Why is heat flow over a mid-ocean ridge higher than over an oceanic trench?

To understand the answer to this question we must first consider the dynamics and structure of each of these plate margins. 

Oceanic trenches form at subduction zones. At a subduction zone dense oceanic crust is subducting beneath less dense continental crust. This oceanic crust is dense because it is old. As this crust is old, in many cases upto 200million years old, it has had a long time to cool and thus has become very cold. This is why the heat flow over oceanic trenches is so low. 

Interestingly, it is for this reason that subduction occurs in the first place. Old, cold oceanic crust reaches a threshold of density at which point it begins to subduct beneath the less dense, warmer continental crust.

Now we will consider mid-ocean ridges. At mid-ocean ridges two oceanic plates are diverging from one another. This effectively leaves a void within the Earth. Obviously, on a planet governed by gravity this is not a stable situation and this void must be infilled.

This void is infilled by asthenospheric mantle. The asthenosphere, the layer of the mantle upon which tectonic plates rest, is rheid (can slowly flow over geological time) due to a combination of its temperature and composition.

This asthenospheric mantle upwells into the void formed by diverging plates. As the source of this material is deeper within the Earth it is much warmer, thus produces a signature of high heat flow. 

Volcanism also occurs at MOR's unlike oceanic trenches. This contributes to high heat flow also. Volcanism occurs as asthenosphere undergoes decompression melting. This is a process whereby a package of hot material melts as it is brought towards the surface and decompresses.

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