Velocity Spectra Research Articles

Generalized vortex flow of nanoparticle shapes over a permeable disc surface with generalized slip conditions

This paper investigates the generalized vortex flow of nanofluid consisting of titanium dioxide (TiO2) with base fluid (H2O) over a permeable disk surface that generates a heat transfer process in the thermal boundary layer of the disk. Four types of non-spherical shapes of nanoparticles (blade, brick, cylinder and platelet) are considered for the research. The motion is produced when the fluid is far from the disk surface and rotates like a solid body with a constant angular velocity [Formula: see text]. The partial differential equations (PDEs) are obtained using boundary layer approximations and then converted into ordinary differential equations (ODEs) using suitable similarity transformations. These nonlinear ODEs are solved using the bvp4c MATLAB solver. The effect of different parameters (n, A, [Formula: see text], [Formula: see text], R and Pr) on the velocity components and temperature profile is shown graphically and in tabular results. This analysis concludes that for all non-spherical shapes, the velocity spectrum of all nanoparticles decreases when the values of factors such as power-law, suction, volume fraction and slip parameter increase. All non-spherical shapes of a nanofluid experience a decrease in fluid temperature due to the Prandtl number, while radiation numbers have the opposite effect.

Very-large-scale motions in open-channel flow: Insights from velocity spectra, correlations, and structure functions

We employ spectral-space and real-space quantities to characterize large-scale motions (LSMs) and very-large-scale motions (VLSMs) in open-channel flows (OCFs). The obtained data on velocity auto- and co-spectra, spectral correlation coefficient, and structure functions of the second and third orders are consistent with the hypothesis that VLSMs are a special type of coherent structure rather than an alignment of LSMs, even if active interactions between LSMs and VLSMs most likely take place. The Letter is concluded with a conceptual velocity spectrum for OCFs and an explanation of the mechanism of VLSMs' contribution to the Reynolds shear stress.

Статистические характеристики расчета приращений сейсмической интенсивности при сейсмическом микрорайонировании горных территорий

Introduction. During seismic micro-zoning of a territory, one of the main instrumental methods for calculating intensity increments is the registration of earthquakes in areas with different soil conditions. In this case, the calculation of increments is based on the amplitudes of oscillations simultaneously in the given and reference sections. The purpose of the work is to compare the increments in various expressions obtained for powerful vibration and pulse sources, as well as for an instrumental measure of seismic intensity. Materials and methods. Processing of instrumental data, their systematization and description using basic statistical indicators. The analysis was carried out using data from the Vladikavkaz network - records of earthquakes obtained simultaneously at four seismic stations located in areas with different ground conditions. Results. To check the formulas for calculating seismic intensity increments and comparing them with the obtained data, distribution histograms and the corresponding probability density functions were constructed, under the assumption that the distribution is normal. All methods for calculating vibration amplitudes gave identical results in ranking areas according to the degree of seismic hazard, however, taking into account rounding, an increase in intensity by 1 point is not always achieved in an area where, due to the worst soil conditions. The best results for calculating intensity increments were obtained for the area values of the velocity and acceleration spectra. At the same time, the areas of the acceleration spectra provide better differentiation of soil conditions according to the degree of danger (with the same coefficient in front of the logarithm). Discussion. The task of developing an instrumental method of seismic micro-zoning is to clarify the coefficients in front of the logarithmic scale of the quantities under consideration, the reduction of which, in turn, determines the clarification of the level of impacts for given intensities. On the other hand, smaller coefficients lead to worse differentiation of soil conditions of areas when calculating increments of seismic intensity. The solution to the problem may lie somewhere at the junction of both approaches. Conclusion. Analysis of the results shows a connection between the magnitude of the distribution dispersion and the complexity of the engineering-geological conditions of the site. The more factors influence the wave processes in the soils, the greater the scatter of values around the average. Main factor is a multi-layered environment, i.e. the presence of reflective boundaries, and the other one is the very properties of the medium, i.e. its dispersion. Resume. Results are useful in differentiating soil conditions by an additional parameter that affects the probability of failure to exceed the calculated values of seismic impacts - the same average values of seismic intensity can be characterized by different confidence intervals; from a practical point of view, one should set the same probability of not exceeding the design impact values. Suggestions for practical applications and directions for future research. For practical applications it is necessary to distinguish the influence of the layering of the medium and the direct dispersion of the soils composing the stratum, which represents the next research task.

