Theory of Parcel Vorticity Evolution in Supercell-Like Flows

In a prior paper, insights into tornadogenesis in supercell storms were gained by discovering analytical formulas for vorticity variations along streamlines in idealized, steady, frictionless, isentropic inflows of dry air imported from a horizontally uniform environment. This work is simplified and extended to the evolution of parcel vorticity in unsteady, nonisentropic flows by integrating the vorticity equation using nonorthogonal Lagrangian coordinates. The covariant basis vectors (e) over bar( 1), (e) over bar (2), and (e) over bar( 3) are material line elements attached to each parcel. Initially they form an orthonormal set with (e) over bar( 1) in the direction of and (e) over bar( 2) left normal to the storm-relative wind at each level in the environment, and e over bar 3 upward. The surface containing all parcels with the same initial height constitutes a material surface, within which initially streamwise and transverse material lines are reoriented and stretched or shrunk. The basis vectors propagate a parcel's barotropic vorticity through time by factoring in the "frozen-field " effect. With a horizontally uniform environment, the barotropic vorticity of a parcel depends on its initial streamwise vorticity times its current (e) over bar (1) plus its initial crosswise vorticity times its current (e) over bar( 2). For baroclinic and frictional vorticity, each contravariant component is the integral from initial to current time of the corresponding contravariant component of the generation vector. The "river-bend " effect acting on all parts (baroclinic, frictional, and barotropic) of transverse vorticity produces streamwise vorticity (parallel to 3D wind). In left-turning steady flow, it arises from (e) over bar( 2) rotating toward (e) over bar( 1). For steady, frictionless, dry isentropic flow, previous vorticity formulas are recovered.