Observation of Horizontal Gravity Wave Activity in the Upper Stratosphere Using Monostatic Rayleigh Lidar

Abstract

The prediction accuracy of the General Circulation Model (GCM) is influenced by the effectiveness of gravity wave activity parameterization. Although research focuses on small-scale horizontal gravity wave activity as a carrier for energy and momentum coupling between atmospheric layers, routine observations of horizontal gravity wave activity on scales less than a dozen kilometers are scarce due to limitations in observational instruments. This paper presents a method for observing small-scale horizontal gravity waves using monostatic Rayleigh lidar, along with the associated data processing workflow. The data processing results indicate that the observed gravity waves generally exhibit wavelengths less than 3 km and phase velocities less than 0.5 m/s. Furthermore, the annual variation in small-scale horizontal gravity waves displays a semi-annual oscillation (SAO), like that observed in medium- and large-scale waves. This suggests that the observed gravity waves originate from secondary gravity waves resulting from saturation dissipation or breaking.

1. Introduction

Gravity wave saturation theory provides a critical dynamical physical model for explaining various phenomena in the middle atmosphere, with its effectiveness validated in GCM [1,2,3]. Advances in middle atmospheric observation technologies have further refined gravity wave theory and confirmed the reliability of numerical models. Due to the coarse grid sizes of atmospheric circulation models, detailed gravity wave parameterization is essential to quantitatively enhance the predictive accuracy of these models [4,5]. Conducting observations and studies on the origin and evolution of full-spectrum gravity waves is a key approach to deepening our understanding of their impacts [6].

Recent in situ, ground-based, and airborne observational studies have provided more detailed insights into middle atmospheric gravity waves, including their origins, scales, amplitudes, fluxes, and spectra [7,8,9]. Hamilton et al. [10] enhanced the spatial resolution of an atmospheric circulation model to 30 km without incorporating gravity wave parameterization. The model results indicated that gravity wave parameterization remains necessary to improve operational effectiveness. Fritts and Alexander [2] found that the widely assumed role of large-scale horizontal waves as the primary drivers of the Middle and Lower Thermosphere (MLT) was not confirmed by GCM simulations with gravity wave parameterization. Ern et al. [11] compared the impacts of large-scale and small-scale horizontal wave spectral segments, revealing that both are equally significant in driving mean flows. Alexander and Dunkerton [12] utilized a gravity wave-breaking parameterization model and found that small-scale horizontal gravity waves exhibit rapid vertical group velocities, enabling swift propagation from the troposphere to MLT. Furthermore, the vertical flux of gravity waves is proportional to their horizontal wavenumbers, indicating that small-scale horizontal gravity waves in the atmosphere can carry substantial momentum.

Small-scale horizontal gravity waves are thought to originate from deep convection and topographic forcing. However, different deep convection models can generate either small-scale or large-scale horizontal gravity waves [13,14,15,16]. Gravity waves in convective regions are often excited near the tropopause, exhibiting slow phase-speed waves and refracted to longer vertical wavelengths and higher vertical group velocities if they propagate against the strong solstitial midlatitude stratospheric jets 3 [17,18]. Observations from the High-Resolution Dynamics Limb Sounder (HIRDLS) indicate that the momentum flux of these waves significantly influences and correlates with the driving of Quasi-biennial Oscillation (QBO) [19,20,21,22]. Orographic gravity waves, or mountain waves, can propagate to the upper troposphere and stratosphere, with wavelengths ranging from a few kilometers to hundreds of kilometers, depending on the topographic scale. Observational studies have shown that the activity intensity of orographic gravity waves alone is sufficient to drive MLT [23,24,25].

Gravity wave breaking is strongly nonlinear and produces a cascade and transfer of energy to smaller-scale spectral components. The deposition of momentum and energy that occur during and after wave breaking and the cascade to turbulence can generate a second type of secondary gravity waves [26]. They can transfer energy and momentum more effectively and propagate to higher altitudes in the upper atmosphere [27,28]. Generation of secondary gravity waves is an important process that can strongly modify the wave’s energy and its momentum transfer and transformation mechanisms. Gavrilov et al. [29] employs a high-resolution three-dimensional nonlinear numerical model to simulate tunneling of the propagation of gravity waves through critical levels. Tunneling facilitates secondary gravity wave generation via nonlinear interactions at critical levels, which contribute ~40% of total momentum flux in the thermosphere, highlighting tunneling as a key enabler of upper atmospheric impacts despite substantial amplitude losses. Vadas et al. [30] investigates the generation, propagation, and impacts of secondary and higher-order gravity waves in the northern polar region during sudden stratospheric warming. Using high-resolution whole-atmosphere simulations with the High Altitude Mechanistic General Circulation Model shows that ~70% of secondary GWs are generated on the winter side of the vortex. Secondary GWs propagate upward into the mesosphere, where they undergo further instability, generating tertiary gravity waves. During SSW-weakened vortex conditions, secondary/tertiary gravity waves dominate thermospheric dynamics, influencing ionospheric variability and space weather. This study underscores the need for resolved small-scale gravity waves in global models to improve forecast accuracy.

