1. Introduction
Acoustical and electromagnetic waves can be dramatically perturbed by the atmosphere through which they propagate. Physically, the variations of the refractive index n along the propagation path relate to variations in the aerological properties (velocity u, temperature T, humidity q, pressure P), notably caused by atmospheric turbulence. At large scattering angles, turbulence causes the penetration of signals into shadow zones. At small angles (line-of-sight propagation), it can induce scintillation, blurring, fluctuations in the angle of arrival, etc. In a general sense, these perturbations affect the performance of optical and acoustical systems used for detection, astronomy, communication, and so on (e.g., Fried 1966; Daigle et al. 1986; Naz et al. 1992; Holst 1995). They allow one to remotely sense the atmosphere with radars (Ottersten 1969), sodars (Thomson et al. 1978), or scintillometers (Ochs and Wang 1978).
Tatarskiĭ (1961) theoretically describes these effects in terms of aerological quantities. Specifically, he derives a number of wave propagation diagnostics in terms of Φn, the three-dimensional spatial spectrum of the refractive index. In electromagnetism, Φn depends on ΦT, Φq, and the three-dimensional spatial cross-spectrum ΦT,q (Andreas 1988). In acoustics, Φn combines ΦT, Φq, ΦT,q, and the turbulent kinetic energy (TKE) spectrum ΦTKE (Ostashev 1994).
The necessary characterization of ΦT, Φq, ΦT,q, and ΦTKE raises particular issues. First, most of the above perturbations result from fluctuations of n in the inertial–convective range, in which the hypothesis of three-dimensional isotropy is valid in a rough approximation. In this range, ΦTKE and Φs (with s a scalar: s = T, q, etc) are proportional to κ−11/3. Thus one can write ΦTKE and Φs as a product of the shape function κ−11/3 by a term that gives the intensity of the small-scale fluctuations (Tatarskiĭ 1961; Lawrence and Strohbehn 1970). Second, the actual volume over which the spectra should be considered is fixed by the wave propagation scenario. For example, in radar and sodar applications, this should be the sampling volume (e.g., Wilson et al. 1996; Pollard et al. 2000). As argued by Muschinski et al. (2004), in the inertial–convective range the actual shape of the averaging volume is commonly assumed to be of minor importance; only the characteristic size r should matter.
From the experimental point of view, a number of methods can document some aspects of the distribution of CT,r2, Cq,r2, CT,r2, and Cu,r2 in the atmosphere. In situ sensors directly sample the atmospheric structure functions with airplanes, radiosondes, or ground platforms (Antonia and Chambers 1980; Gurvich and Makarova 1988; Kukharets 1988; Muschinski et al. 2004). Scintillometry uses the scintillation of an artificial source near the surface (Ochs and Wang 1978; Frehlich 1992; Andreas et al. 2003). Sodars measure the scattering of their acoustical emission (Thomson et al. 1978; Petenko and Shurygin 1999). Their electromagnetic counterparts are the atmospheric radars (e.g., Hardy and Ottersten 1969; Muschinski 2004).
Numerical simulation offers a complementary alternative. It provides controlled, reproducible, four-dimensional fields. Several decades ago, atmospheric large-eddy simulation (LES) became available. Although the boundary layer dynamics of LES generally compare satisfactorily to experimental data, there are documented deficiencies in the first few tens of meters due to unresolved motions (Wyngaard et al. 1998; Khanna and Brasseur 1997). This limitation stands where experimental characterization is more accessible. Peltier and Wyngaard (1995) demonstrate the feasibility and the relevance of the LES diagnostics of Cn,r2. Muschinski et al. (1999) use LES to simulate radar signals. Wave propagation solvers have also been operated through LES atmospheric fields (e.g., Gilbert et al. 1999; Wilson et al. 2007). None of these studies has specifically addressed the statistical distribution and the spatial organization of Cn,r2 with height in the atmosphere.
The present work is an attempt to document the variability of Cn,r2 from LES in a typical convective boundary layer. It is organized as follows: In the second section, the basics of LES are introduced, together with our method to diagnose the local inertial–convective range parameters in 3D + time. Section 2 also describes the specific model and weather regime analyzed in the following sections. In section 3, we analyze the simulated statistical distribution and the spatial variability of Cn,r2. Section 4 discusses the implication of our findings on the averaging of inertial–convective range parameters. The last section summarizes the results.
