## 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 Φ_{n}, the three-dimensional spatial spectrum of the refractive index. In electromagnetism, Φ_{n} depends on Φ_{T}, Φ_{q}, and the three-dimensional spatial cross-spectrum Φ_{T}_{,q} (Andreas 1988). In acoustics, Φ_{n} combines Φ_{T}, Φ_{q}, Φ_{T}_{,q}, and the turbulent kinetic energy (TKE) spectrum Φ_{TKE} (Ostashev 1994).

The necessary characterization of Φ_{T}, Φ_{q}, Φ_{T}_{,q}, and Φ_{TKE} 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, Φ_{TKE} and Φ_{s} (with *s* a scalar: *s* = *T*, *q*, etc) are proportional to *κ*^{−11/3}. Thus one can write Φ_{TKE} and Φ_{s} 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.

**x**,

*t*) is

*κ*and

*r*in the inertial–convective range (e.g., Muschinski et al. 2004, p. 324). The numerical coefficients in Eq. (1) are used for consistency with the standard literature. The physical variables C

_{T}

_{,r}

^{2}, C

_{q}

_{,r}

^{2}, C

_{T}

_{,r}

^{2}, and C

_{u,r}

^{2}hold the space and time dependence of small-scale fluctuations. They are hereafter referred to as local (instantaneous) structure parameters. They can greatly vary in time and space as a result of synoptic-scale disturbances, gravity waves, or large-scale turbulence (especially in the lower atmosphere). This variability is sometimes referred to as large-scale intermittency (e.g., Muschinski et al. 2004). The importance of accounting for it in wave propagation applications is discussed by Tatarskiĭ and Zavorotnyi (1985), Tatarskiĭ (1987), and Wilson et al. (1996), among others.

From the experimental point of view, a number of methods can document some aspects of the distribution of C_{T}_{,r}^{2}, C_{q}_{,r}^{2}, C_{T}_{,r}^{2}, and C_{u}_{,r}^{2} 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 *C _{n}*

_{,r}

^{2}. 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

*C*

_{n}_{,r}

^{2}with height in the atmosphere.

The present work is an attempt to document the variability of *C _{n}*

_{,r}

^{2}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

*C*

_{n}_{,r}

^{2}. 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

_{s}and ε

_{TKE}the dissipation rates of scalar variance and TKE, respectively. The theory of Kolmogorov (1962; see also Van Atta 1971) introduces ε

_{s}

_{,r}and ε

_{TKE,r}, the local averages of ε

_{s}and ε

_{TKE}over a volume of characteristic dimension

*r*in the inertial–convective range. We follow Peltier and Wyngaard (1995; see also Muschinski et al. 2004, 325–326) and use the above theoretical framework to relate the local structure parameters to the local dissipation rates:

*α*

_{1,loc}and

*β*

_{1,loc}are constant, that is, should not depend on

*r*. A similar relation holds for the structure parameters that enter cross-correlation spectra. Gurvich and Makarova (1988) directly evaluate Eq. (2) in the surface layer. They obtain that

*α*

_{1,loc}≈ 0.57, with a slight dependence on

*r*and a scatter of 20% around this value.

Equation (2), sometimes referred to as a local similarity form, is commonly used in wave propagation studies. Thomson et al. (1978) use *α*_{1,loc} = 0.5 and *β*_{1,loc} = 0.4 for *r* ≈ 18 m. Pollard et al. (2000) use *β*_{1,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 *α*_{1,loc} = 0.52 and *β*_{1,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 *α*_{1,loc} and *β*_{1,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 *α*_{1,loc} and *β*_{1,loc}. Some results also question the validity of Eq. (2) (Szilagyi et al. 1996; Schmidt and Schumann 1989).

