Part 5, Chapter 2

Neoclassical Transport

Banana orbits, trapped particles, and bootstrap current in toroidal geometry

2.1 Why Toroidal Geometry Matters

Classical transport theory (Chapter 1) treats the plasma as uniform and the magnetic field as straight. In a real tokamak, the magnetic field varies along a field line due to the toroidal geometry: the field is stronger on the inboard side (high-field side) and weaker on the outboard side (low-field side):

$$B(\theta) \approx B_0 \frac{R_0}{R_0 + r\cos\theta} \approx B_0 \left(1 - \epsilon\cos\theta\right)$$

where $$\epsilon = r/R_0$$ is the inverse aspect ratio. This variation creates a magnetic mirror that traps particles with small parallel velocities. The resulting neoclassical transport is qualitatively different from and much larger than classical transport, dominating the irreducible minimum transport level in tokamaks.

2.2 Trapped Particles and Banana Orbits

A particle is trapped when its parallel kinetic energy is insufficient to overcome the magnetic mirror. The trapping condition is:

$$\frac{v_\parallel^2}{v^2} < \frac{2\epsilon}{1+\epsilon} \approx 2\epsilon$$

For a Maxwellian distribution, the trapped particle fraction is:

$$f_t \approx 1.46\sqrt{\epsilon} = 1.46\sqrt{r/R_0}$$

For a typical tokamak with $$\epsilon \sim 0.3$$, about 80% of particles are passing and 20% are trapped. Trapped particles execute banana orbits in the poloidal plane, bouncing back and forth between mirror points. The width of a banana orbit is:

$$\Delta_b \approx \frac{q\rho_i}{\sqrt{\epsilon}}$$

where q is the safety factor and $$\rho_i$$ is the ion gyroradius. Since$$q/\sqrt{\epsilon} \gg 1$$, the banana width greatly exceeds the gyroradius, and the effective step size for cross-field diffusion is enhanced accordingly.

Physical Picture: In the banana regime, the effective diffusion step is the banana width rather than the gyroradius. Since the banana width is q/sqrt(epsilon) times larger, neoclassical diffusion exceeds classical by a factor $$\sim q^2/\epsilon^{3/2}$$.

2.3 Collisionality Regimes

Neoclassical transport depends critically on the collisionality parameter:

$$\nu_* = \frac{\nu_{ii}\,qR}{\epsilon^{3/2}\,v_{th,i}}$$

This measures the ratio of the collision frequency to the banana bounce frequency. Three distinct regimes emerge:

Banana Regime

$$\nu_* < 1$$

Low collisionality. Trapped particles complete full banana orbits. Diffusion scales as:

$$D_{\text{banana}} \sim q^2\rho_i^2\,\frac{\nu_{ii}}{\epsilon^{3/2}}$$

Plateau Regime

$$1 < \nu_* < \epsilon^{-3/2}$$

Intermediate. Trapped particles are scattered before completing orbits. Diffusion is independent of collisionality:

$$D_{\text{plateau}} \sim \frac{q\rho_i^2 v_{th,i}}{R}$$

Pfirsch-Schluter

$$\nu_* > \epsilon^{-3/2}$$

High collisionality. No trapped particles; transport arises from Pfirsch-Schluter currents:

$$D_{\text{PS}} \sim q^2\,D_\perp^{\text{classical}}$$

2.4 Bootstrap Current

One of the most important consequences of neoclassical theory is the bootstrap current, a self-generated toroidal current driven by the pressure gradient in the banana regime. Trapped particles carry a net toroidal precession current due to the asymmetry of their banana orbits in the presence of a density or temperature gradient.

The bootstrap current density is approximately:

$$j_{\text{BS}} \approx -\frac{\sqrt{\epsilon}}{B_\theta}\,\frac{dp}{dr}$$

where $$B_\theta$$ is the poloidal magnetic field and dp/dr is the radial pressure gradient. This current flows in the same direction as the Ohmic current and can provide a significant fraction (30-80%) of the total plasma current in advanced tokamak scenarios, reducing the need for external current drive and enabling steady-state operation.

The fraction of bootstrap current scales as:

$$f_{\text{BS}} = \frac{I_{\text{BS}}}{I_p} \propto \sqrt{\epsilon}\,\beta_p$$

where $$\beta_p$$ is the poloidal beta. High-beta, high-bootstrap-fraction scenarios are the basis for advanced tokamak and spherical tokamak designs.

