A Wavenumber–Frequency Spectrum Model for Sheared Convective Atmospheric Boundary Layer Flows

Naseem Ali; Juan Pedro Mellado; Michael Wilczek

doi:10.1175/JAS-D-22-0079.1

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Journal of the Atmospheric Sciences

Abstract
1. Introduction
2. Direct numerical simulations
3. Linear random advection model
4. Parameterizations of the wavenumber–frequency spectrum: von Kármán model
5. Conclusions
Acknowledgments.
Data availability statement.
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Fig. 1.

Vertical cross section of the logarithmic buoyancy gradient for (a) Fr₀ = 10 and (b) Fr₀ = 20 cases.

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Fig. 2.

Space–time plots of the (left) streamwise velocity, (center) vertical velocity, and (right) buoyancy fluctuations along the streamwise direction for Fr₀ = 10 case. (a)–(c) Mixed layer (z/z_enc = 0.33) and (d)–(f) the entrainment zone (z/z_enc = 1.23). Here and in the following, b₀ = B₀/(N₀L₀).

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Fig. 3.

Comparison of the wavenumber–frequency spectra obtained from DNS and the LRAM for streamwise velocity, vertical velocity, and buoyancy in the mixed layer (z/z_enc = 0.33) of Fr₀ = 20 case. (a)–(c) Contours from DNS (colored shades) and the model (dashed black lines). (d)–(f) Normalized cuts through the k₁–ω spectra from DNS (colored lines), along with the results from the LRAM (black lines).

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Fig. 4.

Comparison of wavenumber–frequency spectra obtained from DNS and the LRAM for streamwise velocity, vertical velocity, and buoyancy in the mixed layer (z/z_enc = 0.33) of Fr₀ = 10 case. (a)–(c) Contours from DNS (colored shades) and the model (dashed black lines). (d)–(f) Normalized cuts through the k₁–ω spectra from DNS (colored lines), along with the results from the LRAM (black lines).

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Fig. 5.

Comparison of wavenumber–frequency spectra obtained from DNS and the LRAM for streamwise velocity, vertical velocity, and buoyancy in the entrainment zone (z/z_enc = 1.23). (a)–(c) The Fr₀ = 20 case and (d)–(f) the Fr₀ = 10 case.

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Fig. 6.

The Jensen–Shannon divergence between LRAM- and DNS-based conditional spectra for (a) streamwise velocity, (b) vertical velocity, and (c) buoyancy.

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Fig. 7.

Two-dimensional wavenumber spectra for the Fr₀ = 20 case. Contours from DNS (colored shades) and the model (dashed black lines). (a)–(c) Spectra from the mixed layer (z/z_enc = 0.33) and (d)–(f) spectra from the entrainment zone (z/z_enc = 1.23).

View raw image

Fig. 8.

Two-dimensional wavenumber spectra for the Fr₀ = 10 case. Contours from DNS (colored shades) and the model (dashed black lines). (a)–(c) Spectra from the mixed layer (z/z_enc = 0.33) and (d)–(f) spectra from the entrainment zone (z/z_enc = 1.23).

View raw image

Fig. 9.

Comparison between the one-dimensional wavenumber spectra obtained from the model and DNS for the streamwise velocity, vertical velocity, and buoyancy. (a)–(c) Mixed layer and (d)–(f) entrainment zone for the Fr₀ = 20 case.

View raw image

Fig. 10.

Comparison of the wavenumber–frequency spectra for streamwise velocity, vertical velocity, and buoyancy in the mixed layer (z/z_enc = 0.33) for the Fr₀ = 20 case. (a)–(c) Contours from DNS (colored shades) and the model (dashed black lines). (d)–(f) Normalized cuts through the k₁–ω spectra from DNS (colored lines), along with the results from the LRAM (black lines).

View raw image

Fig. 11.

Comparison of the wavenumber–frequency spectra for streamwise velocity, vertical velocity, and buoyancy in the entrainment zone (z/z_enc = 1.23) for the Fr₀ = 20 case.

View raw image

Fig. 12.

Comparison of the wavenumber–frequency spectra for streamwise velocity, vertical velocity, and buoyancy in the mixed layer (z/z_enc = 0.33) for the Fr₀ = 10 case.

View raw image

Fig. 13.

Comparison of the wavenumber–frequency spectra for streamwise velocity, vertical velocity, and buoyancy in the entrainment zone (z/z_enc = 1.23) for the Fr₀ = 10 case.

View raw image

Fig. 14.

The Jensen–Shannon divergence between von Kármán–based LRAM- and DNS-based conditional wavenumber–frequency spectra for (a) streamwise velocity, (b) vertical velocity, and (c) buoyancy.

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Article Type:: Research Article

A Wavenumber–Frequency Spectrum Model for Sheared Convective Atmospheric Boundary Layer Flows

Naseem Ali

Naseem Ali ^aMax Planck Institute for Dynamics and Self-Organization, Göttingen, Germany

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Juan Pedro Mellado

Juan Pedro Mellado ^cDepartment of Earth System Sciences, University of Hamburg, Hamburg, Germany

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Michael Wilczek

Michael Wilczek ^aMax Planck Institute for Dynamics and Self-Organization, Göttingen, Germany
^bTheoretical Physics I, University of Bayreuth, Bayreuth, Germany

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Online Publication:: 16 Feb 2023

Print Publication:: 01 Mar 2023

DOI:: https://doi.org/10.1175/JAS-D-22-0079.1

Page(s):: 763–776

Received:: 02 Apr 2022

Accepted:: 15 Sep 2022

Published Online:: 16 Feb 2023

Displayed acceptance dates for articles published prior to 2023 are approximate to within a week. If needed, exact acceptance dates can be obtained by emailing amsjol@ametsoc.org.