2. Small-scale turbulence in LES
a. Local similarity
Equation (2), sometimes referred to as a local similarity form, is commonly used in wave propagation studies. Thomson et al. (1978) use α1,loc = 0.5 and β1,loc = 0.4 for r ≈ 18 m. Pollard et al. (2000) use β1,loc = 0.4 with r ≈ 15 m. As shown in the appendix, some standard subgrid closures of LES are also derived based on Eq. (2), again with similar constants. Hereafter, we will use the widespread values α1,loc = 0.52 and β1,loc = 0.4. However, despite this apparent consensus, it must be recognized that the extent to which Eq. (2) holds remains to be ascertained. Revealingly, the constants α1,loc and β1,loc are not well determined so far [e.g., see Hill (1997), his appendix A]. Wang et al. (1996, 1999) find a large scatter of α1,loc and β1,loc. Some results also question the validity of Eq. (2) (Szilagyi et al. 1996; Schmidt and Schumann 1989).
b. Local dissipation rates in LES
The characteristic size of the grid box–averaged values is the LES spatial resolution Δ. The local dissipation rates εTKE,Δ and εs,Δ can be computed from the LES with Eqs. (4)–(6). One can then obtain the local structure parameters CT,Δ2, Cq,Δ2, CT,Δ2, and Cu,Δ2 with Eq. (2). Peltier and Wyngaard (1995) demonstrate the relevance of such LES diagnostics.
c. Model description
Hereafter, we use the Dutch Atmospheric LES (DALES), described in Cuijpers and Duynkerke (1993) and Siebesma and Cuijpers (1995), which implicitly filters the subgrid scales with a top-hat spatial filter. The subgrid fluxes are represented with K-diffusion, a prognostic equation for TKE, and turbulent closures at the first order. This provides consistency with the derivation of the subgrid closures (see appendix). There is no prognostic equation for scalar variances, so Eq. (6) is used to compute the scalar structure parameters.
In DALES, Δc is generally taken as 2.5Δ, with Δ taken as the power one-third of the product of the spatial resolutions in each direction (Cuijpers and Duynkerke 1993). Other definitions can be found in the literature for Δ as well as for Δc. For example, Schmidt and Schumann (1989) define Δ as one-third the sum of the spatial resolutions in each direction and they use Δc = 2Δ (i.e., a truncation wavenumber of π/Δ). Under stable conditions, Δc is taken proportional to the square root of TKE times τBV, the inverse of the Brunt–Väisälä frequency (Cuijpers and Duynkerke 1993). With Δc = 2.5Δ, Eq. (8) differs from a direct spatial differencing by a factor of 1.22 for a homogeneous grid.
d. Case description
The present study focuses on the case of a convective boundary layer: no clouds, no mean wind, no large-scale advection, and upward surface heat flux. This weather is representative of fair weather boundary layers with calm winds. The initial potential temperature gradient is constant with height, set to 3 K km−1. The turbulence is driven by the prescribed surface fluxes of sensible heat (0.1 K m s−1) and momentum (−0.073 m2 s−2). There is no humidity in the simulation, so the sensible heat and buoyancy fluxes are equal. The LES domain is 10 km × 10 km × 2 km, and the run lasts 3 h. The model has 256 × 256 × 64 grid points with a resolution of 39 m on the horizontal and 32 m on the vertical. The time step is 1 s.
In the quasi-steady state, the mean vertical profile of many parameters follows the same behavior, once normalized with the surface forcings and some height scaling. This similarity principle allows one to compare our LES predictions with observations. In the surface layer, the characteristic height is the Monin–Obukhov length (LMO = −15.3 m in our case). Above, the mixed layer similarity holds with the mixed layer height (Zi = 1000 m in our case; see below). The corresponding mixed layer velocity and temperature scales w* and Θ* are 1.43 m s−1 and 0.07 K (Kaimal et al. 1976). As already discussed, the LES is relevant typically above 50–100 m. At such heights, the surface layer similarity already matches its local free convection asymptote [Z/LMO → −∞, Wyngaard et al. (1971b)].
In a search for conciseness, we mainly focus on CT,Δ2. This restricts the field of applicability of our results to the propagation of electromagnetic waves at visible wavelengths (Andreas et al. 2003) and to the acoustical backscatter in the dry atmosphere (Ostashev 1994). Being proportional, CT,Δ2 and the electromagnetic Cn,Δ2 will be used interchangeably to ease comparison with other data.
3. Analysis of LES results
a. Probability density functions
Following a standard practice [e.g., Schmidt and Schumann(1989), p. 518], we approximate the ensemble average A by the average over all LES grid points in a horizontal plane at a given time, here t = 10 000 s. Figure 1 shows the horizontally averaged profile of various quantities at that time. [N.B. hereafter ln stands for the Napierian logarithm and log(x) denotes ln(x)/ln(10).] The boundary layer has reached a quasi-steady state (Fig. 1d). The mean potential temperature is characterized by an unstable stratification in the surface layer, a well-mixed profile in the bulk boundary layer, and an inversion at the mixed layer top (Zi ≈ 1000 m). Other standard characteristics of the simulation, not presented here, show the same degree of agreement with observations illustrated in a number of studies.
