The physical mechanism behind the tokamak density limit scaling is described in terms of a non-linear theory of tearing mode growth in cylindrical geometry coupled to a model for thermal transport in the island. Important new physics features of the model include: (1) island asymmetry due to finite island width in cylindrical geometry, (2) a model of radiation based on local coronal equilibrium including impurity radiation, (3) current perturbations due the perturbed resistivity, and (4) numerical solution of the cylindrical eigenfunctions and Δ′. The semi-analytic cylindrical model is then solved for a wide range of current profiles, magnetic field values, and plasma currents using reasonable assumptions for impurity densities and the Greenwald limit [M. Greenwald et al., Nucl. Fusion 28, 2199 (1988)] is reproduced. The limit is shown to be only weakly dependent on variations in the assumed parameters.