Perpendicular Closure Scheme

As we have seen, the neoclassical cross-flux-surface particle, heat, and momentum diffusivities are all much smaller than the experimentally observed diffusivities. The additional, or anomalous, cross-flux-surface transport that is found in tokamaks is due to the action of small-scale plasma turbulence [53]. Turbulent eddies in a tokamak plasma are the nonlinearly saturated states of micro-instabilities driven by temperature gradients [14]. As was mentioned previously, turbulent eddies are localized on toroidal magnetic flux-surfaces in flux-tubes that run parallel to the magnetic field and whose radial extents are a few ion gyro-radii. It follows that turbulent eddies are characterized by the so-called flute ordering, $\vert k_\parallel\vert/\vert k_\perp\vert \ll 1$, where $k_\parallel$ and $k_\perp$ are the wavenumbers of the underlying micro-instabilities parallel and perpendicular to the equilibrium magnetic field, respectively. An immediate consequence of the flute ordering is that turbulent eddies generate comparatively little transport parallel to magnetic field-lines. Consequently, we shall assume that turbulence does not upset the parallel force and heat flux balance described in Equations (2.175) and (2.176). This implies that the expressions for the in-flux-surface neoclassical ion and electron flows given in Sections 2.18 and 2.19, well as the expression for the parallel current density given in Section 2.20, remain valid in the presence of turbulence. We also expect the expressions for the cross heat fluxes and the gyro-viscous tensors given in Section 2.6 to remain valid, because these effects are merely a consequence of the rapid gyration of charged particles perpendicular to magnetic field-lines, which is not significantly affected by turbulence. On the other hand, we shall replace our previous expressions for the perpendicular viscous force densities and the perpendicular heat fluxes by phenomenological terms of the forms

$$\nabla\cdot \mbox{\boldmath$\pi$} _{\perp\,s} = - \nabla\cdot(n_e\,m_s\,{\mit\Xi}_{\perp\,s}\,{\bf W}_s),$$ (2.303)

$${\bf q}_{\perp\,s} = -n_e\,\chi_{\perp\,s}\,\nabla_\perp T_s,$$ (2.304)

respectively, where ${\mit\Xi}_{\perp\,s}$ and $\chi_{\perp\,s}$ are taken from experimental measurements.

Finally, if axisymmetric tokamak plasmas are to retain their freedom to rotate in the toroidal direction then anomalous transport needs to be intrinsically ambipolar. Fortunately, good theoretical arguments can be made that this is the case [8,29,49].