Employing Equation (2.193), and only retaining terms up to first order in the small parameter , we obtainwhere Here, use has been made of Equations (2.204)–(2.206). Incidentally, is only a small parameter in the large aspect-ratio limit .
It follows from Equations (2.165), (2.166), (2.170), (2.181)–(2.185), (2.214), and (2.216) thatand
Consider the circular magnetic flux-surface limit. In this limit, and , so
According to Equation (2.227), the poloidal component of the ion heat flux is equivalent to the poloidal component of the diamagnetic heat flux specified in Equation (2.137), except that it is reduced by a factor . The explanation for this reduction is that trapped ions cannot carry a net poloidal heat flux around a magnetic flux-surface, because they are forced to periodically reverse their poloidal drift direction. Hence, the diamagnetic ion heat flux is reduced by a factor that is roughly equal to the fraction of passing particles. [See Equation (2.201).] This reduction is accomplished by the parallel heat flux specified in Equation (2.229).
Equation (2.226) shows that the poloidal component of the ion fluid velocity is directly proportional to the poloidal component of the ion heat flux. The most surprising feature of Equation (2.226) is the absence of a contribution from the E-cross-B velocity, despite the fact that this velocity possesses a non-zero poloidal component. [See Equation (2.138).] The explanation for this absence is that parallel friction forces drive an ion flow parallel to the magnetic field that cancels out the poloidal component of the E-cross-B velocity, as well as the poloidal component of the ion diamagnetic velocity. The requisite flow, which is specified in Equation (2.228), is quite large, being a factor larger than a typical diamagnetic velocity. Equation (2.226) implies that a tokamak plasma is not free to rotate in the poloidal direction. In fact, any deviation of the ion poloidal flow velocity from the so-called neoclassical velocity specified by Equation (2.226) is opposed by friction between trapped and passing ions, and consequently damps away on a timescale that is proportional to . This effect is known as poloidal flow damping . (See Section 2.24.)
Incidentally, in the large aspect-ratio limit, Equation (2.189) reduces tobecause the omitted term is smaller than the other terms. Thus, it follows from Equations (2.130), (2.165), (2.166), (2.181), and (2.186) that