# Neoclassical transport

The Neoclassical Transport Model is one of the pillars of the physics of magnetically confined plasmas. [1] [2] It provides a model for the transport of particles, momentum, and heat due to Coulomb collisions in confined plasmas in complex magnetic geometries, assuming that the plasma is in a quiescent state. Thus, transport due to fluctuations lies outside of the scope of the theory. The difference between the Neoclassical and the Classical models lies in the incorporation of geometrical effects, which give rise to complex particle orbits and drifts that were ignored in the latter.

## Brief summary of the theory

The theory starts from the Kinetic Equation for the mean particle distribution function $f_\alpha(x,v,t)$:

$\frac{\partial f_\alpha}{\partial t} + v\cdot \nabla f_\alpha + F \frac{\partial f_\alpha}{\partial v} = C_\alpha(f) + S_\alpha$

where $\alpha$ indicates the particle species, $v$ is the velocity, $F$ is a force (the Lorentz force, $F = q(E + v \times B)$ acting on the particle), $S_\alpha$ a source and $C_\alpha$ the collision operator. If the chosen collision operator is the Fokker-Planck operator, the equation is called the Fokker-Planck Equation. The derivation of this collision operator is highly non-trivial and requires making specific assumptions; in particular it must be assumed that a single collision has a small random effect on the particle velocity, and that the collisions are sufficiently frequent for the resulting particle trajectory to be described as a random walk. The collision operator must also satisfy some obvious conservation laws (conservation of particles, momentum, and energy).

Once the collision operator is decided, the moments of the Kinetic Equation can be computed. These fluid moments are:

$n = \int{f d^3v}$

(particle density)

$n u = \int{v f d^3v}$

(particle flux)

$P = \int{m v \cdot v f d^3v}$

(stress tensor)

$Q = \int{\frac{m v^2}{2} v f d^3v}$

(energy flux)

$P' = \int{m (v-u) \cdot (v-u) f d^3v}$

(pressure tensor)

$q = \int{\frac{m (v-u)^2}{2} (v-u) f d^3v}$

(heat flux)

As an example, the evolution equation of the first moment becomes:

$\frac{\partial n}{\partial t} + \nabla \cdot (n u) = S_n$

Similar conservation-type equations can be written down for the higher moments.

The main goal of Neoclassical transport theory is to provide a closed set of equations for the time evolution of these moments, for each particle species. Since the determination of any moment requires knowledge of the next order moment, this requires truncating the set of moments (closure of the set of equations). [3]

It is customary to make a number of additional assumptions to facilitate the analysis: e.g., small gyroradius, nested magnetic surfaces, large parallel transport, Maxwellian distribution functions, etc. As a consequence of such assumptions, the equations can be restated to reflect the 'radial' transport (normal to the flux surfaces, and averaged over flux surfaces). Thus, the magnetic geometry is incorporated at an essential level in the theory.

The theory takes account of all particle motion associated with toroidal geometry; specifically, ∇ B and curvature drifts, and passing and trapped particles (banana orbits). The theory is valid for all collisionality regimes, and includes effects due to resistivity and viscosity. An important prediction of the theory is the bootstrap current.

## Predictive and interpretative modelling

The derived transport equations can be used in several ways.

In predictive modelling, the transport is computed on the basis of the magnetic geometry, the collision operator, sources, and boundary conditions. The predicted transport and the resulting profiles can then be compared to experimental data.

In interpretative modelling, experimentally measured profiles are used to infer the corresponding sources or transport coefficients.

## Achievements

Neoclassical models have been used with success to predict transport under certain specific conditions. (Citation needed) The bootstrap current and radial electric field predicted by the theory are confirmed experimentally, both qualitatively and quantitatively in most scenarios. (Citation needed) In experimental studies, Neoclassical transport estimates are often used as a "baseline" transport level - even though experimental values often exceed Neoclassical estimates by an order of magnitude or more. In any case, this "baseline" level facilitates the comparison between devices. Neoclassical theory is also used in the process of machine design and optimisation. [4]

## Limitations

Neoclassical theory is based on a set of assumptions that limit its range of applicability and explain why it is not capable of predicting transport in all magnetic confinement devices and under all circumstances. These are:

• Maxwellianity. This assumption implies that a certain minimum level of collisionality is needed in order to ensure that Maxwellianisation is effective. The strong drives and resulting gradients that characterise fusion-grade plasmas often lead to a violation of this assumption.
• A fixed geometry. Neoclassical transport is calculated in a static magnetic geometry. In actual experiments (especially Tokamaks), the magnetic field evolves along with the plasma itself, leading to a modification of net transport. While a slow evolution (with respect to typical transport time scales) should not be problematic, rapid changes (such as magnetic reconnections) are outside of the scope of the theory.
• The linearity of the model. Neoclassical theory is a linear theory in which profiles are computed from sources, boundary conditions, and transport coefficients (that depend linearly on the profiles). No non-linear feedback of the profiles on the transport coefficients is not usually contemplated. However, there are many experimental studies that show that the profiles feed back non-linearly on transport (via turbulence), leading to some degree of self-organisation.
• Locality. Neoclassical theory is a theory of diffusion, and therefore it assumes that radial particle motion between collisions is small with respect to any other relevant spatial scales. This assumption then allows writing down differential equations, expressing the fluxes in terms of local gradients. This basic assumption is violated under specific conditions, which may include: (a) the low-collisionality limit, (b) any situation in which the gradient scale length is very small (e.g., Internal Transport Barriers), (c) locations close to the plasma edge[5][6], and (d) particles transported in streamers. Such phenomena could give rise to super-diffusion. Points (a) through (c) can be handled by using a Monte Carlo or Master Equation approach instead of deriving differential equations.
• Markovianity. A second assumption underlying diffusive models (including Neoclassics) is Markovianity, implying that the motion of any particle is only determined by its current velocity and position. However, there are situations, such as stochastic magnetic field regions, persistent turbulent eddies, or transport barriers, where this assumption may be violated (due to trapping effects, so that the preceding history of the particle trajectory becomes important). Typically, this would then give rise to sub-diffusion.

## References

1. F.L. Hinton and R.D. Hazeltine, Rev. Mod. Phys. 48, 239 (1976)
2. P. Helander and D.J. Sigmar, Collisional Transport in Magnetized Plasmas, Cambridge University Press (2001) ISBN 0521807980
3. T.J.M. Boyd and J.J. Sanderson, The physics of plasmas, Cambridge University Press (2003) ISBN 0521459125
4. M. Hirsch et al. Major results from the stellarator Wendelstein 7-AS, Plasma Phys. Control. Fusion 50, 5 (2008) 053001
5. T. Fülöp, P. Helander, Phys. Plasmas 8, 3305 (2001)
6. V. Tribaldos and J. Guasp, Neoclassical global flux simulations in stellarators, Plasma Phys. Control. Fusion 47 (2005) 545