Theory of fully developed turbulence in buoyancy-driven fluids and pressure-gradient-driven plasmas

A new theoretical method is presented to analyse turbulence and associated transport in far-non-equilibrium fluids and plasmas. First, direct nonlinear interactions with background turbulence are renormalized into nonlinear dielectric form. The relation between the turbulent intensity spectra of energy and temperature, E(k) and E_θ(k), and the nonlinear transfer rates (dielectric) of momentum and energy, ν_N and κ_N, are obtained as recurrent formulae of integral equations. Second, nonlinear marginal stability conditions are examined by introduction of dressed test mode analysis. Solutions have a power law which are analogous to critical exponents in renormalization group theory. The same paradigm is first applied to neutral fluids to recover conventional results. For two-dimensional (2D) buoyancy-driven turbulence, where a supercritical turbulence appears, spectral forms of E(k), E_θ(k) ∝ |g·∇T|k^-3 and ν_N(k), κ_N(k) ∝ (|g·∇T|)^1/2k^-2 are obtained in the energy containing range (or subrange). (∇T is the temperature gradient and g is the gravity.) The relation between global turbulent transport coefficients, such as ν_T and κ_T, and nonlinear transfer rates ν_N and κ_N is obtained. A global spatial structure of the turbulent fluid, which is consistent with the spectrum, is solved. The Nusselt number is obtained as Nu ≃ 0.4(R_a/R_ac)^1/3 and the relations κ_T ∝ (R_a/R_ac)^4/9 and ν_T ∝ (R_a/R_ac)^4/9 are obtained. (R_a is the Rayleigh number and R_ac is the critical Rayleigh number.) To turbulence in the magnetized plasma, where a subcritical turbulence appears, this paradigm is applied. The combination of the pressure gradient (∇p) and magnetic field gradient (Ω′), G₀ = ∇p·Ω′, characterizes the non-equilibrium form of the plasma. The spectral intensity of the fluctuating fields forthe potential, current and pressure E(k), E_J(k), Eθ(k) ∝ G₀k^-3 and the nonlinear transfer rates μ_N(k), λ_N(k), χ_N(k) ∝ (G₀)^1/2k^-2 are first obtained (symbols μ, λ and χ correspond to the ion viscosity, current diffusivity and heat diffusivity, respectively) in the energy containing range. The turbulence level, W, and the transport coefficients, χ_T, are derived as W ∝ G₀², and χ_T ∝ G₀^3/2. The dissipation balance is also examined. These analyses demonstrate that this method is applicable to both the supercritical and subcritical turbulences in neutral fluid and plasma turbulences, i.e. in systems which are far from the thermal equilibrium.