Investigating the Outer Scale of Atmospheric Turbulence with a Hartmann Sensor

Abstract

A Hartmann Turbulence Sensor (HTS) system has been used to study the outer scale of turbulence. The atmospheric turbulence power spectrum is usually presumed to obey the Kolmogorov power law within some inertial range, while at spatial frequencies outside this range, the power spectrum is expected to fall away from this curve. The outer scale is the spatial frequency where the low frequency side of this roll-off occurs. In length units the outer scale is just the inverse of this spatial frequency. In the free atmosphere, this outer scale is presumed to be on the order of a hundred meters, but near the ground, the outer scale is expected to be on the order of the height above ground. The HTS used for this study has an aperture of 16“ and employed a beam path which was around 5’ above the ground, thus the effects of the outer scale are expected to be minimal within the telescope aperture. However by relying on the cross wind to move turbulence across the telescope aperture much longer baselines can be achieved and outer scale effects can be sought. The presumption that the dominant temporal variation in turbulence is wind driven translation is called the Taylor Frozen Flow Hypothesis. When an outer scale is introduced into the Kolmogorov power spectrum the resulting power spectrum is called the von Karman power spectrum. The wave structure functions due to these two power spectra are very different, as the Kolmogorov spectrum leads to a structure function which increases without bound as the separation between points increases, whereas the structure function due to the von Karman spectrum rolls-over near the outer-scale and becomes constant. Unfortunately, the structure function itself isn't measured by the HTS. Instead, the HTS can observe the tilt differences between subapertures separated in space or time. The Taylor Frozen Flow Hypothesis can then be used to switch between time and space. It is clear in the experimental data that this presumption is largely correct for some of the cases studied. Some of the data sets were collected with the fortuitous condition that the wind was approximately perpendicular to the path, with product between the wind speed and frame rate nearly matching the subaperture spacing. The expected differential tilt variance between subaperture pairs rolls over as the subaperture spacing increases and approaches a constant value for both the Kolmogorov and von Karman power spectra, however in the case of the von Karman spectrum this roll-over happens more clearly and the differential tilt variance exhibits a broad weak peak near the outer scale. Also, in the case of the von Karman spectrum the constant value approached is smaller. Comparisons between measured differential tilt variances and those predicted by theory allows some estimates to be made about the size of outer scale.