The ability of gravity waves to transfer momentum flux from the lower atmosphere to the middle atmosphere varies with their horizontal scales and phase velocities. It remains unclear which wavelength spectral segments are most associated with momentum flux deposition in the middle atmosphere, but it is generally believed that gravity waves with horizontal scales less than 50 km can transport significant momentum flux to the middle atmosphere [18]. Achieving full-spectrum gravity wave observations has been a significant challenge. While satellite remote sensing, radio radar, and optical radar technologies enable observations of gravity waves on scales of tens of kilometers, observing smaller-scale horizontal gravity waves remains difficult. This paper presents a method for observing small-scale horizontal gravity waves using monostatic Rayleigh lidar, along with the associated data processing workflow. Through the processing of one year of observational data, the feasibility of this method is validated through comparative analysis.

2. Methods

Middle atmospheric lidar, characterized by parameters such as laser energy, divergence angle, and gated response time, emits a vertical laser beam that illuminates atmospheric molecules within a cylindrical volume during the gated sampling interval. The photon counts received by the photodetector represent the accumulation of backward-elastic-scattering photons from atmospheric molecules within this illuminated cylinder. Within this cylindrical region, perturbations in atmospheric density primarily arise from gravity wave activity. Therefore, the collection of temporally continuous photon count profiles observed by the middle atmospheric lidar system encompasses all information related to gravity wave activity. The fluctuating echo photon count profiles represent the superposition of various low-signal-to-noise-ratio monochromatic spatiotemporal fluctuations against the background atmosphere. In theory, as long as the frequencies among the constituent monochromatic waves differ and the signal-to-noise ratio is not excessively low, it is possible to extract information about each monochromatic fluctuation from the mixed signals.

As shown in Figure 1, the atmospheric lidar repeatedly emits light pulses at fixed time intervals. Due to the divergence of the laser beam, it exhibits divergence. The time-gated memory sequentially records the photon counts received by the photomultiplier tube at fixed time intervals, triggered by the light pulse. The atmospheric molecular region illuminated by the laser during each sampling time interval (depicted as disks in Figure 1) approximates a cylinder. The gated memory records the backward-elastic-scattering photon counts from atmospheric molecules within this approximate cylinder. The collection of consecutive gated memory counts forms a vertical observation profile for the middle atmospheric lidar. The operating principle of the middle atmospheric lidar is analogous to computed tomography, achieving observations of atmospheric molecular activity at different altitudes through time gating. If gravity wave activity is present in the laser-illuminated atmospheric region, the atmospheric molecular density in that region exhibits a periodic spatial distribution (as shown by the laser slice in Figure 1). Since the receiving device, the photomultiplier tube, is a scalar recording instrument, the gravity wave-perturbed atmospheric molecular activity in the laser-illuminated region is condensed into a single scalar value. However, the collection of scalar records over continuous time within the illumination cross-section conceals spatial information about the gravity wave activity. It is common knowledge that middle atmospheric lidar can facilitate observational studies of vertical-scale gravity waves. However, it is generally believed that it cannot be used for observations of horizontal-scale gravity waves.

2.1. Principle

Assume that monochromatic horizontal gravity wave activity exists in the lidar illumination cross-section. The atmospheric density perturbation induced by this fluctuation can be expressed as

$$\rho ={\rho }_{BG}+{\rho }_d\mathrm{c}\mathrm{o}\mathrm{s}({\omega }_{s}t+kx+{\phi }_0)$$

