1. Introduction
The structure function of a random turbulent field represents the intensity of fluctuations with spatial length scales that are smaller than, or on the order of, a prescribed separation distance (Kolmogorov 1941a,b; Tatarskii 1961). Examples of random fields in the atmosphere include spatial distributions of meteorological variables such as temperature, humidity, velocity, and refractive index of the air. For many applications associated with atmospheric boundary layer processes, it is customary to characterize turbulent fluctuations within the inertial subrange of turbulence scales through a single representative parameter called the structure-function (or structure) parameter. It enables the description of the turbulence fluctuations in terms of a single quantity by removing the explicit dependence on a separation distance.
Properties of structure parameters for scalars have been historically discussed in the literature more than those for velocity (e.g., Wyngaard et al. 1971; Burk 1980; Andreas 1988; Peltier and Wyngaard 1995; Frederickson et al. 2000; Wilson and Fedorovich 2012; Wainwright et al. 2015), although the topic has been addressed (e.g., Lesieur and Metais 1996; Rizza et al. 2006, 2010). This is apparently because the parameters of scalar fluctuations (particularly, fluctuations of temperature and humidity) are essential for designing atmospheric measurement techniques, such as scintillometry, and for understanding the physics of atmospheric acoustic and electromagnetic wave propagation, which is fundamental for remote sensing applications. The idea of using structure functions to describe atmospheric turbulence was first put forward by Kolmogorov (1941a,b). In these seminal works, Kolmogorov established fundamental properties of turbulence dynamics and energy transfer across turbulent scales through the use of velocity structure functions. The structure functions and associated structure parameters for velocity are important for assessing boundary layer mixing (Moulsley et al. 1981), turbulence intermittency (Gaudin et al. 1998), and acoustic wave scattering (Little 1969).
In this paper, we extend the work of Gibbs et al. (2016), which dealt with structure functions of potential temperature fluctuations, and focus on second-order velocity structure functions and corresponding structure parameters in the atmospheric convective boundary layer (CBL). In Gibbs et al. (2016), three methods to compute the potential temperature structure parameter for two atmospheric CBL regimes were evaluated. Those methods included the direct method (DM), the true spectral method (TSM), and the conventional spectral method (CSM). Hereinafter, we modify the naming of the CSM to the approximate spectral method (ASM) for reasons discussed in section 2. The DM implies the evaluation of the structure function by directly applying its mathematical formulation to the gridded numerical simulation data. The structure parameter is then found by limiting the structure function to spatial increments that lie within the inertial subrange of turbulence scales. According to the TSM, the structure function is calculated through an integral relationship between the spectral density and second-order structure function under the assumption of turbulence isotropy. The computational burden of the TSM may be reduced as compared with the DM due to the numerical efficiency of the fast Fourier transform (FFT) technique used for computing spectra, although the numerical evaluation of the integral may offset such gains. The structure parameter is then retrieved from the structure function in the same manner as with the DM. Last, the ASM is based on an analytical relationship between the structure parameter and spectral density of velocity by assuming that the velocity spectrum follows the inertial-subrange scaling over the entire wavenumber range (from 0 to ∞). This simplification makes the ASM the least computationally intensive procedure and is the main reason for its popularity in practical evaluation of structure parameters, at least when applied to fluctuations of atmospheric scalars (e.g., Kaimal 1973; Wyngaard and LeMone 1980; Green et al. 1994; Cheinet and Siebesma 2009; Maronga 2014).
In Gibbs et al. (2016), potential temperature structure parameters evaluated by the DM and TSM were found to be nearly identical except, most notably, in the near-surface region, where turbulence anisotropy over horizontal planes manifests most strongly. Conversely, the potential temperature structure parameters computed according to the ASM were comparatively overpredicted by as much as an order of magnitude. It was hypothesized that this discrepancy was a result of assumptions underlying the ASM, and that such behavior was further exacerbated in regions prone to deviations from local isotropy. Accordingly, the DM and TSM were recommended as the preferred methods to evaluate structure parameters using gridded numerical simulation data.
