## Abstract

The methods to obtain atmospheric refractive index structure constant $({C}_{n}^{2})$ by instrument measurement are limited spatially and temporally and they are more difficult and expensive over the ocean. It is useful to forecast ${C}_{n}^{2}$ effectively from Weather Research and Forecasting Model (WRF) outputs. This paper introduces a method that WRF Model is used to forecast the routine meteorological parameters firstly, and then ${C}_{n}^{2}$ is calculated based on these parameters by the Bulk model from the Monin-Obukhov similarity theory (MOST) over the ocean near-surface. The corresponding ${C}_{n}^{2}$ values measured by the micro-thermometer which is placed on the ship are compared with the ones forecasted by WRF model to determine how this method performs. The result shows that the forecasted ${C}_{n}^{2}$ is consistent with the measured ${C}_{n}^{2}$ in trend and the order of magnitude as a whole, as well as the correlation coefficient is up to 77.57%. This method can forecast some essential aspects of ${C}_{n}^{2}$ and almost always captures the correct magnitude of ${C}_{n}^{2}$, which experiences fluctuations of two orders of magnitude. Thus, it seems to be a feasible and meaningful method that using WRF model to forecast near-surface ${C}_{n}^{2}$ value over the ocean.

© 2016 Optical Society of America

## 1. Introduction

The electromagnetic waves propagating through the atmosphere changes as it encounters the inhomogeneities of refractive index caused by the atmospheric turbulence, which degrades the image quality of the photoelectric system dramatically [1–3]. Therefore, the knowledge of ${C}_{n}^{2}$ is essential to evaluate the performance of photoelectric system. The intensity of the atmospheric turbulence is usually described by refractive index structure constant ( ${C}_{n}^{2}$, m^{−2}^{/}^{3}), and there are a variety of methods to measure it [4–6]. As almost all of methods using the instruments to obtain ${C}_{n}^{2}$ are limited in spatial-temporal domain, researchers propose a way to forecast ${C}_{n}^{2}$ from Numerical Weather Prediction (NWP) model outputs. In previous years, some representative studies have been carried out to forecast the parameters of atmospheric turbulence with Mesoscale Non-hydrostatic (Meso-Nh) model. Specially, Masciadri’s group devotes themself to simulate optical turbulence with numerical model: The ability of Meso-Nh model to reproduce a correct spatial distribution of ${C}_{n}^{2}$ is supported by a comparison between Scidar measurements and simulations above Cerro Paranal (Chile) [7,8]. However, a set of 3D ${C}_{n}^{2}$ simulation which is provided by Meso-NH model shows that the horizontal distribution of the ${C}_{n}^{2}$ is not necessarily uniform, especially in the first 10 km above the ground [9]. Soon afterwards, a new calibration technique for the optical turbulence parameterization is proposed in order to reduce some systematic errors at astronomical site of Roque de los Muchachos (Canary Islands) [10]. Apply this calibration technique to a sample of 10 nights of the San Pedro Martir site testing campaign (Mexico), and the result shows that the dispersion between measurements and simulations is comparable to that obtained between different instruments measurement (≤30%) [11–12]. Furthermore, a subsample of the ${C}_{n}^{2}$ profiles obtained with a generalized scidar from 41 nights is used for calibrating and quantifying the Meso-NH model’s ability to reconstruct the optical turbulence above Mt Graham (Arizona, USA), site of the large binocular telescope, and the result shows a good ability in reconstructing the shape of the vertical distribution of ${C}_{n}^{2}$ as well as the seeing (*ε*), the isoplanatic angle (*θ*_{0}) and the wavefront coherence time (*τ*_{0}) [13]. On the one hand, the study employs a wide variety of measurements obtained with different instruments running simultaneously to constrain and validate the model at European Southern Observatory ground-based sites. Results obtained are very promising showing that the numerical simulation technique is already mature for an operational implementation in present and forthcoming observatories [14]. On the other hand, the ability of the Meso-Nh model to discriminate different types of optical turbulence behavior has been investigated above Dome C, Dome A and South Pole, and it is evident that three sites have different characteristics in view of the seeing and the surface-layer thickness [15,16]. Along with Cherubini’s group work: The two papers [17,18] present the problem of modeling seeing using the fifth-generation Pennsylvania State University-National Center for Atmospheric Research Mesoscale Model (MM5) model and the preliminary results from a validation study. Furthermore, the papers [19,20] explains how the system evolved into an operational prediction system where the ${C}_{n}^{2}$ or seeing (*ε*) algorithm is implemented within WRF model at the Mauna Kea Weather Center (MKWC), is operationally used at MKWC for prediction of both weather and optical turbulence, referring to website http://mkwc.ifa.hawaii.edu/models/index.cgi?model=wrf-hi. In addition, the validation of using WRF model to forecast atmospheric turbulence parameters has been performed above Observatory del Roque de Los Muchachos site and Gaomeigu Observatory (China). The result shows that this method is sturdy enough to be applied to potential astronomical sites where no instruments are available [21,22]. Besides, the method of using European Center for Medium Range Weather Forecasts (ECMWF) model products to calculate near-surface optical turbulence is evaluated against scintillometry measurements at climatologically distinct sites in western Europe, and the results show that the model can forecast some essential aspects of near-surface ${C}_{n}^{2}$ [23].