According to the local free convection and mixed layer similarities, A(εTKE,Δ) only depends on the surface buoyancy flux Fb; that is, A(εTKE,Δ) = Bε g Fb/Θs, with Θs a reference temperature and Bε a constant. The balance between buoyancy production and dissipation of TKE, integrated over height, gives Bε ≤ 0.5. Our LES predicts that A(εTKE,Δ) is rather constant with Bε ≈ 0.3–0.5, in agreement with the LES results of Nieuwstadt et al. (1991). A comparison with observational data is difficult. In the data of Kaimal et al. (1976) and Caughey and Palmer (1979), Bε exceeds 0.5 at all heights, which suggests that shear alters their TKE budget. Furthermore, there is a large scatter in the values of Bε obtained from various local free convection formulations (Edson and Fairall 1998; Hill 1997; Kader and Yaglom 1990). This uncertainty may point back to a possible dependence of this diagnostic on Z/Zi (Johansson et al. 2001).
The predicted decrease of A(εT,Δ) with height agrees with the similarity theory. Whereas our LES underestimates A(εT,Δ) compared to the average of Caughey and Palmer (1979), it is within the scatter of their measurements. Similarly, the decrease of A(CT,Δ2) with (Z/Zi)−4/3 in the lower boundary layer is captured, in agreement with observations and theory (Wyngaard et al. 1971a,b). The peak at the inversion is also simulated (Kaimal et al. 1976). Again, there is a discrepancy between the LES and observations of a factor of 2.
There is only a weak consensus on A(εT,Δ) among LES; however, a similar underestimation of A(εT,Δ) and A(CT,Δ2) compared to observations appears with other LES (e.g., Peltier and Wyngaard 1995). The spatial filtering of LES is known to cause an excessive decrease of the spectra at the largest resolved wavenumbers, which can lead to an underestimation of the local structure parameters (e.g., Moeng and Wyngaard 1988; Schmidt and Schumann 1989). Besides, one salient feature of DALES is its relatively low Prandtl number—a feature that also points back to the uncertainties in α1,loc and β1,loc (see above). It is difficult to anticipate the impact of increasing Pr or decreasing KT as such a modification may affect the simulated dynamics. Moeng and Wyngaard report that it enhances the small-scale temperature fluctuations of their LES-resolved flow. A similar sensitivity in our LES would increase our estimates of A(εT,Δ) and A(CT,Δ2).
The above underestimation stresses that our LES predictions are sensitive to the formulation of the subgrid scales and are subject to a comparable degree of uncertainty or arbitrariness (e.g., Schmidt and Schumann 1989, p. 556; Muschinski 1996). This sensitivity suggests that wave propagation experiments have the potential to provide some insight on the subgrid-scale parameterization in LES. Furthermore, the magnitude of our mean diagnostics may be affected, although their variability may not. In that perspective, we note that the probability density function (PDF) of Cn,Δ2 spans over more than two orders of magnitude at all heights (Fig. 2a). Hence, the horizontal variability of C n,Δ2 is much larger than the discrepancy on the mean levels. In fact, the large width of this PDF by itself justifies the need to account for this variability in wave propagation studies.
The lognormal distribution is the reference PDF of local structure parameters in wave propagation studies (Gurvich and Kukharets 1986; Tatarskiĭ 1987; Wilson et al. 1996). According to Fig. 2b, in the upper boundary layer the PDF of log(Cn,Δ2) is very close to Gaussian. Our LES shows a physical continuity below 200 m (Fig. 2a). Keeping in mind the possible deficiency of LES near the surface (see above), we note that the predicted PDF of log(Cn,Δ2) at Z = 80 m departs from the Gaussian model, with a negative skewness and a more abrupt decrease at high values. Andreas et al. (2003) find a similar trend in their surface layer seasonal observations. Frehlich (1992) and Frehlich et al. (2004) find that the lognormal distribution fits to their measurements in the stable surface layer.
