Abstract <p>A two-parameter family of vortices is studied in which the air motion near the vortex axis differs from solid-body rotation and the tangential velocity increases according to a power law. It is shown that such vortices satisfy the angular momentum balance equation augmented with a model of turbulent viscosity, including both the traditional mechanism of eddy viscosity and the mechanism of negative diffusion of angular momentum in the vortex. Emphasis is placed on reconstructing the radial profile of the pressure drop in the vortex in an explicit analytical form. Examples of two-parameter representation of vortices are given, both for discrete, integer parameter values and for a continuous spectrum of their changes. The results are applied to supercell tornadoes. It is shown that the width of the strip on the surface of the earth, swept by the vortex during its movement and determined from the condition that the wind has hurricane force, systematically decreases when the air motion near the vortex axis deviates from solid-body rotation. Using the downward flux of helicity in a vortex as a measure of tornado intensity, as well as, to a certain extent, their “destructive power,” confirms these results. The question of the best approximation, within the framework of a generalized two-parameter family of vortices, of the radial profile of the tangential velocity in the well-known Sullivan vortex is discussed.</p>