Scaling Law

Empirical formulas derived from multi-machine databases that predict plasma energy confinement time from engineering parameters — the primary tool for designing next-step fusion devices.

What Is a Scaling Law?

In fusion plasma physics, a scaling law is an empirical power-law expression that relates the energy confinement time of a plasma to measurable machine and plasma parameters such as plasma current, magnetic field strength, density, heating power, and device size. These formulas are regression fits to large, multi-machine databases and serve as the principal tool for projecting the performance of future devices.

The most widely used scaling law in magnetic confinement fusion is IPB98(y,2), developed by the ITER Physics Basis group in 1999. It takes the form:

τ_E = 0.0562 × I_p^0.93 B_T^0.15 P^−0.69 n_e^0.41 M^0.19 R^1.97 κ^0.78 ε^0.58

where I_p is the plasma current, B_T the toroidal field, P the loss power, n_e the line-averaged electron density, M the isotopic mass, R the major radius, κ the elongation, and ε the inverse aspect ratio.^[1]

Historical Development

Early scaling laws emerged in the 1980s from single-machine datasets. The Goldston scaling (1984) first captured the degradation of confinement with increasing heating power in L-mode plasmas, showing τ_E ∝ P^−0.5.^[2] As the H-mode was discovered and characterized across multiple tokamaks, international databases pooled results from JET, DIII-D, JT-60U, ASDEX Upgrade, and other machines to produce progressively refined scalings — ITER89-P for L-mode, and the ELMy H-mode family culminating in IPB98(y,2).

The ITER project's entire design point — its size, field, current, and expected Q ≥ 10 performance — rests on the validity of the IPB98(y,2) scaling extrapolated roughly a factor of two beyond existing data in the dimensionless parameter ρ*.^[1]

Methodology and Limitations

Scaling laws are constructed by log-linear regression of confinement data from the International Tokamak Physics Activity (ITPA) global databases. The databases apply strict selection criteria: steady-state intervals, ELMy H-mode, no impurity accumulation events, and sufficient diagnostics. Despite these controls, several well-known limitations persist:

• Collinearity — Machine parameters are not independently varied across devices. Current, field, and size tend to increase together, making it difficult to isolate individual exponents.
• Extrapolation risk — ITER sits outside the convex hull of the existing database in several parameters, so the prediction is a genuine extrapolation, not an interpolation.
• Physics opacity — Power-law fits do not encode the underlying transport physics. They cannot predict transitions between confinement regimes or capture the effect of plasma shape and rotation in detail.

Dimensionless Scaling

An alternative approach expresses confinement in terms of dimensionless plasma physics parameters — ρ* (normalized Larmor radius), β (ratio of plasma to magnetic pressure), and ν* (collisionality). This framework, grounded in the Kadomtsev-Connor-Taylor similarity principle, enables same-shape experiments at different sizes to test whether the scaling exponents are physically consistent. Dedicated ρ*-scaling experiments on DIII-D and JET have broadly confirmed the IPB98(y,2) size dependence, though residual uncertainties remain at the 10–20% level.^[3]

Sources

1. ITER Physics Basis Editors et al., "Chapter 2: Plasma confinement and transport," Nuclear Fusion, vol. 39, no. 12, pp. 2175–2249, 1999.
2. R. J. Goldston, "Energy confinement scaling in Tokamaks: some implications of recent experiments with Ohmic and strong auxiliary heating," Plasma Physics and Controlled Fusion, vol. 26, no. 1A, pp. 87–103, 1984.
3. C. C. Petty et al., "Gyroradius scaling of electron and ion transport," Physical Review Letters, vol. 74, no. 10, pp. 1763–1766, 1995.

Related

Confinement Scaling H-Mode Energy Confinement Time ITER Turbulence in Plasmas