The Greenwald Density Limit

An empirical ceiling on the line-averaged density set by the plasma current and minor radius.

The Limit

The Greenwald limit is an empirical bound on the line-averaged electron density in a tokamak: n_G = I / (pi a^2), with the density in units of ten to the twentieth per cubic meter, current I in mega-amperes, and minor radius a in meters. Discharges pushed above the Greenwald fraction (density divided by n_G) near one tend to develop a cold radiative edge, detached plasma, and often disruptions.

Why It Matters

Fusion power density scales with the square of the density, so operating at high density is desirable. The Greenwald limit therefore constrains achievable performance directly. Because n_G is proportional to current over area, higher current or more compact plasmas raise the accessible density, which is another reason high-current, high-field devices are attractive.

Physical Mechanism

Unlike the beta and current limits, the Greenwald limit is not a clean ideal-MHD boundary; it reflects edge physics. As density rises the edge cools, radiation losses grow, the current profile contracts, and MHD instability or a radiative collapse follows. Careful fueling, edge control, and high edge temperature allow operation modestly above the classic limit, so it is a soft, physics-dependent boundary rather than a hard wall.

Relevance

The Greenwald fraction is a standard planning parameter for any tokamak scenario, traded against confinement and power. For the Hyperion breeder concept, the high plasma current relative to its compact size raises the Greenwald density, supporting high fusion power density in design-stage modeling; these are simulation values for a machine not yet built.