In the current work, we use high-resolution large-eddy simulation (LES) to reproduce 3D velocity fields in a shear-free and a shear-driven atmospheric CBL. Turbulence properties are explored using velocity structure functions, and the velocity structure parameters are computed using the three methods summarized above, with comparisons made between results obtained by each method. Descriptions of the methods used to evaluate structure functions and structure parameters are given in section 2. Details of the numerical simulations and data processing methodology are presented in section 3. Results are shown in section 4, while discussion and conclusions are provided in section 5.
2. Structure function and structure parameter formulations
Spatial variability of velocity component fluctuations associated with atmospheric turbulence may be described in terms of the second-order velocity structure function (Tatarskii 1961; Pope 2000; Wyngaard 2010), given by the tensor
where x is the position vector, r is the separation vector, and ui and uj (i = 1, 2, 3; j = 1, 2, 3; u1, u2, u3 = u, υ, w) are velocity fluctuation components associated with the coordinate directions xk (k = 1, 2, 3; x1, x2 = x, y: horizontal, x3 = z: vertical). Here,
where the scalar functions of separation distance DLL and DNN are referred to as the longitudinal and transverse structure functions. If the coordinate system is oriented such that the separation vector is in the x direction (i.e., r = e1r), then
Based on the similarity hypotheses presented in Kolmogorov (1941a), the velocity structure function may be written as
where C2 is a universal constant, ε is the turbulence kinetic energy dissipation rate, and
The longitudinal and transverse velocity structure parameters are related as
The velocity structure function may be directly computed using (1) applied to numerical LES gridded data if the turbulence is assumed to be isotropic, and ensemble averaging is approximated by evaluating means over statistically homogeneous spatial directions. Using such directly computed structure functions, the velocity structure parameter may be evaluated for a given separation distance by normalizing the inertial subrange value of Dij by the corresponding inertial subrange separation distance r according to (4). We adopt the naming convention in Gibbs et al. (2016) and refer to this procedure as the direct method. As noted in Gibbs et al. (2016), this method has certain drawbacks despite its straightforwardness as compared to the other methods presented below. By design, the DM requires data on velocity from multiple locations within the flow. Furthermore, while this limitation may be overcome by applying Taylor’s frozen turbulence hypothesis to high-frequency temporal data at a single location, the generalizability of the method is compromised when it is applied to heterogeneous atmospheric flows.
We explore two alternative methods discussed in Gibbs et al. (2016) based on the mathematical relationship between Dij and one-dimensional spectral density of velocity Φij. The designation of these methods is to evaluate the velocity structure parameter from simulation (and potentially, also observational) data in the most efficient manner while retaining as much of the original physical interpretation as possible. The first of the alternatives to the DM is called the true spectral method. It is based on the following relationship between Dij and Φij under the assumption of turbulence isotropy (Tatarskii 1961; Wyngaard 2010):
where k ≡ |k| is the wavenumber. The “true” part of the method name points to the lack of any assumption made about the form of the spectral density function Φij in the integral. Using the TSM, Dij is numerically computed from LES gridded data through (6) once Φij is known. Employing the numerical FFT technique for calculation of the spectral function makes the TSM technique attractive compared to the numerical overhead associated with the DM, although potential gains may be offset by the combined effects of computing spectral density and the numerical evaluation of the integral in (6).
