Part 6, Chapter 4

Current Drive

Bootstrap current, NBCD, LHCD, and ECCD

4.1 Bootstrap Current

The bootstrap current is a self-generated toroidal current driven by the pressure gradient in a tokamak. It arises from the neoclassical transport of trapped particles in the inhomogeneous magnetic field. Trapped particles on banana orbits have a net toroidal precession, and collisional friction between trapped and passing particles generates a parallel current.

The bootstrap current density is approximately:

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

where epsilon = r/R is the inverse aspect ratio and B_theta is the poloidal field. The fraction of total current provided by bootstrap scales as:

$$f_{BS} = \frac{I_{BS}}{I_p} \approx C\,\sqrt{\epsilon}\,\beta_p$$

where beta_p = 2 mu_0 <p> / B_theta^2 is the poloidal beta and C is a coefficient of order unity that depends on the density and temperature profile shapes. For ITER, f_BS approximately 0.2-0.3. For advanced tokamak scenarios and spherical tokamaks, f_BS can reach 0.5-0.8, significantly reducing the external current drive requirements.

A fully detailed neoclassical calculation shows that the bootstrap current has contributions from both density and temperature gradients:

$$j_{BS} = -\frac{p\sqrt{\epsilon}}{B_\theta}\left[L_{31}\frac{1}{n}\frac{dn}{dr} + L_{32}\frac{1}{T_e}\frac{dT_e}{dr} + L_{34}\frac{1}{T_i}\frac{dT_i}{dr}\right]$$

The coefficients L_31, L_32, L_34 are neoclassical transport coefficients that depend on the collisionality regime and trapped particle fraction.

4.2 Neutral Beam Current Drive (NBCD)

When neutral beams are injected tangentially to the magnetic field, the resulting fast ions carry a net toroidal momentum, driving a parallel current. The current drive efficiency is characterized by the normalized figure of merit:

$$\eta_{CD} = \frac{n_e\,R_0\,I_{CD}}{P_{inj}} \quad \text{[A/W/m}^2\text{]}$$

For NBCD, typical efficiencies are eta_CD approximately 0.2-0.4 x 10^20 A/(W m^2). The efficiency increases with beam energy and electron temperature because faster particles slow down primarily on electrons, which generates less pitch-angle scattering and preserves the directed momentum. However, the beam energy must not be so high that shine-through occurs (beam crosses the plasma without being absorbed).

The driven current density depends on the competition between momentum input and collisional friction. For a beam with velocity v_b much greater than v_th,i, the fast-ion current is partially cancelled by an electron shielding current. The net current is:

$$I_{CD} = e n_b v_b \left(1 - \frac{Z_{eff}}{Z_b}\frac{1}{1 + (v_c/v_b)^3}\right)$$

where the second term in brackets represents the electron back-current (Ohkawa effect). At high beam energies (v_b much greater than v_c), the shielding factor approaches Z_eff/Z_b, reducing the net driven current.

4.3 Lower Hybrid Current Drive (LHCD)

Lower hybrid waves (frequency 1-8 GHz) propagate into the plasma and are absorbed by electrons via Landau damping when the wave phase velocity matches the electron thermal velocity. The resonance condition is:

$$v_\parallel = \frac{\omega}{k_\parallel} \approx (2.5\text{-}5)\,v_{th,e}$$

When the wave is absorbed asymmetrically (preferentially by electrons moving in one direction), it creates a plateau in the electron distribution function at high parallel velocities, driving a net current. LHCD is the most efficient RF current drive method, with eta_CD approximately 0.2-0.3 x 10^20 A/(W m^2), because fast electrons have long collision times.

The current drive efficiency from quasi-linear theory scales as:

$$\eta_{LHCD} \propto \frac{T_e}{n_e} \cdot \frac{4}{5 + Z_{eff}} \cdot \frac{v_\parallel^2}{v_{th,e}^2}$$

LHCD is particularly effective for off-axis current profile control and is used to sustain reversed-shear q-profiles in advanced tokamak scenarios. However, at high electron temperatures (above 10 keV), the wave accessibility condition limits penetration to the plasma core, as the parallel refractive index must satisfy n_parallel greater than n_parallel,acc for the wave to reach the center.

4.4 Electron Cyclotron Current Drive (ECCD)

ECCD drives current through the Fisch-Boozer mechanism: electron cyclotron waves preferentially heat electrons moving in one direction (selected by the wave launch angle), reducing their collisionality and thereby creating an asymmetry in the resistivity-weighted current.

$$j_{ECCD} \propto \frac{T_e}{n_e} \cdot \frac{\partial f_e}{\partial v_\parallel}\bigg|_{v_\parallel = v_{res}}$$

The Fisch-Boozer effect can be understood as follows: electrons heated by the wave at v_parallel = v_res have a longer collision time (tau approximately v^3), so they carry current for longer before being scattered. The opposing electrons at -v_res are not heated and scatter normally. This asymmetry produces a net current in the direction of the heated electrons.

ECCD efficiency is lower than LHCD (eta_CD approximately 0.05-0.15 x 10^20 A/(W m^2)) but has the crucial advantage of extremely localized deposition. This makes ECCD the method of choice for neoclassical tearing mode (NTM) stabilization, where current must be driven precisely at the O-point of a magnetic island to replace the "missing" bootstrap current.