To our knowledge, using WRF model to forecast near-surface ${C}_{n}^{2}$ over the ocean is seldom reported. The Bulk model which is based on MOST provides a theoretical basis to calculate near-surface ${C}_{n}^{2}$ from routine meteorological parameters over the ocean [24–26]. The verification of the Bulk model for calculating ${C}_{n}^{2}$ overwater has been performed yet [27–30]. However, the instrument application is limited in lots of environment, such as the ocean surface as well as the complexity underlying surface. To mitigate the above limitation, in this study, we introduce a method of forecasting ${C}_{n}^{2}$ from routine meteorological parameters which are forecasted by WRF model coupled with Bulk model over the ocean near-surface. The ${C}_{n}^{2}$ values measured by the micro-thermometer which is placed on the ship that drops anchor close to the center grid of simulation domain (17.5°N, 109.5°E) used for a comparison.

In Section 2, we give a description of WRF model. In Section 3, we show the theoretical background of estimating ${C}_{n}^{2}$. In Section 4, we briefly describe the principle of the micro-thermometer. In Section 5, we compare the diurnal feature of ${C}_{n}^{2}$ between forecast and measurement. In Section 6, we discuss the possible causes of the deviation and the improvements. Finally, the conclusion of this study is drawn in Section 7.

## 2. WRF model configuration

WRF model is a mesoscale numerical weather prediction model used for both professional forecasting and atmospheric research. It is developed by the National Center of Environment Prediction (NCEP) of the United States and the National Center of Atmospheric Research (NCAR) of the United States. Details of governing equations, transformations and grid adaptation are given in Modeling System User’s Guide, referring to http://www2.mmm.ucar.edu/wrf/users/docs/user_guide_V3/contents.html. In this study, WRF is initialized with the Global Forecast System (GFS) data that has a horizontal resolution of 1° × 1° (in longitude and latitude), downloaded from the NCEP website http://www.ncep.noaa.gov/. The basic simulation parameters setting are listed in Table 1, and the simulation area is shown in Fig. 1.