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Abstract

Parameterizing turbulence in the atmospheric boundary layer as a function of space and time is essential for weather and climate models. Here, we explore a model for wavenumber–frequency spectra based on a linear random advection approach to characterize sheared convective atmospheric boundary layer flows. Building on previous works, we obtain the wavenumber–frequency spectrum as a product of the wavenumber spectrum and a Gaussian frequency distribution, whose mean and variance are given by the mean advection and random sweeping velocities, respectively. The applicability of the model is tested with direct numerical simulation data in the mixed layer and the entrainment zone for the streamwise and vertical velocity components and buoyancy. To obtain a fully analytical model, we propose using a von Kármán wavenumber spectrum parameterized by the characteristic variances and integral length scales. These parameters are height dependent and vary considerably with the relative balance of buoyancy and shear forces. The introduced analytical model relies on fitting parameters obtained from numerical data in the relevant range of scales. The comparison of the von Kármán–based spectra for velocity and buoyancy to simulation results shows that the main features of the measured spectra are captured by the model.

Corresponding author: Naseem Ali, naseem@pdx.edu

Abstract

Corresponding author: Naseem Ali, naseem@pdx.edu

Keywords: Convection; Boundary layer; Spectral analysis/models/distribution

1. Introduction

With increasing resolution, weather and climate models are starting to resolve spatiotemporal scales on which turbulence plays an important role. The representation of the atmospheric boundary layer (ABL) in this so-called gray-zone regime poses new challenges for weather and climate models (Zhou et al. 2014). As major challenges, the energy-containing motions are only partly resolved, and the hypothesis of quasi equilibrium and statistical homogeneity inside the grid cells used for classical parameterizations are not necessarily satisfied (Wyngaard 2004; Zhou et al. 2014; Berner et al. 2017; LeMone et al. 2019). To correctly represent the variability of flow features on the small unresolved scales, a good characterization of the spatiotemporal statistics of boundary layer properties becomes pertinent, preferably distinguishing between different regions with different characteristic scales, such as the mixed layer and the entrainment zone.

While space–time spectra or correlation functions provide this statistical characterization of the turbulent motion (Favre 1965; Wallace 2014; He et al. 2017; Wu et al. 2017), there is the inherent problem that they can only be obtained from datasets resolved in both space and time (de Kat and Ganapathisubramani 2015). Indeed, this may be challenging for both numerical simulations and experimental measurements. While simulations in principle provide a complete spatiotemporal record of the flow, they may be limited in length due to computational resources. Experimental investigations, on the other hand, are often limited in terms of the available number of measurement probes and their spatiotemporal resolution (de Kat and Ganapathisubramani 2015). Hence, the majority of studies regarding high Reynolds number flows and particularly atmospheric flows have focused on either spatial or temporal spectra to characterize the spectral distributions of temperature and velocity; for example, see Taylor (1938), Bolgiano (1959), Kaimal et al. (1972), Grossmann and L’vov (1993), McNaughton et al. (2007), Mishra and Verma (2010), and Glazunov and Dymnikov (2013).

The challenges of obtaining spectra at all spatiotemporal scales naturally give rise to the need for conceptual, physics-based models. In this respect, there have been several attempts to introduce and apply qualitative models for spatiotemporal spectra. For example, He and Zhang (2006) and Zhao and He (2009) introduced the elliptic model for turbulent shear flows, in which the isocorrelation lines of space–time velocity correlations are ellipses quantified by space correlations and the mean and sweeping velocities. Wilczek and Narita (2012) introduced the linear random advection model based on the random sweeping approach put forward by Kraichnan and Tennekes with additional mean flow (Kraichnan 1964; Tennekes 1975).

Modeling space–time statistics using the elliptic model or the advection model is a promising approach for investigating canonical turbulent flows. For example, He et al. (2010) explored the shape of space–time correlation functions for turbulent Rayleigh–Bénard convection using the elliptic model. Moreover, using large-eddy simulation (LES) data, the validity of the linear random advection model has been established for neutral atmospheric boundary layers and wall turbulence by Wilczek et al. (2014, 2015a,b), which can be utilized for different practical applications such as wind farm modeling (Bossuyt et al. 2017; Lukassen et al. 2018). Recently, Everard et al. (2021) used the elliptic model to represent the space–time correlation function of temperature in the roughness sublayer of a sloped vineyard canopy at different atmospheric stability conditions to quantify spatiotemporal scales of turbulent eddies in this flow.

In comparison to the body of work that has focused on neutral atmospheric flows, however, less attention has been given to model the space–time spectra for sheared-convective atmospheric boundary layer flows. The variation of driving regimes from the dominant shear force at the surface layer to the dominant buoyancy force at the outer layer (LeMone 1973; Hunt et al. 1988; McNaughton 2004; Lothon et al. 2006; McNaughton et al. 2007; Tong and Nguyen 2015) renders CBL flows rather complex. This complexity causes the development of models for CBL space–time spectra challenging and a subject of inquiry.