Interactive Simulations

Current Drive Efficiency vs Electron Temperature

Python

script.py113 lines

import numpy as np

# Current Drive Efficiency Comparison
print("Non-Inductive Current Drive Efficiency vs Temperature")
print("=" * 58)
print()

e = 1.602e-19
m_e = 9.109e-31
eps0 = 8.854e-12
ln_Lambda = 17.0

# Temperature scan
T_e_arr = np.array([1, 2, 3, 5, 8, 10, 15, 20, 30, 50])  # keV
n_e = 1.0e20   # m^-3
R0 = 6.2       # m (ITER)
Z_eff = 1.5

print("Current drive efficiencies eta_CD = n_e * R * I_CD / P [10^20 A/(W m^2)]:")
print()
print(f"{'T_e (keV)':>10} {'NBCD':>10} {'LHCD':>10} {'ECCD':>10} {'Bootstrap':>10}")
print(f"{'':>10} {'':>10} {'':>10} {'':>10} {'f_BS':>10}")
print("-" * 52)

# Scalings (based on established fusion physics results):
# NBCD: eta ~ 0.06 * T_e / (1 + 0.05*T_e) [saturates at high T]
# LHCD: eta ~ 0.12 * T_e / (5 + Z_eff) * correction
# ECCD: eta ~ 0.02 * T_e^0.5 * (1/(1+Z_eff))
# Bootstrap fraction: f_BS ~ C * sqrt(eps) * beta_p

eta_NBCD_arr = []
eta_LHCD_arr = []
eta_ECCD_arr = []
f_BS_arr = []

for T_e in T_e_arr:
    # NBCD efficiency (Hirshman-Sigmar scaling)
    # Increases with T_e, saturates due to electron shielding
    E_beam = 500  # keV (ITER NBI)
    E_c = 14.8 * T_e  # critical energy for D beam in DT
    v_ratio = (E_beam / E_c)**1.5
    shielding = 1 - Z_eff / (1 + v_ratio)
    eta_NB = 0.22 * (T_e / 10)**0.5 * shielding
    eta_NBCD_arr.append(eta_NB)

# LHCD efficiency (Fisch scaling)
    # eta_LH ~ 0.12 * T_e[keV] / (5 + Z_eff)
    # Normalized to ITER reference
    eta_LH = 0.12 * T_e / (5 + Z_eff)
    # Accessibility limit reduces efficiency above ~10 keV
    if T_e > 10:
        eta_LH *= (10 / T_e)**0.3  # reduced penetration
    eta_LHCD_arr.append(eta_LH)

# ECCD efficiency (Fisch-Boozer)
    # eta_EC ~ 0.035 * T_e^0.5 / (1 + 0.5*Z_eff)
    eta_EC = 0.035 * T_e**0.5 / (1 + 0.5 * Z_eff)
    eta_ECCD_arr.append(eta_EC)

# Bootstrap fraction estimate
    # f_BS ~ 0.5 * sqrt(epsilon) * beta_p
    # beta_p ~ 2*mu0*n*T / B_theta^2
    # Assume B_theta ~ 0.3 T for ITER
    epsilon = 2.0 / 6.2  # a/R for ITER
    mu0 = 4 * np.pi * 1e-7
    p = n_e * T_e * 1e3 * e  # Pa
    B_theta = 0.3  # T (typical)
    beta_p = 2 * mu0 * p / B_theta**2
    f_BS = 0.5 * np.sqrt(epsilon) * beta_p
    f_BS = min(f_BS, 0.9)  # physical limit
    f_BS_arr.append(f_BS)

print(f"{T_e:10.0f} {eta_NB:10.3f} {eta_LH:10.3f} {eta_EC:10.3f} {f_BS:10.3f}")

print()
print("Notes:")
print("- NBCD: best for core current drive, limited by shine-through at low n_e")
print("- LHCD: most efficient RF method, best for off-axis, limited by accessibility")
print("- ECCD: lowest efficiency but most localized, ideal for NTM control")
print("- Bootstrap: free current from pressure gradient, increases with beta_p")
print()

# Current drive requirements for ITER
print("ITER Steady-State Scenario (I_p = 9 MA):")
print("=" * 50)
I_p = 9e6    # A
T_e_ITER = 13  # keV (volume average)

# Bootstrap
eps_ITER = 2.0/6.2
p_ITER = n_e * T_e_ITER * 1e3 * e
beta_p_ITER = 2 * 4*np.pi*1e-7 * p_ITER / 0.3**2
f_BS_ITER = 0.5 * np.sqrt(eps_ITER) * beta_p_ITER
I_BS = f_BS_ITER * I_p
print(f"Bootstrap current: I_BS = {I_BS/1e6:.1f} MA (f_BS = {f_BS_ITER:.2f})")

# NBCD: 33 MW at eta = 0.25
eta_NB_ITER = 0.25
P_NB = 33.0  # MW
I_NB = eta_NB_ITER * P_NB * 1e6 / (n_e * R0)
print(f"NBCD: P = {P_NB} MW => I_NB = {I_NB/1e6:.2f} MA (eta = {eta_NB_ITER})")

# ECCD: 20 MW at eta = 0.08
eta_EC_ITER = 0.08
P_EC = 20.0  # MW
I_EC = eta_EC_ITER * P_EC * 1e6 / (n_e * R0)
print(f"ECCD: P = {P_EC} MW => I_EC = {I_EC/1e6:.2f} MA (eta = {eta_EC_ITER})")

I_total = I_BS + I_NB + I_EC
print(f"\nTotal non-inductive: {I_total/1e6:.1f} MA")
print(f"Required: {I_p/1e6:.0f} MA")
print(f"Gap (needs inductive or LHCD): {(I_p - I_total)/1e6:.1f} MA")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

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