Steady but generally unstable solutions of the 2D Boussinesq equations are obtained for no-slip boundary conditions and Prandtl number 7. The primary solution that bifurcates from the conduction state at Rayleigh number Ra ≈ 1708 has been calculated up to Ra ≈ 5.10 6 and its Nusselt number is N u ∼ 0.143 Ra 0.28 with a delicate spiral structure in the temperature field. Another solution that maximizes N u over the horizontal wavenumber has been calculated up to Ra = 10 9 and scales as N u ∼ 0.115 Ra 0.31 for 10 7 < Ra ≤ 10 9 , quite similar to 3D turbulent data that show N u ∼ 0.105 Ra 0.31 in that range. The optimum solution is a simple yet multi-scale coherent solution whose horizontal wavenumber scales as 0.133 Ra 0.217 . That solution is unstable to larger scale perturbations and in particular to mean shear flows, yet it appears to be relevant as a backbone for turbulent solutions, possibly setting the scale, strength and spacing of elemental plumes.

The system of equations describing the shallow-water limit dynamics of the interface between two layers of immiscible fluids of different densities is formulated. The flow is bounded by horizontal top and bottom walls. The resulting equations are of mixed type: hyperbolic when the shear is weak and the behavior of the system is internal-wave like, and elliptic for strong shear. This ellipticity, or ill-posedness is shown to be a manifestation of large-scale shear instability. This paper gives sharp nonlinear stability conditions for this nonlinear system of equations. For initial data that are initially hyperbolic, two different types of evolution may occur: the system may remain hyperbolic up to internal wave breaking, or it may become elliptic prior to wave breaking. Using simple waves that give a priori bounds on the solutions, we are able to characterize the condition preventing the second behavior, thus providing a long-time well-posedness, or nonlinear stability result. Our formulation also provides a systematic way to pass to the Boussinesq limit, whereby the density differences affect buoyancy but not momentum, and to recover the result that shear instability cannot occur from hyperbolic initial data in that case.

Please cite only the published version using the reference above.See http://opus.bath.ac.uk/ for usage policies.Please scroll down to view the document. A Model for Strongly Nonlinear Long Interfacial Waves with Background Shear By Anakewit Boonkasame and Paul A. MilewskiThe Miyata-Choi-Camassa (MCC) system of equations describing long internal nonhydrostatic and nonlinear waves at the interface between two layers of inviscid fluids of different densities bounded by top and bottom walls is mathematically ill-posed despite the fact that physically stable internal waves are observed matching closely those of MCC. A regularization to the MCC equations that yields a computationally simple well-posed system for time-dependent evolution is proposed here. The regularization is accomplished by keeping the full hyperbolic part of MCC and exchanging spatial and temporal derivatives in one of the linearized dispersive terms. Solitary waves of MCC over a wide range of parameters are used as a benchmark to check the accuracy of the model. Our model includes the possibility of a background shear, and we show that, contrary to the no shear case, solitary waves can cross the midlevel between the top and the bottom walls and may have different polarity from the case with no background shear. Time-dependent solutions of the regularization stable model are presented, including interactions of its solitary waves, and classical and modified Korteweg-de Vries equations for small amplitude waves with the inclusion of background shear are derived. Throughout the paper, the Boussinesq approximation is taken, although the results can be extended to the non-Boussinesq case.