Extending a surface-layer Cn2 model for strongly stratified conditions utilizing a numerically generated turbulence dataset

Ping He; Sukanta Basu

doi:10.1364/OE.24.009574

1. Introduction

Accurate estimation of optical turbulence in the lower atmosphere is essential for various ground-based optical applications (e.g., astronomical observation, laser communication and target detection) [1]. The intensity of optical turbulence is commonly quantified by the refractive index structure parameter ( $C_{n}^{2}$ , m⁻²^/³). The Monin-Obukhov similarity theory (MOST) [2] has found widespread usage for estimating $C_{n}^{2}$ in the surface layer (typically within tens of meters above the ground; see a brief review in [3]). Traditionally, the application of MOST requires information of turbulence statistics (e.g., momentum and sensible heat fluxes [4, 5]); however, measurements of these quantities require research-grade instruments and are not readily available. To circumvent this limitation, the gradient-based MOST approach can be utilized, which only requires routine meteorological information (wind speed and temperature) at two heights. Essentially, the gradient-based approach connects turbulence statistics and mean gradients using the so-called flux-gradient similarity relationship (e.g., the Businger – Dyer relationship [6, 7]). In light of this concept, a number of gradient-based formulations have been proposed to calculate $C_{n}^{2}$ in the surface layer ([8–15], to name a few). For example, in Wyngaard et al. [8] (hereinafter refer to as W71), a simple $C_{n}^{2}$ model was proposed:

\frac{C_{T}^{2}}{z^{4 / 3} {(\frac{\partial \bar{θ}}{\partial z})}^{2}} = g_{T} (R i_{g}) .

Here

C_{T}^{2} (K^{2} m^{- 2 / 3})

is the temperature structure parameter. z (m) is the height above the ground.

\bar{θ} (K)

is the mean potential temperature. g_T is the similarity function. Ri_g is the gradient Richardson number defined as:

R i_{g} = \frac{g}{\bar{θ}} \frac{\partial \bar{θ} / \partial z}{S^{2}} .

Here g (m s⁻²) is the gravitational acceleration.

S = \sqrt{{(\partial \bar{u} / \partial z)}^{2} + {(\partial \bar{v} / \partial z)}^{2}}

is the mean wind shear with

\bar{u} ({m s}^{-}^{1})

and

\bar{v} ({m s}^{-}^{1})

being the horizontal velocity components. Ri_g is a stability parameter which reflects the local balance between buoyancy and shear generation. Generally, over land, Ri_g is positive during nighttime (stable condition), and it becomes negative during daytime (unstable condition).

Given the gradient information (which is more easily available than turbulent fluxes) and the expression for g_T, $C_{T}^{2}$ can be directly calculated using Eq. (1). For visible and near-infrared light, $C_{T}^{2}$ can be easily converted to $C_{n}^{2}$ using:

C_{n}^{2} = {(7.9 \times 10^{- 5} \frac{P}{T^{2}})}^{2} C_{T}^{2} {(1 + \frac{0.03}{β})}^{2},

where P (hPa) and T (K) are pressure and temperature, respectively, and β is the ratio of sensible and latent heat fluxes (known as the Bowen ratio) [4, 5].

Although the gradient-based model can be used to estimate $C_{n}^{2}$ in a practical way, the most challenging part is a robust construction of similarity function g_T in Eq. (1). In W71, g_T was constructed using the observational data collected during a field campaign in Kansas [6, 16]:

\frac{C_{T}^{2}}{z^{4 / 3} {(\frac{\partial \bar{θ}}{\partial z})}^{2}} = g_{T} (R i_{g}) = g_{T} (f_{T} (ζ)) = {\begin{array}{l} 1.07 {(\frac{1 - 9 ζ}{1 + 0.5 | ζ |^{2 / 3}})}^{1 / 2}, & ζ \leq 0, & (4 a) \\ \frac{0.79}{(0.74 + 4.7 ζ) {(1 + 2.5 ζ^{3 / 5})}^{1 / 2}}, & ζ > 0. & (4 b) \end{array}

Here ζ is the flux-based stability parameter, which can be related to Ri_g through:

R i_{g} = f_{T} (ζ) = {\begin{array}{l} 0.74 ζ {(\frac{1 - 15 ζ}{1 - 9 ζ})}^{1 / 2}, & ζ \leq 0, & (5 a) \\ \frac{ζ (0.74 + 4.7 ζ)}{{(1 + 4.7 ζ)}^{2}}, & ζ > 0. & (5 b) \end{array}

An example of g_T based on Eqs. (4) and (5) is shown in Fig. 1 where the similarity function (g_T) is plotted against the stability parameter (Ri_g).

