Greenwald Density Limit Plasma Calculator
Tokamak Magnetic Confinement: Evaluate the empirical Greenwald density limit ($n_G = I_p / (\pi a^2)$), operational density fraction ($n_e/n_G$), and plasma disruption safety margin.
Tokamak Geometry & Plasma Current
Density Limit & Stability Metrics
The Greenwald Limit & Tokamak Operational Boundaries
In magnetic confinement fusion, achieving high fusion power requires maximizing fuel density ($P_{fus} \propto n_i^2$). However, tokamak density is constrained by magnetohydrodynamic (MHD) and radiative limits.
1. Mathematical Formulation
n_G = I_p / (π · a²) [10²⁰ m⁻³] f_G = n̄_e / n_G [Greenwald Fraction] q_cyl = (5 · a² · B_T) / (R₀ · I_p) · ((1 + κ²) / 2)
2. Key Regimes
- f_G < 0.70: Standard ELMy H-mode regime with reliable pedestal energy confinement.
- 0.70 ≤ f_G ≤ 1.00: Pedestal degradation, increased divertor detachment, and edge Marfe formation.
- f_G > 1.00: Radiative collapse, $m/n=2/1$ mode locking, and major thermal disruption unless stabilized by high core peaking.
Frequently Asked Questions
What is the Greenwald density limit in tokamak physics?
The Greenwald density limit is an empirical upper bound on the line-averaged electron density in tokamaks: $n_G = I_p / (\pi a^2)$ [expressed in units of $10^{20}\,\text{m}^{-3}$, with plasma current $I_p$ in $\text{MA}$ and minor radius $a$ in $\text{m}$]. Discovered by Martin Greenwald in 1988, attempting to operate standard unpelletized H-mode or L-mode plasmas above $f_G = n_e / n_G > 1.0$ triggers intense edge radiation (Marfes), magnetic tearing modes ($m/n = 2/1$), and catastrophic plasma disruptions.
Can advanced tokamaks exceed the Greenwald limit?
Yes. Cryogenic core pellet injection (high-field-side launch) and central neutral beam fueling can deposit fuel deep inside the magnetic core without swelling the scrape-off layer (SOL) edge density. Tokamaks like DIII-D, ASDEX Upgrade, and EAST have sustained peaking factors enabling $n_e/n_G \approx 1.2\sim 1.5$ while maintaining an edge density safely below the disruptive boundary.
Why does exceeding the Greenwald limit cause a plasma disruption?
At high edge electron densities, atomic impurity radiation (carbon, tungsten, beryllium) exceeds the local ohmic and auxiliary heating input power in the edge plasma. This cools the edge, shrinking the current profile and steepening the radial current gradient $dj/dr$ near the $q=2$ rational flux surface. The resulting resistive tearing mode grows rapidly, locking to the wall and triggering a thermal quench followed by a vertical displacement event (VDE).