Energy Confinement Scaling Laws
Empirical power-law fits that predict confinement time from machine size, field, current, and heating.
Why empirical laws
Turbulent transport is too complex to predict confinement time tau_E from first principles alone across all regimes. Instead, the community fits multi-machine databases to power laws in the engineering parameters. The most cited is the ITER98(y,2) H-mode scaling:
tau_E = 0.0562 * I_p^0.93 B^0.15 P^-0.69 n^0.41 M^0.19 R^1.97 eps^0.58 kappa^0.78with plasma current I_p, toroidal field B, loss power P, density n, isotope mass M, major radius R, inverse aspect ratio eps, and elongation kappa. The strong positive exponents on current and size, and the negative exponent on power (power degradation), are the key lessons.
Confinement modes
Different regimes have different scalings: ohmic (L-mode) confinement differs from high-confinement H-mode, which forms an edge transport barrier. Separate fits (L-mode, H-mode, I-mode) exist because the underlying turbulence changes.
How the fits are built
- Assemble a database of validated discharges across many tokamaks
- Perform log-linear least-squares regression on the engineering variables
- Report confidence intervals and check dimensional consistency against gyro-Bohm scaling
Dimensionless-parameter scalings (in rho-star, beta, nu-star) are preferred for extrapolation because they respect the physics constraints that raw engineering fits can violate.
Using them for design
Confinement scalings give the first estimate of tau_E for a proposed machine, feeding the Lawson and power-balance calculations. For a compact, high-field spherical tokamak like the Hyperion breeder, low-aspect-ratio scalings and gyro-Bohm expectations are used and cross-checked against transport simulation, since standard fits are sparse at very low aspect ratio.