## 1. Introduction

The physical origin of the optical effects of atmospheric turbulence is in the random index-of-refraction fluctuations, also known as optical turbulence. The energy source for optical turbulence is derived from larger-scale wind shear or convection. Because an analytic solution of the equations of motion is not possible for turbulent flow, statistical treatments are used.

In general, turbulent flow in the atmosphere is neither homogeneous nor isotropic. However, it can be considered to be locally homogeneous and isotropic in small subregions of the atmosphere.^{1} These regions are those whose scale lies between that of the larger eddies that make up the energy source for the turbulence and the small-scale eddies for which viscous effects become important. This region of locally isotropic turbulence is known as the inertial subrange (Fig. 1).

The fundamental statistical description of atmospheric turbulence in the inertial subrange was developed by Kolmogorov (1941) in terms of velocity field fluctuations. Kolmogorov assumed that the velocity fluctuations can be represented by a locally homogeneous and isotropic random field for scales smaller than the large eddies that provide the energy source for the turbulence. This implies that the second- and higher-order statistical moments of the turbulence depend only on the distance between any two points in the turbulent layer.

*n*along the direction

*r*are described by the refractive index structure

*D*(

_{n}*r*). For locally isotropic turbulence fields, the structure function of the velocity field can be written aswhere

*l*≪

*r*≪

*L*, with

*l*being the inner scale, or the size below which viscous effects are important and energy is dissipated into heat, and

*L*is the outer scale, or the size above which isotropic behavior is violated (Fig. 1). For eddies with sizes between the inner and outer scales, fluctuations in the refractive index are correlated. A detailed review of this formulation can be found in Tatarski (1961, 1971), Roddier (1981), and Vernin (2011). Astroparameters of importance for ground-based astronomy can be derived once

*L*

_{tp}is the total pathlength.

Fluctuations in the refractive index are related to corresponding fluctuations in temperature, pressure, and humidity. At high-altitude locations such as Mauna Kea, Hawaii, the humidity fluctuations in the infrared range of the spectrum account for less than 1% of the value of the index of refraction and pressure fluctuations are negligible. Therefore, the refractive index fluctuations associated with the visible and near-infrared region of the spectrum are caused primarily by random temperature fluctuations.

*L*

_{0}is the turbulent mixing length that characterizes the turbulent eddies,

*K*and

_{H}*K*are the exchange coefficients for heat and momentum, and

_{M}*a*is an empirical constant.

*M*that takes into account the fact that in the free atmosphere (i.e., above the ground layer), the adiabatic lapse rate γ

_{a}could be comparable to the environmental temperature gradient:The derivation of (9) in Tatarski (1961, 1971) assumes that the atmosphere is in hydrostatic equilibrium and that the temperature change of a displaced parcel will follow an adiabatic lapse rate. Tatarski (1971) pointed out that this formulation is not valid for temperature fluctuations associated with larger vertical air motions. In Tatarski (1971),

*H*is obtained by expanding

*θ*in series and using the barometric equation, and this approximation is most valid in the lower troposphere.

*H*as a “potential temperature,” which might have led to substitute

*θ*for

*H*in the structure function, and the following is found often in the literature:

As a result there is a lack of clarity in the literature regarding the derivation of *z*^{5/3}; therefore the upper profile is more important.

## 2. Another look at the refractive index structure function

*θ*instead of

*H.*For application with electromagnetic waves, the refractive index

*n*can be expressed as follows:where

*T*is temperature (K),

*p*is pressure (hPa), and

*e*is water vapor pressure (hPa). Because

*T*and

*e*are not conservative additives, (11) can best be expressed as a function of the potential temperature

*θ*and the specific humidity

*q*, which are both conservative variables. The potential temperature is defined aswhere

*p*

_{0}is the reference pressure at 1000 hPa,

*R*is the ideal gas constant, and

*C*

_{υ}is the heat capacity at constant volume. The specific humidity is defined byExpression (11), in terms of

*θ*and

*e*, becomesthat is,

*z*

_{1}to

*z*

_{2}. The value of

*N*for this parcel will undergo the following:with

*θ*and

*q*conserving their values. Therefore the variation of the refractivity at level

*z*

_{2}between the environment and the raised parcel isBy applying (15) to (14) the following expression is found:Using the potential temperature definition (16) becomes

*H*. Accordingly, the expression for

*H*and

*ϑ*and the possible impact that using one versus the other might have in the estimation of optical turbulence.

### Evaluating the difference between H and ϑ

A sample plot of *H* and *ϑ* as a function of height shows that *H* is a good approximation for *ϑ* for most of the atmosphere at and below the temperature inversion (Figs. 2, 3), where the difference between the two curves is at most 1°–1.5°C. However, the values of *H* and *ϑ* start to differ significantly at about 6–7 km above sea level; consequently their derivatives will also differ.

