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We investigate the question of how plasma currents circulate and close in the scrape-off-layer (SOL) of convection-limited tokamak plasmas. A simplified two-fluid model describes how currents must evacuate charge at the sheaths due to cross-field currents that are not divergence-free. These include turbulence-driven polarization currents and poloidally asymmetric equilibrium diamagnetic currents. The theory provides an estimate for the radial profile of the floating potential, which reveals a dipolar structure like the one observed experimentally. Simulations with a fluid turbulence code provide evidence for the predicted behaviour of currents and floating potential.