The Sauter Bootstrap Formula
A widely used analytic fit for the neoclassical bootstrap current valid across collisionality regimes.
The Bootstrap Current
The bootstrap current is a toroidal current generated by the plasma itself, driven by the radial pressure gradient acting through trapped-particle physics. It flows without external drive and can supply a large fraction of the total current in high-pressure, low-collisionality plasmas, which is why it is central to steady-state tokamak scenarios. Predicting it accurately across the profile requires a formula valid in all collisionality regimes.
The Sauter Fit
The Sauter formula expresses the bootstrap current density as a sum of terms proportional to the logarithmic gradients of pressure, density, and temperature, each multiplied by transport coefficients L31, L32, and L34 (and alpha) that depend on the trapped fraction and on collisionality nu-star. It is an analytic fit to numerical drift-kinetic solutions, extending earlier collisionless results to finite collisionality and arbitrary aspect ratio.
Schematically j_bootstrap is proportional to -(I p / ) [ L31 (dln p/dpsi) + ... ], where I is the poloidal current function and psi the poloidal flux. The trapped fraction f_t enters every coefficient, so accurate f_t evaluation at each aspect ratio is essential.
Accuracy and Use
The Sauter formula and its later refinements are standard in transport and equilibrium codes for self-consistent current-profile evolution. Because it covers the banana, plateau, and Pfirsch-Schluter regimes with a single expression, it is convenient for whole-discharge modeling, though at very low aspect ratio and high collisionality it is checked against full drift-kinetic calculations.
Relevance
Spherical tokamaks such as the Hyperion breeder concept have large trapped fractions and can reach high bootstrap fractions. The Sauter formula is used at the design stage to estimate the self-driven current and its alignment with the desired q profile; results are computational for a machine not yet built.