turn lowers the rate of tidal dissipation. This looks like a stable feedback. It follows that, while tidal dissipation was important, the base of the mantle was solid but the rest of it was fluid, and tidal heating generated almost all of the thermal energy radiated to space. In the limit of an asymptotically thick steam atmosphere, tidal dissipation would have been regulated to generating heat at the runaway greenhouse limit of ∼140 W/m2.
In a caption for a figure the authors also note:
Tidal heating plays an important role in prolonging the magma ocean. Tidal forcing wanes as the Moon evolves away from the Earth. Thereafter heat flow is controlled by convection of the solid mantle. By 4.4 Ga the global average heat flow would have been ∼0.5 W/m2. Later in the Hadean typical heat flows would have been 0.2–0.3 W/m2, not enormously larger than what they are now. For comparison heat flow today is 0.065 W/m2 through the continents and 0.1 W/m2 through the ocean crust.
Jeffronicus's answer appears to be a good one re the influence of lunar tidal heating on the early Earth. To answer the other question that has emerged in comments, as to what proportion of total heating is from tidal effects now, we can turn to Munk & Wunsch 1998. They tell us that the lunar tidal power going into earth tides (as opposed to into the sea or atmosphere) is about 170 GW.
The same paper reports that the total heat flow from the earth's interior to the surface is about 30 TW, so the proportion that comes from the moon's influence now is perhaps around half a percent.
