# Re-Visiting Acoustic Sounding to Advance the Measurement of Optical Turbulence
## 2 Materials and Methods
### Structure Parameters
This expression was originally derived from first principles and presented by [PERSON] [8] (p. 102) but was also arrived at through empirical measurement of fine scale temperature fluctuations in the course of optical experiments [10]. Several key assumptions underlie the application of Equation (3) [11]. These include statistically homogeneous and isotropic Kolmogorov turbulence; moisture fluctuations are of small consequence in both the optical (visible) and near infra-red portions of the spectrum, which generally is acknowledged as reasonable except over water surfaces [12]; and pressure fluctuations that are presumed to rapidly dissipate as compared to the temperature fluctuations, and generally thought to be inconsequential. Interestingly then, though it is widely known that the original theory underpinning optical turbulence and its effects originated with regard to velocity fluctuations (pulsations) [7], the velocity structure constant is not explicitly represented in Equation (3). Indeed, very satisfactory results have been obtained by simply calculating \(C_{T}^{2}\) using two high-speed, fine-wire thermometers to record temperature fluctuations as a function of time. With comparable accuracy, 3D sonic anemometers have also been used extensively to record temperature fluctuations [13]. Yet, for instances of neutral stratification, \(C_{T}^{2}\) will trend towards zero, and one would presume optical turbulence strengths would trend likewise. However, observations of such tendencies, as might be anticipated during the diurnal quiescent periods after sunrise or before sunset or on windy or cloudy days, have not been so absolute [14]. [PERSON]'s original development attempted to quantitatively characterize optical turbulence even for these instances of neutral stratification by accounting for the vertical gradients in wind velocity, pressure, and moisture, as well as temperature--which led to fluctuations in the index of refraction and its structure constant [8]. As will be discussed shortly, sonic anemometers are inherently designed to measure speed of sound fluctuations caused by moisture, temperature, and velocity fluctuations and, hence, pressure fluctuations. Turbulent eddies, each with characteristic moisture, temperature, and pressure, cross the sonic anemometer's acoustic propagation path at a velocity linked to that of the localized mean wind velocity as well as fluctuations in the flow's velocity. In fact, [PERSON]'s principle suggests the latter fluctuations can lead to pressure fluctuations. These velocity fluctuations can be characterized according to a derived structure parameter for velocity. As earlier, assuming [PERSON] turbulence and [PERSON]'s frozen turbulence hypothesis, the velocity structure parameter, \(C_{v}^{2}\), which characterizes the eddy and turbulent flow's velocity fluctuations, can be written as follows [11]:  
\[C_{v}^{2}=\left[V_{\perp}(t+\Delta t)-V_{\perp}(t)\right]^{2}(V_{\perp}\Delta t )^{-2/3}, \tag{4}\]  
where \(V_{\perp}\) represents the measurement of the velocity fluctuations transverse to the field instrumentation's characteristic observation path(s). Returning to [PERSON]'s original derivation, where \(C_{v}^{2}\) was shown to describe the turbulence flow's total energy, an explicit expression showing the link between \(C_{n}^{2}\) and \(C_{v}^{2}\) is derived in Appendix A and is as follows:  
\[C_{n}^{2}=1.4C_{v}^{2}[\
abla(n)/\left(\partial(V(z))\partial z\right)]^{2}, \tag{5}\]where \(\
abla\langle n\rangle\) is the gradient of the ensemble average of the fluctuations of the index of refraction, and \(\partial\langle V(z)\rangle\partial z\) is the ensemble average of the vertical gradient of the wind velocity fluctuations. As will be made clear shortly, this result accounts for fluctuations in temperature as well as moisture, by way of \(\
abla n\). This expression brings forward the centrality of the turbulent flow's velocity gradients and fluctuations, which anchored [PERSON], [PERSON], [PERSON], and others' optical turbulence theory.