Analysis of the radar reflectivity of aircraft vortex wakes

Abstract

Radar has been proposed as a way of tracking wake vortices to reduce aircraft spacing and tests have revealed radar echoes from aircraft wakes in clear air. The mechanism causing refractive index gradients in these tests is thought to be the same as that for homogeneous and isotropic atmospheric turbulence in the Kolmogorov inertial range, for which there is a scattering analysis due to Tatarski. In reality, however, the structure of aircraft wakes has a significant coherent part superimposed with turbulence, about whose structure very little is known. This work adopts a picture of a coherent (in fact two-dimensional) wake to perform a scattering analysis and calculate the reflected power. In particular, two simple mechanisms causing refractive index gradients are considered: (A) radial pressure (and therefore density) gradient in a columnar vortex arising from the rotational flow; (B) adiabatic transport of atmospheric fluid within a descending oval surrounding a vortex pair. In the scattering analysis, Tatarski's weak scattering approximation is kept but the usual assumptions of a far field and a uniform incident wave are dropped. Neither assumption is generally valid for a wake that is coherent across the radar beam. For analytical insight, an approximate analysis that invokes, in addition to weak scattering, the far-field and wide cylindrical beam assumptions, is also developed and compared with the more general analysis. Reflectivities calculated for the oval (mechanism B) are within 2–13 dB m2 of the measurements (≈−70 dB m2) of MIT Lincoln Laboratory at Kwajalein atoll. However, the present predictions have a cut-off away from normal incidence which is not present in the measurements. This implies that the two-dimensional picture is not entirely complete. Estimates suggest that the thin layer of vorticity which is baroclinically generated at the boundary of the oval is turbulent and this may account for reflectivity away from normal incidence. The reflectivity of a vortex (mechanism A) is comparable to that of the oval (mechanism B) but occurs at a frequency (about 50 MHz) that is lower than those considered in all the experiments to date. This result may be useful because: (i) existing atmospheric radars (known as ST radars) already operate at this frequency and so the present prediction could be verified; (ii) rain clutter is not a problem at this frequency; (iii) mechanism A is more robust because it is independent of atmospheric stratification.