Relationships between ground motion parameters and energy demands for regular low-rise RC frame buildings

AbstractThe present study aims to analyze the correlation between ground motion parameters and energy demands of low-rise RC buildings without shear walls. Two regular 4- and 7-story residential buildings were seismically designed to represent low-rise RC buildings. In order to establish the requirements of single degree of freedom (SDOF) systems as well as multi degree of freedom (MDOF) systems, the dynamic features of “equivalent” SDOF systems were defined by using MDOF systems. The correlation of 20 ground motion parameters (GMPs) of 44 records with the energy demands obtained from a total of 176 nonlinear time history analyses was examined for the SDOF and MDOF systems within the scope of this study. The ground motion parameters (GMPs) were taken as intensity measures (IMs). In contrast, maximum input energy was used as a demand measure (DM) and these energy values were normalized with the masses of the buildings for cases, where the buildings were evaluated together. Parameters related to acceleration and velocity are generally found to yield better results in comparison to the ones related to frequency and displacement. Velocity Spectrum Intensity (VSI) and Arias Intensity (Ia) were obtained to have the highest correlation values as a single parameter. This study suggested new equations by combining multiple ground motion parameters for SDOF and MDOF systems to better reflect damage potential than a single parameter. The use of multiple parameters in combination results in better correlation coefficients.

Determination of the VS profile at a “noisy” industrial site via active and passive data: The critical role of Love waves and the opportunities of multicomponent group velocity analysis

To define the procedures necessary to unambiguously define the subsurface model, a comprehensive set of active and passive seismic data was collected in an industrial area characterized by an extremely high level of background microtremors. Passive data are recorded to define three observables: the dispersion curve of the vertical component of Rayleigh waves via miniature array analysis of microtremors, the Love-wave dispersion curve via extended spatial autocorrelation, and the horizontal-to-vertical spectral ratio (HVSR). Active data used for the holistic analysis of surface waves are extracted from data recorded through a hybrid acquisition procedure accomplished with only two 3C geophones used to simultaneously define the HVSR at two points. Defined observables are combined according to three different approaches: the joint analysis of Rayleigh waves and HVSR, the joint analysis of Rayleigh and Love waves together with the HVSR, and the joint analysis of multicomponent group velocities together with the HVSR and Rayleigh-wave particle motion (RPM) curves. In agreement with the theory, data indicate that, in general, surface-wave modeling cannot be performed considering modal dispersion curves: dispersion obtained from passive data needs to be modeled considering the effective curve, whereas group velocity obtained from active data can be analyzed using the full velocity spectrum technique. Results indicate that joint inversion of Rayleigh-wave dispersion and HVSR does not necessarily ensure the correctness of the obtained S-wave velocity ([Formula: see text]) profile and that Love waves represent a key observable to fully constrain an unambiguous inversion procedure. However, the joint analysis of multicomponent group velocity spectra (from active multicomponent single-offset data) together with the HVSR and RPM curves is a further efficient way to obtain robust [Formula: see text] profiles through the active and passive data obtained by a single 3C geophone.

A numerical framework to investigate isotropic turbulent inflow interacting with an airfoil’s leading edge

This paper presents a numerical framework to study the interaction of isotropic turbulence with airfoils. Specifically, the developed numerical framework is used to investigate the distortion of the turbulent structures interacting with an airfoil’s leading edge. For turbulence modeling, Large Eddy Simulation (LES) is used. The isotropic turbulent inflow for the Computational Fluid Dynamics (CFD) simulations is synthetically generated using the turbulent digital filter method. The case studied with the numerical framework is a NACA 0012 airfoil with a chord-based Reynolds number of Rec=2×105 and an angle of attack of α=5°. The numerical simulation results are compared to high-speed particle image velocimetry (PIV) measurements performed for a NACA 0012 airfoil at the equivalent chord-based Reynolds number and turbulent inflow conditions. The CFD results of the numerical framework compare well with the experiments in terms of velocity spectra and RMS values upstream of the airfoil’s leading edge. The spectra and correlations of the velocity field generated by the turbulent digital filter demonstrate its ability to generate isotropic turbulent inflow. Instantaneous velocity fields show that the airfoil suppresses large-scale turbulent structures of the incoming turbulent flow. The size of the incoming turbulent structures decreases as they approach the leading edge due to the presence of the airfoil. The pre-multiplied spectra of the different velocity components show that downstream of the airfoil’s leading edge, the turbulent structures are stretched in the streamwise direction. The streamwise turbulent integral length scales and the velocity RMS values upstream of the airfoil’s leading edge indicate that the velocity components most affected by distortion are the streamwise and wall-normal components.