### b. Local dissipation rates in LES

_{c}, in the inertial–convective range. Note that Δ

_{c}differs from the LES spatial resolution—denoted Δ below. The subgrid eddies must be represented in LES. The

*K*-diffusion parameterization writes the subgrid transport as a diffusive process; that is, it takes the subgrid flux of

*s*, denoted

*F*, to be opposed and proportional to the gradient of

_{s}*s*. Subsequently, the local subgrid flux and stress tensor are parameterized as

*i*,

*j*= 1,3. A prognostic equation is used for TKE. Among the calculated terms, the local dissipation rate of TKE is parameterized according to

*σ*

_{s}^{2}of a scalar and its local dissipation rate can be related through

*σ*

_{s}^{2}is not prognostic, one may alternatively use a simple local budget equation between the gradient production and dissipation of variance:

*i*= 1,3. Note also that the potential temperature Θ replaces

*T*in the right side of the above equation for ε

_{T}, to account for pressure relaxation. The appendix derives the LES closures

*N*,

_{s}*N*,

_{u}*A*, and

_{u}*A*in terms of the local similarity constants

_{s}*α*

_{1,loc}and

*β*

_{1,loc}.

The characteristic size of the grid box–averaged values is the LES spatial resolution Δ. The local dissipation rates ε_{TKE,Δ} and ε_{s}_{,Δ} can be computed from the LES with Eqs. (4)–(6). One can then obtain the local structure parameters *C _{T}*

_{,Δ}

^{2},

*C*

_{q}_{,Δ}

^{2},

*C*

_{T}_{,Δ}

^{2}, and

*C*

_{u}_{,Δ}

^{2}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.

*A*and

_{u}*N*follow Eqs. (A1) and (A4). Equation (A4) gives

_{u}*β*

_{1,loc}and compute

*N*and Pr from it. With

_{s}*β*

_{1,loc}= 0.4, one has Pr = 0.42 (Schmidt and Schumann 1989). Another closure is to specify Pr and derive

*N*from it. In DALES, Pr is set to 0.33 (Cuijpers and Duynkerke 1993; Moeng and Wyngaard 1988). Equations (2), (4), and (6) then yield

_{s}*α*

_{1,loc}and

*β*

_{1,loc}and Pr follow Eq. (7), the above structure parameters do not depend on any of them. The choice of

*β*

_{1,loc}= 0.4,

*α*

_{1,loc}= 0.52, with Pr = 0.33 does not provide this property. The gradients in Eqs. (6) and (8) are approximated with finite differences on the staggered grid on which the subgrid fluxes are computed. From Eq. (8), we note that our formulation is equivalent to a direct spatial differencing of the temperature field (Muschinski et al. 1999). This equivalence may not be guaranteed with more elaborated parameterizations of the dissipation rates.

In DALES, Δ_{c} 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 Δ_{c}. For example, Schmidt and Schumann (1989) define Δ as one-third the sum of the spatial resolutions in each direction and they use Δ_{c} = 2Δ (i.e., a truncation wavenumber of *π*/Δ). Under stable conditions, Δ_{c} is taken proportional to the square root of TKE times *τ*_{BV}, the inverse of the Brunt–Väisälä frequency (Cuijpers and Duynkerke 1993). With Δ_{c} = 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 m^{2} 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 (*L*_{MO} = −15.3 m in our case). Above, the mixed layer similarity holds with the mixed layer height (*Z _{i}* = 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*/

*L*

_{MO}→ −∞, Wyngaard et al. (1971b)].

In a search for conciseness, we mainly focus on *C _{T}*

_{,Δ}

^{2}. 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,

*C*

_{T}_{,Δ}

^{2}and the electromagnetic

*C*

_{n}_{,Δ}

^{2}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 (*Z _{i}* ≈ 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*(ε_{TKE,Δ}) only depends on the surface buoyancy flux *F _{b}*; that is,

*A*(ε

_{TKE,Δ}) =

*B*

_{ε}

*g F*/Θ

_{b}_{s}, with Θ

_{s}a reference temperature and

*B*a constant. The balance between buoyancy production and dissipation of TKE, integrated over height, gives

_{ε}*B*≤ 0.5. Our LES predicts that

_{ε}*A*(ε

_{TKE,Δ}) is rather constant with

*B*≈ 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),

_{ε}*B*exceeds 0.5 at all heights, which suggests that shear alters their TKE budget. Furthermore, there is a large scatter in the values of

_{ε}*B*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

_{ε}*Z*/

*Z*(Johansson et al. 2001).