The second alternative to the DM considered in Gibbs et al. (2016) is what we call the approximate spectral method, which is referred to by Gibbs et al. (2016) as the conventional spectral method. However, to our knowledge, this method is not as widely reported in the literature for velocity as for refractive index and temperature, which makes the term “conventional” potentially misleading in this context. According to ASM, the integral in (6) is evaluated analytically for Φij assumed to have the inertial-subrange representation Φij = Ak−5/3 across the entire range of turbulence scales (Tatarskii 1961; Essenwanger and Reiter 1969; Wyngaard et al. 1971), that is, for k from 0 to ∞, which provides
where Γ is the gamma function. Making use of (4) yields
Solving for A provides
which results in
Rearranging (8) leads to the following approximate relationship between the velocity structure parameter and velocity spectrum:
The value 0.125 in the denominator of (9) differs from the widely adopted proportionality coefficient of 0.25 originally presented in Wyngaard et al. (1971). We refer the reader to Gibbs and Fedorovich (2020) for a formal derivation that shows the value of 0.25 is an error. The ASM is advantageous computationally since the estimation of
3. Experimental design
a. Numerical simulation code
Two simulations were conducted using MicroHH (van Heerwaarden et al. 2017), an open-source computational fluid dynamics code, applied in LES mode. The code has proven successful at faithfully reproducing atmospheric flows across a range of environmental regimes (e.g., van Heerwaarden et al. 2014; Gentine et al. 2015; Fedorovich et al. 2017; van der Linden et al. 2019). The filtered Boussinesq-approximated Navier–Stokes equations of motion and scalar transport equations are spatially discretized and solved numerically using second-order, centered finite differencing of the advection and diffusion terms. A third-order Runge–Kutta scheme is applied for time integration of these equations, and a second-order Poisson solver is used for pressure. The subgrid turbulence closure is based on the Lilly–Smagorinsky model (Lilly 1967), in which the subfilter eddy diffusivity is assumed proportional to the strain-rate tensor and the subfilter scalar diffusivity is prescribed using the subgrid turbulent Prandtl number.
b. Simulated convective boundary layers
We applied the three methods described in section 2 (DM, TSM, and ASM) to gridded numerical output from LES of a shear-free (hereinafter “Free”) and shear-driven (hereinafter “Shear”) CBL. The studied CBL flow types were qualitatively similar to those reproduced by the University of Oklahoma LES (OU-LES) code used in Gibbs et al. (2016). One difference is enhanced spatial resolution in our current simulations. Another difference is that we now operate with buoyancy instead of potential temperature. The buoyancy is defined through b = g(Θ − Θenv)/Θr, where g is acceleration due to gravity, Θ is potential temperature, Θenv is the environmental potential temperature, and Θr is a constant reference potential temperature.
Configuration details for each simulation are provided in Table 1. Each simulation was initialized with a vertically constant statically stable background stratification quantified in terms of N2, where N = [(g/Θr)(dΘenv/dz)]1/2 is the Brunt–Väisälä frequency. The Coriolis parameter was set with consideration to midlatitudes in the Northern Hemisphere. The lower and upper boundary conditions for velocity were no-slip and free-slip, respectively. The corresponding conditions for buoyancy were prescribed in the form of buoyancy flux, positive at the lower boundary and zero at the top boundary. Lateral boundary conditions for all prognostic fields were periodic. A Rayleigh damping layer was applied in the upper 20% of the domain. A constant geostrophic wind ug = 10 m s−1 was applied in the x direction in the Shear case, and ug was set to zero in the Free case.
Simulation configuration values for geostrophic wind ug (υg = 0), background stratification N2, surface kinematic buoyancy flux
Properties of each simulation are given in Table 2. Each simulation was run for approximately 30 large-eddy turnover times T* = zi/w* (6 h of physical time), where zi is the boundary layer depth (computed as the level at which the vertical buoyancy flux
Simulation properties, including boundary layer depth zi, Obukhov length L, instability parameter −zi/L, friction velocity u*, buoyancy scale b*, and convective velocity scale w* at the comparison time of 14 400 s.
Potential temperature deviations from means over horizontal planes are shown in Fig. 1, which allow for visualization of dominant CBL flow structures for each regime over vertical and horizontal cross sections of the flow. Potential temperature deviation θ = Θ − Θr is related to buoyancy as θ = (Θr/g)[b − N2(Lz − z)], where Θr is set to 300 K and Lz is the domain length in the vertical direction. The Free CBL displays the expected organized cellular-type convection with vertically oriented structure (e.g., Deardorff 1972; Schmidt and Schumann 1989; Sullivan and Patton 2011), while the Shear CBL shows evidence of elongated flow-aligned rolls that are rotated horizontally due to the joint effect of the Coriolis force and turbulent friction, and tilted vertically due to mean wind shear (e.g., Moeng and Sullivan 1994; Khanna and Brasseur 1998; Salesky et al. 2017).
