Effect of finite radar pulse volume on turbulence measurements.

Abstract

Equations relating the mean of the Doppler spectrum and the distribution of point velocities, and their spectra are derived under the assumption that: (1) the scatters follow the air motion faithfully, (2) the reflectivity is constant, and (3) the beam illumination function is separable. It is found that the threedimensional spectral density function is strongly attenuated at scales small compared to the beam dimensions, and essentially unaffected at scales large compared to the beam dimensions. Relationships between the one-dimensional longitudinal and transverse spectra of the mean velocity and three-dimensional spectrum of the point velocities are derived. Numerical computations with a model Kolomogorov-Obukhov turbulences spectrum are performed to illustrate the effects of filtering. Energy at scales small compared to the beam dimensions is attenuated. Energy at scales large compared to the beam dimensions is also reduced, in the case of the one-dimensional spectrum, because the small scales in the orthogonal directions contributing to the energy are attenuated by the filtering. The energy depleted from the spectrum of the mean velocity appears as an increased variance of the Doppler spectrum. The ratio of the total energy under the measured spectrum to that under the spectrum of the point velocities is computed as a function of beam dimensions. An equivalent rectangular filter approximation is proposed for computing the one-dimensional spectra. Analytical results are obtained for the longitudinal spectrum and are shown to be in excellent agreement with the numerical results of the actual filter. The use of a spherical volume equal to that of the actual radar pulse volume is shown to be invalid. (A)