Fig. 1 Dependence of similarity function (g_T) on stability parameter (Ri_g). The plot is based on the similarity function proposed in W71 [8], i.e., Eqs. (4) and (5).

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It is well known in the literature that $C_{n}^{2}$ estimation for stable conditions is much more challenging compared with unstable conditions (see [13, 15] for example). In the case of W71, there is a limitation that the range of application of g_T is very short under stable conditions. As can be seen in Fig. 1, g_T (normalized $C_{T}^{2}$ ) drops off sharply as Ri_g approaches ~0.2, implying that turbulence diminishes at this limit. However, extensive laboratory and field experiments have shown evidence that turbulence survives when Ri_g is larger than 0.2 (see [17, 18] for example). These findings are also corroborated by the results to be presented later in this study; we observe that Ri_g is larger than 0.2 in about 50% of the stably stratified conditions. It is evident that the relatively short range of g_T significantly limits its applicability for estimating $C_{n}^{2}$ during nighttime.

To mitigate the above limitation, in this study, we construct a new similarity function g_T for stable conditions using an extensive dataset generated by a high-fidelity numerical modeling approach called direct numerical simulation (DNS). In the DNS approach, the Navier – Stokes (N−S) equations are solved without any averaging, filtering, or other approximations for turbulence, and the grid spacing is set to be on the order of Kolmogorov scale (centimeters or less) to capture the smallest scale of turbulent motions. Owing to its precise representation of the N−S equations, DNS is considered to be of extremely high fidelity and is utilized to provide a better physical understanding of various types of turbulent flows [19]. Recently, DNS was shown to have the potential of developing similarity functions for $C_{n}^{2}$ given its low uncertainty compared with the observationally-based approaches [20]. More importantly, by constructing a comprehensive DNS dataset, a wide range of stabilities can be covered which provides a robust estimation of g_T for Ri_g larger than 0.2. We would like to highlight that the novelty of this paper is that it constructs a new surface-layer $C_{n}^{2}$ model using the high-fidelity numerical simulation approach (DNS). Moreover, we focus on the model performance in capturing $C_{n}^{2}$ variations in strongly stratified conditions, such topic has not been addressed in the previous studies [8–15].

The structure of this paper is as follows: In Section 2, we introduce the DNS approach and summarize the computational configurations. Construction of a new similarity function g_T for stable conditions using DNS is reported in Section 3. In Section 4, the validation of the newly proposed g_T is conducted, followed by the summary of this paper in Section 5.

2. Direct Numerical Simulation

In this work, an extensive DNS dataset is created consisting of five simulations of stably stratified open-channel flows at bulk Reynolds number Re_b = u_bh/ν = 20000. Here h, u_b, and ν are the channel height, the bulk (averaged) velocity in the channel, and the kinematic viscosity, respectively. This DNS dataset is an extension of our previous work [20] with a much higher Reynolds number and a wider stability range. Moreover, we utilize a newly developed DNS solver (called HERCULES [21]) for the simulations to achieve a higher numerical accuracy. The simulations are conducted by solving the normalized N−S and temperature equations:

\frac{\partial u_{j}}{\partial x_{j}} = 0,

\frac{\partial u_{i}}{\partial t} + \frac{\partial u_{i} u_{j}}{\partial x_{j}} = - \frac{\partial p}{\partial x_{i}} + \frac{1}{{Re}_{b}} \frac{\partial}{\partial x_{j}} \frac{\partial u_{i}}{\partial x_{j}} + Δ P δ_{i 1} + R i_{b} θ δ_{i 3},

\frac{\partial θ}{\partial t} + \frac{\partial θ u_{j}}{\partial x_{j}} = \frac{1}{{Re}_{b} \Pr} \frac{\partial}{\partial x_{j}} \frac{\partial θ}{\partial x_{j}} .

Here ΔP is the streamwise pressure gradient driving the flow and p is the dynamic pressure.

