Helmholtz decompositions of horizontal structure functions including components associated with cyclone-anticyclone symmetry breaking

In recent years, several studies have been made in which atmospheric and oceanic data were used to decompose horizontal velocity statistics into a rotational component, associated with vertical vorticity, and a divergent component, associated with horizontal divergence. Making the assumption of statistical homogeneity in a horizontal plane, this can be accomplished by relating the rotational and divergent components of the difference between the velocities at two points to the corresponding longitudinal and transverse components, where the longitudinal and transverse directions are parallel respectively perpendicular to the line between the points. In previous studies, the decomposition has most often been made under the assumption of statistical isotropy. Some attempts have also been made to analyse the anisotropic problem. We derive the full anisotropic equations relating the rotational, divergent and the rotational-divergent components of the second order structure functions to the longitudinal, transverse and longitudinal-transverse components and solve the equations analytically. We also derive some results for third order structure functions, with special focus on the components associated with cyclone-anticyclone asymmetry. Based on the analysis of these components and results from previous analyses of aircraft data, it is concluded that there is an exclusively rotational flow component that is giving rise to strong dominance of cyclonic motions in the upper troposphere and a strong dominance of anticyclonic motions in the lower stratosphere in the range of scales from ten to one thousand km