Hitching the subcritical branch of convection

Thermal convection due to temperature differences in fluids drives multiple processes from planetary and stellar interiors, to the casting of metals and heat extraction. In its simplest form, convection sets in when the ratio of buoyancy to viscous forces, the Rayleigh number Ra, exceeds the critical value Rac at which an infinitesimal perturbation to the non-convective equilibrium becomes amplified by the flow dynamics. In more complex examples, subcritical convection may exist, i.e. for Ra below Rac. This not only happens in the ducts of heat exchangers, but also was recently discovered during the continuous casting of alloys, where it could lead to unwanted segregation defects in solidified alloys. Recently, subcritical convection also appeared in numerical simulations of planetary interiors, where a hot solid core is surrounded by a colder liquid metal extracting heat from it through a complex convective process controlled by the interplay between buoyancy, the Coriolis force induced by the planet's rotation and the Lorentz force due to its magnetic field.

In all three cases, the subcritical nature of convection is crucial: igniting convection below Rac significantly enhances heat transfer: this is the holy grail of cooling applications and one of the technological deadlocks in the design of nuclear fusion reactors. The challenge is to perturb a non-convective flow into a subcritical convective state. Conversely, defects incurred by subcritical convection must be avoided in continuous casting. Convection is also the beating heart of planets, driving amongst other processes, the dynamo action that sustains their magnetic field. An excursion away from a potentially subcritical convective state could shut down convection in planetary cores, one of the ways planets may "die".

In these problems the central questions are "how far below criticality can convection exist ?" and "what perturbation either ignites or extinguishes subcritical convection ?". Furthermore, whether subcritical convection even subsists in the presence of planetary magnetic fields is not even known. Straight simulations of the governing equations cannot answer these questions because they cannot reliably tell if convection is stable. Continuation methods can capture convective states regardless of their stability, but do not directly apply as reaching or leaving the convective state requires a discontinuous 'jump', as sought here.

This project will answer these mathematical questions in all three examples, by taking advantage of recent developments in stability theory. For the first question, exact solutions on disconnected branches will be captured from either simulations or distant states by adapting the hook step and Time Delay Control methods currently used to study the transition to turbulence in shear flows. These states can then be traced back to the origin of the subcritical branch using continuation methods. For the second, we will use perturbations with optimal transient energy growth to destabilise the non-convective equilibrium into the subcritical branch (or the reverse) and find paths to the extinction or the ignition of convection.
While the importance of subcritical convection in geophysical and casting problems only came to light very recently, so did the techniques to elucidate its true role. So too did the opportunity to exploit them in industry, as metallurgists increasingly turn to rigorous mathematics to control their processes. Ongoing collaboration with metallurgists and this work's relevance to nuclear fusion reactors offer direct opportunities for these new methods to start replacing current trial-and-error practice in design by tailored optimisation methods in these industries and potentially others. To this end, we will implement these methods into an open-source numerical package capable of finding or igniting the full range of subcritical convective flows in the widest possible range of problems.