On detrending stream velocity time series for robust tidal flow turbulence characterization

We investigated the impact of detrending techniques on turbulence quantities from tidal stream flow data, focusing on the autocorrelation function, ρuu, and velocity spectrum, Φ(f). Standard detrending methods, including high-pass frequency-based and polynomial-based techniques, are examined, alongside a proposed alternative method, the empirical mode decomposition (EMD). Our results highlight that intervals of flow acceleration and deceleration, typical in tidal and riverine flows, significantly affect the estimation of turbulence quantities using high-pass frequency filtering and polynomial detrending of varying orders. These methods can strongly influence ρuu and Φ(f), thereby affecting the accurate estimation of derived quantities. We examine two variations of detrending data using EMD; the first removes only the EMD residue, and the second removes both the residue and the largest scale intrinsic mode function (IMF). By comparing the detrended spectra with the modeled von Kármán spectra, we demonstrate that the second variation (i.e., removing the residue and the largest scale IMF) successfully removed the large-scale trend of the data while retaining the energy of other scales.

A new strategy for prestack time migration velocity analysis to assess and mitigate structural uncertainty

Velocity uncertainty represents a significant contributor to structural uncertainty in seismic imaging. Therefore, improving velocities and enhancing the resolution of velocity spectra are crucial for reducing structural uncertainty. In the present study, a novel approach is introduced to increase velocity resolution and mitigate structural uncertainty. The methodology's impact on spatial and temporal uncertainty of image points is compared with the improved velocity continuation method. The proposed strategy involves modifying the equations of the weighted semblance method and integrating them into the equations of the velocity continuation method. Specifically, the vertical analysis window of the weighted semblance, typically applied on coherency panels is changed to a horizontal window that is applied on common-image gathers. Correspondingly, the weighting factor is adjusted using a time-adaptive function. The velocity uncertainty equation in the velocity continuation method is then replaced by the new weighted semblance eq. A similar substitution can be performed for structural uncertainty equations. These new equations are then used on common-image gathers obtained by the velocity continuation method to increase the accuracy of the velocity model and then investigate changes in structural uncertainty. In the proposed strategy, prestack time migration is initially conducted on seismic common-offset sections using various constant velocities within a predefined range. Subsequently, the weighted semblance technique is used for automatic velocity model updating in all common-image gathers. The new velocity model is then used for final prestack time migration. In the equations pertaining to velocity uncertainty, a width equivalent to one standard deviation on both sides of the picked velocity value is considered as the uncertainty range. By narrowing this width through the proposed method, it is expected to mitigate subsequent structural uncertainty, specifically for 1D media. Results from applying the presented method on synthetic and marine seismic data demonstrate its effectiveness in reducing structural uncertainty for 1D time migrated images compared to the velocity continuation method. It is also shown that horizontal structural uncertainty is significantly more sensitive to velocity errors rather than vertical uncertainty. An added benefit is that the increased resolution in velocity spectra also provides appropriate panels for automatic velocity picking.

Automatic stack velocity picking using a semi‐supervised ensemble learning method

AbstractPicking stack velocity from seismic velocity spectra is a fundamental method in seismic stack velocity analysis. With the increase in the scale of seismic data acquisition, manual picking cannot achieve the required efficiency. Therefore, an automatic picking algorithm is urgently needed now. Despite some supervised deep learning–based picking approaches that have been proposed, they heavily rely on sufficient training samples and lack interpretability. In contrast, utilizing physical knowledge to develop semi‐data‐driven methods has the potential to efficiently solve this problem. Thus, we propose a semi‐supervised ensemble learning method to reduce the reliance on manually labelled data and improve interpretability by incorporating the interval velocity constraint. Semi‐supervised ensemble learning fuses the information of the estimated spectrum, nearby velocity spectra and few‐shot manual picking to recognize the velocity picking. Test results of both the synthetic and field datasets indicate that semi‐supervised ensemble learning achieves more reliable and precise picking than traditional clustering‐based techniques and the currently popular convolutional neural network method.