_{i}The predicted decrease of *A*(ε_{T}_{,Δ}) with height agrees with the similarity theory. Whereas our LES underestimates *A*(ε_{T}_{,Δ}) compared to the average of Caughey and Palmer (1979), it is within the scatter of their measurements. Similarly, the decrease of A(*C _{T}*

_{,Δ}

^{2}) with (

*Z*/

*Z*)

_{i}^{−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*(ε_{T}_{,Δ}) among LES; however, a similar underestimation of *A*(ε_{T}_{,Δ}) and *A*(*C _{T}*

_{,Δ}

^{2}) 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

*α*

_{1,loc}and

*β*

_{1,loc}(see above). It is difficult to anticipate the impact of increasing Pr or decreasing

*K*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

_{T}*A*(ε

_{T}

_{,Δ}) and

*A*(

*C*

_{T}_{,Δ}

^{2}).

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 *C _{n}*

_{,Δ}

^{2}spans over more than two orders of magnitude at all heights (Fig. 2a). Hence, the horizontal variability of

*C*

_{n}_{,Δ}

^{2}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(*C _{n}*

_{,Δ}

^{2}) 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(

*C*

_{n}_{,Δ}

^{2}) 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(*C _{n}*

_{,Δ}

^{2}) can be inferred from the maximum value of the Gaussian fit (Fig. 2b). It is maximum in the bulk boundary layer, where

*A*(

*C*

_{n}_{,Δ}

^{2}) is relatively low, and decreases near the surface and the inversion, where

*A*(

*C*

_{n}_{,Δ}

^{2}) is large. This behavior is confirmed in Fig. 2c, which shows

*F*

_{T}_{,Δ}, the normalized variance of

*C*

_{T}_{,Δ}

^{2}, defined as the ratio between the variance and the squared mean:

*F*

_{T}_{,Δ}increases with height, reaches a maximum as the mean

*C*

_{T}_{,Δ}

^{2}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

*C*

_{n}_{,r}

^{2}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

*C*

_{n}_{,r}

^{2}, its performance improves less with height than suggested by the mean profile of C

_{n}

_{,r}

^{2}.

### b. Spatial organization

This section investigates the spatial organization of *C _{n}*

_{,Δ}

^{2}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}(= 0.35

*w*

_{*}) 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 *C _{n}*

_{,Δ}

^{2}, and Fig. 4 shows some (

*x*,

*y*) example slabs at four heights. In the bulk boundary layer, the low

*C*

_{n}_{,Δ}

^{2}regions correspond to the nonconvective air. Conversely, the maxima in

*C*

_{n}_{,Δ}

^{2}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

*C*

_{n}_{,Δ}

^{2}values. The strong

*C*

_{n}_{,Δ}

^{2}regions in the inversion layer take on a dome-shaped aspect (Fig. 3; see also Kaimal et al. 1976) with relatively low

*C*

_{n}_{,Δ}

^{2}inside the plumes. At this height,

*C*

_{n}_{,Δ}

^{2}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 *C _{n}*

_{,Δ}

^{2}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(ε_{TKE,Δ}), log(ε_{T}_{,Δ}), and log(*C _{T}*

_{,Δ}

^{2}). The ratio between ε

_{TKE,Δ}inside and outside the plumes considerably increases with height, so the variance of ε

_{TKE,Δ}also increases. The PDF of log(ε

_{TKE,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

*C*

_{u}_{,Δ}

^{2}. 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 ε_{T}_{,Δ} and *C _{T}*

_{,Δ}

^{2}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 ε

_{T}

_{,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 ε_{TKE,Δ}, ε_{T}_{,Δ}, and *C _{T}*

_{,Δ}

^{2}. 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(ε_{TKE,Δ}) and log(ε_{T}_{,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 ε_{T}_{,Δ} being proportional to ε_{TKE,Δ} in the environment. A widely used model proposed by Van Atta (1971) assumes that ε_{TKE,Δ} and ε_{T}_{,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 *C _{T}*

_{,Δ}

^{2}for specified values of ε

_{TKE,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.

