## 4. Numerical resultsWe verified numerically that the initially weak magnetic field approaches the equilibrium configuration (15) with accuracy 1% for , and an accuracy of for . As is anticipated in the previous section, the equilibrium magnetic field near to the generation threshold value is more complicated. The threshold value for the nonlinear solution of Eqs. (7) and (8) is , while the linear threshold value is . This is because the nonlinear solution arranges itself so that the term in (see Eq. (6)) is of order 1. Thus, for the nonlinear solution with , the maximal value of is about 1.25, whereas for , the maximal value is about 1.76. For we obtain numerically where is the nonlinear threshold value. As increases towards 10, the slope increases slightly, but Eq. (17) remains a reasonable estimate. (Note that accurately estimating the exponent in Eq. (17), and subsequently, is a quite delicate matter even in this one-dimensional problem, and that the quoted figures may be uncertain in the last digit.) This result is robust under variations of the helicity profile. For we get in the nonlinear case , while the linear threshold value is and near , i.e. again a square root dependence to within the errors of our procedure. Further, with in , and a smooth interpolation to zero in , we find and , again closely the same dependence. (In this case the linear threshold value is .) © European Southern Observatory (ESO) 2000 Online publication: September 5, 2000 |