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We give a short summary of some results of a numerical study of magnetic field concentrations in the solar photosphere and upper convection zone. We have developed a 2D time dependent code for the full MHD equations (momentum equation, equation of continuity, induction equation for infinite conductivity and energy equation) in slab geometry for a compressible medium. A Finite-Element-technique is used. Convective energy transport is described by the mixing-length formalism while the diffusion approximation is employed for radiation. We parametrize the inhibition of convective heat flow by the magnetic field and calculate the material functions (opacity, adiabatic temperature gradient, specific heat) self-consistently. Here we present a nearly static flux tube model with a magnetic flux of ∼ 1018 mx, a depth of 1000 km and a photospheric diameter of ∼ 300 km as the result of a dynamical calculation. The influx of heat within the flux tube at the bottom of the layer is reduced to 0.2 of the normal value. The mass distribution is a linear function of the flux function A: dm(A)/dA = const. Fig. 1 shows the model: Isodensities (a), fieldlines (b), isotherms (c) and lines of constant continuum optical depth (d) are given. The “Wilson depression” (height difference between τ = 1 within and outside the tube) is ∼ 150 km and the maximum horizontal temperature deficit is ∼ 3000 K. Field strengths as function of x for three different depths and as function of depth along the symmetry axis are shown in (e) and (f), respectively. Note the sharp edge of the tube.
Accretion disks are approximated by thin tori and the generation of magnetic fields by torus–dynamos is investigated. Solutions for the general α2ω–dynamo embedded into vacuum and into an ideally conducting medium are presented. Whereas the former solutions are qualitatively similar to those obtained for thin disks, there is a mode for the latter peculiar to torus–geometry. Excitation conditions for torus–dynamos are compared to those realized in accretion disks in cataclysmic variables, around T Tauri stars and in AGN's.
Accretion disks around compact objects as well as the gaseous components in galaxies often have the form of a torus. To study the structure and behaviour of magnetic fields generated in such rings, a dynamo is investigated, which is working inside a torus embedded into vacuum. The equations for the kinematic αω-dynamo are written down in toroidal coordinates (see Figure 1). Besides loss of magnetic flux by Ohmic diffusion (characterized by the magnetic diffusivity D) they describe its production by the inductive effects of differential rotation and of turbulent matter, which we have chosen as ω(r) = ω′0r and α = α0 sin θ, respectively. These equations are solved by series expansion into the exponential decay modes of slender tori, which are available in analytical form. A linear homogeneous system of equations follows for the expansion coefficients; its eigenvalues determine the time-dependence of the solutions, the dynamo modes.
A random superposition of waves in a rotating, stratified, electrically conducting fluid leads to dynamo action in the sense that it yields a mean electric field having a component parallel to the mean magnetic field (‘α-effect’). Using Fourier analysis methods, we derive an explicit expression for the mean electric field. The α-effect has tensor form. We obtain a finite α-tensor even in a case of vanishing mean helicity. The result is discussed in the context of the solar turbulent dynamo.
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