The variance of the PDF of log(Cn,Δ2) can be inferred from the maximum value of the Gaussian fit (Fig. 2b). It is maximum in the bulk boundary layer, where A (Cn,Δ2) is relatively low, and decreases near the surface and the inversion, where A(Cn,Δ2) is large. This behavior is confirmed in Fig. 2c, which shows FT,Δ, the normalized variance of CT,Δ2, defined as the ratio between the variance and the squared mean: FT,Δ increases with height, reaches a maximum as the mean CT,Δ2 is low, and then decreases. Our result is in good agreement with the LES of Peltier and Wyngaard (1995). From this analysis, the probability of large values of Cn,r2 decreases less with height than is suggested by the mean. This has a direct consequence on wave propagation. For example, as illustrated by Cheinet and Siebesma (2007), if a sensor is only affected by the largest values of Cn,r2, its performance improves less with height than suggested by the mean profile of Cn,r2.
b. Spatial organization
This section investigates the spatial organization of Cn,Δ2 in the boundary layer in the light of the spatial organization of the resolved motions. An outstanding feature in the convective boundary layer is the presence of plumes that drive the upward transport of heat (Schmidt and Schumann 1989). Following Cheinet (2003), in the surface layer the plumes are characterized with a buoyancy lower limit of 0.1 K (= 1.43Θ*). In the bulk boundary layer, the buoyancy is converted into vertical motion. The plumes are characterized with a lower limit of 0.5 m s−1 (= 0.35w*) on the vertical velocity. At the inversion the convective plumes overshoot in the warm free troposphere, so a buoyancy upper limit of −0.1 K is used. This method to detect plumes is not very sensitive to the thresholds in buoyancy and vertical velocity.
Figure 3 shows an (x,z) example slab of Cn,Δ2, and Fig. 4 shows some (x,y) example slabs at four heights. In the bulk boundary layer, the low Cn,Δ2 regions correspond to the nonconvective air. Conversely, the maxima in Cn,Δ2 strikingly match the convective plumes. The surface layer shows smoother contrasts and variations on smaller scales. Still, we find a good correlation between the plume roots and the high Cn,Δ2 values. The strong Cn,Δ2 regions in the inversion layer take on a dome-shaped aspect (Fig. 3; see also Kaimal et al. 1976) with relatively low Cn,Δ2 inside the plumes. At this height, Cn,Δ2 is maximum in the plume surroundings in which the entrainment process takes place. The circular patterns on the horizontal match the edges of the convective plumes penetrating the inversion.
The plume structure is general to the convective boundary layer, and the reliability of LES in simulating it has been extensively demonstrated; therefore, the reported distribution of Cn,Δ2 is presumably quite common on a meteorological sense. Many experimental results converge to support this conclusion. Radar measurements in the clear convective boundary layer show similar structures (Hardy and Ottersten 1969; Lohou et al. 1998). Kropfli (1983) discusses the echo-free region at the central core of convective cells. Time series of the returns of vertically pointing sodars are also very comparable (Thomson et al. 1978).
Figure 5 shows the contributions from plumes and their environment to the centered PDF of log(εTKE,Δ), log(εT,Δ), and log(CT,Δ2). The ratio between εTKE,Δ inside and outside the plumes considerably increases with height, so the variance of εTKE,Δ also increases. The PDF of log(εTKE,Δ) can be approximated by a sum of two normal PDFs, notwithstanding the environmental negative skewness. By virtue of Eq. (2), most of the above comments apply to Cu,Δ2. Near-surface observations emphasize the bimodal structure of the PDF of the local dissipation rate of TKE, with larger values inside the plumes (Khalsa 1980). Siebert et al. (2006) find that it is maximum inside their convective shallow cumuli, the dynamics of which are analogous to plumes (Cheinet 2004).
Roughly speaking, the PDF of εT,Δ and CT,Δ2 are similar. The convective parcels have the largest values in the lower and bulk boundary layer, by an order of magnitude. Petenko and Shurygin (1999) also report a bimodal PDF of the local temperature structure parameter in the lower convective boundary layer. Note that after Eq. (6), this suggests that in addition to momentum fluctuations, strong buoyancy contrasts also exist inside the plumes (see below). Near the inversion, the plume and environmental contributions are almost equivalent. As seen above, the dominant gradients near the inversion are at the edges of plumes; that is, the plume versus nonplume partitioning becomes less relevant. Also, note that the variance of εT,Δ does not systematically increase with height.
From Fig. 5, the internal variability (inside plumes, inside the environment) brings contributions to the overall variance that are never negligible and dominant at some heights. Nevertheless, the contrast between plumes and their environment also contributes to the variance of εTKE,Δ, εT,Δ, and CT,Δ2. These so-called top-hat contributions cannot be neglected on a systematic basis. In particular, as the internal variability is rather constant with height, the change of the variances with height mainly results from changes in the top-hat contributions.
c. Joint probability density
Figure 6 shows the centered joint probability distribution (JPD) of log(εTKE,Δ) and log(εT,Δ) at four heights, with the contribution of plumes. As expected, there is a general positive correlation. The upper boundary layer shows a clearly bimodal JPD, with εT,Δ being proportional to εTKE,Δ in the environment. A widely used model proposed by Van Atta (1971) assumes that εTKE,Δ and εT,Δ have a joint lognormal statistical distribution (e.g., Wilson 1997; Muschinski et al. 2004). Our LES predictions undermine the applicability of this model in the convective boundary layer, especially in its upper part.
