The Current Diffusion Equation
The resistive-diffusion equation that evolves the poloidal flux and hence the safety-factor profile.
How the current profile evolves
The plasma current does not redistribute instantly; the poloidal flux psi diffuses resistively on the slow current-relaxation timescale. Combining Faraday's law, Ampere's law, and Ohm's law gives a diffusion equation for the poloidal flux:
d(psi)/dt = (eta / mu0) * (geometric operator) psi + source terms (bootstrap, current drive)The diffusivity is set by the neoclassical resistivity; the source terms are the non-inductive currents (bootstrap and externally driven). Solving it gives the time evolution of the q-profile, which controls stability.
Timescales
Because resistivity is small at high temperature, the current-diffusion (resistive) time is long, often seconds to many seconds in large hot plasmas. This means the q-profile evolves slowly and can be actively shaped, but also that reaching a fully relaxed current profile takes a long pulse.
What it couples to
- Neoclassical resistivity (temperature-dependent) sets the diffusion rate
- Bootstrap current enters as a distributed source, tied to the pressure profile
- External current drive (beams, radio-frequency, electron-cyclotron) adds localized sources
How it is solved numerically
The current-diffusion equation is a one-dimensional parabolic equation in the flux-surface radius, solved implicitly and coupled at each step to the equilibrium (Grad-Shafranov) so the geometry and q-profile stay consistent. It is a core module of every transport and scenario-modeling code.
Why it matters
The q-profile governs sawteeth, tearing modes, and the accessible confinement regime, so controlling current-profile evolution is central to scenario design. Steady-state operation requires the non-inductive sources (bootstrap plus drive) to sustain the current profile against resistive diffusion. Current-profile evolution and control are modeled for the Hyperion breeder as part of its scenario design study.