To address this challenge, we use direct numerical simulations (DNS) datasets of different sheared convection scenarios to explore their wavenumber–frequency spectra and test the validity of the linear random advection model for the velocity and buoyancy fluctuations in the mixed layer and the entrainment zone. Since it helps to understand the flow dynamics and statistics, it is of practical significance to investigate a robust model that captures the space–time flow features in these regions. We find that the main features of the wavenumber–frequency spectra, such as a frequency shift due to mean-flow advection and a frequency broadening due to random sweeping effects are captured by the model.

The paper is organized as follows. Section 2 describes the DNS data, flow parameterization, and numerical simulations. The modeling approach for determining the temporal aspects of the space–time spectrum model is outlined and compared to DNS data in section 3. Section 4 then provides a step toward a fully analytical model based on von Kármán spectra and shows tests of the quality of reconstruction for velocity components and buoyancy. The main conceptual findings of the current work are summarized in section 5.

2. Direct numerical simulations

We consider a cloud-free barotropic CBL over an aerodynamically smooth surface that penetrates into a linearly stratified free atmosphere. Convection is forced by a constant and homogeneous surface buoyancy flux. Details of the simulations can be found in Garcia and Mellado (2014) and Haghshenas and Mellado (2019), but we include a brief summary in this section for convenience. We use the Boussinesq approximation for the conservation equations for mass, momentum, and energy:

\nabla \cdot u = 0,

(1)

\frac{\partial u}{\partial t} + \nabla \cdot (u \otimes u) = - \nabla p + ν \nabla^{2} u + b e_{z},

(2)

\frac{\partial b}{\partial t} + \nabla \cdot (u b) = κ_{b} \nabla^{2} b,

(3)

where u represents the velocity vector, p is the kinematic pressure, b is the buoyancy, the parameters κ_b and ν represent thermal diffusivity and kinematic viscosity, respectively, and e_z is the unit vector in vertical direction. The buoyancy b is linked to the virtual potential temperature θ_υ by

b \approx \frac{g (θ_{υ} - θ_{υ, 0})}{θ_{υ, 0}},

(4)

where g is the gravitational acceleration, and θ_υ_,0 is a constant. The lateral directions are periodic, implying statistical homogeneity in horizontal planes. At the bottom, we use no-slip boundary conditions for the velocity and a Neumann boundary condition for the scalar. At the top, we use free-slip boundary conditions for the velocity and a Neumann boundary condition for the buoyancy. The top boundary is placed far enough in the vertical direction to mitigate finite-domain effects on the turbulent boundary layer at the bottom, and it includes a sponge layer to reduce the reflection of gravity waves. The wind velocity and the buoyancy frequency in the free atmosphere are U₀ and N₀, respectively. The surface buoyancy flux is B₀. The corresponding nondimensional numbers are the Reynolds number defined as

{Re}_{0} = \frac{B_{0}}{ν N_{0}^{2}},

(5)

and the Froude number defined as

{Fr}_{0} = \frac{U_{0}}{N_{0} L_{0}} .

(6)

The reference Ozmidov scale, a length scale that characterizes the vertical thickness of the entrainment zone, is defined as

L_{0} = {(\frac{B_{0}}{N_{0}^{3}})}^{1 / 2} .

(7)

The Prandtl number, defined as

Pr = \frac{ν}{κ_{b}},

(8)

is set to unity. The reference Reynolds number is 42, while the Froude numbers of the two cases are 10 and 20, respectively. Table 1 shows the most important parameters for the considered cases CBL1 and CBL2. The spatial resolutions are slightly different from those listed in the study by Haghshenas and Mellado (2019) since the simulations have been extended on a different supercomputer cluster.

Table 1

Flow parameters for the two simulation cases. Here, L_x and L_y represent the domain size in the streamwise direction and spanwise direction, respectively. Also, N_x, N_y, and N_z denote the grid points in the streamwise, spanwise, and vertical directions, respectively.

Figure 1 illustrates the logarithmic buoyancy gradient in a vertical cross section for two different Froude numbers. Here, we focus on the quasi-steady state of the CBL, meaning that the time over which the mean properties change is large compared to the large-eddy turnover time. The encroachment length provides a measure of the CBL depth and can be determined from the mean buoyancy profile according to (Carson and Smith 1975; Haghshenas and Mellado 2019)

z_{enc} = {2 N_{0}^{- 2} \int_{0}^{z_{\infty}} [⟨ b ⟩ (z, t) - N_{0}^{2} z] d z}^{1 / 2} .