Our findings contrast with those of Frehlich et al. (2004), who find that the joint lognormal distribution is an appropriate fit to their measurements in the near-surface shear-driven turbulence. Based on this model, Muschinski et al. (2004) analyze the conditional average of CT,Δ2 for specified values of εTKE,Δ (their Figs. 8–11). Figure 7 shows the same analysis based on our LES predictions at two heights. The existence of distinct linear regimes at each height is a further evidence of a qualitative difference between our simulation of convective turbulence and the results of Frehlich et al. (2004) and Muschinski et al. (2004) in nonconvective turbulence.
The emerging picture is as follows: the larger the vertical velocity, the larger the shear production of TKE, and the larger εTKE,Δ. Comparably, the larger the buoyancy, the larger the gradient production of temperature variance, and the larger εT,Δ and CT,Δ2. This explains the match between plumes and high CT,Δ2 regions in the bulk boundary layer (Figs. 3 and 4). The correlation between εT,Δ and εTKE,Δ in the plumes stems from the relation between buoyancy and vertical velocity (Cheinet 2003). As the buoyancy is converted into vertical motion along the ascent, the plume excess in εT,Δ decreases and the excess in εTKE,Δ increases (Fig. 5,6). Near the inversion, the largest temperature gradients are at the edges of plumes, and so are the largest values of εT,Δ and CT,Δ2. Consequently, the contrast between plumes and their environment is less apparent.
According to this analysis, the dynamics particular to convective boundary layers explain the enhanced width of the PDF of εTKE,Δ near the inversion. The plumes bring a major contribution to the overall correlation between εT,Δ and εTKE,Δ. This correlation is approximately 0.6 throughout the lower bulk boundary layer; see below (Fig. 9). One may speculate that the nonconvective turbulence may result in a different behavior. Notably, the PDF of εTKE,Δ might not widen, and the correlation coefficient between εT,Δ and εTKE,Δ could also be smaller. Some data support these speculations. Shaw and Businger (1985) report a constant plume contribution to the dissipation rate of TKE in their near-neutral boundary layers. This supports the view of a constant width of the PDF of εTKE,r with height. Frehlich (1992) operates scintillometer measurements at night, presumably under stable stratification. For each of his datasets, we calculate cor(εT,r; εTKE,r) from his fit to data. We then average to obtain cor(εT,r; εTKE,r) = 0.13. Wang et al. (1999) report that cor(εT,r; εTKE,r) = 0.16 in their shear-driven turbulence simulation.
Even at a single location, the turbulence can change from a convective type to a stably stratified shear-driven structure in time scales of a few hours. The present analysis suggests that the small-scale fluctuations statistics may dramatically change, not only quantitatively but also qualitatively (e.g., from lognormal to bimodal distributions), as a response to such changes.
d. Sensitivity to the averaging size
At each grid point, we locally average the LES fields over a horizontal disk of radius r′= (j − 0.5)Δ, with j = 1,30. For example, with j = 30, one averages over disks of diameter greater than 2Zi. Then we calculate the variance of Fj over the domain. Figure 8 shows the dependence of the variances of ln(εT, r), ln(εTKE,r), and ln(Cn,r2) with ln(r′) at four heights. The general decrease with ln(r′) is expected, as the larger the area of the disk, the smaller the variability from disk to disk. The dimensions of the plumes and the distance between plumes are thought to explain the two transitions of slopes in Fig. 8. The first regime shows a moderate decrease. As the averaging diameter exceeds the width of the plumes, the plumes and environmental contributions are mixed together in the averaging process, so the decrease is enhanced. When it exceeds the distance between plumes, the average hardly varies from disk to disk, and the variance converges to zero. Hence, the effect of large-scale fluctuations is efficiently smoothed out when the considered wave propagation application uses r > 2Zi.