(1)

where ρ_BG is the background atmospheric density; ρ_d is the amplitude of the horizontal gravity wave; ω_s is the angular frequency; φ₀ is the initial phase; and k is the horizontal wavenumber. Let the radius of the illumination cross-section be R, and perform a circular area integration over radius R in Equation (1):

$$2{\int }_{-R}^R[{\rho }_{BG}+{\rho }_d\mathrm{c}\mathrm{o}\mathrm{s}({\omega }_{s}t-kx+\phi \left)\right]\sqrt[2]{R^2-x^2}dx={\rho }_{BA}+{\displaystyle \frac{2{\rho }_d\pi R{J}_1\left(Rk\right)}{k}}\mathrm{c}\mathrm{o}\mathrm{s}({\omega }_{s}t+\phi )$$

(2)

where ρ_BA = πR²ρ_BG is the illuminated area integral of the background atmospheric density, and J₁ is the first-order Bessel function of the first kind. Applying the relative perturbation process to Equation (2) yields the following:

$${\displaystyle \frac{[{\rho }_{BA}+\frac{{\rho }_d\pi R{J}_1(Rk)}{k}\mathrm{c}\mathrm{o}\mathrm{s}({\omega }_{s}t+\phi )]-{\rho }_{BA}}{{\rho }_{BA}}}={\displaystyle \frac{2{\rho }_d\pi R{J}_1\left(Rk\right)\mathrm{c}\mathrm{o}\mathrm{s}({\omega }_{s}t+\phi )}{{\rho }_{BA}k}}$$

(3)

The modulus of the discrete Fourier transform of the above equation is

$$P_s={\displaystyle \frac{2{\rho }_d{\pi RJ}_1\left(Rk\right)}{{\rho }_{BA}k}}$$

(4)

In the above equation, ρ_d/ρ_BA represents the relative atmospheric density perturbation caused by the gravity wave. In theory, if the radius of the laser illumination cross-section, the relative perturbation, and the spectral peak amplitude P_S are known, the wavelength of the monochromatic horizontal fluctuation can be calculated using Equation (4).

2.2. Data Processing

As shown in Figure 2, data processing is divided into three parts: preprocessing, processing methods, and parameter estimation. Preprocessing first adjusts the resolution of the original echo photon count profiles based on the spatial-temporal signal-to-noise ratio, followed by routine background noise removal, R² correction (which refers to multiplying the photon count by the square of the distance), profile fitting, and finally yielding the relative perturbation dataset. Processing methods consist of detrending, wavelet denoising, and spectral analysis. The vertical echo profile observation of middle atmosphere lidar is one of the most direct methods for studying the vertical activity of gravity waves. The evolution of vertical activity introduces periodic noise into the horizontal time series. A detrending process was applied during data processing to remove the influence of these vertical fluctuations. Therefore, the regular periodic oscillations observed in the vertically detrended time series are regarded as spatial sampling of horizontal spatiotemporal waves. The relative perturbation data undergoes wavelet denoising to improve the signal-to-noise ratio, followed by fast Fourier transform to obtain spectral information, and then estimation of parameters such as perturbation frequency and spectral amplitude.

Evidently, an iterative Formula (5) for solving the wavenumber k can be constructed from Equation (4), where P_S is the spectral peak value.

$$k=f(k):k={\displaystyle \frac{2{\rho }_d{\pi RJ}_1\left(Rk\right)}{{\rho }_{BA}{P}_S}}$$

(5)

In actual observations, gravity waves often manifest as quasi-monochromatic waves with modulation characteristics. Based on the assumption that gravity waves are generated from a certain full-spectrum source, it can be deduced that the apparent modulated fluctuations after signal processing result from the superposition of multiple monochromatic waves with equal phase velocities. Therefore, any two monochromatic waves in the fluctuation signal have the following relationship:

$${\displaystyle \frac{k_2}{k_1}}={\displaystyle \frac{T_1}{T_2}}$$

(6)

where T is the period. Using Equation (4), the amplitude ratio of the two monochromatic waves from spectral analysis is

$${\displaystyle \frac{P_S^1}{P_S^2}}={\displaystyle \frac{{{\rho }_d^{1}J}_1\left(R{k}_1\right)}{{{\rho }_d^{2}J}_2\left(R{k}_2\right)}}{\displaystyle \frac{k_2}{k_1}}$$

(7)

Substituting this into Equation (6) yields the following:

$${\displaystyle \frac{P_S^1}{P_S^2}}={\displaystyle \frac{{\rho }_d^1{J}_1\left(R{k}_1\right)}{{{\rho }_d^{2}J}_2\left(R{k}_1\frac{T_1}{T_2}\right)}}{\displaystyle \frac{T_1}{T_2}}⟹{\displaystyle \frac{P_S^1{\rho }_d^2{T}_2}{P_S^2{\rho }_d^1{T}_1}}={\displaystyle \frac{J_1\left(x\right)}{J_2\left(x\alpha \right)}}$$

(8)

where x = Rk₁, α = T₁/T₂. The wavelength of the horizontal gravity wave is estimated by solving the transcendental Equation (8). Compared to Equation (5), Equation (8) does not require the background atmospheric density ρ_BA and utilizes multiple pieces of fluctuation information obtained from spectral analysis to enhance estimation reliability.