_{c}under stable stratification leads to ε

_{TKE,Δ}= TKE/

*τ*

_{BV}and

*K*= TKE

_{h}*τ*

_{BV}, so Eq. (6) gives

_{TKE,Δ}and ε

_{T}

_{,Δ}. Note that with this assumption, Eq. (9) also becomes very similar to Muschinski et al. (2004), their Eq. (4.28).

*K*-diffusion model. If the vertical gradient dominates over the lateral ones, then Eq. (6) gives

*C*

_{T}_{,Δ}

^{2}should be proportional to (

*C*

_{u}_{,Δ}

^{2})

^{2}inside the thermals and to

*C*

_{u}_{,Δ}

^{2}in the stable environment.

The emerging picture is as follows: the larger the vertical velocity, the larger the shear production of TKE, and the larger ε_{TKE,Δ}. Comparably, the larger the buoyancy, the larger the gradient production of temperature variance, and the larger ε_{T}_{,Δ} and *C _{T}*

_{,Δ}

^{2}. This explains the match between plumes and high

*C*

_{T}_{,Δ}

^{2}regions in the bulk boundary layer (Figs. 3 and 4). The correlation between ε

_{T}

_{,Δ}and ε

_{TKE,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 ε

_{T}

_{,Δ}decreases and the excess in ε

_{TKE,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 ε

_{T}

_{,Δ}and

*C*

_{T}_{,Δ}

^{2}. 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 ε_{TKE,Δ} near the inversion. The plumes bring a major contribution to the overall correlation between ε_{T}_{,Δ} and ε_{TKE,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 ε_{TKE,Δ} might not widen, and the correlation coefficient between ε_{T}_{,Δ} and ε_{TKE,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 ε_{TKE,r} with height. Frehlich (1992) operates scintillometer measurements at night, presumably under stable stratification. For each of his datasets, we calculate cor(ε_{T}_{,r}; ε_{TKE,r}) from his fit to data. We then average to obtain cor(ε_{T}_{,r}; ε_{TKE,r}) = 0.13. Wang et al. (1999) report that cor(ε_{T}_{,r}; ε_{TKE,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

*r*′ =

*r*/2 the radius of the local volume of averaging. The estimate of the variances of ln(ε

_{T}

_{,r}), ln(ε

_{TKE,r}), and ln(

*C*

_{n}_{,r}

^{2}) in terms of

*r*or

*r*′ is useful in wave propagation applications. For example, Gurvich and Yaglom (1967; after Kolmogorov 1962) predict that for

*r*′ in the inertial–convective range the variances of ln(ε

_{TKE,r}) and ln(ε

_{T}

_{,r}) vary according to

*L*

_{0}the upper bound of the inertial–convective range,

*M*

_{TKE}(

**x**,

*t*) and

*M*(

_{T}**x**,

*t*) some functions of the large-scale flow, and

*μ*

_{TKE}and

*μ*two universal constants. Equation (11) was originally introduced with spherical volumes but, as noted above, the volume shape should not matter. The need for further quantitative evaluation of Eq. (11) is recognized (e.g., Wilson et al. 1996, p. 3398).

_{T}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 2*Z _{i}*. Then we calculate the variance of

*F*over the domain. Figure 8 shows the dependence of the variances of ln(ε

_{j}_{T, r}), ln(ε

_{TKE,r}), and ln(

*C*

_{n}_{,r}

^{2}) 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*> 2

*Z*.