We interpret the first regime as pertaining to the inertial–convective range; thus, we can test Eq. (11) over the uppermost portion of the inertial–convective range. With three data points in this regime, the validity of this model cannot be ascertained rigorously. Nonetheless, Fig. 8 shows that the first rhs term in Eq. 11 cannot be neglected in this portion [Wilson et al.(1996), their Eq. 12; see also Cheinet (2008)]. Furthermore, it is noteworthy that the initial slopes in Figs. 8a–c do not notably change with height. The asymptotic slope is μTKE ≈ 0.2, in surprisingly good agreement with surface layer measurements (Antonia and Chambers 1980) and numerical experiments (Wang et al. 1996). The analysis of the variance of ln(εT,r) shows a similar behavior (Fig. 8b) and gives μT ≈ 0.5, again in fair agreement with the simulations of Wang et al. (1999). The boundary layer measurements of Kukharets (1988) yield μT ≈ 0.8 ± 0.34. The variance of ln(Cn,r2) also shows a very similar picture, with μC ≈ 0.4 (Fig. 8c).
4. Discussion
Our LES predictions for a convective turbulence can be compared with Frehlich’s analysis of nonconvective turbulence. Both applications are based on a comparable value of r and assume that α1,loc and β1,loc are constant. Frehlich (1992) reports that RTKE,r ≈ 1.16 [his Eq. (78)]. In our LES, RTKE,r ≈ 1.1 holds in the lower and bulk convective boundary layer. From Eq. (13), our results obtained in convective turbulence extend the argument of Frehlich that α1,ave ≈ 0.8–0.9 α1,loc. Alternatively, α1,locA(εT,r)2/3 produces a biased estimate of A(Cu,r2), by a factor of 1.1–1.2 [see Eq. (12)].
We find that RT,Δ > 1 in the lower and bulk boundary layer. This is due to the strong correlation between εT,Δ and εTKE,Δ, a feature that we have related to the convective dynamics. Based on his measurements, Frehlich (1992) reports that RT,Δ ≈ 0.75 [his Eq. (76)]. This is expected since, in nonconvective turbulence, cor(εT,r; εTKE,r) is supposedly smaller (see above) and RT,Δ < 1 [Eq. (14)]. Therefore, the ratio β1,loc/β1,ave shows some stability-dependent fluctuations; that is, one among β1,loc and β1,ave is not a constant. Following Frehlich, β1,ave ≈ 1.33β1,loc under stable stratification, whereas our convective case yields β1,ave ≈ 0.83β1,loc. Alternatively, Eq. (12) cannot correctly estimate A(CT,r2) at all ranges of stability. For example, β1,locA(εT,r)[A(εTKE,Δ)]−1/3 produces a biased estimate of A(CT,r2) by a factor that changes with stability: it is 0.75 in the nonconvective turbulence configuration of Frehlich and approximately 1.2 in the local free convection.
There is an interest in deriving A(Cn,r2) from atmospheric models that use coarser spatial resolutions than LES (e.g., Businger et al. 2002). If one accepts the above limitations of Eq. (12), the remaining issue is to parameterize A(εTKE,r) and A(εT,r) for r within the inertial–convective range. The issue is critical, though, because at such resolutions, the subgrid turbulence occasionally includes some contributions from eddies outside the inertial–convective range (Cheinet 2003). The parameterization of Masciadri et al. [1999, Eqs. (13) and (16)] does not filter out these contributions [e.g., their Eq. (8)], so it may not be representative of inertial–convective range fluctuations. This casts some doubt on the applicability of this direct approach, for example, in the convective boundary layer.
5. Conclusions and perspectives
The propagation of optical and acoustical waves is affected by the atmospheric turbulence through the local instantaneous refractive index structure parameter in a volume of characteristic size r, denoted Cn,r2. In many standard wave propagation applications, r is within the inertial–convective range. In the generic case, Cn,r2 is a combination of Cu,r2, CT,r2, Cq,r2, and CT,q,r. In this study, we have used a large-eddy simulation to document the statistical distribution and the spatial organization of Cn,Δ2 in the case of a convective boundary layer, with Δ the LES spatial resolution.
First, we have introduced Eq. (2), a local version of the refined theory of inertial–convective range turbulence, which gives the local structure parameters of wind and scalars in terms of the local dissipation rates of TKE and scalar variances. We have argued that this equation supports the parameterization of subgrid turbulence in LES (see the appendix). Specifically, we have demonstrated that some standard subgrid closures of LES are mathematically related to the constants α1,loc and β1,loc that appear in Eq. (2). Our method to diagnose the local structure parameters is based on the derivation of the local dissipation rates of TKE and scalar variances from the LES variables. With a first-order closure to parameterize the subgrid turbulence, this method has been found to be equivalent to a direct spatial differencing of the resolved fields. We have argued that this method may not generally apply as such with atmospheric models that have a coarser spatial resolution than LES.
