  The "density limit" is one of the fundamental bounds on tokamak operating
space, and is commonly estimated via the empirical Greenwald scaling. This
limit has garnered renewed interest in recent years as it has become clear that
ITER and many tokamak pilot plant concepts must operate near or above the
widely-used Greenwald limit to achieve their objectives. Evidence has also
grown that the Greenwald scaling - in its remarkable simplicity - may not
capture the full complexity of the disruptive density limit. In this study, we
assemble a multi-machine database to quantify the effectiveness of the
Greenwald limit as a predictor of the L-mode density limit and identify
alternative stability metrics. We find that a two-parameter dimensionless
boundary in the plasma edge, $\nu_{*\rm, edge}^{\rm limit} = 3.0 \beta_{T,{\rm
edge}}^{-0.4}$, achieves significantly higher accuracy (true negative rate of
97.7% at a true positive rate of 95%) than the Greenwald limit (true negative
rate 86.1% at a true positive rate of 95%) across a multi-machine dataset
including metal- and carbon-wall tokamaks (AUG, C-Mod, DIII-D, and TCV). The
collisionality boundary presented here can be applied for density limit
avoidance in current devices and in ITER, where it can be measured and
responded to in real time.