(9)

Here, ⟨⋅⟩ indicates an average over the horizontal plane. The upper limit of integration z_∞ is set far enough into the free atmosphere to ensure that the integration is approximately independent of z_∞. Integration of the evolution equation for the buoyancy provides the following relation between the encroachment length and time:

z_{enc} = L_{0} {[2 (1 + {Re}_{0}^{- 1}) N_{0} (t - t_{0})]}^{1 / 2},

(10)

where t₀ is an integration constant. The locations of local features in the entrainment zone, like the point of maximum mean temperature gradient or the point of minimum heat flux, are commensurate with the encroachment length. As highlighted in Garcia and Mellado (2014), the height of the maximum flux is about 1.1z_enc and the height of maximum mean temperature gradient is 1.2z_enc. However, the integral definition of the encroachment length makes it more robust than pointwise quantities (Haghshenas and Mellado 2019). The convective velocity can be obtained from the surface buoyancy flux and the CBL depth as (Deardorff 1970)

w_{*} = {(B_{0} z_{enc})}^{1 / 3} .

(11)

Fig. 1.

Vertical cross section of the logarithmic buoyancy gradient for (a) Fr₀ = 10 and (b) Fr₀ = 20 cases.

Citation: Journal of the Atmospheric Sciences 80, 3; 10.1175/JAS-D-22-0079.1

Figure 2 shows an example of space–time plots for the streamwise and vertical velocities as well as buoyancy. Here, the encroachment length scale z_enc, and the buoyancy frequency N₀ are used to normalize the spatial and temporal coordinates, respectively. The z_enc used in the normalization corresponds to the value at the initial time, which is z_enc/L₀ = 20.61. Note that the z_enc range mentioned in Table 1 refers to the part of the simulations analyzed in this work. The figure distinctly shows marks of mean-flow advection and random sweeping effects in the mixed layer and the entrainment zone. The departures from the ideal case, where the footprint of the advection are perfectly straight lines, originate from the temporal evolution of the small-scale structures as well as large-scale random sweeping effects (Wilczek et al. 2015a). Further, the space–time plots highlight differences in the structure between the CBL zones. In the mixed layer, the shear and buoyancy mechanisms conventionally preserve the flow structures in smaller scales compared to those at the entrainment zone. The large-scale features modify the appearance of turbulent patches in the entrainment zone. These observations agree well with previous studies of the CBL flows (Hunt et al. 1988; Lothon et al. 2006; Fedorovich and Conzemius 2008).

Fig. 2.

Citation: Journal of the Atmospheric Sciences 80, 3; 10.1175/JAS-D-22-0079.1

3. Linear random advection model

To proceed with the key points of the current work, which is modeling the wavenumber–frequency spectra for the velocity components and buoyancy, we first restate the main steps of the linear random advection model (LRAM) introduced in Wilczek and Narita (2012) and Wilczek et al. (2015a) based on the Kraichnan (1964)–Tennekes (1975) random sweeping hypothesis. We present the main steps for the streamwise velocity, but they are also valid for the vertical velocity and buoyancy. In the framework of the linear advection model, we assume that velocity and buoyancy fluctuations are primarily advected with the mean flow and a random sweeping velocity in horizontal planes at height z. We furthermore assume statistical homogeneity in the horizontal planes. The random sweeping velocity is assumed to be slowly varying in space and time compared to the fluctuations such that it can be considered approximately constant. All other dynamical effects are neglected. We, therefore, consider the advection equation in Fourier space (Kraichnan 1964; Wilczek and Narita 2012; Wilczek et al. 2015a):

\frac{\partial}{\partial t} {\hat{u}}^{'} (k, z, t) + i (U + V) \cdot k {\hat{u}}^{'} (k, z, t) = 0,

(12)

where k = (k₁, k₂) denotes the wave vector, composed of the streamwise and spanwise wavenumbers, z is the height, t is the time, i is the imaginary unit, and

{\hat{u}}^{'}

is the streamwise fluctuating velocity. Also, U and V represent the height-dependent mean velocity and random sweeping velocity in the plane, respectively. Equation (12) can be straightforwardly solved yielding

{\hat{u}}^{'} (k, z, t) = {\hat{u}}^{'} (k, z, 0) \exp [- i (U + V) \cdot k t] .

(13)

The Fourier coefficients can be used to obtain an expression for the two-time wavenumber spectrum of the streamwise velocity component as a function of the horizontal wave vector, time lag, and height:

E_{u} (k, τ; z) = E_{u} (k; z) ⟨ \exp [- i (U + V) \cdot k τ] ⟩ .

(14)

With the assumptions that the mean velocity is constant across all realizations, and the distribution of the large-scale random advection velocity is Gaussian, the ensemble average in Eq. (14) can be evaluated. By a Fourier transform into frequency space the wavenumber–frequency spectrum is then obtained as (Wilczek et al. 2015a)

E_{u} (k, ω; z) = \frac{E_{u} (k; z)}{{[2 π ⟨ {(V \cdot k)}^{2} ⟩]}^{1 / 2}} \exp [- \frac{{(ω - k \cdot U)}^{2}}{2 ⟨ {(V \cdot k)}^{2} ⟩}] .

(15)

That means, it is a product of the wavenumber spectrum and a Gaussian frequency distribution, where the mean velocity introduces a frequency shift whereas random sweeping effects introduce a frequency broadening. In the following section, we compare the wavenumber–frequency spectrum model to spectra obtained from direct numerical simulations of CBLs.

