Azimuthally symmetric medium

For numerical computations, an explicit form of the phase function is required. A Legendre polynomial expansion of the phase function is the most common form. For an azimuthally symmetric medium (i.e., one independent of azimuthal angle <(>), the phase function is written as ... 

Isotropic scattering indicates that the radiant energy incident on a volume element is uniformly distributed to all directions. For an isotropically scattering medium, all a, coefficients of the phase function are zero, except a0. If only a0 and the first coefficient a, are considered, then one obtains the linearly anisotropic phase function, which means that the phase function is a linear function of cos 0 (or, in the case of an azimuthally symmetric medium, a linear function of p = cos 9). 

Let us assume that, in a plane-parallel and azimuthally symmetric medium, the intensities in positive and negative z-directions are given as (see Fig. 7.14) ... 

The symmetry is further broken, and the effect of the rotation translated into a poloidal field, through the combined action of circulation and turbulence. An initially axisymmetric field is sheared by differential rotation, and if it is initially cylindrical (Bz) or poloidal Br, Bg), then an azimuthal field (B ) results. Here r and 9 are the radius and latitude, respectively. A poloidal field results from a toroidal potential field. Bp = A x A, so that the toroidal magnetic field results from a distortion of the poloidal field. Finally, to convert the toroidal field back into a toroidal potential, some additional symmetry breaking is required. Turbulence in a rotating medium has vor-ticity, or handedness, which is parallel to the local angular-velocity vector and neither radial nor even hemispherically symmetric. 

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