Building Dynamos Differential Rotation and Turbulence

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. 

In an electrically conducting fluid, buoyant turbulent cells produce a helical twist to the toroidal field and induce a poloidal conversion. This is the basis of the a — dynamo model. The electromotive force is = aB, where a is related to the velocity correlation function and essentially 

Then a /o ( 2 h and therefore depends on the velocity fluctuation spectrum. The turbulence therefore controls the small-scale structure, and the differential rotation (shear) controls the large-scale, ordered field and provides the symmetry breaking necessary to generating the dipole. 

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