Current weather models that include algorithms to model optical turbulence extend well into the stratosphere. For these applications, *ϑ* is the better choice of conservative variable to use in these algorithms. The results from a case study of the impact of using one formulation versus the other in model calculations are presented in the next section.

## 3. Numerical model application of the new formulation

The model used in this study is the Weather Research and Forecasting (WRF) model (Klemp et al. 2007; http://www.wrf-model.org). The model configuration chosen for this case study is the same operational configuration used at the Mauna Kea Weather Center (MKWC; http://mkwc.ifa.hawaii.edu; Businger et al. 2002). The configuration of WRF is the same as detailed in Cherubini et al. (2011), and the nested domains are shown in Fig. 4. The WRF model is initialized with the National Centers for Environmental Prediction Global Forecasting System (GFS) analyses. Boundary conditions are updated every 6 h also using the GFS analyses.^{2}

In this implementation, the optical turbulence algorithm is parameterized following (18)–(20). The exchange coefficients for heat and momentum, *K _{H}* and

*K*, are parameterized within the model planetary boundary layer scheme [Mellor–Yamada–Janjic scheme (MYJ); Janjic 2002], while the outer length scale of turbulence is parameterized as described in Masciadri et al. (1999). The full details regarding the optical turbulence algorithm are not included here for the sake of brevity and can be found in Cherubini et al. (2011).

_{M}In order for turbulent production to begin under conditions of a stable atmosphere, the turbulent scheme requires a nonzero background for the turbulent kinetic energy (TKE). Within the WRF MYJ boundary layer scheme, which solves the TKE budget equation, the background TKE is set to *E*_{min} = 0.1 m^{2} s^{−2}. For optical turbulence purposes, however, this value is too large to produce realistic values of *E*_{min} = 1 × 10^{−4} m^{2} s^{−2}. Masciadri and Jabouille (2001) and Masciadri et al. (2004) proposed to calibrate the background TKE in the turbulent scheme. In the operational model setting, a calibration of *E*_{min} is included (see Cherubini et al. 2011). In this particular experiment, no calibration on *E*_{min} is performed since doing so is beyond the scope of this paper. Instead *E*_{min} is set as a constant with height: *E*_{min} = 1 × 10^{−4} m^{2} s^{−2}.

### Case study from the 2002 campaign

The vertical distribution of turbulence over Mauna Kea was measured as a part of a site characterization campaign held during October and December 2002. For the purpose of this work, only the data from the Generalized Scintillation Detection and Ranging (G-SCIDAR) for the October portion of the campaign are used. G-SCIDAR is an instrument that remotely measures the vertical distribution of the atmospheric turbulence by analyzing the stellar scintillation of a binary start target produced by the turbulent layers present in the atmosphere. For more details on the G-SCIDAR that was operated during the 2002 Mauna Kea campaign, the reader can refer to Cherubini et al. (2008).

Once the algorithm to calculate *T* and *θ* and given the differences, already described in section 3, between *H* and *θ*. This same experiment has been repeated for 22 and 24 October 2002, and the results show similar findings (not shown).

Clearly, the choice of the denominator in the

A seeing monitor was installed at the summit of Mauna Kea in September 2009. The seeing monitor is composed of two parts: a multiaperture scintillation sensor (MASS) and a differential image motion monitor (DIMM). MASS relies on the analysis of scintillation of single stars (Kornilov et al. 2003) and provides an integrated value of

The optical turbulence algorithm implemented in the current version of the operational WRF model running at the MKWC includes these latest findings. Figure 6 shows an example of good agreement between the seeing observed by the seeing monitor and predicted seeing. In this particular case, the WRF algorithm was able to capture not only the average nightly seeing value but also the variability through the night. The value of this kind of graphic, which is produced operationally and posted on the MKWC Internet site, is that it allows the forecaster to quickly verify the performance of the seeing algorithm on a daily basis.

## 4. Conclusions and discussion

A review of the derivation of the refractive index structure function *ϑ* is used as the passive conservative variable instead of the pseudopotential temperature *H = T + γ _{a}*, which presumes that the atmosphere is in hydrostatic equilibrium. The difference between

*H*and

*ϑ*is illustrated through an example in section 3. Results from a sample case study show a positive impact for the upper troposphere when using the newer formulation of

Moreover, other observational systems can benefit from a corrected formulation of

Work to construct a robust calibration of the revised optical turbulence algorithm is currently in progress. The use of data from the MASS–DIMM system, which has been operating at the summit of Mauna Kea since September 2009, and data from the Thirty Meter Telescope monitoring campaign will allow an accurate calibration that is based on data from a large sample of nights to more completely represent the range of turbulence conditions associated with the naturally occurring atmospheric variability.

## Acknowledgments

We thank René Racine for his constructive comments on an early draft. We also thank Elena Masciadri for insightful discussion on this subject and comments on an early draft.

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