The Greenwald Density Limit
An empirical ceiling on tokamak line-averaged density set by plasma current and minor radius.
An empirical ceiling
Tokamaks cannot run at arbitrarily high density: above a threshold the edge cools, radiation and MARFEs (localized radiative condensations) grow, and the discharge disrupts. Greenwald found that the maximum line-averaged density follows a simple empirical law:
text
n_GW (10^20 /m^3) = I_p (MA) / (pi a^2)with plasma current I_p in mega-amperes and minor radius a in meters. Densities are usually quoted as the Greenwald fraction, the ratio of the actual density to n_GW; most conventional operation stays below one, though edge conditioning and pellet fueling can exceed it modestly.
Why a limit exists
The limit is tied to edge physics: as density rises, the edge temperature drops, radiation losses climb, the current channel contracts, and MHD stability degrades. The exact mechanism is still an area of study, but the empirical scaling is robust across machines.
Why density matters
- Fusion power scales as density squared, so operators want density as high as possible
- The Greenwald limit thus caps achievable fusion power for a given current and size
- Peaked density profiles and good particle control help approach the limit safely
How it constrains design
Together with the Troyon beta limit and the safety-factor (current) limits, the Greenwald limit defines the operating window of a tokamak. Design points are chosen with margin below all three to allow control and avoid disruptions.
Spherical-tokamak note
Spherical tokamaks carry high current in a compact volume, which raises the Greenwald density and helps reach high fusion power density. The Hyperion breeder design point (9.66 MA plasma current) is set consistent with the Greenwald and beta limits, checked as part of the design study.