The empirical ceiling on plasma density in tokamaks, linking maximum safe operation to plasma current and minor radius.
In 1988 Martin Greenwald, working at MIT's Alcator C-Mod, codified an observation that tokamak operators had long recognized: there is a practical upper limit to the line-averaged electron density a tokamak can sustain before a disruptive termination occurs.1 The limit is expressed remarkably simply:
The formula contains no explicit dependence on magnetic field, heating power, or plasma shape—an astonishing simplification that nonetheless fits data across dozens of machines spanning three decades of operation.2
The Greenwald limit is not a hard boundary but a zone of increasing susceptibility to radiative collapse and MARFE (multifaceted asymmetric radiation from the edge) formation. As density rises, edge cooling intensifies, the current profile contracts, and the plasma becomes vulnerable to tearing modes that trigger disruptions.3
Several mechanisms contribute. Higher density increases resistivity, narrowing the current channel and destabilizing the 2/1 tearing mode. Simultaneously, impurity radiation scales as ne2, so the edge power balance tips toward a thermal collapse. Pellet fuelling and peaked density profiles can push the central density well above nG while keeping the edge density below it, exploiting the fact that the limit applies most strictly to the edge.2
Fusion power scales as n2, so reactor designers want the highest density possible. The Greenwald limit constrains the operating space, especially for steady-state scenarios with modest plasma current. ITER's baseline scenario operates at a Greenwald fraction fG = n̄e/nG ≈ 0.85, deliberately staying below the boundary.4
Spherical tokamaks, with their high current density, naturally have elevated nG values, which is one reason they are attractive for compact, high-power-density fusion systems.