Before presenting the results of the LRAM, we first describe the procedure to estimate the spectra from DNS. The simulations are conducted in a frame moving with half of the reference mean velocity U₀/2. This allows larger time steps in the time marching scheme as determined by the CFL constraint. Therefore, we transform the horizontal velocity and buoyancy planes onto a stationary grid using linear interpolation. The sampling time is also interpolated linearly to be constant. To obtain the spectra, we use a fast Fourier transform (FFT) algorithm. In the time domain, a Hanning window is used to impose periodicity in time implicit in the discrete Fourier transform (Oppenheim and Schafer 1975). The wavenumber–frequency spectrum is acquired by evaluating the spectrum of a space–time segment in the streamwise direction, and then averaging over the spanwise direction to enhance the statistical convergence. For the two-dimensional wavenumber spectrum, the computation of the FFT is achieved by first computing the one-dimensional FFT along the streamwise direction and then applying the FFT to the output of the first step along the spanwise direction. For the evaluation of the LRAM model presented in Eq. (15), we calculate the wavenumber spectrum from the DNS data and multiply it with the frequency distribution. The required mean velocity is straightforwardly computed from the DNS data. For the mean-square random sweeping velocities we also take the mean squared velocity fluctuations from the DNS data. The mean and sweeping velocities are defined as $U = ⟨ u ⟩ e_{x}$ and $⟨ {(V \cdot k)}^{2} ⟩ = ⟨ u^{' 2} ⟩ k_{1}^{2} + ⟨ υ^{' 2} ⟩ k_{2}^{2}$ , respectively. The spectra are evaluated from 4000 time steps of the simulation, which corresponds to approximately 10 large-eddy turnover times based on the z_enc and w_* scales.

Figures 3 –5 show the spectra of the velocity components and buoyancy as a function of the streamwise wavenumber (k₁) and the frequency (ω) for the considered cases. The figures also show normalized cuts of the conditional spectra [E(k₁,ω)/E(k₁)], which enable a quantitative comparison of the frequency distributions from the model and the DNS data. The spectra are shifted toward higher frequencies with increasing wavenumber as a result of the Doppler shift induced by the mean velocity. Additionally, the spectra exhibit a Doppler broadening caused by the random advection. The contour lines of the modeled spectra qualitatively resemble the data for the considered ranges of wavenumbers and frequencies, indicating that the decorrelation trends rooted in the Doppler broadening are captured by the model.

Fig. 3.

Citation: Journal of the Atmospheric Sciences 80, 3; 10.1175/JAS-D-22-0079.1

Fig. 4.

Citation: Journal of the Atmospheric Sciences 80, 3; 10.1175/JAS-D-22-0079.1

Fig. 5.

Citation: Journal of the Atmospheric Sciences 80, 3; 10.1175/JAS-D-22-0079.1

The prediction of the model spectra is examined further quantitatively by comparing normalized cuts through the spectrum, i.e., the frequency distributions at fixed wavenumbers. The peaks of the conditional spectra are larger in magnitude at lower frequencies. The model reproduces the broadening of the frequency distributions with increasing wavenumbers observed from the DNS data accurately. The evolution from Gaussian to non-Gaussian conditional spectra is also precisely captured by the model. The non-Gaussian frequency distribution is a consequence of integrating out the spanwise wavenumber direction leading to a superposition of different Gaussian frequency distributions. A possible interpretation is that this variation in the amplitudes between the large and small scales is caused by the stretching of the large-scale features that adjust the amplitudes of small scales, and swift distortion of small-scale eddies (Wu and He 2020). The applicability of the LRAM for vertical velocity and buoyancy supports the assumption that the small scales are approximately advected as a scalar with mean and random sweeping velocity. In addition to advection by the mean flow, the advection might be related to longitudinal roll plumes that align with the mean flow and sustain aloft fine-scale eddies and scalars embedded within it (Sykes and Henn 1989; Moeng and Sullivan 1994; Salesky et al. 2017).

To quantify the differences between the LRAM- and DNS-based spectra, we test the Jensen–Shannon divergence between their conditional wavenumber–frequency spectra. The Jensen–Shannon divergence is the symmetric version of the Kullback–Leibler divergence (also called relative entropy) which always results in a finite value (Lin 1991). The Jensen–Shannon divergence is a measure of the similarity of the two distributions and is constrained to the interval [0, 1] for base-2 logarithms (Lin 1991), where lower values indicate greater similarity. The divergence for discrete distributions for any conditional wavenumber–frequency spectra (ϵ) can be calculated as

ϵ (P | | Q) = \frac{1}{2} \sum_{l = 1}^{N} P_{l} \log [\frac{P_{l}}{\frac{1}{2} (P_{l} + Q_{l})}] + \frac{1}{2} \sum_{l = 1}^{N} Q_{l} \log [\frac{Q_{l}}{\frac{1}{2} (P_{l} + Q_{l})}] .

(16)

Here, l refers to the bin of the discrete histograms. P and Q present conditional spectra based on DNS and LRAM, respectively. Now to better understand the performance of the LRAM, we evaluate the divergence for the considered cases. Figure 6 presents the divergence between the reconstructed spectra based on the LRAM and those from DNS data. As shown in the figure, the maximum divergence occurs at the large scale, and the Jensen–Shannon divergence asymptotically decreases with increasing wavenumber.

Fig. 6.

The Jensen–Shannon divergence between LRAM- and DNS-based conditional spectra for (a) streamwise velocity, (b) vertical velocity, and (c) buoyancy.

