Multi-alpha analysis of the mean-field Boussinesq equations

The quasilinear (QL) approximation, in which fluctuation/fluctuation nonlinearities are suppressed except where they feed back upon a suitably defined mean field, has a long history in fluid mechanics, dating back at least to the work of Stuart (1958) and Herring (1963, 1964) in their investigations of shear and convective turbulence. Interestingly, the Euler--Lagrange equations arising from variational analyses that yield rigorous bounds on the achievable momentum and heat transport in these flows have a similar QL form. Busse (1969, 1970, 1978) leveraged this mathematical structure to construct optimal momentum and heat transporting fields that include a hierarchy of horizontal wavenumbers. The resulting 'multi-alpha' optimal solutions exhibit nested boundary layers, reminiscent of Townsend's attached eddies. Here, we investigate the extent to which Busse's innovative multi-alpha asymptotic analysis can be adapted to obtain multiscale solutions of QL dynamical models. We pursue this investigation in the context of Rayleigh--Benard convection between stress-free isothermal boundaries, thereby extending an analysis by Howard (1965), who constructed steady single-wavenumber solutions to the mean-field (QL) Boussinesq equations in the large Rayleigh number limit.