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Abstract The determination of the temperature in and above the slab in subduction zones, using models where the top of the slab is precisely known, is important to test hypotheses regarding the causes of arc volcanism and intermediate-depth seismicity. While 2D and 3D models can predict the thermal structure with high precision for fixed slab geometries, a number of regions are characterized by relatively large geometrical changes over time. Examples include the flat slab segments in South America that evolved from more steeply dipping geometries to the present day flat slab geometry. We devise, implement, and test a numerical approach to model the thermal evolution of a subduction zone with prescribed changes in slab geometry over time. Our numerical model approximates the subduction zone geometry by employing time dependent deformation of a Bézier spline that is used as the slab interface in a finite element discretization of the Stokes and heat equations. We implement the numerical model using the FEniCS open source finite element suite and describe the means by which we compute approximations of the subduction zone velocity, temperature, and pressure fields. We compute and compare the 3D time evolving numerical model with its 2D analogy at cross-sections for slabs that evolve to the present-day structure of a flat segment of the subducting Nazca plate. 

Abstract Molnar and England (1990,https://doi.org/10.1029/JB095iB04p04833) introduced equations using a semianalytical approach that approximate the thermal structure of the forearc regions in subduction zones. A detailed new comparison with high‐resolution finite element models shows that the original equations provide robust predictions and can be improved by a few modifications that follow from the theoretical derivation. The updated approximate equations are shown to be quite accurate for a straight‐dipping slab that is warmed by heat flowing from its base and by shear heating at its top. The approximation of radiogenic heating in the crust of the overriding plate is less accurate but the overall effect of this heating mode is small. It is shown that the previous and updated approximate equations become increasingly inaccurate with decreasing thermal parameter and increasing variability of slab dip. It is also shown that the approximate equations cannot be extrapolated accurately past the brittle‐ductile transition. Conclusions in a recent paper (Kohn et al., 2018,https://doi.org/10.1073/pnas.1809962115) that modest amount of shear heating can explain the thermal conditions of past subduction from the exhumed metamorphic rock record are invalid due to a number of compounding errors in the application of the Molnar and England (1990,https://doi.org/10.1029/JB095iB04p04833) equations past the brittle‐ductile transition. The use of the improved approximate equations is highly recommended provided their limitations are taken into account. For subduction zones with variable dip and/or low thermal parameter finite element modeling is recommended.