_{i}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 *μ*_{TKE} ≈ 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(ε_{T}_{,r}) shows a similar behavior (Fig. 8b) and gives *μ _{T}* ≈ 0.5, again in fair agreement with the simulations of Wang et al. (1999). The boundary layer measurements of Kukharets (1988) yield

*μ*≈ 0.8 ± 0.34. The variance of ln(

_{T}*C*

_{n}_{,r}

^{2}) also shows a very similar picture, with

*μ*≈ 0.4 (Fig. 8c).

_{C}## 4. Discussion

*α*

_{1,ave}and

*β*

_{1,ave}constants [Wyngaard et al. (1971a), Eq. (5); Edson and Fairall (1998), Eq. (56), among others]. Equation (12) apparently originates with the theory of Kolmogorov (1941) and Corrsin (1951) [e.g., Wang et al. (1996), Eq. (1.3)]. This equation has also been used to estimate (

*α*

_{1,loc};

*β*

_{1,loc}), with the a priori assumption that (

*α*

_{1,ave};

*β*

_{1,ave}) = (

*α*

_{1,loc};

*β*

_{1,loc}) (e.g., Champagne et al. 1977). Whereas this assumption is rarely discussed in the literature, the difference between the theories of Kolmogorov (1941) and Kolmogorov (1962) is well known (e.g., Van Atta 1971; Peltier and Wyngaard 1995, p. 3642; Wyngaard et al. 2001, p. 664). Comparing the ensemble average of Eq. (2) with Eq. 12 gives

*R*

_{T}_{,Δ}and

*R*

_{TKE,Δ}from the LES statistical distributions of ε

_{T}

_{,Δ}and ε

_{TKE,Δ}.

*R*

_{TKE,Δ}≥ 1, an inequality that increases with the variance of ε

_{TKE,Δ}:

*R*

_{TKE,Δ}is constant throughout the boundary layer (Fig. 9a), in agreement with an indirect diagnostic by Moeng and Wyngaard (1988), it increases near the inversion owing to the widening of the PDF of ε

_{TKE,Δ}(Fig. 5). Comparably,

*R*

_{T}_{,Δ}also varies with height. From the definition of the correlation coefficient of ε

_{T}

_{,Δ}and ε

_{TKE,Δ}

^{−1/3}, we obtain, with simple algebra,

*σ*is the root-mean-square and

*B*(ε

_{TKE,Δ}) =

*A*[(ε

_{TKE,Δ})

^{−1/3}] − [

*A*(ε

_{TKE,Δ})]

^{−1/3}. The two rhs terms largely compensate (Fig. 9b):

*B*is positive, whereas cor(ε

_{T}

_{,r}; ε

_{TKE,r}

^{−1/3}) is negative owing to a conspicuous positive correlation between ε

_{T}

_{,Δ}and ε

_{TKE,Δ}(Fig. 9c). The second rhs term is almost proportional to

*σ*(ε

_{T}

_{,Δ})/

*A*(ε

_{T}

_{,Δ}), the profile of which resembles that of −

*F*

_{T}_{,Δ}(see above). This ratio causes the variations of

*R*

_{T}_{,Δ}with height. For example, the minimum in the upper boundary layer renders the rhs of Eq. (14) positive, so

*R*

_{T}_{,Δ}< 1.

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 *α*_{1,loc} and *β*_{1,loc} are constant. Frehlich (1992) reports that *R*_{TKE,r} ≈ 1.16 [his Eq. (78)]. In our LES, *R*_{TKE,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 *α*_{1,ave} ≈ 0.8–0.9 *α*_{1,loc}. Alternatively, *α*_{1,loc}*A*(ε_{T}_{,r})^{2/3} produces a biased estimate of *A*(*C _{u}*

_{,r}

^{2}), by a factor of 1.1–1.2 [see Eq. (12)].

