Superconducting depairing current density ($j_d={\Phi }_0/(3\sqrt{3}\pi {\mu }_0{\lambda }^2\xi )$)

For uniform flow in local Ginzburg-Landau theory of superconductivity, a phase gradient $q$ suppresses the equilibrium condensate density by $1-{\xi }^2{q}^2$. Hence $j(q)={\Phi }_{0}q(1-{\xi }^2{q}^2)/(2\pi {\mu }_0{\lambda }^2)$, maximized at $q=1/(\sqrt{3}\xi )$. This intrinsic local limit differs from vortex entry, depinning and the cross-section average in a thick wire. A Meissner surface shell has average-to-surface current-density ratio approximately $2\lambda /a$ for wire radius $a\gg \lambda$.