H. Tasso

Renormalization group in magnetohydrodynamic turbulence

The renormalization group (RNG) theory is applied to magnetohydrodynamic (MHD) equations written in Elsasser variables, as done by Yakhot and Orszag for Navier–Stokes equations. As a result, a system of coupled nonlinear differential equations for the ‘‘effective’’ or turbulent ‘‘viscosities’’ is obtained. Without solving this system, it is possible to prove their exponential behavior at the ‘‘fixed point’’ and also determine the effective viscosity and resistivity. Strictly speaking, the results do not allow negative effective viscosity or resistivity, but in certain cases the effective resistivity can be continued to negative values, but not the effective viscosity. In other cases, the system tends to zero effective viscosity or resistivity. The range of possible values of the turbulent Prandtl number is also determined; the system tends to different values of this number, depending on the initial values of the viscosity and resistivity and the way the system is excited.

Hamiltonian Formulation of Two-Dimensional Gyroviscous MHD

Gyroviscous MHD in two dimensions is shown to be a Hamiltonian field theory in terms of a non-canonical Poisson bracket. This bracket is of the Lie-Poisson type, but possesses an unfamiliar inner Lie algebra. Analysis of this algebra motivates a transformation that enables a Clebsch-like potential decomposition that makes Lagrangian and canonical Hamiltonian formulations possible.

Three‐dimensional toroidal magnetohydrodynamics equilibria with ‘‘canal’’ surface currents

Three‐dimensional magnetohydrodynamics equilibria with surface currents on a toroidal ‘‘canal’’ surface are constructed. Inside the surface the pressure is constant and the magnetic field vanishes. The vacuum magnetic field lines on the surface are closed in the poloidal section which corresponds to infinite rotational transform.

On the Spectra of Turbulent Fluids at Large k

Abstract The turbulence spectra of continuous fluids at large k are discussed. A basic difference in behaviour appears between conservative and dissipative systems. Only in the latter case can one eliminate higher order ultraviolet divergences.

Energy principle for dissipative two-fluid plasmas

Abstract An energy principle for the dissipative two-fluid theory in Lagrangian form is given. It represents a necessary and sufficient condition for stability allowing the use of test functions. It is exact for two-dimensional disturbances but is still correct in terms of perturbation theory for long wavelengths along the magnetic field. This may well find application in tokamak plasmas. A discussion of the general case and its relation to the stability of flows in hydrodynamics is given. This energy principle may be applied for estimating the magnitude of residual tearing modes in tokamaks.

Self-similar statistics in MHD turbulence

Abstract The fully developed decaying turbulence of 3-d resistive, viscous, incompressible magnetohydrodynamics is investigated using Elsasser variables and Hopf equation for probability distributions. The method is an extension of a previous work for Navier Stokes equations done by Foias et al. based on a suggestion by Hopf. It uses essentially self-similar properties of the statistics, which almost determine the turbulence spectrum up to a mild assumption on an unknown function. This spectrum is the well known Kolmogorov spectrum

On the integrability of an attracting system of nonlinear oscillators

The problem of the integrability of a peculiar system of nonlinear oscillators is considered. While the case of two oscillators is integrable, the case of many is open. The Lax pair method is not applicable to such a system.

Hamiltonian Formulation and Statistics of an Attracting System of Nonlinear Oscillators

An attracting system of r nonlinear oscillators of an extended van der Pol type is investigated with respect to Hamiltonian formulation. The case of r = 2 is rather simple, though nontrivial. For r > 2 the tests with Jacobi’s identity and Frechet derivatives are negative if Hamiltonians in the natural variables are looked for. Independently, a Liouville theorem is proved and equilibrium statistics is made possible, which leads to a Gaussian distribution in the natural variables.

On Phase Space and Statistics of Continua

Problems in introducing suitable phase space and statistics occur for continua and degenerate discrete systems. The solution of these problems for the Korteweg-de Vries equation is discussed. The classical removal of the ultraviolet catastrophe in this case is contrasted with Planck’s black-body radiation spectrum.

Generalized Hamiltonians, Functional Integration and Statistics of Continuous Fluids and Plasmas

Generalized Hamiltonian formalism including generalized Poisson brackets and Lie-Poisson brackets is presented in Section II. Gyroviscous magnetohydrodynamics is treated as a relevant example in Euler and Clebsch variables. Section III is devoted to a short review of functional integration containing the definition and a discussion of ambiguities and methods of evaluation. The main part of the contribution is given in Sect. IV, where some of the content of the previous sections is applied to Gibbs statistics of continuous fluids and plasmas. In particular, exact fluctuation spectra are calculated for relevant equations in fluids and plasmas.