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Abstract Fine-resolution computer models of supercell storms generate realistic tornadic vortices. Like real tornadoes, the origins of these virtual vortices are mysterious. To diagnose the origin of a tornado, typically a near-ground material circuit is drawn around it. This circuit is then traced back in time using backward trajectories. The rate of change of the circulation around the circuit is equal to the total force circulation. This circulation theorem is used to deduce the origins of the tornado’s large vorticity. However, there is a well-known problem with this approach; with staggered grids parcel trajectories become uncertain as they dip into the layer next to the ground where horizontal wind cannot be interpolated. To circumvent this dilemma, we obtain a generalized circulation theorem that pertains to any circuit. We apply this theorem either to moving circuits that are constrained to simple surfaces or to a ‘hybrid’ circuit defined next. Let A be the horizontal surface at one grid spacing off the ground. Above A the circuit moves as a material circuit. Horizontal curve segments that move in A with the horizontal wind replace segments of the material circuit that dip below A . The circulation equation for the modified circuit includes the force circulation of the inertial force that is required to keep the curve segments horizontal. This term is easily evaluated on A . Use of planar or circular circuits facilitates explanation of some simple flows. The hybrid-circuit method significantly improves the accuracy of the circulation budget in an idealized supercell simulation.