Here n is the plasma density and vT is the thermal velocity.
It is useful to transform Eq. 3.5.9 to a form more convenient for applications. It is possible to invert the integral operator and obtain the following equation:
One should solve Eq. 3.5.14 in order to use Eq. 3.5.12, except in the case in which the collision frequency is large enough and a 'plateau' is absent. In this case one can neglect fo in Eq. 3.5.12 and obtain the well known unsaturated nonlinear damping rate (Lee and Völk, 1973; Kulsrud, 1978; Achterberg, 1981):
In the opposite case of small vone should use Eq. 3.5.14 and put v= 0:
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