Citation: Journal of the Atmospheric Sciences 80, 3; 10.1175/JAS-D-22-0079.1

4. Parameterizations of the wavenumber–frequency spectrum: von Kármán model

So far, we have tested whether the temporal decorrelation trends of the flow are captured by the frequency distribution obtained from the LRAM model. To that end, we have been using the wavenumber spectrum from DNS data. Next, we additionally model the wavenumber spectrum to obtain a fully analytical model for the wavenumber–frequency spectrum. To derive an analytical expression for the spectrum, we propose a von Kármán function to model the wavenumber spectrum. Subsequently, we combine the von Kármán function with the frequency part of the linear advection model to construct the entire wavenumber–frequency spectrum.

The von Kármán spectrum is an empirical model introduced for homogeneous and isotropic turbulence, which interpolates between a k⁴ increase for the energy-containing range at low wavenumbers followed by a k^−5/3 decay in the inertial subrange. The von Kármán spectrum can be written as (von Kármán 1948)

E (k) = \frac{A {(k / k_{e})}^{4}}{{[1 + {(k / k_{e})}^{2}]}^{17 / 6}},

(17)

where k_e is the wavenumber corresponding to the integral length scale, and A is a constant fixed by the total energy that can be found through the integration of E(k) (Glegg and Devenport 2017). Here, we assume that k_e is determined by the longitudinal integral length scale and given by (Glegg and Devenport 2017)

k_{e} = \frac{\sqrt{π}}{L_{f}} \frac{Γ (5 / 6)}{Γ (1 / 3)},

(18)

where Γ is the Gamma function. The integral length scale L_f is obtained from the energy spectrum through (Pope 2000)

L_{f} = \frac{π}{2 ⟨ u^{' 2} ⟩} \int_{0}^{\infty} \frac{E (k)}{k} d k = \frac{π E_{u} (0)}{2 ⟨ u^{' 2} ⟩} .

(19)

This approach is justified by the natural scaling of the prevailing large-scale flows, based on which we expect that the eddies that sustain the individual plumes have scales proportional to L_f. For the subsequent modeling, we first assume isotropic statistics which is expected to be approximately valid at sufficiently small scales. The three-dimensional velocity spectral tensor for isotropic turbulence has the general form (Pope 2000)

Φ_{i j} (\hat{k}) = \frac{E (k)}{4 π k^{2}} [δ_{i j} - \frac{k_{i} k_{j}}{k^{2}}],

(20)

where δ_ij is the Kronecker delta, and

\hat{k}

is the three-dimensional wave vector;

\hat{k} = (k_{1}, k_{2}, k_{3})

. Based on that we can calculate the wavenumber spectrum in the plane as

E_{i j} (k_{1}, k_{2}) = \int_{- \infty}^{\infty} Φ_{i j} (\hat{k}) d k_{3} .

(21)

As a result, the planar wavenumber spectra based on the von Kármán model for streamwise velocity (E₁₁ = E_u) and vertical velocity (E₃₃ = E_w) take the form (Glegg and Devenport 2017)

E_{u} (k_{1}, k_{2}) = \frac{1}{18 π} \frac{⟨ u^{' 2} ⟩}{k_{e}^{2}} \frac{[3 + 3 {(\frac{k_{1}}{k_{e}})}^{2} + 11 {(\frac{k_{2}}{k_{e}})}^{2}]}{{[1 + {(\frac{k_{1}}{k_{e}})}^{2} + {(\frac{k_{2}}{k_{e}})}^{2}]}^{7 / 3}},

(22)

E_{w} (k_{1}, k_{2}) = \frac{4}{9 π} \frac{⟨ w^{' 2} ⟩}{k_{e}^{2}} \frac{[{(\frac{k_{1}}{k_{e}})}^{2} + {(\frac{k_{2}}{k_{e}})}^{2}]}{{[1 + {(\frac{k_{1}}{k_{e}})}^{2} + {(\frac{k_{2}}{k_{e}})}^{2}]}^{7 / 3}} .

(23)

While we assumed isotropic statistics for the derivation of these spectra, we will relax this assumption for comparisons with DNS data and allow the streamwise and vertical velocity components to have a different mean-square values. We simply assume that the von Kármán spectrum is scaled with the variance of the velocity component of interest and the local integral scale L_f.

For the CBLs under consideration, we expect the dynamics of buoyancy and vertical velocity to be similar. This phenomenon is very well known in atmospheric flows (Tong and Nguyen 2015). Applying the same approach as for the vertical velocity to the buoyancy, we start from the (zeroth-order) von Kármán–based spectral tensor for a scalar

Φ_{b} (\hat{k}) = E_{b} (k) / (4 π k^{2})

and integrate with respect to the vertical wavenumber to obtain the spectrum for the buoyancy. The corresponding two-dimensional wavenumber spectrum model for the buoyancy is

E_{b} (k_{1}, k_{2}) = \frac{1}{27 π} \frac{⟨ b^{' 2} ⟩}{k_{e}^{2}} \frac{[3 + 11 {(\frac{k_{1}}{k_{e}})}^{2} + 11 {(\frac{k_{2}}{k_{e}})}^{2}]}{{[1 + {(\frac{k_{1}}{k_{e}})}^{2} + {(\frac{k_{2}}{k_{e}})}^{2}]}^{7 / 3}} .