We find that *R _{T}*

_{,Δ}> 1 in the lower and bulk boundary layer. This is due to the strong correlation between ε

_{T}

_{,Δ}and ε

_{TKE,Δ}, a feature that we have related to the convective dynamics. Based on his measurements, Frehlich (1992) reports that

*R*

_{T}_{,Δ}≈ 0.75 [his Eq. (76)]. This is expected since, in nonconvective turbulence, cor(ε

_{T}

_{,r}; ε

_{TKE,r}) is supposedly smaller (see above) and

*R*

_{T}_{,Δ}< 1 [Eq. (14)]. Therefore, the ratio

*β*

_{1,loc}/

*β*

_{1,ave}shows some stability-dependent fluctuations; that is, one among

*β*

_{1,loc}and

*β*

_{1,ave}is not a constant. Following Frehlich,

*β*

_{1,ave}≈ 1.33

*β*

_{1,loc}under stable stratification, whereas our convective case yields

*β*

_{1,ave}≈ 0.83

*β*

_{1,loc}. Alternatively, Eq. (12) cannot correctly estimate

*A*(

*C*

_{T}_{,r}

^{2}) at all ranges of stability. For example,

*β*

_{1,loc}

*A*(ε

_{T}

_{,r})[

*A*(ε

_{TKE,Δ})]

^{−1/3}produces a biased estimate of

*A*(

*C*

_{T}_{,r}

^{2}) 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*(*C _{n}*

_{,r}

^{2}) 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*(ε

_{TKE,r}) and

*A*(ε

_{T}

_{,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 *C _{n}*

_{,r}

^{2}. In many standard wave propagation applications,

*r*is within the inertial–convective range. In the generic case,

*C*

_{n}_{,r}

^{2}is a combination of

*C*

_{u}_{,r}

^{2},

*C*

_{T}_{,r}

^{2},

*C*

_{q}_{,r}

^{2}, and

*C*

_{T}_{,q,r}. In this study, we have used a large-eddy simulation to document the statistical distribution and the spatial organization of

*C*

_{n}_{,Δ}

^{2}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 *α*_{1,loc} and *β*_{1,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 ε_{TKE,Δ}, ε_{T}_{,Δ}, and C_{T}_{,Δ}^{2} 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 *C _{T}*

_{,Δ}

^{2}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

*C*

_{T}_{,Δ}

^{2}may be a poor indicator of the performance bounds of a sensor sensitive to turbulence. The probability density function of

*C*

_{T}_{,Δ}

^{2}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 *C _{T}*

_{,Δ}

^{2}with a physically based criterion to sort the convective plumes. In the lower boundary layer, the plumes have some excesses in ε

_{TKE,Δ}, ε

_{T}

_{,Δ}, and

*C*

_{T}_{,Δ}

^{2}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 ε

_{T}

_{,Δ}at the edges of plumes. Consequently, the plumes’ excess in ε

_{TKE,Δ}increases, and plumes show comparable ε

_{T}

_{,Δ}and

*C*

_{T}_{,Δ}

^{2}as their environment. Our results support a bimodal description of the PDF of log(ε

_{TKE,Δ}), log(ε

_{T}

_{,Δ}), and log(

*C*

_{T}_{,Δ}

^{2}). 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 ε

_{TKE,Δ}and ε

_{T}

_{,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 *C _{u}*

_{,r}

^{2}is of 1.1–1.2 almost irrespective of stability (see above). The bias for

*C*

_{T}_{,r}

^{2}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(ε_{TKE,r}), ln(ε_{T}_{,r}), and ln(*C _{T}*

_{,r}

^{2}) linearly decrease with ln(

*r*′), independently of height, for

*r*′ =

*r*/2 small enough. The intermittency parameters are

*μ*

_{TKE}≈ 0.2 and

*μ*≈ 0.5, in good agreement with various experimental data. For

_{T}*C*

_{T}_{,Δ}

^{2}, we obtain

*μ*≈ 0.4.

_{C}The present method and results can be extended in various directions. First, one may use the LES distribution of *C _{n}*

_{,Δ}

^{2}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

*C*

_{n}_{,r}

^{2}. 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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,*J. Opt. Soc. Amer.***61****,**1646–1650.Wyngaard, J. C., O. R. Coté, and Y. Izumi, 1971b: Local free convection, similarity, and the budget of shear stress and heat flux.

