In this study we report on a 2D fluid model of a gliding arc discharge in argon. Despite the 3D nature of the discharge, 2D models are found to be capable of providing very useful information about the operation of the discharge. We employ two models—an axisymmetric and a Cartesian one. We show that for the considered experiment and the conditions of a low current arc (around 30 mA) in argon, there is no significant heating of the cathode surface and the discharge is sustained by field electron emission from the cathode accompanied by the formation of a cathode spot. The obtained discharge power and voltage are relatively sensitive to the surface properties and particularly to the surface roughness, causing effectively an amplification of the normal electric field. The arc body and anode region are not influenced by this and depend mainly on the current value. The gliding of the arc is modelled by means of a 2D Cartesian model. The arc–electrode contact points are analysed and the gliding mechanism along the electrode surface is discussed. Following experimental observations, the cathode spot is simulated as jumping from one point to another. A complete arc cycle is modelled from initial ignition to arc decay. The results show that there is no interaction between the successive gliding arcs.