A nonlinear formula for updraft motion in supercell storms is derived from Petterssen's formula for the motion of systems and the vertical equation of motion, and tested on an exact Beltrami flow solution. At each level, continuous propagation of an updraft maximum is determined largely by the horizontal gradient of the nonhydrostatic vertical pressure-gradient force (NHVPGF) at the updraft center. The NHVPGF is deduced from the formal solution of the Poisson equation for nonhydrostatic pressure subject to homogeneous Neumann boundary conditions at horizontal boundaries. The solution is partitioned into parts that arise from forcing terms involving buoyancy, linear interaction between the environmental shear and updraft, and nonlinear dynamical effects. Recourse also is made to published fields of partitioned vertical pressure-gradient force. The dynamics of supercell storms in nearly straight and highly curved hodographs are found to be different. Nonlinear rotationally induced propagation is important during storm splitting in unidirectional shear where the vortex pair, formed at mid levels by lifting of environmental vortex tubes, straddles the initial updraft. After the initial storm splits into severe right-moving (SR) and left-moving (SL) supercells, anomalous motion is maintained by the distribution of nonlinear NHVPGF. For the SR storm, the NHVPGF is upward below the cyclonic vortex on the right forward side of the updraft and downward on the left rear side. The anticyclonic vortex on the left side of this storm migrates to the downdraft and so does not affect updraft propagation. For a storm in shear that turns markedly clockwise with height, the cyclonic vortex is nearly coincident with the updraft while the anticyclonic vortex is located in downdraft so that the horizontal gradient of nonlinear NHVPGF at the updraft center associated with rotationally induced propagation is relatively small. Linear shear-induced propagation now becomes the dominant mechanism. At each level, propagation off the hodograph to the concave side increases with updraft width. Once the propagation has been deduced, linear streamwise-vorticity theory adequately predicts the sense of overall updraft rotation in all cases.