Bounding Heat Transfer in Internally Heated Convection

Rigorously bounding emergent properties of turbulent flows provides a means for improving our mathematical understanding of turbulence, while yielding results with real world applications. Using the background method formulated in terms of quadratic auxiliary functions we prove bounds on the mean convective heat transport < wT > in internally heated (IH) convection. Boundary driven Rayleigh-Bénard convection has seen extensive analysis with the background method, yet an extension of the same analysis to IH convection is incomplete. The change in mechanism driving convection presents a problem of importance in geophysical and astrophysical phenomena such as convection in the mantle or the Venusian atmosphere. This talk will demonstrate that Rayleigh dependent bounds on < wT > can be obtained, the scaling of which can be verified with numerical optimisation and shown to be optimal within our bounding framework.