Potential temperature perturbations reproduced by MicroHH for the (a),(b) Free case and (c),(d) Shear case. (left) Horizontal x–y cross sections are taken at z/zi = 0.1, where zi is the depth of the boundary layer, and (right) x–z vertical cross sections are taken at the midpoint in the y direction. Cross-sectional locations are denoted by dashed white lines.
Citation: Journal of the Atmospheric Sciences 77, 10; 10.1175/JAS-D-20-0038.1
c. Data processing
Before computing the structure functions and spectra, horizontal planar means (denoted by overbars) were subtracted from each field to produce perturbation quantities
Let us consider the DM, which implies direct evaluation of the velocity structure function. We choose a coordinate system where the separation vector is in the x direction (r = e1r). As described in (3), Dij is only defined for i = j = 1, 2, 3 under this alignment. Accordingly, we start by summing the square differences of each velocity component in the x direction at a given height and for a given separation distance using (1) {i.e., summing [ui(x + r, y, z) − ui(x, y, z)]2}. The sum is only modified if x + r ≤ Lx, where Lx is the domain length in the x direction. Last, we divide the sum by the number of incremental squared differences to arrive at the structure function Dij(z, r). This process is repeated for all separation distances and vertical levels. We separately repeated this process for a separation vector oriented in the y direction (r = e2r) and found results to be consistent with the change of separation vector orientation (not shown). Additionally, Wilson and Fedorovich (2012) describe a procedure to compute the structure function through separations in the z direction (r = e3r), where the squared differences for a particular separation distance are evaluated as the averages of the differences in the positive (upward) and negative (downward) directions. However, the number of physically meaningful separation distances is severely limited near the upper and lower boundaries when using this procedure. These flow regions, especially the one near the lower boundary, are also prone to anisotropy and non-Kolmogorov turbulence spectral behavior. This affects the robustness and interpretability of computed structure functions. For these reasons, we only consider structure functions based on separations in the x direction.
Spectral densities (spectra) of velocity components are needed for both the TSM and ASM. Consistent with Gibbs et al. (2016), we followed the spectrum computation algorithm described in Gibbs and Fedorovich (2014). For a given horizontal plane, spectral density was computed for each velocity component in the x direction (Φ11, Φ22, Φ33) and then averaged in the y direction. This process was repeated for each vertical level. For the TSM, the structure function was computed at each height and for all separation distances using (6). Next, the inertial subrange values of r (risr) and k (kisr) need to be evaluated in order to compute the structure parameters for each method.
We first attempted an objective method (hereinafter the polyfit method) very similar to that used in Gibbs et al. (2016), which itself was based on a method suggested by Hartogensis and De Bruin (2005). At each height, a polynomial regression was fit to the spectra since the raw spectra were very noisy locally in wavenumber space. Before doing so, we cut structure functions and spectra over the 100 largest wavenumbers. These large wavenumber parts are practically never within the inertial subrange, and their absence from the regression calculation does not affect the identification thereof. Using the regression line, the inertial subrange was established objectively as the widest contiguous region of wavenumbers over which the slope in logarithmic space followed the −5/3 power law within a 20% error tolerance. An analogous procedure was used to determine the inertial subrange for structure functions by applying the 2/3 power-law criterion to the logarithmic slope. The geometric means of r and k within the identified subranges were taken to compute risr and kisr. If the identified inertial subrange extended over fewer than three wavenumbers (separation distances), then that vertical level was ignored. At the end of the procedure, missing values were computed via interpolation. We found this method to be very sensitive to the choice of regression, error tolerance, and smoothing filter applied to the data. Such issues with ad hoc methods are well known (Ortiz-Suslow et al. 2020).
