Differential rotation stretches the frozen-in field lines and after several rotation periods the field lines are wound-up with a strong toroidal component. The field lines become very close to each other, meaning that the field has been amplified. Some energy of the differential rotation is converted into magnetic energy. Let us assume  that the initial field is weak, i.e., the average Alfven frequency rnA (13.17) is small compared to the rotation frequency Q,
where B and Q are the mean field and density, respectively, R is the radius. The above condition is verified for the Sun (if B < 106 G) and for early-type stars.
Let us consider only the azimuthal motions due to rotation and ignore magnetic diffusivity in the induction equation (13.5) as well as the thermal effects. The initial poloidal field Bp is assumed axisymmetric around the rotation axis. The components (Br,BV,0) do not change with time, but a new toroidal component Bq can be generated by the differential rotation. Equation V • B = 0 implies that B = V x ^, where ^ is a function called the stream function, the value of ^ is constant on a given field line. Thus, for an axisymmetric field the two components of B are
Br = (V x f )r = -Tl-dV , Bv = (V x f =--^ dr . (13.50)
r2sin V dV rsin V or
The components of the induction equation (13.5) become then, since Br and Bft are constant and only v9 = 0, dBr = 0 dBft = 0 dB dt ' dt ' dt
The equation for the toroidal component can be integrated in time rt ( VQ \
nb. of differential revolutions^^
The initial field can be considerably amplified. Since the number of differential revolutions (i.e., the number of rotations due to differential rotation) can be huge, even a minute initial field can become large after a certain time . A rough estimate of the minimum initial poloidal field is obtained by considering a field such that its Alfven velocity (13.17) is equal to the velocity necessary for crossing the stellar radius during the star lifetime to, i.e., vA = R/t0. For the Sun, this gives an Alfven velocity of 2.2 x 10~7 cm and an initial field of the order of 10~6 G. This is a rough estimate, since the growing field and the differential rotation would change with time. Nevertheless, it shows that the amplification of even a very small poloidal field may occur and influence star evolution. In general, the initial field may be larger than the above minimum.
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