ITER H98(y,2) scaling

Overview

The ITER H98(y,2) scaling is an empirical relationship used to predict the thermal energy confinement time (τ_E) in the high-confinement mode (H-mode) of tokamak plasmas. Developed by the International Thermonuclear Experimental Reactor (ITER) Confinement Database and Modeling Expert Group, this scaling law is a cornerstone of modern fusion science, providing the primary basis for performance predictions for next-generation devices like ITER and DEMO. It statistically correlates τ_E with several global engineering and plasma parameters, such as plasma current, magnetic field, heating power, and device size.

The formula is derived from a log-linear regression analysis of a large, multi-machine database of experimental results from tokamaks worldwide. Its primary function is to extrapolate from the performance of existing machines to the expected performance of larger, future ones. The quality of a plasma's confinement relative to this standard prediction is quantified by the confinement enhancement factor, H_H98. An H_H98 value of 1.0 indicates that the plasma's confinement matches the scaling law's prediction, while values greater than 1.0 signify superior performance. Achieving H_H98 ≥ 1.0 is a critical design goal for ITER, as it is necessary to reach the target fusion power gain (Q_plasma ≈ 10).

Physics and Mechanism

The H98(y,2) scaling law is a power-law expression based on experimental data, not derived from first-principles plasma theory. It represents a statistical best fit to observations of how energy confinement behaves across a range of operational parameters. The standard form of the equation is:

τ_E,H98(y,2) = 0.0562 × I_p^0.93 × B_T^0.15 × P_L^-0.69 × n_e^0.41 × M^0.19 × R^1.97 × ε^0.58 × κ_a^0.78

Where the parameters are:

• τ_E,H98(y,2): The predicted thermal energy confinement time in seconds (s).
• I_p: The plasma current in mega-amperes (MA).
• B_T: The toroidal magnetic field at the geometric major radius in tesla (T).
• P_L: The loss power in megawatts (MW), defined as P_L = P_Ohm + P_aux - dW/dt, where P_Ohm is ohmic heating, P_aux is auxiliary heating, and dW/dt is the rate of change of stored energy.
• n_e: The line-averaged electron density in units of 10^19 m^-3.
• M: The effective isotopic mass of the plasma fuel ions in atomic mass units (amu); e.g., M=2 for pure deuterium, M=2.5 for a 50-50 D-T mix.
• R: The major radius of the plasma in meters (m).
• ε: The inverse aspect ratio, defined as a/R, where 'a' is the plasma minor radius.
• κ_a: The plasma elongation, measured at the separatrix or the 95% flux surface.

Several key physical dependencies are captured by the exponents. The strong positive dependence on plasma current (I_p^0.93) and major radius (R^1.97) indicates that larger, higher-current devices are expected to have significantly better confinement. Conversely, the strong negative dependence on heating power (P_L^-0.69) reflects the well-observed phenomenon of confinement degradation with increased power input. The weaker positive dependencies on magnetic field, density, and isotopic mass also reflect established experimental trends. This degradation with power is a primary challenge for achieving ignition, as the alpha particle heating in a burning plasma contributes to P_L and thus reduces τ_E.

Historical Development

The development of confinement scaling laws has been an iterative process, evolving as more experimental data became available and statistical methods improved. Early scaling laws in the 1980s, such as Goldston scaling for L-mode (low-confinement mode), provided initial frameworks. The discovery of the H-mode in 1982 on the ASDEX tokamak necessitated the development of new scaling laws to describe this improved confinement regime.

Throughout the 1990s, an international collaboration under the auspices of the ITER project began compiling a standardized database of confinement data from various tokamaks, including JET, DIII-D, ASDEX Upgrade, and JT-60U. This effort led to the ITER89-P scaling law, one of the first widely accepted H-mode scalings. As the database grew and was refined, new versions were released.

The ITER H98(y,2) scaling was published in 1999 by the ITPA (International Tokamak Physics Activity) Topical Group on Confinement Database and Modeling. It was based on an updated and more rigorously filtered dataset known as DB3v13.1, which included a wider range of plasma shapes, aspect ratios, and operating conditions. The "(y,2)" designation refers to specific choices in the regression analysis: the inclusion of hydrogen (y=1) and deuterium (y=2) data and the use of a particular statistical model. This scaling law demonstrated a better statistical fit and lower uncertainty than its predecessors, and it quickly became the international standard for projecting H-mode performance, a status it largely retains.

Current Status (as of 2026)

The ITER H98(y,2) scaling remains the baseline for performance predictions for ITER and other future reactor designs. Its validity has been continually tested on existing machines as they push into new operational regimes. Experiments on devices like JET and DIII-D operating with ITER-relevant parameters (e.g., low torque, high beta) have generally shown good agreement with the scaling, typically achieving H_H98 factors between 0.9 and 1.15. These results provide confidence in the extrapolations to ITER's operational space.

