Heating Methods

3.1 Ohmic Heating

The simplest heating method in a tokamak exploits the resistivity of the plasma itself. The induced toroidal current heats the plasma through Joule dissipation:

where eta is the Spitzer resistivity and j is the current density. The Spitzer resistivity decreases strongly with temperature:

Because eta scales as T_e^(-3/2), ohmic heating becomes increasingly ineffective as the plasma temperature rises. For Z_eff = 1 and ln Lambda approximately 17, the resistivity at T = 1 keV is about 3 x 10^-8 Ohm m, but at 10 keV it drops to 10^-9 Ohm m. Consequently, ohmic heating alone can only reach temperatures of about 2-3 keV in typical tokamaks. Auxiliary heating is essential to reach fusion-relevant temperatures of 10-20 keV.

The total ohmic power scales as P_OH approximately eta_0 I_p^2 / (pi a^2) x Volume, where I_p is the plasma current. For ITER-scale devices, P_OH is roughly 1-2 MW -- far below the 50-100 MW of auxiliary heating power needed.

3.2 Neutral Beam Injection (NBI)

NBI is the workhorse of plasma heating. High-energy neutral atoms (typically deuterium at 80-1000 keV) are injected across the magnetic field. Because they are neutral, they penetrate the confining field and are ionized inside the plasma by charge exchange, impact ionization, or electron stripping. The resulting fast ions transfer energy to the bulk plasma through Coulomb collisions.

The slowing-down process is characterized by a critical energy where the beam ion transfers energy equally to electrons and ions:

where A_b is the beam mass number and A_i is the background ion mass number. For deuterium beams in a deuterium plasma (A_b = A_i = 2), E_c approximately 14.8 T_e. Above E_c, the beam predominantly heats electrons (drag force dominated by electron friction); below E_c, ion heating dominates (nuclear scattering). At T_e = 10 keV, the critical energy is about 148 keV.

The beam slowing-down distribution function is:

where S_b is the beam source rate, tau_s is the Spitzer slowing-down time on electrons, and E_0 is the injection energy. This distribution is constant (flat in energy) for E much greater than E_c (electron drag dominated) and rises as E^(1/2) for E much less than E_c (ion drag dominated), before thermalizing at the bulk temperature.

3.3 Ion Cyclotron Resonance Heating (ICRH)

ICRH uses radio-frequency waves in the range of 30-120 MHz to heat ions at their cyclotron frequency or its harmonics. The resonance condition is:

where Omega_i = eB/(m_i) is the ion cyclotron frequency, n is the harmonic number, and the k_parallel v_parallel term accounts for Doppler broadening. The fundamental (n=1) resonance is the strongest but cannot directly heat the majority ion species in a single-species plasma because the left-hand polarized component of the fast wave vanishes at the ion-ion hybrid resonance. Instead, minority heating is employed: a small concentration (1-5%) of a minority species (e.g., H in a D plasma) absorbs energy at its fundamental cyclotron resonance.

The absorbed power per unit volume for minority heating scales as:

where E_+ is the left-hand circularly polarized component of the wave electric field. ICRH creates energetic ion tails that can reach MeV energies, making it useful for simulating alpha particle behavior and for synergistic NBI+ICRH scenarios.

3.4 Electron Cyclotron Resonance Heating (ECRH)

ECRH uses millimeter waves at the electron cyclotron frequency (typically 100-170 GHz for modern tokamaks). The resonance condition is omega = n Omega_ce, where Omega_ce = eB/m_e. Because the magnetic field varies across the plasma cross-section (B approximately B_0 R_0/R), the resonance location can be precisely controlled by varying the wave frequency or the toroidal field.

ECRH has several unique advantages: it provides highly localized power deposition (the absorption layer is only a few centimeters wide), can be launched from outside the vessel with simple waveguide systems, and can be steered to target specific flux surfaces. The ordinary mode (O-mode) is absorbed at the fundamental resonance when launched from the low-field side, while the extraordinary mode (X-mode) can access the second harmonic.

For ITER at B = 5.3 T, the fundamental ECRH frequency is 170 GHz. The installed ECRH power for ITER is 20 MW from 24 gyrotrons, each delivering approximately 1 MW CW at 170 GHz. ECRH is also the primary tool for neoclassical tearing mode (NTM) stabilization, where localized current drive at the rational surface suppresses the magnetic island.