R i_{b} = (θ_{t o p} - θ_{b o t}) g h / u_{b}^{2} θ_{t o p}

is the bulk Richardson number with the subscripts bot and top denoting the bottom and top walls of the channel, respectively. Pr = ν/k = 0.7 is the Prandtl number with k being the thermal diffusivity. The simulation domain size is chosen to be L_x × L_y × L_z = 18h × 10h × h with 2304 × 2048 × 288 grid points in the x (streamwise), y (spanwise), and z (wall normal) directions, respectively. The other simulation configurations are similar to our previous work [20] with two exceptions. First, the constant mass flow rate condition is imposed instead of the constant pressure gradient condition. This modification allows a decreasing friction velocity during the simulations. Second, instead of fixed bottom wall temperature, all the simulations use fully developed neutrally stratified flows as initial conditions. A gradual temperature decrease is then applied at the bottom wall (by changing Ri_b) to mimic the cooling of the ground during nighttime. It is found that the second modification is critical for obtaining similarity function g_T for Ri_g larger than 0.2. In this study, five normalized “cooling rates” are imposed which cover both weakly and strongly stable conditions, i.e., Ri_b/t = 1 × 10⁻³ to 5 × 10⁻³ with t being the non-dimensional time. Finally, the simulations are run for t = 100 and the simulation results are saved every 10 non-dimensional time. The simulations were conducted on the Army Research Laboratory High-Performance Computing (ARL HPC) system using 4096 CPU processors, and each simulation took about 80 hours.

3. Construction of New Similarity Function g_T

In order to construct the dependence of g_T with respect to Ri_g using the DNS output, we first calculate the left hand side of Eq. 1. Here $C_{T}^{2}$ can be calculated using the Corrsin’s expression [22]:

C_{T}^{2} = 1.6 ε^{- 1 / 3} χ,

where ε and χ are the energy and temperature dissipation rate, respectively. ε and χ can be directly related to the DNS output through:

ε = 〈 ν \frac{\partial u_{i}^{'}}{\partial x_{j}} \frac{\partial u_{i}^{'}}{\partial x_{j}} 〉, χ = 〈 2 k \frac{\partial θ_{i}^{'}}{\partial x_{j}} \frac{\partial θ_{i}^{'}}{\partial x_{j}} 〉 .

Here < · > denotes horizontal (planar) averaging, and the prime symbol represents the fluctuation of a variable with respect to its planar averaged value. Therefore, the left hand side of Eq. 1 is calculated as:

1.6 {〈 ν \frac{\partial u_{i}^{'}}{\partial x_{j}} \frac{\partial u_{i}^{'}}{\partial x_{j}} 〉}^{- 1 / 3} 〈 2 k \frac{\partial θ_{i}^{'}}{\partial x_{j}} \frac{\partial θ_{i}^{'}}{\partial x_{j}} 〉 z^{- 4 / 3} {(\frac{\partial 〈 θ 〉}{\partial z})}^{- 2} .

Ri_g can be calculated using the DNS output in a similar manner.

In this paper, g_T is constructed based on the DNS-generated data at different times (50 < t < 100) and height locations (0.1 < z < 0.9H_b), resulting in about 4000 samples. Here H_b is defined as the height where the local sensible heat flux drops to 10% of its surface value. Please refer to [20] for more details on ensemble calculation using the DNS output.

Figure 2 shows the dependence of similarity function (g_T) on stability parameter (Ri_g) constructed from the DNS dataset. One can see that the DNS-based g_T (black symbols) agrees reasonably well with that proposed by W71 (red dashed line) for Ri_g < 0.1. However, in the range Ri_g > 0.1, the W71 g_T drops off sharply and goes to zero around Ri_g ≈ 0.2. In contrast, a relatively wider range of Ri_g is covered in the DNS-based g_T. Moreover, the DNS-based g_T exhibits an asymptotic behavior as Ri_g increases. Therefore, we fit an exponential curve based on the DNS data (50th percentile), and a new g_T expression is proposed for stably stratified conditions:

\frac{C_{T}^{2}}{z^{4 / 3} {(\frac{\partial \bar{θ}}{\partial z})}^{2}} = g_{T} (R i_{g}) = 0.05 + 1.02 e^{- 14.49 R i_{g}}, R i_{g} > 0.

Fig. 2 Dependence of similarity function (g_T) on stability parameter (Ri_g) for stable conditions. The symbols with error bars are the ensemble results constructed from the DNS dataset. The lower error bars, the circles, and the higher error bars represent 25th, 50th, and 75th percentile values of the ensemble results, respectively. The blue dash-dot line is the regression fit, i.e., Eq. (12), based on the 50th percentile of the DNS data (r² = 0.981 in logarithmic coordinates). The red dashed line is the similarity function proposed in W71 [8], i.e., Eqs. (4b) and (5b).

