Growth of Circulation around Supercell Updrafts

A formula is derived for the rate of change of circulation around an updraft perimeter at a constant elevation. This quantity depends on the continuous propagation of points on the edge, so an expression for local propagation of the edge is obtained from Petterssen's formula for the motion of an isopleth and the vertical equation of motion. On the edge of an updraft in inviscid anelastic flow, the local propagation velocity along the outward normal is equal to the local nonhydrostatic vertical pressure-gradient force (NHVPGF) divided by the magnitude of the local vertical-velocity gradient. Circulation around an updraft perimeter increases at a rate equal to the line integral around the edge of vertical vorticity times the outward propagation velocity. Formulas are also found for the propagation of an updraft's centroid at a given height and for the acceleration of an updraft's vertical helicity. All of the formulas are tested on exact Beltrami-flow solutions of the governing equations. The relevance of two paradigms of supercell dynamics to local edge propagation and circulation growth of updrafts is evaluated by decomposing the NHVPGF into linearly and nonlinearly forced parts and examining results of supercell simulations in different types of shear. Propagation across the shear and rate of increase of circulation depend mostly on the nonlinear part of the NHVPGF (as in the vertical-wind-shear paradigm) for updrafts in nearly unidirectional shear and on the linear part (as in the helicity paradigm) for updrafts in shear that turns markedly with height.