WRF model exports a large number of routine meteorological parameters (such as temperature, absolute humidity, pressure, wind speed, etc.) which depend upon the physical schemes that have been chosen for the simulation. The main physical schemes used for the simulation are listed in Table 2, and the detailed physical schemes are explained as follows [31]:

- The micro-physics process uses the WRF Single-Moment 3 class (WSM-3) scheme which is suitable for medium-scale grid dimension, and contains three kinds of water materials: water vapor, cloud water or cloud ice, rainwater or snow.
- The Rapid Radiative Transfer Model (RRTM) scheme is used for the longwave radiation. The longwave process is caused by water vapor, ozone, carbon dioxide and other gases, as well as by the optical depth of cloud.
- The shortwave radiation uses the Goddard scheme which is suitable for cloud-resolution models, and contains the effect of graupel.
- The planetary boundary layer uses the Yonsei University (YSU) scheme which is suitable for simulating over the ocean, and adds the process of dealing with entrainment at the top of planetary boundary layer.
- The surface layer uses the Monin-Obukhov scheme which is based on the Monin-Obukhov theory.
- The cumulus parameterization uses the Kain-Fritsch scheme which consists of a cloud model concerning the water vapor lifting and subsidence, with the phenomena, such as, entrainment, detrainment, air-current ascension and subsidence covered.

## 3. Theoretical background

#### 3.1. The atmospheric refractive index structure constant $({C}_{n}^{2})$

For visible and near-infrared wavelength (*λ*), the atmospheric refractive index fluctuation is mainly caused by temperature fluctuation. The refractive index for air (*n*) can be defined as follows [32]

*P*is the atmospheric pressure (hPa),

*T*is the absolute temperature (K),

*q*is the absolute humidity (

*g/kg*),

*ε*= 0.622, and

*γ*= 1 + 0.61

*q*. The functions

*m*

_{1}(

*λ*) and

*m*

_{2}(

*λ*) are given by

*λ*is in micrometers. For the wavelength of interest in this study,

*λ*= 0.55

*μm*,

*m*

_{1}(

*λ*) = 79.0 and

*m*

_{2}(

*λ*) = 66.7.

Because pressure fluctuations are very small, assuming pressure fluctuations are negligible [32]. The atmospheric refractive index fluctuation (*n*′) can be expressed as

*A*and

*B*are defined as

For Kolmogorov turbulence, the atmospheric refractive index structure constant in the inertial-subrange is defined as

*r*is the magnitude of $\overrightarrow{r}$, 〈

*…*〉 represents the ensemble average,

*l*

_{0}and

*L*

_{0}are the inner and outer scales of the atmospheric turbulence and have units of m.

Being similar to the atmospheric refractive index structure constant ${C}_{n}^{2}$, the temperature structure constant $({C}_{T}^{2})$, absolute humidity structure constant $({C}_{q}^{2})$ and the temperature-humidity cross-structure constant (*C _{Tq}*) can be defined as

Applying Eqs. (5) and (6) to Eq. (3), the atmospheric refractive index structure constant ${C}_{n}^{2}$ can be defined in terms of ${C}_{T}^{2}$, ${C}_{q}^{2}$ and *C _{Tq}*, as follows

#### 3.2. The Bulk ${C}_{n}^{2}$ model in near-surface layer

According to MOST, the dimensionless function (*ξ*) is used to express the dynamic near-surface layer property. The *ξ* value is often referred to simply as the “stability”, and is negative in unstable conditions, zero in neutral conditions, and positive in stable conditions, defined as

*z*is the height above the sea surface (m),

*k*is the von-Karman constant (0.35),

*g*is the gravitational acceleration,

*ϑ*is the virtual potential temperature,

_{v}*T*

_{*},

*q*

_{*}and

*u*

_{*}are the scaling parameters for temperature, absolute humidity and wind speed, respectively.

By using the MOST in near-surface layer, the ${C}_{T}^{2}$, ${C}_{q}^{2}$ and *C _{Tq}* can be expressed in terms of

*T*

_{*}and

*q*

_{*}. The expressions ${C}_{T}^{2}$, ${C}_{q}^{2}$ and

*C*are given as follows [33]

_{Tq}*f*(

_{T}*ξ*),

*f*(

_{q}*ξ*) and

*f*(

_{Tq}*ξ*) are determined by experiment, Andreas [34] demands

*f*(

_{T}*ξ*) =

*f*(

_{q}*ξ*) =

*f*(

_{Tq}*ξ*).