Multimodal frequency and phase velocity spectrum of shear wave in microstructural flexomagnetic plate loaded with complex fluid

Multimodal frequency and phase velocity spectrum of shear wave in microstructural flexomagnetic plate loaded with complex fluid

Spectral characteristics of particle preferential concentration in turbulence computed by Eulerian and Lagrangian methods

This study conducts a comprehensive analysis of the Fourier spectra characteristics of particle preferential concentration in turbulent flows obtained by Eulerian and Lagrangian modelling approaches for different Stokes numbers. Particle preferential concentration is characterized by clusters and voids in the spatial particle distributions formed in turbulent flows and can significantly influence processes such as droplet coalescence, evaporation, and gas-particle reactions. The research primarily focuses on comparing the performance of two Eulerian models—one with and one without second-order velocity moments—and one Lagrangian model, which are used to predict particle dispersion. Key findings include the impact of artificial diffusion in Eulerian methods, the superiority of methods incorporating second-order velocity moments contributions at higher Stokes numbers, and the sensitivity of the Lagrangian method to Poisson statistics. Notably, the study reveals minimal variation in mean velocity spectra across different Stokes numbers, as opposed to the marked variations in particle concentration, momentum, and energy spectra. Grid resolution emerges as a crucial factor in enhancing spectral energy predictions in Eulerian methods. The research underscores the nuanced distinctions between Eulerian and Lagrangian methods in modelling preferential concentration, providing a detailed spectral comparison of particle concentration, velocity, momentum, and energy, highlighting the importance of method selection based on specific modelling needs.

Wind noise due to flow induced around a windscreen in large-scale turbulence

Turbulent wind noise on microphones is a ubiquitous problem for sound measurement and recording outdoors. Windscreens, such as open-cell foam spheres, are effective at reducing wind noise. The unsteady pressure sources induced on the windscreen surface must contribute less noise on average than the stagnation pressure on a bare microphone, or windscreens would not be effective. To quantify this noise reduction mechanism, a model is presented for the unsteady pressure within an impermeable windscreen of arbitrary shape immersed in an initially isotropic turbulent flow. Only turbulent scales larger than the windscreen are considered, which contribute the most intense low-frequency band of the noise spectrum in most practical situations. The inhomogeneous velocity field around the windscreen is modeled in terms of an induced irrotational blocking field that meets the kinematic boundary condition. The interior pressure is expressed as a convolution of the surface pressure field with a filter function in wavevector space. The wind noise wavevector spectrum is then shown to be only a function of the inflow isotropic velocity spectrum and the windscreen shape. Solutions are obtained for the pressure frequency spectrum within a spherical windscreen for a von Kármán turbulence model and compared with prior results in literature.

Improved response spectrum analysis procedure for an adaptive variable friction pendulum isolated structure considering frictional dependency

The effectiveness of the conventional friction pendulum isolator (FPI) is limited in complex geographical environments and unpredictable earthquake events due to its complex frictional properties and the requirement for a unified setup target. With a view to addressing the aforementioned issues, in conjunction with an adaptive variable friction pendulum isolator (AVFPI), a simple and practical mechanical model is proposed. After applying the mechanical model with a friction-dependent influence model, an improved response spectrum analysis procedure (IRS) is proposed for variable friction pendulum isolation systems. Notably, the proposed procedure allows for the consideration of the effects of variations in velocity, temperature, and pressure on the friction characteristics without the necessity for nonlinear response history analysis (NRHA). The procedure commences with obtaining a numerical model of friction dependency through friction material examinations, then proceeds to verifying the reliability of both the friction dependency and mechanical model through prototype tests of AVFPIs. The following study presents a proposed correction method that depends on random vibration theory to establish a functional relationship between the pseudo-velocity spectrum and the velocity spectrum. Additionally, an association is established between the peak velocity and the generation of frictional heat. Finally, the procedure is evaluated based on the results of an NRHA that takes friction dependence into account through a numerically variable friction pendulum isolation system. The results of the study indicate that the IRS provides a more accurate and secure assessment of the isolator displacement at various temperatures than the response spectrum method, which does not account for the influence of frictional dependency. The failure to consider the influence of friction dependency in high-temperature environments produces an underestimation of isolator slip displacement and a subsequent increase in failure rates. The Polytetrafluoroethylene (PTFE) friction materials that are currently in widespread use exhibit a strong correlation with friction dependence and, as a consequence, cause relatively large displacements at temperatures that are both low and high. The results of this study present an opportunity to expand how response spectrum analysis techniques can be applied to friction pendulum isolation systems.

