Useful, Formulas for Computing Divergence, Vorticity, and Their Errors from Three or More Stations

Abstract Given wind data from three noncollinear observing stations, divergence and vorticity can be computed very efficiently by fitting a linear velocity field to the observed wind components. The four wind gradients and the four kinematic quantities (divergence, vorticity, and stretching and shearing deformation) can be expressed as simple algebraic functions of the station coordinates and the observed wind components. Computation of all eight quantities requires only 31 arithmetic operations. All the methods for computing divergence from three stations (linear fitting, Bellamy's graphical method, the line-integral method, and the linear vector point function method) are shown to be equivalent. The fitting method is extended to a six-station network, using a quadratic velocity field to fit the data. Apart from the inversion of a 6×6 matrix, which needs to be performed only once for a fixed network geometry, the solution is again simple. It is shown that the 6×6 matrix is singular when the stations all ...