In the remainder of the study, we have analyzed the predictions of a LES in a purely convective boundary layer. It has been found that the mean profiles of εTKE,Δ, εT,Δ, and CT,Δ2 predicted by our LES agree with the local free convection and mixed layer similarity theories. They differ from observational predictions by a rather constant factor of 2. Possible causes have been suggested: (i) the impact of the LES spatial filtering and (ii) a low Prandtl number closure. Further investigations are needed to better understand the relation between the LES formulation (filter of the subgrid scales, subgrid parameterizations) and the LES predictions for the subgrid dissipation rates.
Our LES predicts that the variance of CT,Δ2 is low where the mean is high and vice versa, in good agreement with a previous LES estimate. This behavior implies that the mean value of CT,Δ2 may be a poor indicator of the performance bounds of a sensor sensitive to turbulence. The probability density function of CT,Δ2 involves two orders of magnitude at all heights. It is nearly lognormal in the upper boundary layer, in agreement with a common assumption, but tends to depart from this in the lower boundary layer. Again, given the potential deficiencies of LES near the surface, further research is needed to confirm this prediction at very low levels.
Next, we have analyzed the spatial distribution of CT,Δ2 with a physically based criterion to sort the convective plumes. In the lower boundary layer, the plumes have some excesses in εTKE,Δ, εT,Δ, and CT,Δ2 compared to their environment as they correlate buoyancy and vertical velocity. In the upper boundary layer, the buoyancy of the plumes is converted into vertical motion, whereas the entrainment enhances εT,Δ at the edges of plumes. Consequently, the plumes’ excess in εTKE,Δ increases, and plumes show comparable εT,Δ and CT,Δ2 as their environment. Our results support a bimodal description of the PDF of log(εTKE,Δ), log(εT,Δ), and log(CT,Δ2). They are in qualitative agreement with many observational data, of which they offer a comprehensive picture. The widespread model of a jointly lognormal statistical distribution of εTKE,Δ and εT,Δ, for example, is challenged in the convective boundary layer.
After Frehlich (1992), the average dissipation rates are known to produce a biased estimate of the average structure parameters in nonconvective turbulence [Eq. (12)]. We have investigated this issue with our LES predictions in convective conditions, also comparing with his data. The bias for Cu,r2 is of 1.1–1.2 almost irrespective of stability (see above). The bias for CT,r2 has been found to range from 0.75 to 1.2 depending on stability. This stability dependence challenges the applicability of surface layer similarity scalings based on this estimate. A physical explanation has been offered that emphasizes the role of the convective dynamics. This explanation highlights the connection between buoyancy and motion and may not extend to purely passive scalars.
We have investigated the sensitivity of these diagnostics to r by averaging them over disks of increasing diameter. One famous law of the refined theory asymptotically holds in our LES—namely, the variances of ln(εTKE,r), ln(εT,r), and ln(CT,r2) linearly decrease with ln(r′), independently of height, for r′ = r/2 small enough. The intermittency parameters are μTKE ≈ 0.2 and μT ≈ 0.5, in good agreement with various experimental data. For CT,Δ2, we obtain μC ≈ 0.4.
The present method and results can be extended in various directions. First, one may use the LES distribution of Cn,Δ2 to quantify the variability of wave propagation. Cheinet and Siebesma (2007) and Cheinet (2008) find that this variability may have a considerable impact on the performance of, respectively, optical and acoustical sensors. Second, we here focus on temperature fluctuations in the convective boundary layer. The humidity and wind fluctuations could be examined, and other meteorological forcings may lead to other behaviors of Cn,r2. Third, we have argued that many wave propagation diagnostics primarily depend on the inertial–convective range turbulence, which is investigated in this study. Other wave propagation diagnostics, like the wave phase fluctuations, may be directly sensitive to the large turbulent motions resolved in LES (e.g., Kallistratova 2002).
Last, both large-scale and small-scale fluctuations of atmospheric fields modulate the mixing of momentum and scalars (thermodynamic properties, aerosol concentration, etc.). Our diagnostics may also have implications in related areas, like aeronautical turbulence, dispersion, or the parameterization of finescale mixing and cloud processes in the atmosphere.
Acknowledgments
This study was performed under the ISL “Exploratory Research” framework and also received support from the KNMI visiting scientist program. We are grateful to P. Naz and to the anonymous reviewers for their valuable comments and suggestions. The figures were realized with the Grid Analysis and Display System (GrADS) from IGES/COLA. Some referenced papers are collected in Turbulence in a Refractive Medium (1990, E. Andreas, Ed., SPIE Milestone Series, MS25).
