A System of Equations Describing Quasi-Symmetric Mesoscale Phenomena

Abstract

An approximated system of equations describing mesoscale (horizontal length scale 10⁵ m) phenomena is constructed. The wind speed of the mesoscale phenomena considered is assumed to be quasi-symmetric along the horizontal stream lines. The quasi-symmetry means that the variation of wind speed in the tangential direction of horizontal stream lines is one order smaller than that of the mesoscale.
As a result of the quasi-symmetry, the primary flow is in a generalized gradient balance with geopotential φ. The primary flow (i.e., the generalized gradient flow) is parallel to the iso-φ lines, and determined by the derivatives of φ up to the second.
The secondary flow, which constitutes the secondary circulation in the vertical plane perpendicular to the primary flow, is algebraically determined by the derivatives of φ up to the third.
The constructed system of equations, which include the primary and secondary flows, exactly conserves the energy, while the potential vorticity is only approximately conserved. The potential vorticity is exactly conserved, when or where the local directional angle of iso-φ lines is vertically and temporally uniform.
When or where the flow is large-scale, then or there the present system is reduced to the quasi-geostrophic system. Furthermore, for quasi-circular flows, the present equations are reduced to the balanced vortex equations.