# Re-Visiting Acoustic Sounding to Advance the Measurement of Optical Turbulence
## 2 Materials and Methods
### Sonic anemometry and Optical Turbulence Point Measurement
 \tag{11}\]  
where \(c_{pa}=1005\) J kg\({}^{-1}\) K\({}^{-1}\) and \(c_{ua}=718\) J kg\({}^{-1}\) K\({}^{-1}\) are the dry air values of the specific heats and \(q\) is, again, the specific humidity. When the air is 100% water vapor, \(q=1\) and Equation (11) provides the specific heat values for pure water vapor, \(c_{pw}=1847\) J kg\({}^{-1}\) K\({}^{-1}\), \(c_{vw}=1386\) J kg\({}^{-1}\) K\({}^{-1}\), and \(\gamma_{w}/\gamma_{a}=1.33/1.4\). However, air in the free atmosphere rarely exceeds about 2% water vapor content by mass; with \(q=0.02\)--which is equivalent to a near sea level dew point temperature of \(\sim\)300 K--\(\gamma_{w}/\gamma_{a}=1.39/1.4\approx 1\). Thus, the leading coefficient to the ratio of \(e/P\) is effectively only \((1-M_{w}/M_{a})\). Accordingly, it had become common practice to assume that the sonic virtual temperature very closely approximates the meteorological virtual temperature and can for all intents and purposes be treated as equivalent; the arguments herein remove any concern about the equivalency of these two parameters. The virtual temperature simplifies the treatment of moisture contributions to many local atmospheric processes including buoyancy and index of refraction gradients, which will be discussed further below. In sum, the sonic virtual temperature, derived through processing of time-delayed acoustic wave propagation, provides ready-access to the meteorological atmospheric virtual temperature. The virtual temperature, in turn, provides a means to quantify the local index of refraction's dependence on both temperature and moisture. Returning to Equation (5), while the results for \(C_{v}^{2}\) and \(\partial(\mathbf{V}(z))\partial z\) are regularly retrieved directly from the sonic anemometer outputs, those for the gradient of the index of refraction, \(\
abla n\), require consideration of the contributing factors. Beginning with Tatarskii, the potential temperature (really the potential virtual temperature)--a conserved quantity--naturally is referenced quite frequently when attempting to infer or evaluate the latter gradient (8; 19; 20). With the advent of new optical techniques to deduce the index of refraction fluctuations, some have chosen \(T_{v}\) as an alternative to using a conservative quantity to assist with quantifying these fluctuations through optical measurements ([PERSON] and [PERSON], 2018). By opting to take the alternative path, separately obtaining accurate water vapor fluctuation measurements is not necessary since the moisture quantification is contained in the virtual temperature. Proceeding in this manner, one begins by selecting an expression for air's index of refraction. Given its lineage, an attractive option is that developed by Edlen, which culminated in what has been labelled as modified-Edlen or Edlen66 ([PERSON], 2018). Amongst other enhancements, Edlen66 accounted for the refractivity of water vapor. Consistent with the development's underlying principle, the air's refractive index is linked to its density according to the following:  
\[n-1=a\rho, \tag{12}\]  
where, according to Edlen66 with air density normalized to 1 kg-m\({}^{-3}\)([PERSON], 2018),  
\[n-1=10^{-6}\left[83.4212+24060.3/\left(130-1/\lambda^{2}\right)+159.97/\left(3 8.9-1/\lambda^{2}\right)\right] \tag{13}\]  
On combining Equations (12) and (13), one can solve for \(a\), which is found to be \(0.227\times 10^{-3}\) m\({}^{3}\)-kg\({}^{-1}\) for visible light (\(\lambda=0.550\)\(\mu\)m) for a standard atmosphere at sea level where the air density is 1.225 kg-m\({}^{-3}\). Consequently, for an air parcel displacedfrom level \(z_{1}\) to level \(z_{2}\), the gradient of the displaced parcel's index of refraction can be approximated as follows:  
\[\
abla n_{z_{1}\to z_{2}}=\overline{\rho}\frac{\partial a}{\partial z}+a \frac{\partial\rho}{\partial z}\approx a\frac{\partial\rho}{\partial z}, \tag{14}\]  
where the first term in Equation (14) is, by [PERSON]'s premise, zero: \(a\) is normalized to air density.