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The Earth is emitting about 47 terawatts of heat energy, producing a heat flow of 0.0921 W/m$^2$ due to radiogenic and primordial heat energy. The heat flow drives mantle convection which in turn drives plate movement. The radiogenic energy declines as the radioisotopes decay and the primordial energy is not replenished; presumably the sum can be approximated as roughly a power-law $\sim (t-t_0)^{-\alpha}$.

At what heat flow will mantle convection become too weak to maintain plate movement and Earth will shift to a stagnant lid mode?

I am interested in estimating the threshold heat flow needed to maintain plate tectonics on an Earthlike world. So we can ignore the sun going red giant in 5-6 Gyr. Another complicating factor is of course water "lubrication" which might change the threshold, presumably reducing it based on past subduction. But for my purposes (an astrobiological argument) a rough estimate is good enough if it can be supported.

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    $\begingroup$ I would say that the answer to this question is very poorly constrained. We don't fully understand why the Earth has plate tectonics as opposed to a stagnant lid situation like Venus. It is also not obviously correct to suggest that a stagnant lid is a low temperature state, since Venus' internal temperature is much higher than that of the Earth (precisely because it has a stagnant lid). So in short I think this is still an open research question that doesn't have a definitive answer. $\endgroup$ – bon Mar 13 '18 at 12:08
  • $\begingroup$ There are rough astronomical estimates usually based on planet size and mass, and, as you mention, oceans. I've never seen an estimate based on heat-flow. Also, about half (very ballpark) of Earths internal heat is from radioactive decay and half is the heat of formation and that ratio would vary with the mass of the planet and composition of the core (a fluid transfers heat more efficiently than a solid). I would guess that the amount of granite on the surface would be a factor too so there's a lot of variables and as @bon points out, lots of unknowns. I like his Venus example too. $\endgroup$ – userLTK Mar 13 '18 at 14:07
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    $\begingroup$ @bon I am amazed by Venus having more internal heat than Earth, but not having significant tectonics. This summons questions... $\endgroup$ – thymaro Mar 17 '18 at 9:59
  • $\begingroup$ @thymaro Yes it's a very interesting question. It's almost certainly related to the fact that Venus has much less water than Earth but whether it had any water in the first place and if so how it lost it is still an open question. $\endgroup$ – bon Mar 17 '18 at 10:59
  • $\begingroup$ @bon I was thinking of opening "How much geothermal energy can we extract from Earth?", but then found there are already answers on the site. We'll kill ourselves off with other means, before we get anywhere near a point where it could be a relevant question. $\endgroup$ – thymaro Mar 17 '18 at 11:33

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