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In this study, we only focus on constructing new g_T for Ri_g > 0. For the range Ri_g < 0, the g_T proposed in W71, i.e., Eqs. (4a) and (5a) is used in order to obtain a complete g_T expression for $C_{n}^{2}$ estimation in the following section. Note that our newly proposed g_T is consistent with W71 at the neutral limit, i.e., g_T = 1.07 as Ri_g →0.

4. Validation of the Newly Proposed g_T

To validate the applicability of the newly proposed g_T for a wide range of atmospheric stabilities (especially for very stable conditions), we utilize approximately four weeks of $C_{n}^{2}$ data (July 1 to 28, 2006) collected during the MLO_CN2 campaign [23] at the Mauna Loa Observatory, Hawaii. This dataset provides 5-minute averaged time-series of wind speed, temperature, and $C_{n}^{2}$ , measured by sonic anemometers on a meteorological tower at three heights (6 m, 15 m, and 25 m above the ground). The framework for calculating $C_{n}^{2}$ from sonic anemometers can be found in [24]. The $C_{n}^{2}$ data at 15 m will be used for validation.

The procedure to estimate $C_{n}^{2}$ using Eq. (12) and Eqs. (4a) and (5a) is as follows:

Calculate the gradients of potential temperature and wind speed at 15 m using the second-order central-difference approach. Potential temperature can be related to temperature by θ = T(P₀/P)^0.286 with P₀ = 1000 hPa. Since pressure was only measured at 2 m during the MLO_CN2 campaign, the hypsometric equation [25] is used to calculate P at 6 m, 15 m, and 25 m. All the data are averaged over 30 minutes. In order to avoid too small values for the wind speed and temperature gradients, thresholds are set to ensure S > 0.001 s⁻¹ and $\partial \bar{θ} / \partial z > 0.001 {K m}^{- 1}$ .
Based on the gradient information, Ri_g is calculated using Eq. (2). $C_{T}^{2}$ is then calculated using Eq. (12) for Ri_g > 0. For the range Ri_g < 0, the g_T proposed in W71 is used, i.e., Eqs. (4a) and (5a). Since analytically solving g_T as a function of Ri_g in Eqs. (4a) and (5a) is cumbersome, we numerically calculate g_T for a given Ri_g by interpolation.
$C_{T}^{2}$ is converted to $C_{n}^{2}$ using Eq. (3). Note that we ignore the 0.03/β term in Eq. (3) following [24], since about 99% of the time the Bowen ratio (β) is found to be larger than 0.3.

Figure 3 shows the comparison of observed and estimated $C_{n}^{2}$ at 15 m at the Mauna Loa Observatory, Hawaii. In order to illustrate the advantage of the DNS-based formulation in predicting $C_{n}^{2}$ for a wide range of atmospheric stabilities, we use the shaded areas to denote the time periods when Ri_g is larger than 0.2. It is found that Ri_g > 0.2 in about 50% of the nighttime data. Although both W71 and DNS-based formulations predict similar $C_{n}^{2}$ variations in Ri_g < 0.2 during nighttime, the former is not applicable for the strongly stratified conditions, i.e, Ri_g > 0.2. In the following, we will only focus on the DNS-based $C_{n}^{2}$ model.

Fig. 3 Time-series of $C_{n}^{2}$ (15 m) at the Mauna Loa Observatory during July 1 to 28, 2006. The circles are the observational data measured at Mauna Loa Observatory, Hawaii. The black solid and hollow circles correspond to the cases U_m > 2 m s⁻¹ and U_m < 2 m s⁻¹, respectively, with U_m being the magnitude of wind speed. The blue solid lines are the $C_{n}^{2}$ values predicted using the DNS-based formulation, i.e., Eq. (12) for stable conditions and Eqs. (4a) and Eqs. (5a) for unstable conditions. The green dashed lines are the predicted $C_{n}^{2}$ using the W71 formulation, i.e., Eqs. (4) and (5). The shaded areas denote the time periods when Ri_g is larger than 0.2. The mean correlation coefficients (R_D and R_N are for daytime and nighttime, respectively, calculated based on $\log_{10} C_{n}^{2}$ ) between observation and DNS-based estimation during the four weeks are also shown in the bottom-left corner of each sub-figure. The data with U_m < 2 m s⁻¹ are excluded from the calculation of correlation coefficients. In addition, the data during July 28−29 are excluded due to the occurrence of rainfall.