We next sought a more robust objective method that required far less human intervention. We rescaled structure functions and spectral densities by multiplying Dij by
Rescaled structure functions of velocity for the (a)–(c) Free case and (d)–(f) Shear case. Values are normalized by the local (in height) maxima. The values of risr found using the polyfit (blue dotted line) and flattop (red dotted line) methods are shown.
Citation: Journal of the Atmospheric Sciences 77, 10; 10.1175/JAS-D-20-0038.1
Rescaled one-dimensional spectral density of velocity for the (a)–(c) Free case and (d)–(f) Shear case. Values are normalized by the local (in height) maxima. The values of kisr found using the polyfit (blue dotted line) and flattop (red dotted line) methods are shown.
Citation: Journal of the Atmospheric Sciences 77, 10; 10.1175/JAS-D-20-0038.1
Last, the risr identified by the flattop method and the structure function associated with risr were applied using (4) to compute the structure parameters according to the DM and TSM. The structure parameter according to the ASM was found at each height by applying the flattop procedure to the relevant spectra to identify kisr, which were then used in (9). Standard error values were very small compared with the structure parameter at each height due to a large effective sample size. Thus, the corresponding confidence intervals are not shown in the respective figures.
4. Results
Here we present both structure functions and structure parameters for velocity components computed using the methods outlined in section 3c.
a. Structure functions
To evaluate representative turbulence characteristics of the considered CBL flow types, we first examine the structure functions computed using the DM. The longitudinal and transverse structure functions, as well as the theoretical relationship between the two, are shown in Fig. 4 as functions of separation distance at height z/zi = 0.5. The transverse structure function values are larger than those of the longitudinal counterparts for horizontal flow components, as expected, in both the Free and Shear cases. In the Free case, the inertial subrange is identified as the range of scales where D11 and D22 follow closely the 2/3 power-law slope. This region is approximately coincident with, and slightly wider than, the range of scales where the local isotropy relationship D22 = (4/3)D11 holds (Pope 2000). On the other hand, D33 has a much narrower inertial subrange and its values are noticeably larger than (4/3)D11. Additionally, the D33 slope actually increases at larger separations (r ≈ 0.1–0.6zi). We suspect that this feature reflects the cellular convection organization of the CBL flow, where the narrow bands of updrafts cause the slopes of structure functions to steepen. Visual inspection of Fig. 1 indicates that the characteristic spacing between these bands is roughly within the range of increased slope. A more detailed objective measure of these effects is needed, however, and is beyond the scope of the current work. One potential future avenue is to explore the “roll factor” introduced in Salesky et al. (2017), which is based on the two-point correlation of vertical velocity in polar coordinates across horizontal planes. In the Shear case, there is again a range of scales, albeit narrower than in the Free case, where the 2/3 power law and local isotropy hold for D11 and D22. In contrast, D33 follows the 2/3 power-law slope for a range scales in the Shear case and its value does not exceed (4/3)D11 by as much as it does in the Free case. Apparently, assumptions of local isotropy for D33 are better upheld in boundary layers with the presence of mean shear than in those without.
Structure functions of velocity computed using the DM for the (a) Free and (b) Shear cases. Solid black lines represent the theoretical 2/3 power law.
Citation: Journal of the Atmospheric Sciences 77, 10; 10.1175/JAS-D-20-0038.1
b. Structure-function parameters
We now show and discuss the structure parameters for velocity components. First, we will examine turbulence isotropy by comparing longitudinal and transverse structure parameters. Next, we compare profiles of the structure parameters as computed by the DM, TSM, and ASM.
