Check for Consistency

Let us make a check for consistency. The rate of magnetic energy production WB per unit of time and volume must be equal to the rate Wv of dissipation of rotational energy by the magnetic viscosity v. We assume here that the whole energy dissipated is converted into magnetic energy, considering thermal dissipation as negligible. The differential motions are those of the shellular rotation Q(r), so that the velocity difference at radius r is dv = rdQ. The amount of energy corresponding to a velocity difference dv during a time dt for an element of matter dm in a volume dV is

because the viscous time dt over a distance dr is given by dt = (dr)2/v (Appendix B.4.1). We also use the parameter of differential rotation q (Sect. 13.4.5). This gives with the expression (13.96) for the kinematic viscosity,

With (13.82), which defines the field amplitude (aA/Q) = Q q/N, the dissipation rate of the differential energy of rotation finally becomes

We now turn to the rate WB of magnetic energy creation by units of volume and time. The magnetic energy density is B2/(8n) (13.10), it is produced within a characteristic time given by O-1 = (®A/Q)-1,

where we have used (13.91) to express the magnetic field, because B9 is the main field component. If we express (da/Q (as recalled after 13.100), we get the same expression as (13.101), thus

The rate of magnetic energy creation in a volume element is equal to the rate of dissipation of differential rotation by the magnetic viscosity. This shows the consistency of the field expression for B9, of the transport coefficient v together with the energy conservation in the process of field creation.

Alternatively, if we impose the equality of the dissipation rates (13.100) and (13.102) and if we use the value of cB, the induction equation and the maximum lr from (13.64), we get the condition (aA/Q) = Q q/N which fixes the field amplitude.

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