*γ*is the temperature-humidity correlation coefficient. We use a value of 0.5 for $\frac{\mathrm{\Delta}T}{\mathrm{\Delta}q}\le 0$, and a value of 0.8 when $\frac{\mathrm{\Delta}T}{\mathrm{\Delta}q}\ge 0$ in this work, following [34,35]. A typical expression

_{Tq}*f*(

_{T}*ξ*) is given by Wyngaard [24]

The scaling parameters for wind speed (*u*_{∗}), temperature (*T*_{∗}) and absolute humidity (*q*_{*}) within the near-surface are defined as the integrated forms of the empirically determined dimensionless functions*Ψ _{U}*(

*ξ*),

*Ψ*(

_{T}*ξ*) and

*Ψ*(

_{q}*ξ*) below

*U*(

*z*),

*T*(

*z*) and

*q*(

*z*) are the wind speed, temperature and absolute humidity at height

*z*, respectively.

*z*,

_{oU}*z*and

_{oT}*z*are the “roughness lengths” of wind speed, temperature and absolute humidity, respectively. The expression

_{oq}*z*can be parameterized as follows [36] where

_{oU}*v*is the kinematic viscosity of air (1.4607×10

^{−5}

*m*

^{2}

*/s*), and we have used a value of 0.0185 for

*α*which is determined by experiment [37].

Seeing that the scalar roughness lengths behave quite similarly, the expression *z _{oT}* =

*z*is assumed [38]. The temperature roughness lengths

_{oq}*z*can be parameterized as

_{oT}*R*is the roughness Reynolds numbers for temperature.

_{T}*R*is parameterized in terms of the roughness Reynolds number for wind speed, ${R}_{U}={z}_{oU}\frac{{u}_{*}}{v}$. The expression

_{T}*R*can be expressed as

_{T}Substituting the Eq. (12) and Eq. (14) into the Eq. (13), we obtain the expression for the temperature roughness lengths *z _{oT}* as follows

In this study, we follow the usual assumption that *Ψ _{T}*(

*ξ*) =

*Ψ*(

_{q}*ξ*) [39,40]. The expression

*Ψ*(

_{U}*ξ*) and

*Ψ*(

_{T}*ξ*) are given as follows

We can express *ξ* by solving Eq. (11) for the scaling parameters *u*_{*}, *T*_{*} and *q*_{*}, and combining with Eq. (8). The expression of stability function *ξ* can be expressed as

Therefore, substituting the scaling parameters *u*_{*}, *T*_{*}, *q*_{*} into Eq. (7) and Eq. (9). The expression ${C}_{n}^{2}$ can be expressed as below

*T*, Δ

*q*, Δ

*U*) are calculated by WRF model outputs. Finally, the functions

*ξ*,

*f*(

*ξ*),

*Ψ*and ${C}_{n}^{2}$ are all expressed by Δ

_{ξ}*T*, Δ

*q*and Δ

*U*, then ${C}_{n}^{2}$ can be calculated by solving Eqs. (17) and (18) with an iterative process (see [41] for details).

## 4. Micro-thermometer

#### 4.1. The principle of micro-thermometer

For visible and near-infrared light, the refractive index fluctuation is mainly caused by temperature fluctuation. The relationship between refractive index structure constant ${C}_{n}^{2}$ and temperature structure constant ${C}_{T}^{2}$ is given by [1, 20]

where*T*is air temperature (K), and

*P*is air pressure (hPa). The process for calculating ${C}_{T}^{2}$ involves the measurement of the square and average of the temperature difference given by two sensors which are separated by a known distance

*r*in the inertial region. The ${C}_{T}^{2}$ is defined as the constant of proportionality in the inertial subrange form of the temperature structure function