2.5 Computational Example

The following simulation plots the neoclassical diffusion coefficient across all three collisionality regimes, showing the characteristic banana, plateau, and Pfirsch-Schluter scalings.

Neoclassical Diffusion vs Collisionality

Python

script.py69 lines

import numpy as np
import matplotlib.pyplot as plt

# Tokamak parameters
R0 = 1.7        # Major radius (m)
a = 0.5         # Minor radius (m)
epsilon = a/R0   # Inverse aspect ratio
q = 2.0          # Safety factor
B0 = 2.5         # Toroidal field (T)
Ti_eV = 1000     # Ion temperature (eV)
mi = 1.67e-27    # Proton mass (kg)
e = 1.602e-19
Ti_J = Ti_eV * e
vth_i = np.sqrt(2*Ti_J / mi)
rho_i = mi * vth_i / (e * B0)

# Collisionality scan
nu_star = np.logspace(-3, 3, 500)

# Effective collision frequency from nu_star
nu_ii = nu_star * epsilon**1.5 * vth_i / (q * R0)

# Classical diffusion
D_classical = rho_i**2 * nu_ii

# --- Three regimes ---
# Banana: D ~ q^2 * rho_i^2 * nu_ii / epsilon^(3/2)
D_banana = q**2 * rho_i**2 * nu_ii / epsilon**1.5

# Plateau: D ~ q * rho_i^2 * v_thi / R  (independent of nu)
D_plateau = q * rho_i**2 * vth_i / R0 * np.ones_like(nu_star)

# Pfirsch-Schluter: D ~ q^2 * D_classical = q^2 * rho_i^2 * nu_ii
D_PS = q**2 * rho_i**2 * nu_ii

# Combined neoclassical (take minimum of banana/plateau at low nu, PS at high nu)
D_neo = np.where(nu_star < 1, D_banana,
        np.where(nu_star < epsilon**(-1.5), D_plateau, D_PS))

fig, ax = plt.subplots(figsize=(10, 6))

ax.loglog(nu_star, D_banana, '--', color='#fb923c', alpha=0.5, linewidth=1.5, label='Banana scaling')
ax.loglog(nu_star, D_plateau, '--', color='#38bdf8', alpha=0.5, linewidth=1.5, label='Plateau scaling')
ax.loglog(nu_star, D_PS, '--', color='#a78bfa', alpha=0.5, linewidth=1.5, label='Pfirsch-Schluter scaling')
ax.loglog(nu_star, D_neo, color='#f59e0b', linewidth=3, label='Neoclassical D (combined)')

# Regime boundaries
ax.axvline(x=1, color='white', linestyle=':', alpha=0.4)
ax.axvline(x=epsilon**(-1.5), color='white', linestyle=':', alpha=0.4)
ax.text(0.01, D_plateau[0]*3, 'Banana', fontsize=13, color='#fb923c', fontweight='bold')
ax.text(3, D_plateau[0]*3, 'Plateau', fontsize=13, color='#38bdf8', fontweight='bold')
ax.text(epsilon**(-1.5)*2, D_plateau[0]*3, 'P-S', fontsize=13, color='#a78bfa', fontweight='bold')

ax.set_xlabel(r'Collisionality $\nu_*$', fontsize=13)
ax.set_ylabel(r'Diffusion Coefficient $D$ (m$^2$/s)', fontsize=13)
ax.set_title('Neoclassical Diffusion: Three Collisionality Regimes', fontsize=14, fontweight='bold')
ax.legend(fontsize=10, loc='lower right')
ax.grid(True, alpha=0.3)
ax.set_xlim(1e-3, 1e3)

plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#0f172a')
plt.show()

print(f"Inverse aspect ratio epsilon = {epsilon:.3f}")
print(f"Trapped fraction ~ 1.46*sqrt(eps) = {1.46*np.sqrt(epsilon):.3f}")
print(f"Banana width ~ q*rho_i/sqrt(eps) = {q*rho_i/np.sqrt(epsilon)*100:.1f} cm")
print(f"Plateau D = {D_plateau[0]:.3e} m^2/s")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

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