On the Shear and Curvature Vorticity Equations

Abstract

The tendency equations for shear and curvature vorticity are interpreted as a function of the terms that modify speed and direction on in a fluid element. The tendency equations consistent with this interpretation do not contain time derivative on the right-hand side, and the interchange terms are kinematically independent of the shear and curvature vorticity tendencies. It is shown that an understanding of the anholonomic reference frame in which these equations are formulated, and the directional derivatives in this frame, is fundamental for the correct formulation and interpretation of the equations. Previous formulations, none of which have the above properties, are discussed and compared with those proposed here. Since shear and curvature vorticity and their rate of change are not Galilean invariant quantities, the above equations only represent relationships between kinematic and dynamic quantities that hold when the different terms are referred to the same reference system. When the equations are referred to a system of axes fixed to the earth, the new results show that both shear and curvature vorticity tendencies depend explicitly on the earth's rotation, although only the curvature tendency depends on the beta effect. The authors define the interchange between shear and curvature vorticity as the amount of vorticity that is cancelled when the shear and curvature tendencies are added. Except for special cases (e.g., when the flow is horizontally nondivergent and therefore relative vorticity is conserved) this interchange between shear and curvature vorticity cannot be identified with a unique collection of interchange terms on the right-hand side of the tendency equations.