Gaussian curvature and the relationship between the shape and the deformation of the Tonga slab

SUMMARY We investigate a particular potential cause of deformation within the subducting Tonga slab: that associated with material that moves over a template while remaining in contact with it. In such a situation both the location and the style of deformation within the material depend, in a predictable way, on the shape of the template, and in particular on its Gaussian curvature. We look for such an association in the Tonga slab, using earthquake locations to define the slab shape and their focal mechanisms to indicate the style of deformation. Only in one place, at 25°S and 500-600 km depth, does the style of the faulting in the earthquakes demonstrably correspond with that required by the Gaussian curvature if the slab were moving over a template. Although the Gaussian curvature in other parts of the slab, particularly near the ‘hook’ at its northern end, would also require deformation if the slab were moving over a template, the pattern of earthquake mechanisms were in those places is not clear enough to confirm the association. Although we are limited by our ability to resolve only the coarsest features (≳ 300 km) of the slab shape, we reach the important conclusion that deformation in response to motion over a template is not the main cause of the intermediate and deep seismicity in the Tonga slab. Most of the earthquakes, and all the biggest ones, occur where, even if the slab were moving over a template, it would not need to deform to do so. Some other explanation is required for these earthquakes.