Here, f is the molecular transfer coefficient,and Fγ is a source term for heat or substance release (or absorption). The main problem in solving the Reynolds equations and the averaged equation of scalar substance transfer is deriving the relations for,and, i.e., closing the equations for averaged values. By closure methods, the models may be divided into the models utilizing the mean velocity field and the models employing the field of mean turbulence characteristics. The first group methods (of Prandtl, Kantian, van Driest and Cebeci) were constructed based on the analogy between turbulence and molecular chaos. They involve such notions as mixing length as well as turbulent viscosity, thermal conductivity and diffusion coefficients. They presume a linear relationship between the tensor of turbulent stresses and the tensor of average deformation rates (the ), as well as between the turbulent heat (or passive admixture) flux and the average temperature (admixture concentration) gradient. The Boussinesq hypothesis has the form:
here, H(ω) = FT[H(t)] is the transfer function of the system. The model construction consists in determining the transfer function of the system, or in computing the convolution-type integral, or in employing the Fourier series with random coefficients to represent random signals. The random coefficients may be prescribed in such a way that the statistical moments of the signal have preset values. With the help of quick Fourier transform methods, the random input signal and, subsequently, the Fourier spectrum are formed. Afterwards, the Fourier spectrum is multiplied by the transfer function and, using the inverse transformation, the sought output signal is obtained.
We re-examine the Boussinesq hypothesis of an effective turbulent viscosity within the context of simple closure considerations for models of strong magnetohydrodynamic turbulence. Reynolds-stress and turbulent Maxwell-stress closure models will necessarily introduce a suite of transport coefficients, all of which are to some degree model-dependent. One of the most important coefficients is the relaxation time for the turbulent Maxwell stress, which until recently has been relatively ignored. We discuss this relaxation within the context of magnetohydrodynamic turbulence in steady high Reynolds-number, high magnetic-Reynolds-number shearing flows. The relaxation time for the turbulent Maxwell stress is not limited by the shear time scale, in contrast with Reynolds-stress closure models for purely hydrodynamical turbulence. The anisotropy of the turbulent stress tensor for magnetohydrodynamic turbulence, even for the case of zero mean-field considered here, can therefore not be neglected. This shear-generated anisotropy can be interpreted as being due to an effective turbulent elasticity, in analogy to the Boussinesq turbulent viscosity. We claim that this turbulent elasticity should be important for any astrophysical problem in which the turbulent stress in quasi-steady shear has been treated phenomenologically with an effective viscosity. Key words: 1
()We re-examine the Boussinesq hypothesis of an effective turbulentviscosity within the context of simple closure considerations for modelsof strong magnetohydrodynamic turbulence.