Nature of driving force on an isolated moving vortex in dirty superconductors

We reconsider the force-balance relation on an isolated vortex in the flux flow state within the scheme of time-dependent-Ginzburg-Landau (TDGL) equation. We define force on the vortex by the total force on superconducting electrons in the region $S$ surrounding the vortex. We derive the local momentum balance relation of superconducting electrons and then find the force-balance relation on isolated vortex with taking account of the fact that the transport current in charged superconductors are inherently spatially varying with the scale of penetration depth $\lambda$. We also find that nature of the driving force is hydrodynamic when $S$ is the disk with radius $R$ satisfying $\xi\ll R\ll \lambda$ ($\xi$ is the coherence length) while the hydrodynamic and magnetic parts contribute equally to the driving force for $\lambda \lesssim R$.