*D*(

_{T}*r*). As defined by Obukhov [42], the temperature structure constant $\left({C}_{T}^{2}\right)$ for a Kolmogorov type spectrum is related to the temperature structure function (

*D*(

_{T}*r*)) as below

The platinum probe has a linear resistance-temperature coefficient, and responds to an increase in atmospheric temperature with an increase in resistance. In our case, the probes used to measure the temperature difference at two points which are horizontally separated by 1.0 m apart. The two probes are legs of a Wheatstone bridge, and the resistance of the probe is very nearly proportional to temperature and thus temperature changes are sensed as an imbalance voltage of the bridge. The micro-thermometer system provides ${C}_{T}^{2}$ data by measuring meansquare temperature fluctuations and thus ${C}_{n}^{2}$ data can be acquired.

#### 4.2. Brief description of the micro-thermometer measurement

The ${C}_{n}^{2}$ values measured by micro-thermometer are used to validate the ${C}_{n}^{2}$ values forecasted by WRF model. A 10 *μm* diameter, 20 mm long platinum wire is used as micro-thermometer probe which has the frequency response range of 0.05–30 Hz (within the inertial subrange of the atmospheric turbulence spectrum) and the equivalent rms noise is about 0.002 K. The micro-thermometer is placed on the stern stretch out about 5 m above the sea-level, shown in Fig. 2, and the ship drops anchor close to the center grid of simulation domain (17.5°N, 109.5°E).

The micro-thermometer measured ${C}_{n}^{2}$ values are sampled at 4 seconds time interval, while WRF model outputs at 30 minutes time interval owing to the model configuration limitation. The measured values are averaged over the same time interval (30 minutes) to match with the simulated ones for a meaningful comparison. We should note that WRF model outputs are saved as UTC time format while the micro-thermometer values are saved as Local time format (Local time = UTC time + 08:00). WRF outputs and micro-thermometer data available for comparison are listed in Table 3. In addition, the sky conditions from 8 Jul 2014 to 12 Jul 2014 are all in sunny day, while from 13 Jul 2014 to 14 Jul 2014 are in cloudy or overcast day during the maritime experiment. Furthermore, there is wind field perturbation in early days, while it evolves into stabilization soon afterwards. The high pressure ridge transports from Hainan Island to ocean through time, and the experiment area is located in the control of a high pressure system. The atmospheric condition over the ocean near surface is relatively stable in the view of the real meteorological conditions during the experiment.

Besides, we all know that model error increases through time, so it is fair to expect that a 60-h (3* ^{rd}* day) forecast might have a larger error than a 12-h (1

*day) forecast. The model is run on 2 different nights using the procedure described before in order to avoid model error increasing through a long time-series. Each WRF run is initialized with the GFS data which is released at 12:00 UTC of each day before the running date (7 Jul, 2014 and 11 Jul, 2014). All the simulation times are listed in Table 4.*

^{st}## 5. Result

#### 5.1. The diurnal feature comparison of ${C}_{n}^{2}$ between WRF model and micro-thermometer

The diurnal feature of the near-surface ${C}_{n}^{2}$ forecasted by WRF model and measured by micro-thermometer over the ocean is shown in Fig. 3. To facilitate direct comparison, measured values are also overlaid in Fig. 3. One can see that the forecasted ${C}_{n}^{2}$ is consistent with the measured ${C}_{n}^{2}$ in trend and the order of magnitude in general, and it has the diurnal variation feature of the near-surface optical turbulence over the ocean. The ”sharp drop-offs” of ${C}_{n}^{2}$ during the morning and evening transitions are hardly any visible in this plot. The diurnal variation of the forecasted ${C}_{n}^{2}$ is relatively stable, while the diurnal variation of measured ${C}_{n}^{2}$ is obvious. Moreover, the forecasted ${C}_{n}^{2}$ values almost always correctly capture the order of magnitude which experiences fluctuations of two orders of magnitude, although the forecasted ${C}_{n}^{2}$ slightly underestimates the measured ${C}_{n}^{2}$ at times. Note that the discontinuity (Jul 9 to Jul 12) naturally occurs with different simulations, as it is no data assimilation performed.