(24)

In these simplified models, the only remaining parameters that require specification are the velocity and buoyancy variances. To model the wavenumber–frequency spectrum, we assume that the mean velocity and variances are already available or that they can be simulated or modeled. In this work, the local mean velocity and variances required for the model are directly obtained from the DNS data to avoid any assumption regarding the scaling properties of the considered flows. Providing comprehensive models for the mean velocity and variances is left for future work.

Finally, to ensure the von Kármán model for wavenumber spectra captures also the dissipation range, we multiply E_u(k₁, k₂) and E_w(k₁, k₂) by an exponential function defined as (Pope 2000)

f_{k_{h} η} = \exp {- β {[{(k_{h} η)}^{4} + c_{η}^{4}]}^{1 / 4} - β c_{η}} .

(25)

Here, the

k_{h} = \sqrt{k_{1}^{2} + k_{2}^{2}}

, and η represents the dissipative scale. The constants β and c_η are determined by fitting the proposed equation to the DNS data and are shown in Table 2 for streamwise and vertical wavenumber spectra, respectively. It is worth noting that the specific form of the exponential cutoff has been introduced by Pope (2000) for the three-dimensional energy spectrum function. We assume here that it also holds for the streamwise and vertical velocity components.

Table 2

Spectrum parameters for all simulation cases in the mixed layer (z/z_enc = 0.33) and the entrainment zone (z/z_enc = 1.23).

The buoyancy wavenumber spectrum E_b(k₁, k₂) is multiplied by a Gaussian function (Goedecke et al. 2006; Wang et al. 2007),

G_{k_{h} η} = \exp [- α {(k_{h} η)}^{2}],

(26)

to include an exponential decay in the high-wavenumber region. Here, α is also determined from the data and listed in Table 2. For the considered cases of Prandtl number equal to unity, for which dissipative and diffusive influences are comparable, we expect that the effective cutoff scale of the buoyancy field should be proportional to the one of the vertical velocity (Corrsin 1951; Mills et al. 1958). However, to introduce a general parameterization for different cases of different Prandtl numbers, we use a general high-wavenumber cutoff model for the buoyancy. The exponential cutoff also needs to be taken into account when determining the numerical constants specifying the model spectrum. Here, we compare the integral of the modeled spectrum to the one obtained from the DNS to check if we need to rescale the numerical constant of the spectrum. The rescaling constant is determined as

C_{u} = \frac{\int E_{u}^{DNS} d k}{\int E_{u}^{Model} d k} .

(27)

These constants are inserted in the von Kármán spectrum formulas for rescaling. The values of the rescaling constants are given in Table 2.

Figures 7 and 8 present the two-dimensional wavenumber spectra for the streamwise velocity, wall-normal velocity, and buoyancy obtained from the von Kármán model and simulations. The presented wavenumber spectra correspond to the mixed layer and the entrainment zone for the Fr₀ = 20 and Fr₀ = 10 cases, respectively. The results show an agreement between the model and data, especially for the vertical velocity and the buoyancy. Small deviations are visible in the streamwise case, especially toward the small streamwise wavenumber k₁, which is the result of underestimating the anisotropy in the model. The plane anisotropy forces the contours to be elliptical and compressed in the k₁ direction. This is also related to considering the anisotropy only in horizontal planes and excluding the wall-normal direction that is possibly influenced by convection. The E_u(k₁, k₂) spectra are considerably expanded in the k₂ direction. For vertical velocity and buoyancy, the contours are approximately circular, and the spectra thus approximately depend only on the horizontal wavenumber value, indicating that the assumption of isotropy is valid. The spectra for different CBL cases appear not to diverge considerably from each other. Also, the model shows an agreement with the data for all cases at the high wavenumbers as shown in Fig. 9.

Fig. 7.

Citation: Journal of the Atmospheric Sciences 80, 3; 10.1175/JAS-D-22-0079.1

Fig. 8.

Citation: Journal of the Atmospheric Sciences 80, 3; 10.1175/JAS-D-22-0079.1

Fig. 9.

Citation: Journal of the Atmospheric Sciences 80, 3; 10.1175/JAS-D-22-0079.1

Equipped with this model spectrum, the analytical form for the two-dimensional wavenumber spectra and the necessary parameters are combined with the LRAM to construct the wavenumber–frequency spectra. The obtained results are shown in Figs. 10 –13 for the considered cases. The model spectra are compared with the original ones obtained from the DNS data, showing an agreement. As expected, the von Kármán model does not alter the broadening of the Doppler shift of the spectra, indicating that we can fully reproduce the wavenumber–frequency spectra with few parameters. We would suppose this scaling to match wavenumber spectra only if the convective plumes rely only on the convective velocity and integral length scales of the carrying eddies.

Fig. 10.

Citation: Journal of the Atmospheric Sciences 80, 3; 10.1175/JAS-D-22-0079.1

Fig. 11.

Comparison of the wavenumber–frequency spectra for streamwise velocity, vertical velocity, and buoyancy in the entrainment zone (z/z_enc = 1.23) for the Fr₀ = 20 case.