Evaluating the Effects of Raindrop Motion on the Accuracy of the Precipitation Inversion Algorithm by X-SAR

Precipitation has a profound impact on both human life and the natural environment. X-band synthetic aperture radar (X-SAR) utilizes high-resolution microwave remote-sensing technology, providing opportunities for global precipitation measurements. The current precipitation inversion algorithms from X-SAR measurements assume that precipitation particles remain relatively stationary with the ground. However, the motion of raindrops could potentially reduce the accuracy of these algorithms. In this study, we first established a functional relationship between raindrop motion and SAR echoes based on the standard deviation of the raindrop Doppler velocity spectrum. Secondly, an exploratory algorithm was proposed to retrieve rainfall distribution under the raindrop motion error model (RMM) and quantitatively calculate the precipitation inversion error caused by raindrop motion. In comparison to conditions where the atmosphere is stationary, when the standard deviation of the Doppler velocity spectrum of raindrops is 1.1 m/s, the relative error of the retrieved surface rain rate increases from 2.1% to 35.8%. Numerical simulations show that SAR echoes are sensitive to changes in the standard deviation of the Doppler velocity spectrum, and the impact of raindrop motion on the accuracy of X-SAR precipitation measurements cannot be neglected.

Recognition of Tunnel Fracture Zones in Seismic Waves and Ground-Penetrating Radar Data

Fracture zones in front of tunnel faces can easily cause falling blocks and landslides during the construction process. Using seismic waves and ground-penetrating radar (GPR) data, we extracted the features of fracture zones and achieved the advanced prediction of tunnel fracture zones. The energy variation in the reflected waves propagated by seismic waves at interfaces with different impedances of contact waves was found to manifest as positive and negative reflections, and the amplitude of reflected signals within the fracture zone areas thus increased. We designed a superimposed velocity spectrum, divided the areas of variation in wave velocity, and constructed the three-dimensional spatial distribution of the tunnel fracture zones. Based on the phase change, increase in amplitude, and increase in the center-frequency characteristics of the one-dimensional time waveform of the electromagnetic waves in the fault zone area (A-scan), we located the characteristic points of the fracture zones and observed the occurrence of in-phase axis misalignment in two-dimensional scanning (B-scan). We then implemented the identification of fracture zones. This method predicted the fractured area in the rock surrounding the Liangwangshan Tunnel, and during the tunnel excavation, the fracture zones appeared in the recognition area.

Ground-Motion Intensity Measures for the Seismic Response of the Roof-Isolated Large-Span Structure

Ground-motion intensity measures (IMs), which quantify and describe the characteristics of earthquake ground motion, are of utmost importance in the assessment of seismic risk and the design of resilient structures with large spans. The appropriate selection of a ground-motion IM is crucial in establishing a reliable and robust correlation between seismic hazards and structural demands. The current study presents a novel ground-motion IM that incorporates the influence of multiple vibration modes and period elongation resulting from isolation based on the velocity spectrum. A comprehensive study has been conducted to examine the efficiency of 37 different ground-motion IMs on a roof-isolated large-span structure with engineering demand parameters (EDPs), using far-field ground-motion data. The initial examination of the proposed intensity measure involves a planar lumped-mass model. Subsequently, a numerical model of a large-span roof-isolated structure, specifically the Beijing Workers’ Stadium, is constructed and examined. The results suggest that the proposed intensity measure (IM) demonstrates satisfactory adequacy and achieves optimal efficiency when considering three different engineering demand parameters among 37 other ground-motion intensity measures.

Evidence of strong wave turbulence and of Bolgiano temperature spectra in katabatic winds on steep slopes