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APPENDIX
LES Closures with Local Similarity
We see from this derivation that the choice of Au, As, Nu, and Ns as (constant) closures in the local relations (3)–(6) assumes that α1,loc and β1,loc are constant. In other words, the local form Eq. (2) backs up the subgrid parameterization of atmospheric LES. The present derivation is directly inspired from a previous one, tied up to the original theory of inertial convective range turbulence (see above; Moeng and Wyngaard 1988; Schmidt and Schumann 1989). Even with an assumption to derive local relations at each LES grid box, this previous method leads to Eqs. (A1) and (A4), but with RTKE α1,ave and RT β1,ave instead of α1,loc and β1,loc. It has next to be assumed that RTKE ≈ 1 and RT ≈ 1 [Eq. (B10) in Schmidt and Schumann 1989]. The present derivation offers a more transparent assessment of the local nature of the standard LES closures Au, As, Nu, and Ns.
(a) Horizontal average of θ at t = 0 and t = 10 000 s. (b) Log of the horizontal average of εTKE,Δ normalized by (εTKE)* = w*3/Zi. (c) As in (b) but for εT,Δ with (εT)* = θ*2w*/Zi. (d) As in (b) but for CT,Δ2 (four profiles, every 200 s) with (CT2)* = θ*2/Zi2/3. Observations include the local free convection asymptote of Edson and Fairall (1998, thick dashed), and some mixed layer profiles (Sorbjan 1988; Kaimal et al. 1976, thin dashed).
Citation: Journal of the Atmospheric Sciences 66, 4; 10.1175/2008JAS2790.1
(a) PDF of log(Cn,Δ2) with height (mean as full white line); (b) centered PDF of log(Cn,Δ2) at four heights with Gaussian of same variance (dashed); (c) normalized variance FT,Δ with the LES prediction (dots) of Peltier and Wyngaard (1995).
Citation: Journal of the Atmospheric Sciences 66, 4; 10.1175/2008JAS2790.1
Cross section (x,z) of log(Cn,Δ2) at t = 10 000 s, y = 5000 m. On the rhs of the figure, the vertical velocity is contoured at w = 0.5 m s−1 (thick full lines) and the potential temperature excess is contoured at −0.1 K (thin dotted lines).
Citation: Journal of the Atmospheric Sciences 66, 4; 10.1175/2008JAS2790.1
Horizontal (x,y) distribution of log(Cn,Δ2) at t = 10 000 s at four heights. The thresholds for plumes are contoured on the left half of the panels, according to potential temperature excess of 0.1 K at 80 m and −0.1 K at 950 m and vertical velocity excess of 0.5 m s−1 at 310 and 630 m.
Citation: Journal of the Atmospheric Sciences 66, 4; 10.1175/2008JAS2790.1
Centered PDF of log(εTKE,Δ), log(εT,Δ), and log(Cn,Δ2) at four heights (thick line), together with their partitioning into the contributions of plumes (with symbol P) and their environment.
Citation: Journal of the Atmospheric Sciences 66, 4; 10.1175/2008JAS2790.1
Centered JPD of log(εTKE,Δ) and log(εT,Δ) at four heights, with the contribution of plumes contoured at values 0.1 and 0.3; also indicated are slopes 1 and 2.
Citation: Journal of the Atmospheric Sciences 66, 4; 10.1175/2008JAS2790.1
Log of the conditional average of Cn,Δ2 for specified values of εT,Δ, at Z = 310 m (thick line) and Z = 950 m (thin line).
Citation: Journal of the Atmospheric Sciences 66, 4; 10.1175/2008JAS2790.1
Variance of (a) ln(εTKE,r), (b) ln(εT,r), and (c) ln(Cn,r2) with ln(r′): heights are z = 80 m (solid), z = 180 m (dashed), z = 340 m (close dots), z = 560 m (long–short dashed), and z = 750 m (distant dots). The crosses indicate the actual LES diagnostics; the thick full lines indicate a constant slope of (a) −0.2, (b) −0.5, and (c) −0.4.
Citation: Journal of the Atmospheric Sciences 66, 4; 10.1175/2008JAS2790.1
Vertical profiles of (a) RT,Δ (solid) and RTKE,Δ (dashed), (b) Normalized rhs terms in Eq. (16): lhs term (solid), first rhs term (dashed), second rhs term (alternate dashed), and −0.18σ(εT,Δ)/A(εT,Δ) (dotted). (c) Correlation coefficients between εT,Δ and TKE,Δ (solid) and between εT,Δ and (εTKE,Δ)−1/3 (dashed).
Citation: Journal of the Atmospheric Sciences 66, 4; 10.1175/2008JAS2790.1