,*J. Atmos. Sci.***28****,**1171–1182.Wyngaard, J. C., L. J. Peltier, and S. Khanna, 1998: LES in the surface layer: Surface fluxes, scaling, and SGS modeling.

,*J. Atmos. Sci.***55****,**1733–1754.Wyngaard, J. C., N. Seaman, S. J. Kimmel, M. Otte, X. Di, and K. E. Gilbert, 2001: Concepts, observations, and simulation of refractive index turbulence in the lower atmosphere.

,*Radio Sci.***36****,**643–669.

## APPENDIX

### LES Closures with Local Similarity

*A*,

_{u}*A*,

_{s}*N*, and

_{u}*N*in terms of

_{s}*α*

_{1,loc}and

*β*

_{1,loc}. We note that

*α*

_{loc}= (55/18)

*α*

_{1,loc}and

*β*

_{loc}= (5/3)

*β*

_{1,loc}. Since Δ

_{c}is in the inertial–convective range, the spectra formed using Eq. (2) in Eq. (1) apply over

*κ*= [2

*π*/Δ

_{c}; ∞]. A spectral integration and comparison with Eqs. (4) and (5) gives

*N*and

_{u}*N*, we write the gradient production terms

_{s}*P*and

_{s}*P*

_{TKE}in the scalar variance and TKE budgets with Eq. (3):

*D*and

_{s}*D*

_{TKE}, relate to resolved variables. Due to the squared gradients, their spectral characterization weights Eq. (3) with a factor

*κ*

^{2}, so the highest resolved wavenumbers are the main contributors. Assuming that the spectra follow the inertial-range behavior for those wavenumbers and integrating over

*κ*= [0; 2

*π*/Δ

_{c}] yields

*P*= ε

_{s}_{s}and

*P*

_{TKE}= ε

_{TKE}. This leads to

We see from this derivation that the choice of *A _{u}*,

*A*,

_{s}*N*, and

_{u}*N*as (constant) closures in the local relations (3)–(6) assumes that

_{s}*α*

_{1,loc}and

*β*

_{1,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

*R*

_{TKE}

*α*

_{1,ave}and

*R*

_{T}β_{1,ave}instead of

*α*

_{1,loc}and

*β*

_{1,loc}. It has next to be assumed that

*R*

_{TKE}≈ 1 and

*R*≈ 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

_{T}*A*,

_{u}*A*,

_{s}*N*, and

_{u}*N*.

_{s}(a) PDF of log(*C _{n}*

_{,Δ}

^{2}) with height (mean as full white line); (b) centered PDF of log(

*C*

_{n}_{,Δ}

^{2}) at four heights with Gaussian of same variance (dashed); (c) normalized variance

*F*

_{T}_{,Δ}with the LES prediction (dots) of Peltier and Wyngaard (1995).

Citation: Journal of the Atmospheric Sciences 66, 4; 10.1175/2008JAS2790.1

(a) PDF of log(*C _{n}*

_{,Δ}

^{2}) with height (mean as full white line); (b) centered PDF of log(

*C*

_{n}_{,Δ}

^{2}) at four heights with Gaussian of same variance (dashed); (c) normalized variance

*F*

_{T}_{,Δ}with the LES prediction (dots) of Peltier and Wyngaard (1995).

Citation: Journal of the Atmospheric Sciences 66, 4; 10.1175/2008JAS2790.1

(a) PDF of log(*C _{n}*

_{,Δ}

^{2}) with height (mean as full white line); (b) centered PDF of log(

*C*

_{n}_{,Δ}

^{2}) at four heights with Gaussian of same variance (dashed); (c) normalized variance

*F*

_{T}_{,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(*C _{n}*

_{,Δ}

^{2}) 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

Cross section (*x*,*z*) of log(*C _{n}*

_{,Δ}

^{2}) 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

Cross section (*x*,*z*) of log(*C _{n}*

_{,Δ}

^{2}) 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(C_{n}_{,Δ}^{2}) 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