1) Longitudinal versus transverse
Vertical profiles of normalized
Structure-function parameters of velocity computed using the DM for the (a) Free and (b) Shear cases. Dashed black lines are the theoretical isotropic transverse structure parameter defined by
Citation: Journal of the Atmospheric Sciences 77, 10; 10.1175/JAS-D-20-0038.1
2) Direct versus spectral methods
Last, we compare vertical profiles of
Structure-function parameters of velocity computed using the DM (blue solid lines), TSM (orange dashed lines), and ASM (green dotted lines) for the (a)–(c) Free case and (d)–(f) Shear case.
Citation: Journal of the Atmospheric Sciences 77, 10; 10.1175/JAS-D-20-0038.1
5. Discussion and conclusions
In this paper, we focus on structure functions and structure parameters for velocity fields in two CBL types (Free and Shear). Atmospheric boundary layer literature has historically paid less attention to these velocity functions/parameters in favor of their scalar counterparts (temperature, moisture, refractive index), despite the utility of the former to assess boundary layer mixing, turbulence intermittency, acoustic wave scattering, and more. We demonstrated the feasibility of direct retrieval of the velocity structure functions and parameters from numerically simulated CBL flow fields. Three methods were compared: calculating the structure function and parameter directly according to their mathematical definitions (DM), computing the same quantities using an integral relationship between the structure function and spectral density of velocity (TSM), and employing an analytical approximation of the TSM that relates the structure parameter to spectral density in the inertial subrange (ASM).
In both the Free and Shear cases, the structure functions for horizontal velocity components exhibited extended regions that followed the 2/3 power law in the middle section of the CBL (at z/zi = 0.5). The lower limit of this range was approximately 4–5 times the grid spacing used in the numerical LES mesh, which matches the findings of Wilson and Fedorovich (2012). Additionally, assumptions of local isotropy [D22 = (4/3)D11] were verified over a range of separation distances. The inertial subrange according to the 2/3 power law in the structure function for vertical velocity was rather narrow in the Free case, with D33 exhibiting a steepening slope at larger separation distances. We hypothesize that this is a result of organized cellular convection structure in the flow. Conversely, the inertial subrange for D33 was wider in the Shear case. In both cases, the assumption of local isotropy [D33 = (4/3)D11] was violated, although the effect was less substantial in the Shear case.
Vertical profiles of the associated structure parameters indicated that values were enhanced (reduced) near the surface (boundary layer top) in the presence of shear. Profiles also demonstrated that local isotropy was upheld for
Vertical profiles of the structure parameters as computed by the DM, TSM, and ASM were compared across cases. In all situations, the DM and TSM were nearly identical. Meanwhile, the ASM overestimated these values for all scenarios by a factor of up to 2. Note that (6) places no limits on the velocity spectrum, while (9) assumes that the entire velocity spectrum follows inertial subrange scaling. In other words, the TSM allows for regions of the spectral density beyond the inertial subrange, while the ASM does not. Consider the idealized spectra in Fig. 7, where a comparison is presented of the Kolmogorov spectrum with a spectrum from a typical large-eddy simulation. The integral of the former is notably larger than of the latter. The result is an apparent overestimation of
Idealized velocity spectra. The black line follows Kolmogorov scaling across all wavenumbers, and the gray line is qualitatively representative of velocity spectra in the atmospheric boundary layer.
Citation: Journal of the Atmospheric Sciences 77, 10; 10.1175/JAS-D-20-0038.1
Acknowledgments
Funding was provided by NOAA/Office of Oceanic and Atmospheric Research under NOAA–University of Oklahoma Cooperative Agreement NA11OAR4320072, U.S. Department of Commerce. Valuable local computing assistance was provided by Gerry Creager. We thank Dr. Elizabeth Smith (NOAA National Severe Storms Laboratory) for reviewing the paper and offering helpful feedback related to content, data processing, and visualization. We also thank Drs. Cristian Proistosescu (University of Illinois at Urbana–Champaign), Scott Giangrande (Brookhaven National Laboratory), and James Correia, Jr. (University of Colorado Boulder CIRES, NOAA/NWS/Weather Prediction Center), for fruitful discussions about filtering techniques used in the objective methods to identify the inertial subrange.
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