Plasma Pressure

The ideal gas law p = n k T applies to the random thermal motion of particles, which is independent of the magnetic field. While electromagnetic forces affect trajectories and transport, the equation of state holds as long as the particle distribution is Maxwellian (thermal equilibrium). The magnetic field does not do work on the particles, so it doesn't alter the kinetic pressure—it merely confines it. This pressure is distinct from magnetic pressure (B²/2μ₀), and both coexist in the force balance.

In a tokamak like ITER, the core plasma pressure is about 5–10 atmospheres (0.5–1 MPa), roughly the same as a car tire. This surprises many because the temperature is 150 million °C. In contrast, a pressurized water fission reactor operates at ~150 atmospheres (15 MPa). The relatively low pressure in fusion is offset by the extreme temperature, and it's the magnetic field (which exerts a much larger pressure of ~50–100 atm) that holds this hot plasma away from the walls.

Yes. The total pressure is the sum of partial pressures: p = n_i k T_i + n_e k T_e. Since plasmas are quasi-neutral (n_i ≈ n_e = n), the formula p = n k T is only valid when T_i = T_e = T. In many fusion plasmas, especially during auxiliary heating, T_i can exceed T_e, so you must use the full sum. In the plasma edge or during current drive, T_e might dominate, so using a single T would be inaccurate.

The maximum pressure is set by the dimensionless beta (β = p / (B²/2μ₀)), which cannot exceed a few percent (typically 3–4% for conventional tokamaks, known as the Troyon limit) without triggering magnetohydrodynamic (MHD) instabilities like ballooning modes or kink modes. Beyond this limit, the plasma's outward kinetic pressure overwhelms the magnetic restoring force, causing a violent loss of confinement (disruption). To increase pressure, you must increase the magnetic field strength, since p_max ∝ B².

Pressure is not measured directly but calculated from profiles. Density (n) is measured via laser interferometry or Thomson scattering. Temperature (T) is measured using Thomson scattering (for electrons) and charge-exchange recombination spectroscopy or Doppler spectroscopy (for ions). These diagnostics provide radial profiles. The local pressure is then computed as p(r) = n_e(r) k T_e(r) + n_i(r) k T_i(r). Additionally, the volume-integrated pressure (plasma stored energy W) is measured using diamagnetic loops that detect the tiny reduction in toroidal magnetic flux caused by the plasma pressure.

During a major disruption, the thermal energy (and thus pressure) is lost in milliseconds. The rapid drop in pressure causes the plasma current to decay (current quench), which induces huge eddy currents in the vacuum vessel and structure. These currents produce enormous electromagnetic forces (up to mega-newtons) that can deform the vessel. Simultaneously, the rapid release of thermal energy creates localized heat loads on the divertor that can melt the tungsten tiles. This is why disruption avoidance and mitigation are top priorities for ITER.

Thermal pressure comes from the random thermal motion of all background plasma particles (fuel ions and electrons). It is isotropic and pushes outward against the magnetic field. Radiation pressure is the momentum carried by electromagnetic waves (photons or synchrotron radiation) and is negligible in fusion plasmas. Fusion alphas (helium ash) deposit their kinetic energy via collisions, increasing the thermal pressure of the background plasma rather than exerting direct radiation pressure. The only relevant radiation is from Bremsstrahlung and synchrotron, which cool the plasma but contribute negligibly to the force balance.

In H-mode, an edge transport barrier forms, creating a steep pressure gradient (the pedestal). This high pedestal pressure acts as a buffer that reduces radial transport, doubling τ_E compared to L-mode. The height of the pedestal pressure strongly correlates with the total stored energy W (since W = ∫p dV). Higher pedestal pressure means higher W for the same heating power, thus a longer τ_E = W / P_loss. This is why achieving a high pedestal is essential for ITER to reach its target Q=10.

While a peaked pressure profile maximizes core fusion power density (since S ∝ p² for a given T), it creates strong pressure gradients that drive MHD instabilities like sawteeth and neoclassical tearing modes (NTMs). A broad or 'flat' pressure profile reduces these gradients, improving stability and allowing operation at higher global beta. This stability-optimized approach often yields higher overall fusion power and a longer pulse duration, even if the peak power density is lower, making it preferable for steady-state reactor operation.

The bootstrap current is a self-generated toroidal current that arises from the pressure gradient (∇p). In a tokamak, trapped particles experience a radial drift due to the ∇p, creating a net current. This current can provide a large fraction (30–80%) of the total plasma current needed for stability, reducing the need for external current drive (which consumes recirculating power). Therefore, optimizing the pressure profile to maximize bootstrap current is a key design goal for steady-state reactors to achieve high Q_eng.

Because density is bounded by the Greenwald density limit: n_G ≈ I_p / (π a²), where I_p is the plasma current and a is the minor radius. Exceeding this limit causes a sudden drop in confinement due to the onset of MHD activity and high radiative losses from the edge. To increase density, you must either increase the plasma current (which requires higher magnetic field and engineering) or increase the size (a). Since fusion power scales as n², this limit is a severe constraint: you cannot arbitrarily raise n without hitting the instability threshold.

During the thermal quench, the plasma temperature plummets from ~10–15 keV to ~100 eV in less than a millisecond due to a sudden burst of turbulent transport and radiation (e.g., from impurity influx). Since p = n k T and the density doesn't drop instantly, the pressure collapses by more than 95% as T collapses. This pressure collapse is followed by the slower current quench (milliseconds to tens of milliseconds), where the plasma current decays as the resistivity rises. The rapid loss of pressure produces a large inductive voltage spike, which drives the runaway electron generation that can damage the vessel.