(6.29) | ||

(6.30) |

(6.36) |

Let us again (see Section 5.14) make the Riccati transformation [2,5]

(6.37) |

(6.41) |

Table 6.2 gives estimates for the normalized resonant layer parameters, , , and , that appear in Equations (6.28)–(6.30), in a low-field and a high-field tokamak fusion reactor. These estimates are made using the data shown in Table 6.1. Table 6.2 also gives estimates for the linear layer thicknesses and tearing mode growth-rates in such reactors. These estimates are obtained via numerical solution of the resonant layer equation. It can be seen that the typical radial thickness of a linear tearing layer in a tokamak fusion reactor is only a few millimeters. Furthermore, linear tearing modes in tokamak fusion reactors grow on timescales that typically lie between a tenth of a second and a second [assuming that .] Finally, the real frequencies of such modes, in a frame of reference that co-rotates with the electron fluid at the resonant surface, (i.e., the imaginary components of ) are very much smaller (by a factor of order ) than a typical diamagnetic frequency. (See Table 6.1.) In other words, linear tearing modes do indeed co-rotate with the electron fluid at the resonant surface to a very high degree of fidelity.

Figures 6.2 and 6.3 show values of and , respectively, evaluated numerically as functions of and . In producing these figures, it is assumed that . The fact that for all values of and confirms that tearing modes are linearly unstable for , and stable otherwise [4]. [See Equation (6.25).] It is clear that the layer width increases with increasing normalized diamagnetic frequency, , and also with increasing normalized magnetic Prandtl number, .