Horizontal (*x*,*y*) distribution of log(C_{n}_{,Δ}^{2}) 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

Horizontal (*x*,*y*) distribution of log(C_{n}_{,Δ}^{2}) 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(ε_{TKE,Δ}), log(ε_{T}_{,Δ}), and log(*C _{n}*

_{,Δ}

^{2}) 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 PDF of log(ε_{TKE,Δ}), log(ε_{T}_{,Δ}), and log(*C _{n}*

_{,Δ}

^{2}) 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 PDF of log(ε_{TKE,Δ}), log(ε_{T}_{,Δ}), and log(*C _{n}*

_{,Δ}

^{2}) 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(ε_{TKE,Δ}) and log(ε_{T}_{,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

Centered JPD of log(ε_{TKE,Δ}) and log(ε_{T}_{,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

Centered JPD of log(ε_{TKE,Δ}) and log(ε_{T}_{,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 *C _{n}*

_{,Δ}

^{2}for specified values of ε

_{T}

_{,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

Log of the conditional average of *C _{n}*

_{,Δ}

^{2}for specified values of ε

_{T}

_{,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

Log of the conditional average of *C _{n}*

_{,Δ}

^{2}for specified values of ε

_{T}

_{,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(ε_{TKE,r}), (b) ln(ε_{T}_{,r}), and (c) ln(*C _{n}*

_{,r}

^{2}) 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

Variance of (a) ln(ε_{TKE,r}), (b) ln(ε_{T}_{,r}), and (c) ln(*C _{n}*

_{,r}

^{2}) 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

Variance of (a) ln(ε_{TKE,r}), (b) ln(ε_{T}_{,r}), and (c) ln(*C _{n}*

_{,r}

^{2}) 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) *R _{T}*

_{,Δ}(solid) and

*R*

_{TKE,Δ}(dashed), (b) Normalized rhs terms in Eq. (16): lhs term (solid), first rhs term (dashed), second rhs term (alternate dashed), and −0.18

*σ*(ε

_{T}

_{,Δ})/

*A*(ε

_{T}

_{,Δ}) (dotted). (c) Correlation coefficients between ε

_{T}

_{,Δ}and

_{T}

_{KE,Δ}(solid) and between ε

_{T}

_{,Δ}and (ε

_{TKE,Δ})

^{−1/3}(dashed).

Citation: Journal of the Atmospheric Sciences 66, 4; 10.1175/2008JAS2790.1

Vertical profiles of (a) *R _{T}*

_{,Δ}(solid) and

*R*

_{TKE,Δ}(dashed), (b) Normalized rhs terms in Eq. (16): lhs term (solid), first rhs term (dashed), second rhs term (alternate dashed), and −0.18

*σ*(ε

_{T}

_{,Δ})/

*A*(ε

_{T}

_{,Δ}) (dotted). (c) Correlation coefficients between ε

_{T}

_{,Δ}and

_{T}

_{KE,Δ}(solid) and between ε

_{T}

_{,Δ}and (ε

_{TKE,Δ})

^{−1/3}(dashed).

Citation: Journal of the Atmospheric Sciences 66, 4; 10.1175/2008JAS2790.1

Vertical profiles of (a) *R _{T}*

_{,Δ}(solid) and

*R*

_{TKE,Δ}(dashed), (b) Normalized rhs terms in Eq. (16): lhs term (solid), first rhs term (dashed), second rhs term (alternate dashed), and −0.18

*σ*(ε

_{T}

_{,Δ})/

*A*(ε

_{T}

_{,Δ}) (dotted). (c) Correlation coefficients between ε

_{T}

_{,Δ}and

_{T}

_{KE,Δ}(solid) and between ε

_{T}

_{,Δ}and (ε

_{TKE,Δ})

^{−1/3}(dashed).

Citation: Journal of the Atmospheric Sciences 66, 4; 10.1175/2008JAS2790.1