#### 5.2. Statistical analysis

We evaluate the reliability of ${C}_{n}^{2}$ forecasted by WRF model using four statistical operators: the BIAS, the root mean square error (RMSE), the bias-corrected RMSE (*σ*) and the correlation coefficient (*R _{xy}*). The expression BIAS, RMSE,

*σ*,

*R*are defined as

_{xy}*=*

_{i}*Y*-

_{i}*X*where

_{i}*X*is the individual ${C}_{n}^{2}$ value measured by the micro-thermometer,

_{i}*Y*the individual ${C}_{n}^{2}$ value forecasted by WRF model at the same time and

_{i}*N*is the number of times for which a couple (

*X*,

_{i}*Y*) is available. ${\overline{X}}_{i}$ and ${\overline{Y}}_{i}$ represent the average value of measured and forecasted parameters.

_{i}The correlation of ${C}_{n}^{2}$ between forecasted by WRF model and measured by micro-thermometer is shown in Fig. 4, and the values of BIAS, RMSE, *σ* and *R _{xy}* for near-surface ${C}_{n}^{2}$ over the ocean are labeled in Fig. 4. One can see that the forecasted ${C}_{n}^{2}$ agrees with the ${C}_{n}^{2}$ measured by micro-thermometer overall. The values of BIAS, RMSE and

*σ*are very small, while the

*R*is high relatively. Hence, the near-surface ${C}_{n}^{2}$ values forecasted by WRF model over the ocean are satisfactory.

_{xy}The histograms and cumulative distributions of overall ${\mathrm{log}}_{10}\left({C}_{n}^{2}\right)$ forecasted by WRF model and measured by micro-thermometer is shown in Fig. 5. Note that we use the ${\mathrm{log}}_{10}\left({C}_{n}^{2}\right)$ to replace the $\left({C}_{n}^{2}\right)$ for the histograms and cumulative distribution plot readable. It is apparent that the shape of the cumulative distribution of forecasted ${\mathrm{log}}_{10}\left({C}_{n}^{2}\right)$ is very similar to the measured ${\mathrm{log}}_{10}\left({C}_{n}^{2}\right)$. From the cumulative distribution, we can extract the mean and the median. The mean (−14.40 vs −14.34) and median (−14.36 vs −14.42) values of ${\mathrm{log}}_{10}\left({C}_{n}^{2}\right)$ between forecasted by WRF model and measured by micro-thermometer are close to each other, showing again a good coherence between forecasted ${C}_{n}^{2}$ and measured ${C}_{n}^{2}$. Consequently, we can confirm that the simulated ${C}_{n}^{2}$ values in this method are practical overall.

## 6. Discussion

In Section 5, we have seen the forecasted ${C}_{n}^{2}$ is well consistent with the measured ${C}_{n}^{2}$ in trend and magnitude as a whole, but the forecasted ${C}_{n}^{2}$ slightly underestimates the measured ${C}_{n}^{2}$ at times. Here we discuss the possible causes of the deviation and the improvements which will be presented in advance.