The katabatic winds on steep slopes investigated in the present study reveal a novel spectral behavior, observed in the outer part of the jet. At low wavenumbers, the one-dimensional (1D) velocity spectra show evidence of a kx−1 range for the three components of the velocity vector: Eu(kx),Ev(kx),Ew(kx)∝kx−1 [as well as for the 1D temperature spectrum Eθ(kx)∝kx−1]. This suggests the existence of strong wave turbulence. A necessary condition for strong wave turbulence to be manifest is that the flow direction wavenumber, kx, extends to much lower values than the slope normal one, kz. This is satisfied in the present field experiment where wave energy is injected at wavenumber kx=kN=(Na sin α)/uj¯, while kz∼1/Δz, with Na the ambient stratification, α the slope angle, uj¯ the maximum wind velocity, and Δz the shear layer thickness of the jet. In the inertial range, the velocity spectra exhibit a power law kx−5/3 over two decades, whereas the temperature-buoyancy spectra show evidence of a −7/5 slope in the buoyancy sub-range, followed by a −5/3 slope. The change in spectral slopes occurs at the Bolgiano scale LB that is close to the Dougherty–Ozmidov scale LOZ. The high Reynolds number based on the Taylor micro-scale, Reλ∼103, allows clear identification of the spectral laws.

Widest scales in turbulent channels

The widest spanwise scales in turbulent channel flows are studied through the use of three periodic channel-flow simulations at friction Reynolds number Reτ=550. The length and height of the channels are the same in all cases (Lx/h=8π and Ly/h=2, respectively), while the width is progressively doubled: Lz/h={4π,8π,16π}. The effects of increasing the domain width cannot be determined with statistical significance in our simulations, since the difference in the statistics between the simulations is of the same order as the errors of convergence. A channel flow similar to the smaller one [Del Álamo et al., “Scaling of the energy spectra of turbulent channels,” J. Fluid Mech. 500, 135–144 (2004)], which was averaged over a very long time, was used as a reference. The one-dimensional spanwise spectrum of the streamwise velocity is computed with the aim of assessing the domain-size effect on the widest scales. Our results indicate that 90% of the total streamwise energetic fluctuations is recovered without a significant influence of the size of the domain. The remaining 10% of the energy reflects that the widest scales in the outer layer are the ones most significantly affected by the spanwise length of the domain. The power-spectral density for kz = 0 remains constant even if the size of the domain in the spanwise direction is increased up to four times the standard spanwise length, indicating that wide, spanwise coherent structures are not an artifact of domain truncation.

Comparing the Velocity Spectrum Caused by the Presence of Circular Urban Subway Tunnels With Free‐Field Conditions

For structures in the near field, the velocity spectrum is significant. The aim of this study is to examine, taking into account the soil–structure interaction (SSI), how metropolitan subway tunnels with a circular cross section in alluvial soils affect the ground surface velocity spectrum. The frequency of the soil–tunnel systems was altered by varying the parameters of the soil and the geometrical features of the tunnel, such as the radius, the lining’s thickness, and the depth of burial. The maximum velocity was determined at several sites on the ground surface as well as the tunnel crown. Based on the soil–tunnel system’s period, the spectrum velocity graph was produced by averaging and squaring the standard deviation of the highest velocity for every frequency. Ansys finite element software and Plaxis2D were used in this study. The findings demonstrate that the tunnel’s presence reduces spectral velocity (SV) relative to free‐field (FF) circumstances during the examined period at the position that corresponds to the ground‐based image of the tunnel crown. The tunnel’s existence lowers the ground‐level velocity range by 14.5 percent. By moving away from the image of the tunnel crown on the ground surface, the presence of the tunnel amplifies the SV at more points by 5.5% at long periods. In addition, the tunnel crown velocity spectrum was developed. The findings indicate that the velocity spectrum diagram’s envelope for the tunnel crown is a nonlinear function of the soil–tunnel system’s period.

Statistical Characteristics of Mesoscale Fluctuations of Wind Velocity, Temperature, and Gas Concentrations Obtained from Aircraft Measurements in the Troposphere of the Arctic Region

Abstract The results of airborne measurements and statistical characteristics of mesoscale fluctuations of wind velocity, temperature, and concentrations of gas constituents at different heights of a stably stratified troposphere are presented. The measurements were carried out in September 2022 in the Arctic region of Russia with the aircraft laboratory Tu-134 “Optik.” The obtained spectra and structure functions of the fluctuations are interpreted with the theoretical model of formation of the spectrum of mesoscale wind velocity and temperature fluctuations described in the paper. The presence at high wavenumbers of a steep section in the obtained horizontal wavenumber spectra of the fluctuations of wind velocity and greenhouse gas concentration with a slope close to −3 is discussed. The fluctuation spectra along different slanted tracks of the aircraft crossing the tropospheric layer between altitudes of 1 and 9 km are also obtained and analyzed with the theoretical model.