Plasma Transport Equations
The evolution equations for density, temperature, and current profiles on the slow confinement timescale.
What transport means
Transport describes how particles, momentum, and heat leak across the confining magnetic field. On the slow timescale of confinement, the plasma is in near-equilibrium at each instant, and the profiles of density n and temperature T evolve by flux-surface-averaged conservation laws.
dn/dt + (1/V') d/drho (V' Gamma) = S_n
(3/2) d(nT)/dt + (1/V') d/drho (V' q) = S_EHere rho is a flux-surface label, V' the differential volume, Gamma the particle flux, q the heat flux, and S the sources (fueling, heating, radiation, fusion). The equations are diffusion-advection equations in one radial dimension after flux-surface averaging.
The flux-gradient relation
Fluxes are written as Gamma = -D dn/drho + n V and q = -n chi dT/drho, with diffusivity D, pinch velocity V, and thermal diffusivity chi. The physics lives in these coefficients, which turbulence makes far larger than collisional (classical or neoclassical) predictions.
How it is solved numerically
- Implicit finite differences in radius with time steps far larger than turbulence timescales
- Coupling to equilibrium (Grad-Shafranov) each step so the geometry stays consistent
- Transport coefficients supplied by neoclassical theory plus turbulence models or reduced gyrokinetic surrogates
The stiffness of the equations (chi rising sharply above a critical gradient) forces implicit solvers and careful treatment of the boundary condition at the separatrix.
Why it decides performance
Confinement time tau_E, and therefore whether the Lawson criterion is met, is set by transport. Predicting the density and temperature profiles of the Hyperion breeder from heating power and geometry is a transport calculation, checked against confinement scaling laws.