Greenwald Density Limit Physics T-Shirt

The Greenwald limit: an empirical speed limit for plasma density.

The Physics

In a tokamak, the line-averaged electron density tends to stay below a value set by the plasma current and the size of the plasma. Martin Greenwald and co-authors put it in this form in 1988. In effect it is set by the average plasma current density I_p/(πa²).

n̄ₑ ≲ n_G = I_p / (πa²) [n_G] = 10²⁰ m⁻³ · [I_p] = MA · [a] = m

• n_G: Greenwald density (10²⁰ m⁻³)
• n̄ₑ: line-averaged electron density (10²⁰ m⁻³)
• I_p: plasma current (MA, 10⁶ A)
• a: plasma minor radius (m)
• f_G = n̄ₑ / n_G: Greenwald fraction (dimensionless)

The units matter: the formula only works with n_G in 10²⁰ m⁻³, I_p in MA and a in m. As an illustration, I_p = 15 MA and a = 2.0 m give n_G ≈ 1.19 × 10²⁰ m⁻³.

Honest caveat: The Greenwald limit is an empirical scaling, not a law of nature. In tokamaks, pushing the line-averaged density above it with ordinary edge (gas) fuelling typically ends in a density-limit disruption, often preceded by edge radiative collapse (e.g. MARFEs). It has been exceeded in experiments, especially with peaked density profiles from pellet or core fuelling, for example up to about 1.5 n_G quasi-steadily on ASDEX and about 2 n_G transiently on TFTR. Stellarators routinely exceed it.

Sources

• M. Greenwald, J. L. Terry, S. M. Wolfe, S. Ejima, M. G. Bell, S. M. Kaye, G. H. Neilson, "A new look at density limits in tokamaks," Nucl. Fusion 28, 2199–2207 (1988).
• M. Greenwald, "Density limits in toroidal plasmas," Plasma Phys. Control. Fusion 44, R27–R53 (2002).
• FusionWiki (CIEMAT), "Greenwald limit."

On the shirt

Back: "GREENWALD LIMIT — AN EMPIRICAL SPEED LIMIT FOR PLASMA DENSITY", with n̄ₑ ≲ n_G = I_p / (πa²), the units line [n_G] = 10²⁰ m⁻³ · [I_p] = MA · [a] = m, the Greenwald fraction f_G = n̄ₑ / n_G, ASDEX and TFTR markers, and "GREENWALD ET AL. 1988 · NUCL. FUSION 28, 2199". Printed front and back on a unisex tee.