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In terms of the performance of the DNS-based formulation, overall, the agreement between the estimated and observed $C_{n}^{2}$ is good, especially for the first week, i.e., Fig. 3(a). As expected, the prediction of nighttime $C_{n}^{2}$ is poorer compared with the daytime. Please refer to the correlation coefficients in Fig. 3 for each week and the overall correlation coefficients in Fig. 4(a). The most challenging part is the sharp drop-offs of $C_{n}^{2}$ during the morning and evening transitions when $\partial \bar{θ} / \partial z$ approaches zero. Although these sharp drop-offs are qualitatively captured by the proposed approach, there are cases that their amplitudes are largely overestimated (e.g., ∼18 UTC on July 1 and 6). In some cases, theses sharp drop-offs are predicted earlier by about one hour (e.g., ∼6 UTC on July 9 and 10). In addition, in calm wind conditions, i.e., the magnitude of wind speed (U_m) less than 2 m s⁻¹, as denoted by the black hollow cycles in Fig. 3, the magnitude of observed $C_{n}^{2}$ is found to be very oscillatory in time and hard to predict. Note that the above adverse impacts significantly degrade the correlation between observation and estimation for stable conditions, e.g., see the relatively large scatter in Fig. 4(a), especially in $C_{n}^{2} < 10^{- 14} m^{- 2 / 3}$ .

Fig. 4 (a) Correlation and (b) quantile-quantile plots between the observed and estimated $C_{n}^{2}$ during July 1 to 28, 2006. R_D and R_N are the correlation coefficients (calculated based on $\log_{10} C_{n}^{2}$ ) for daytime and nighttime, respectively. The calm wind data (U_m < 2 m s⁻¹) are excluded.

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In addition to the temporal variation of $C_{n}^{2}$ , it is important to evaluate the performance of the new $C_{n}^{2}$ formulation in capturing the percentile distribution of the observational data. Figure 4(b) shows the quantile-quantile plot of observed and estimated $C_{n}^{2}$ during July 1 to 28, 2006. Quite interestingly, although the correlation coefficient between observation and estimation is satisfactory during daytime, the estimated $C_{n}^{2}$ tends to underestimate the low $C_{n}^{2}$ values (black plus-symbols). In contrast, during nighttime, the new $C_{n}^{2}$ formulation is found to have a relatively better performance in capturing the percentile distribution of the observed $C_{n}^{2}$ (the blue cross-symbols).

5. Summary

In this study, we construct a new surface-layer $C_{n}^{2}$ model utilizing a high-fidelity numerically-generated turbulence dataset. Compared with the $C_{n}^{2}$ model proposed by Wyngaard et al. [8], the most distinguishing feature of the new model is that it covers a wide range of atmospheric stabilities including the very stable regime. To validate this newly proposed $C_{n}^{2}$ model, we utilize approximately four weeks of observed $C_{n}^{2}$ data collected during the MLO_CN2 campaign at Mauna Loa Observatory, Hawaii. This proposed $C_{n}^{2}$ model is shown to have satisfactory performance in estimating $C_{n}^{2}$ in the atmospheric surface layer during nighttime, including the strongly stratified (very stable) conditions. However, accurate prediction of temporal variation of $C_{n}^{2}$ during morning and evening transitions is found to be very challenging, as the potential temperature gradient approaches zero. In addition, we observe that the Wyngaard’s model tends to underestimate the low $C_{n}^{2}$ values for unstable conditions, and improvements are needed in the future work.

Acknowledgments

The authors acknowledge financial support received from the Department of Defense (AFOSR grant under award number FA9550-12-1-0449). Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the Department of Defense. The authors also acknowledge computational resources obtained from the Department of Defense Supercomputing Resource Center (DSRC). In addition, the authors would like to thank Adam DeMarco for his valuable comments and suggestions for improving the quality of the paper, as well as his kind help on the DSRC systems.

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Extending a surface-layer $C_{n}^{2}$ model for strongly stratified conditions utilizing a numerically generated turbulence dataset

Abstract

1. Introduction

2. Direct Numerical Simulation

3. Construction of New Similarity Function g_T

4. Validation of the Newly Proposed g_T

5. Summary

Acknowledgments

References and links

Cited By

Figures (4)

Equations (12)

Optics Express

Abstract

1. Introduction

2. Direct Numerical Simulation

3. Construction of New Similarity Function gT

4. Validation of the Newly Proposed gT

5. Summary

Acknowledgments

References and links

Cited By

Figures (4)

Equations (12)

Optics Express

3. Construction of New Similarity Function g_T

4. Validation of the Newly Proposed g_T