Turbulent transport in mixed states of convective cells and sheared flows

A thermodynamic state space description for mixed states of temperature-gradient-driven turbulent convection and sheared mass flows is derived by introducing a Markovianization of the small scale modes in a well-known low-order mode coupling model of the Rayleigh-Benard system. The mode coupling equations contain the Lorenz attractor on a submanifold and the shear-flow attractor in the full d=6 phase space. The three-dimensional thermodynamic state space follows the three physically distinct energy components: potential energy U, the turbulent energy W, and the kinetic energy F contained in the sheared mass flows. In contrast to weak turbulence theory for the wave-kinetic equation the closure presented here contains nonlinear functions of F and U in the triplet closure that are similar in structure to the Pade approximants of renormalized turbulence theory. The mathematical procedure is to determine a Markovian expansion parameter ΕM1 in the smallest scale modes and to systematically eliminate these damped modes in the limit the ΕM&rarr0.