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 lie on a conic section. For an n-sided (n>3) polygon of stations, simple formulas for divergence and the other quantities are obtained from the generalized, piecewise-linear, Bellamy, and line-integral methods, which again are found to be equivalent. Also presented are least-squares solutions for both the linear and quadratic fitting methods for networks with more than three and six stations, respectively. Apart from the elements of the square matrices and column vectors being summations instead of individual-station quantities, the least-squares solutions have the same form as the exact ones. Simplified versions of the formulas are presented for networks configured in the form of regular polygons with an interior station at the center included to eliminate the singularity of the quadratic method. In this special case, the 6?6 matrix can be inverted analytically. For such networks, all the methods give the same value of mean divergence, which is independent of the central observation. Tests involving observations from two particular networks of an analytical wind field show that the computed divergence is a much better estimate of the mean divergence over the network than of the divergence at the centroid. The general simplicity of the analytical formulas for the derived quantities permits analysis of errors due to random wind observing errors. Truncation errors also are discussed.