Many planetary objects rotate rapidly and  experience persistent perturbations so that the flow is not uniform rotation (e.g precessional and/or tidal forces cause distortions of the rotating flow). If this distortion is large enough the flow can become unstable and ultimately a turbulent state can arise. Appreciating exactly what flow response is generated is  important for understanding many geophysical and astrophysical processes (e.g. the geodynamo and the subterranean ocean on Enceladus - Wilson & Kerswell, EPSL, 2018). The initial instability can be understood as the excitation of normal modes - variously called inertial waves or Poincare modes - on top of uniform rotation. Once they reach finite but still small amplitude, a weak wave turbulence approximation is then applicable. Current work is looking at how to use this approximation to make predictions about the flow response. 

Variational methods provide a unique approach to derive rigorous inequality information on turbulence in the form of bounds on key global properties such as heat flux in convection, momentum flux in shear flow or mass flux in a pressured-driven flow. The hope is that these bounds at least capture the correct asymptotic scaling with the parameter(s) of the problem (e.g. the heat flux with the Rayleigh number scaling in Boussinesq convection as in figure) and at best also produce relevant flow fields. Recently we have been exploring efficient ways to solve the Euler-Lagrange equations which emerge from these problems (e.g. Wen et al. Phys. Rev. E  2015) and pushing known techniques such as the Background method to the limits (e.g. Ding & Kerswell J. Fluid Mech. 2020). One goal of this approach is to establish a bridge between actual solutions of the underlying governing equations and a variational problem which maximises some key quantities subject to just a subset of the dynamical constraints.