In a steady state, the linear scaling laws are confirmed between the intensity characteristics of electroconvective (EC) vortex (including the vortex height and electroosmotic slip velocity) and the applied voltage for the nonshear EC flow with finite vortex height near permselective membranes. This finding in the nonshear EC flow is different from the shear EC flow [Kwak et al., Phys. Rev. Lett. 110, 114501 (2013)10.1103/PhysRevLett.110.114501] and indicates that the local concentration gradient has a significant improvement in the analysis of slip velocity. Further, our study reveals that the EC vortex is mainly driven by the second peak effect of the Coulomb thrust in the extended space-charge layer, and the linear scaling law exhibited by the Coulomb thrust is an essential reason for the linear scaling laws of vortex intensity. The scaling laws proposed in this paper are supported by our direct numerical simulation data and previous experimental observations [Rubinstein et al., Phys. Rev. Lett. 101, 236101 (2008)10.1103/PhysRevLett.101.236101].