Every open Riemann surfaceRof finite genus can be continued to a closed Riemann surface of the same genus. This classical result is usually proved by a local argument: one considers only a planar neighborhood of the ideal boundary ofRand applies the generalized uniformization theorem of Koebe. In the present paper we prove a continuation theorem of a global character: Let there be given a meromorphic functionfonRwith a special boundary behavior. Im(dfshall be a distinguished harmonic differential of Ahlfors. Then there exists a closed Riemann surfaceR*of the same genus asRand a meromorphic extensionf*offontoR*such that (i)R*\Rhas a vanishing area, (ii)f*is holomorphic onR*\R, and (iii) Imf*assumes a constant value on each boundary component ofRwith respect toR*Since ƒ describes a hydrodynamic phenomenon onR, we callR*a hydrodynamic continuation ofRwith respect tof. The ideal boundaryRis then realized onR*as a set of arcs on the streamlines offwith a total vanishing area.