- Pellat will be paid in the following volumes of the series.. - 2.2.2 Calculation of the transport coefficients. - 2.3.1 The nonlinear behaviour of the transport coefficients. - In particular, deeper and wider recognition of the fact that “the shape and size change the property of the matter in the confined plasma”. - Wakatani has been one of the leading figures in the series of the workshop.. - The fourth is the study of the structure formation in turbulent plasmas. - More famous example of his achievements is the investigation of the. - The rate of the nonlinear transfer is summarized in the index,. - Wakatani’s work in the progress of the theory of plasma confinement.. - Wakatani has thus clarified an essence of the problem. - Wakatani has done has been full of the essence for it.. - In other words, the fluid motions are critical to main- tenance of the magnetic field. - Indeed, from the initial formulation of the problem in a. - (2.2) yields an equation for the evolution of the mean magnetic field:. - in the absence of B 0 the small-scale field simply decays to zero. - (2.15) and to a weighted average of the helicity in Eq. - and that β is determined by the energy of the flow. - A very different approach can be taken by the formal neglect of the diffusive terms in the induction equation. - For any given value of z a field of the form Eq. - The nature of the α-effect for the Roberts flow in the limit of Rm. - (2006) have investigated the nature of the α-effect for the flow Eq. - Thus this component of the emf corresponds to α 11 . - The thick line in the upper panel corresponds to the time average up to time t of the signal. - 2.3 Evolution of the magnetic field through one cycle of a kinematic αω-dynamo (from P.H. - The related problem of the influence of lateral boundaries (i.e. - on the scale of the small-scale eddies with short turnover times). - Combining one of the conservation laws with Eq. - The components of the mean emf E = (E x , E y , E z ) are then calculated as a. - (2.37) is a considerably more accurate description of the behaviour of the α-effect in the nonlinear regime.. - little effect on the spatial dependence of the waves. - In general, the amplitude of the energy in the mean magnetic field is an increasing function of dynamo number. - This is due to the interaction of the nonlinear. - interaction of the jet with the ambient medium (see Fig. - 3 The dynamics of the formation of this lobe is not considered in the present review.. - ct v , i.e., of the order of the Schwarzschild radius (R S = 2GM/c 2 ) of Black Holes in the above quoted mass range.. - The size of the “central engine”. - The inferred direction of the magnetic field is either longitudinal (in com- pact sources) or transverse (mostly in extended radio sources) (see, e.g., [14. - counteracting the confining action of the azimuthal magnetic field tension 14 . - The spin down of the star is a priori more. - 3.2 Sketches of the two proposed configurations of protostar-disk magnetic inter- action, presented as cuts in the (r, z) plane (axisymmetry is assumed). - In the initial stage of the ejection process (close to the disk), the field energy density dominates over the matter one. - where ν v is the turbulent viscosity used in the modelling of the turbulent Reynolds stress tensor. - Turning to the vertical component of the momentum equation, one finds, in the thin disk approximation, to leading order in. - 0, as a consequence of the disk rotation and of the outwards radial current in the disk. - (3.7) The jet recollimates if the curvature of the magnetic surface R <. - 3.3 View of the poloidal current (dashed lines) and magnetic surfaces (solid lines) in a self-similar solution of the coupled disk-jet problem (adapted from [21]).. - Also, the assumed role of the kink instability is often oversimplified. - 3.4 Sketch of the mechanism of the Kelvin-Helmholtz instability (see text).. - In this geometry, perturbations of the form exp(iωt−ikz−imφ) are looked for. - This evolution avoids the disruption of the fluid slab (weak field regime).. - L is the length of the column. - 21 Note also that the magnetic shear is the logarithmic derivative of the safety factor.. - In the “astrophysical regime” (R/P o 1), the growth-rate scales like the inverse relative pitch, independently of the details of the current structure. - In what follows, P o refers to the value of the pitch on the axis.. - These authors have followed some of the unstable equilibria discussed above in the non- linear stabilization phase. - where k is the component of the total wavenumber along the unperturbed magnetic field.. - The core of the plasma. - However, none of the scenarios discussed in Sec. - For more extensive discussions of the MHD approximation, see, e.g., [138] pp. - Similarly, the toroidal com- ponent of the induction equation reads. - Similarly, the toroidal component of the induction equation , Eq. - The toroidal component of the momentum equation, Eq. - The only remaining dynamical information is contained in the poloidal part of the momentum equation. - l, E and K), the shape of the magnetic surfaces is determined. - Perturbations of the form exp(iωt − ikz) are assumed (a wavenumber could be added in the direction y). - δT ≡ F(ξ), (3.37) where δT is the perturbation of the magnetic tension, δP. - (3.44) where ξ ⊥ is the component of the displacement perpendicular to the un- perturbed field B, Q. - −B θ 2 /(B 2 r)e r is the curvature vector of the magnetic field (e is the unit vector in the direction of the unperturbed magnetic field.. - In this Appendix, successive analytic reductions of the momentum equation, Eq. - (mB z /r − kB φ )/B (3.50) are the components of the wavenumber k parallel and perpendicular to the unperturbed magnetic field.. - 3.11 Views of the jets of the galaxies M87 (left) and 3C175 (right). - 4.1 Basic cartoon explanation of the Richardson-Kolmogorov cascade. - 4.2 Basic cartoon explanation of the Richardson-Kolmogorov cascade. - 4.3 Basic idea of the Richardson dispersion problem. - The first fundamental hypotheses of the K41 theory is:. - each correspond to one of the two Elsasser populations. - v A is the auto-correlation time of the Alfven spectrum.. - refers to the bandwidth of the k. - The success of the I.-K. - A particularly interesting limit of the anisotropic MHD cascade is the. - Further detailed study of the k. - modulation of the uni-directional Alfven wave train. - ρ (2) /ρ 0 is easily determined by considering of the parallel flow dynamics. - One particularly interesting generalization of the DNLS is the KNLS or KDNLS, i.e. - An essential element of the physics. - The 3D EMF is central to the theory of the turbulent dynamo. - The quantities τ c φ (k) and τ c A (k) are the self-correlation times (lifetimes), at k, of the fluid and field perturbations, respectively. - assuming incompressibility of the flow. - v x A k is the k-component of the flux. - In 2D, several aspects of the problem merit further discussion. - Here D is the dimensionality of the space. - The calibration of the flux is 3.6×10 16 cm −2 s −1. - Measurement of scaling exponent of the structure function, S p (τ. - There are a few underlying aspects which need to be emphasized in the context of the above studies. - ψ 2 dr, where ψ is the component of the magnetic vector potential along the symmetry direction. - C[1 − (1 − x/C) p/g ] (5.10) Here g is related to the basic scaling of the relevant field δz l ∼ l 1/g . - Equation (5.12) gives the convection of the scalar field ψ − d 2 e ∇ 2 ψ in the flow for which the velocity potential is given by b. - The skin depth d e is an important characteristic scale of the model. - As the amplitudes of the individual plasma modes become high and the mode. - Numerical simulations show distinct spectral scaling indices in the two sides of the electron skin depth