## Viscous Coupling by the Field

The Alfven frequency is obtained from (13.85) and n is given by (13.83). The az-imuthal field component largely dominates over the radial one, thus one has the approximation

The radial component of the field is given by (13.68) with lr given by the maximum length (13.64) used with N given by (13.75). If N1 dominates, this gives

Let us now turn toward the transport of angular momentum by the magnetic field. The momentum of force S by volume unity due to the magnetic field is obtained by writing the momentum of the Lorentz force (13.1). The current density j is given by the first of (13.3). Thus, one has

S = r x Fl = 1 r x (j x B) = ^r x ((V x B) x B) , (13.93)

or in modulus

where (13.68) and the maximum length (13.64) are used. The units of S are g s-2 cm-1, the same as for B2 in the Gauss system. The kinematic viscosity v (in cm2 s-1) for the vertical transport of angular momentum can be expressed in terms of S (the viscosity of a fluid represents its ability to transport momentum from one place to another, B.48). According to Appendix B.4.1, one has

V 1 dr 1 dr 1 d ln r v = - = - F— = -F-= -F-, (13.95)

where F according to the definition of the viscosity is a force by surface unity, which also corresponds to a momentum of force by volume unity in g s-2 cm-1. Considering only positive quantities, with q = \dlnQ/dlnr\ and (13.94), one has

This is the general expression of viscosity v with rnA given by the solution of (13.85) and with N by (13.75). If N dominates in (13.75), one has from (13.86)

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