Citation: Journal of the Atmospheric Sciences 80, 3; 10.1175/JAS-D-22-0079.1

Fig. 12.

Comparison of the wavenumber–frequency spectra for streamwise velocity, vertical velocity, and buoyancy in the mixed layer (z/z_enc = 0.33) for the Fr₀ = 10 case.

Citation: Journal of the Atmospheric Sciences 80, 3; 10.1175/JAS-D-22-0079.1

Fig. 13.

Comparison of the wavenumber–frequency spectra for streamwise velocity, vertical velocity, and buoyancy in the entrainment zone (z/z_enc = 1.23) for the Fr₀ = 10 case.

Citation: Journal of the Atmospheric Sciences 80, 3; 10.1175/JAS-D-22-0079.1

As a quantitative test, Fig. 14 shows the Jensen–Shannon divergence between the reconstructed spectra based on the von Kármán model and that of DNS data. The trend of the divergence is similar to that shown in Fig. 6. However, at the largest scales, the divergence is higher due to the von Kármán model not matching the DNS spectra, hence amplifying the divergence.

Fig. 14.

The Jensen–Shannon divergence between von Kármán–based LRAM- and DNS-based conditional wavenumber–frequency spectra for (a) streamwise velocity, (b) vertical velocity, and (c) buoyancy.

Citation: Journal of the Atmospheric Sciences 80, 3; 10.1175/JAS-D-22-0079.1

5. Conclusions

The main body of the current work builds on the advection model developed in Wilczek et al. (2015a) for wall-bounded flows, corresponding to neutral atmospheric boundary layers. Here, we investigate sheared convective boundary layers and compare the linear random advection model against data from direct numerical simulations. In particular, we introduce an analytical form of the wavenumber–frequency spectrum model adapted to sheared convective boundary layers. The model results in a wavenumber–frequency spectrum, which is a combination of the two-dimensional wavenumber spectrum with a Gaussian frequency distribution. We have tested the applicability of the model in the mixed layer and the entrainment zone. To obtain a full parameterization for the wavenumber–frequency spectra, we use a von Kármán spectrum to model two-dimensional wavenumber spectra for velocity components and buoyancy. Despite the simplifying assumptions, the two-dimensional wavenumber spectra based on the von Kármán model capture the spectra of DNS datasets quite well. Systematic differences are visible on the large scales, where the von Kármán spectra neglect the anisotropy of the large-scale turbulence, which can be improved in future studies by using more sophisticated wavenumber spectra. Our results can provide a starting point for parameterizing the gray zone, in which the coherent energy-containing eddies are partially resolved, but their interaction with smaller-scale motions is not fully represented. Although our analytical model relies on fitting parameters obtained from DNS data in the relevant range of spatiotemporal scales, we think that parameterizing the spatiotemporal energy content in the form of wavenumber–frequency spectra is a useful intermediate step from well-resolved idealized settings toward fully parameterized spatiotemporal models.

Acknowledgments.

This work is supported by the Priority Programme SPP 1881 Turbulent Superstructures of the Deutsche Forschungsgemeinschaft and by the Max Planck Society. Partial support is also provided by Grant PID2019-105162RB-I00 funded by MCIN/AEI/10.13039/501100011033. The authors also acknowledge the Gauss Centre for Supercomputing e.V. (www.gauss-centre.eu) for providing computing time through the John von Neumann Institute for Computing (NIC) on the GCS Supercomputer JUWELS at Jülich Supercomputing Centre (JSC).

Data availability statement.

Primary data and scripts used in the analysis and other supporting information that may be useful in reproducing the authors’ work can be obtained by contacting the corresponding author.

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Journal of the Atmospheric Sciences

Abstract
1. Introduction
2. Direct numerical simulations
3. Linear random advection model
4. Parameterizations of the wavenumber–frequency spectrum: von Kármán model
5. Conclusions
Acknowledgments.
Data availability statement.
REFERENCES

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Fig. 1.

Vertical cross section of the logarithmic buoyancy gradient for (a) Fr₀ = 10 and (b) Fr₀ = 20 cases.

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Fig. 2.

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Fig. 3.

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Fig. 4.

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Fig. 5.

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Fig. 6.

The Jensen–Shannon divergence between LRAM- and DNS-based conditional spectra for (a) streamwise velocity, (b) vertical velocity, and (c) buoyancy.

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Fig. 7.

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Fig. 8.

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Fig. 9.

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Fig. 10.

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Fig. 11.

Comparison of the wavenumber–frequency spectra for streamwise velocity, vertical velocity, and buoyancy in the entrainment zone (z/z_enc = 1.23) for the Fr₀ = 20 case.

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Fig. 12.

Comparison of the wavenumber–frequency spectra for streamwise velocity, vertical velocity, and buoyancy in the mixed layer (z/z_enc = 0.33) for the Fr₀ = 10 case.

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Fig. 13.

Comparison of the wavenumber–frequency spectra for streamwise velocity, vertical velocity, and buoyancy in the entrainment zone (z/z_enc = 1.23) for the Fr₀ = 10 case.

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Fig. 14.

The Jensen–Shannon divergence between von Kármán–based LRAM- and DNS-based conditional wavenumber–frequency spectra for (a) streamwise velocity, (b) vertical velocity, and (c) buoyancy.