However, the fusion community recognizes the limitations of an empirical law derived from devices significantly smaller than ITER. Research is focused on developing a more physics-based understanding of confinement to supplement and eventually supersede empirical scaling. This involves extensive work on gyrokinetic simulations, which model the micro-instabilities and turbulence believed to govern heat and particle transport. These simulations are computationally intensive but are beginning to reproduce experimental results and provide insight into the physics underlying the scaling laws. The goal is to create integrated modeling suites, like the one developed by the FAIR program, that can predict plasma performance from first principles, reducing reliance on empirical extrapolation.

Notable Implementations

The H98(y,2) scaling is not implemented in a single device but is a foundational design tool used across the global fusion program.

• ITER Organization: The entire design and operational plan for ITER is predicated on achieving H_H98 = 1.0. This value is required to reach the primary mission goal of Q_plasma = 10, which corresponds to producing 500 MW of fusion power from 50 MW of input heating power. The ITER research plan includes dedicated experimental campaigns to validate the scaling law in its unique size and parameter regime.

• JET (Joint European Torus): As the largest operating tokamak and the one most similar to ITER in terms of plasma shape and wall materials (beryllium and tungsten), JET's experiments are crucial for validating the H98(y,2) scaling. Its D-T campaigns, particularly the DTE2 campaign in 2021, provided critical data on the isotopic mass dependence (the M^0.19 term) and confirmed that high fusion power could be achieved in plasmas with H_H98 ≈ 1.0.

• DIII-D and ASDEX Upgrade: These devices are at the forefront of developing advanced operating scenarios intended to exceed the standard H-mode performance. Research on scenarios like the high-poloidal-beta and I-mode regimes aims to find pathways to achieving H_H98 > 1.2 consistently, which would provide a greater operational margin for ITER and make future power plants more compact and economical.

Open Challenges

Despite its success, the H98(y,2) scaling law faces several scientific and engineering challenges.

1. Extrapolation Uncertainty: The primary challenge is the large extrapolation in size and other dimensionless parameters (like normalized gyroradius, ρ*) from current machines to ITER. While the scaling is based on the best available data, hidden physical dependencies that are not prominent in current devices could become dominant in the ITER regime, leading to deviations from the prediction. The Root Mean Square Error (RMSE) of the regression fit is approximately 15%, implying a significant uncertainty band for the ITER prediction.

2. Physics of Confinement Degradation: The physical mechanism behind the degradation of confinement with heating power (the P_L^-0.69 term) is not fully understood. It is believed to be related to the interaction of different scales of turbulence, but a complete theoretical model is lacking. This gap in understanding limits the ability to design scenarios that might mitigate this degradation.

3. Isotope Effect: The weak positive scaling with isotopic mass (M^0.19) is not well explained by basic turbulence theory, which often predicts a null or slightly negative effect. While experimentally observed, the uncertainty in this exponent is significant, and its physical origin remains an active area of research. This is critical for predicting the performance of D-T plasmas based on data from pure deuterium experiments.

4. Applicability to Advanced Scenarios: The scaling was derived primarily from standard ELMy H-mode plasmas. Its applicability to alternative or advanced scenarios, such as quiescent H-mode (QH-mode), I-mode, or scenarios with strong internal transport barriers (ITBs), is not guaranteed. These scenarios are being developed to avoid the damaging effects of large edge-localized modes (ELMs) on plasma-facing components, but their confinement properties may follow different scaling laws.

Outlook

In the next 5-15 years, the role of H98(y,2) scaling will evolve. It will remain the reference for the initial operational phases of ITER, which are expected to begin in the early 2030s. The first plasma and engineering checkout phases will be followed by dedicated experiments in hydrogen and deuterium plasmas designed to systematically test the scaling of confinement with each of the key parameters in the formula. The results from these experiments will provide the first direct test of the law's validity at the reactor scale.

Simultaneously, progress in high-performance computing will continue to advance first-principles modeling. It is anticipated that within this timeframe, validated, multi-scale simulation codes will become predictive tools capable of calculating confinement with an accuracy comparable to or better than empirical scaling laws. This will enable a shift from reliance on empirical extrapolation to a more physics-based design methodology for future reactors like DEMO.

The ultimate test will come when ITER begins its deuterium-tritium (D-T) operations, projected for the mid-to-late 2030s. These experiments will determine if the target H_H98 = 1.0 can be sustained in a burning plasma environment, where self-heating from alpha particles becomes a significant part of the power balance. The success or failure of the H98(y,2) scaling to predict ITER's performance will be a pivotal moment in the history of fusion energy development.

References

1. Chapter 2: Plasma confinement and transport — Nuclear Fusion (ITER Physics Basis) (1999)
2. ITER Physics Basis Editors, et al. Overview of the ITER Physics Basis — Nuclear Fusion (2007)
3. Scaling of H-mode energy confinement from a multi-machine database — Plasma Physics and Controlled Fusion (1994)
4. On the H-mode confinement scaling — Nuclear Fusion (2013)
5. High-fusion-power plasmas in the Joint European Torus — Physical Review Letters (2022)
6. Physics of the L-H transition — Plasma Physics and Controlled Fusion (2002)
7. Dimensionless parameter scaling of turbulent transport in tokamaks — Physics of Plasmas (1998)