Divertors and Edge Physics

5.1 X-Point Geometry and the Magnetic Separatrix

A divertor configuration is created by adding external poloidal field coils that produce a null point (X-point) in the poloidal magnetic field. At the X-point, B_theta = 0, and the poloidal flux surface that passes through this point defines the last closed flux surface (LCFS) or separatrix. Plasma outside the separatrix flows along open field lines to divertor target plates, while the confined plasma inside remains on closed flux surfaces.

Near the X-point, the poloidal field has a hyperbolic structure:

The field lines form a separatrix with four branches (two entering, two leaving the X-point). The flux expansion near the X-point, defined as the ratio of the distance between flux surfaces at the target to their spacing at the midplane, is a critical parameter:

Larger flux expansion spreads the heat load over a wider area on the target plate. ITER's divertor is designed with a single-null configuration (one X-point at the bottom), with flux expansion of about 5-10 at the strike points. The connection length (distance along field lines from midplane to target) is typically 20-50 m, depending on the geometry.

5.2 Scrape-Off Layer (SOL) Physics

The scrape-off layer is the region of open field lines between the separatrix and the vessel wall. Plasma in the SOL flows parallel to the magnetic field toward the divertor targets at roughly the sound speed. The parallel transport time is much shorter than the perpendicular diffusion time, making the SOL a narrow layer (typically 1-10 mm in the midplane).

The SOL width at the midplane is characterized by the power decay length:

This Eich scaling (from multi-machine regression) shows that lambda_q is alarmingly narrow for large tokamaks. For ITER, lambda_q approximately 1 mm at the midplane, which maps to only about 5-10 mm at the target after flux expansion. The peak heat flux on the target is:

where P_SOL is the power crossing the separatrix, R_strike is the strike-point radius, and alpha is the field-line incidence angle. For ITER with P_SOL = 100 MW, the unmitigated peak heat flux would exceed 100 MW/m^2, far beyond the engineering limit of about 10 MW/m^2 for tungsten. This motivates the need for divertor detachment.

5.3 The Two-Point Model

The two-point model connects upstream (midplane) conditions to downstream (target) conditions along a flux tube. It is derived from parallel momentum balance, energy balance, and the Spitzer-Harm heat conduction along the field:

where T_u is the upstream temperature, T_t is the target temperature, q_parallel is the parallel heat flux density, L is the connection length, and kappa_0 approximately 2000 W/(m eV^(7/2)) is the Spitzer parallel heat conductivity coefficient. This equation follows from integrating the heat conduction equation q_parallel = -kappa_0 T^(5/2) dT/ds along the field line.

Pressure conservation along the flux tube (neglecting friction and momentum loss) gives:

At the sheath entrance, the heat flux is related to the particle flux by:

where gamma approximately 7 is the sheath transmission coefficient and c_s,t = sqrt(2T_t/m_i) is the sound speed at the target. These three equations form a closed system: given the upstream conditions (n_u, T_u) and the connection length L, one can solve for n_t and T_t at the target.

5.4 Detachment Physics

Detachment occurs when volumetric power and momentum losses in the divertor region reduce the target temperature below about 5 eV and the ion flux to the target drops significantly. The primary mechanisms are:

(1) Impurity radiation cooling: injected impurities (N, Ne, Ar) radiate strongly at temperatures of 5-50 eV via line emission, removing power from the plasma before it reaches the target.

(2) Charge exchange and recombination: at T_t below 5 eV, neutral recycling becomes important. Charge-exchange collisions transfer momentum from ions to neutrals, which are not confined and spread the momentum to the walls. Volume recombination (three-body and radiative) converts ions back to neutrals, reducing the particle flux to the target.

For nitrogen seeding, L_z peaks at about 5 x 10^-31 W m^3 near T_e = 10 eV. The detachment threshold is reached when the radiation front moves upstream from the target and the target temperature drops to 1-2 eV. In this regime, the two-point model must be modified to include momentum loss factors f_mom and radiation fractions f_rad, giving reduced target heat fluxes by factors of 10-100.