- In reality, there exists intermittent turbulence or large uncertainties that trigger turbulence so that the value of ${C}_{n}^{2}$ measured by micro-thermometer is not necessarily very small [1]. Moreover, the ${C}_{n}^{2}$ value measured by micro-thermometer could be considered as the “point” measurement at a limited domain, while the ${C}_{n}^{2}$ value forecasted by WRF model coupled with Bulk model should be considered as the statistical average value which is a simplify process of various influencing factor to atmospheric turbulence at 3 km horizontal grid resolution for the center of simulation domain. As far as the impact of the body of a measurement ship to the micro-thermometer value is concerned.
- The empirically determined dimensionless functions
*f*(*ξ*) and*Ψ*(*ξ*) which Bulk method depends upon have been determined with greater certainty in unstable conditions than in stable conditions [43]. The dynamic characteristics of the calculated ${C}_{n}^{2}$ with Bulk model in unstable surface layer are better than that in stable surface layer in general. The ocean near-surface conditions are more complex than other category of underlying surface relatively, and the measurement height or the height where ${C}_{n}^{2}$ value was calculated with the method of Bulk model implement in a WRF model may be above the constant flux layer in a region where MOST is not valid when the ocean surface layer can become very thin in very stable conditions. In the future, we will do more sufficient field experiments to gain a more precise understanding the performances of the model under not perfect conditions (for example lack of homogeneity or different conditions of the ocean) and use this understanding to improve current Bulk model for estimating ${C}_{n}^{2}$. - The surface heterogeneity has a much greater impact on atmospheric turbulence in stable conditions, especially in offshore flow [44]. In this study, the simulation area is in a coastal ocean region where the atmospheric turbulence is impacted by the surroundings greatly and MOST will be invalid if the dynamic atmospheric properties depend upon surface characteristics. It is fair to expect that the ${C}_{n}^{2}$ forecasted by WRF model using Bulk model will perform better in the open ocean where the atmospheric turbulence is impacted by the surroundings slightly.
- In this paper, the lack of measured routine meteorological data which is used to be compared with WRF model is a shortcoming. The functions
*f*(*ξ*) and*Ψ*(*ξ*) used here are the empirical form, and they are probably not appropriate for real ocean surface condition at times. In the future, improvements are expected from WRF model better horizontal grid resolution and outputs time interval, as well as the quite appropriate form of functions*f*(*ξ*) and*Ψ*(*ξ*) would be adopted.

## 7. Conclusion

In this paper, a method that WRF model coupled with Bulk model has been able to forecast routine meteorological parameters (such as temperature, relative humidity, pressure, wind speed, etc.), but also the ${C}_{n}^{2}$ values over the ocean near-surface. It is found that this method almost always captures the correct order of magnitude for ${C}_{n}^{2}$, which experiences fluctuations of two orders of magnitude in view of the comparison between ${C}_{n}^{2}$ forecasted by WRF model and ${C}_{n}^{2}$ measured by micro-thermometer. The forecasted ${C}_{n}^{2}$ is well consistent with the measured ${C}_{n}^{2}$ in trend and the orders of magnitude, as well as the values of the bias, the RMSE and the *σ* for the ${C}_{n}^{2}$ are very small and the correlation coefficient is up to 77.57%. Therefore, we conclude that WRF model coupled with Bulk model performances in reconstructing the ${C}_{n}^{2}$ time series above the ocean near-surface is satisfactory. In future work, we will do more sufficient field experiments to gain a better understanding of how ${C}_{n}^{2}$ behaves with different ocean atmosphere conditions and use this understanding to improve current Bulk model for estimating ${C}_{n}^{2}$. Furthermore, the forms of functions *f*(*ξ*) and *Ψ*(*ξ*) are very important to the accuracy of calculating ${C}_{n}^{2}$ and the quite appropriate form of functions *f*(*ξ*) and *Ψ*(*ξ*) would be adopted in next work.

The result of this study guarantees a concrete practical advantage from the implementation of any potential optical engineering fields where the instruments are difficult to use and it provides the assistance for the equipment of laser atmospheric transmission over the ocean near-surface.

## Acknowledgments

We sincerely acknowledge the reviewers and the editor for their valuable comments and suggestions. We also thank the NCAR and the NCEP for accessing to their initialized meteorological dataset. This work is supported by the National Natural Science Foundation of China (NSFC, Grant Nos. 41275020; 41576185).

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