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

S. I. Itoh, K. Itoh

Research output: Contribution to journalArticle

6 Citations (Scopus)

Abstract

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 Eθ(k), and the nonlinear transfer rates (dielectric) of momentum and energy, νN and κN, 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), Eθ(k) ∝ |g·∇T|k-3 and νN(k), κN(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 νT and κT, and nonlinear transfer rates νN and κN 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(Ra/Rac)1/3 and the relations κT ∝ (Ra/Rac)4/9 and νT ∝ (Ra/Rac)4/9 are obtained. (Ra is the Rayleigh number and Rac 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), EJ(k), Eθ(k) ∝ G0k-3 and the nonlinear transfer rates μN(k), λN(k), χN(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, χT, are derived as W ∝ G02, and χT ∝ G03/2. 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.

Original languageEnglish
Pages (from-to)1729-1766
Number of pages38
JournalPlasma Physics and Controlled Fusion
Volume40
Issue number10
DOIs
Publication statusPublished - Dec 1 1998

All Science Journal Classification (ASJC) codes

  • Nuclear Energy and Engineering
  • Condensed Matter Physics

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