2.3. Influence of Illumination Aperture

Figure 3 presents a contour plot of the normalized spectral amplitude varying with relative perturbation and wavelength for an illumination cross-section aperture of 35 m. The relative disturbance is defined as the ratio ρ_d/ρ_BA in Equation (4), with the value of this ratio falling within the interval (0, 1). Under the condition of known gravity wave wavelength, the spectral amplitude increases monotonically with increasing relative perturbation. This phenomenon arises because the data in the relative perturbation originates from the sampling of spatial plane waves by the illumination aperture. When the illumination aperture remains constant, longer sampled wavelengths result in smaller changes in spectral amplitude. Under the condition of known relative perturbation amplitude, the spectral amplitude exhibits a monotonic change, increasing rapidly with wavelength before gradually plateauing. This indicates that when the illumination aperture exceeds tens of meters, observations of gravity waves on scales of at least several kilometers can be achieved.

3. Results and Discussion

The data in this paper are derived from the 2022 observational results of the dual-wavelength lidar system at the Beijing Yanqing Station (116.68° E, 40.33° N) of the National Space Science Center, Chinese Academy of Sciences. The system features a laser divergence angle of 0.5 milliradians, emits laser beams at wavelengths of 532 nm and 589 nm, and utilizes three-channel receiving technology based on Mie scattering, Rayleigh scattering, and sodium resonance fluorescence scattering to enable effective detection of atmospheric temperature in the 35–80 km altitude range and sodium layer density in the 80–110 km altitude range. The monthly observational data distribution chart presented in Figure 4 indicates that the data volume in winter exceeds that in summer, with a total of 100 days, averaging 8 days per month.

3.1. Case Study

Following the data processing workflow outlined in Figure 2, fluctuation parameters were estimated for observations conducted from the night of 18 November to the early morning of 19 November 2022. The original data had a height resolution of 96 m and a temporal resolution of 3 min. To improve the signal-to-noise ratio, the height resolution was reduced to 192 m during preprocessing, while the temporal resolution remained unchanged. The original relative perturbation signals required the application of a wavelet denoising algorithm to extract horizontal fluctuation signals. We utilized the sym7 wavelet operator from the symlets wavelet family [31]. Compared to other operators such as db, bior, and dmey, this operator more accurately extracts periodic fluctuations from the original data.

Figure 5 presents a comparison of the time–frequency domain and optimized fitting results for data processed with a four-level wavelet operator. The unprocessed original data were selected from the lidar echo time series at a 35 km altitude. In Figure 5a, the scattered points represent the wavelet-filtered echo signal, while the solid black line is the fit obtained by superimposing two sinusoidal components, revealing the presence of a modulated amplitude sine wave (quasi-monochromatic gravity wave). Figure 5b shows the double-side amplitude spectrum of the wavelet-denoised time-domain sequence, indicating that the fluctuation primarily consists of two monochromatic sine waves with periods of 1.99 h (1.39 × 10⁻⁴ Hz) and 1.33 h (2.08 × 10⁻⁴ Hz). The amplitude of the 1.99 h period sine wave is 1.49 times that of the 1.33 h period wave. Generally, the signal-to-noise ratio of horizontal time series in lidar data is significantly lower than that of vertical echo signals. Therefore, we consider the wavelet-denoised time-domain sequence, shown as the black solid line in Figure 5a, to be a quasi-monochromatic sine wave with amplitude modulated by multiple waves.

Through processing and analysis of the prominent sinusoidal fluctuations in the wavelet-denoised data and wavelength estimation using Equation (8),

Table 1. The estimated parameters of horizontal gravity waves.

Period (hour)/ Frequency (Hz)	Relatival Amplitude (%)	Wavelength (m)	Phase Velocity (m/s)
1.99/1.39 × 10−4	3.85	1274	0.17
1.33/2.08 × 10−4	1.18	851	0.17

above provides the fluctuation parameters for the two monochromatic waves. During the night of 18 November to the early morning 19 November 2022, two monochromatic horizontal-scale gravity waves contributing to the primary modulated fluctuation were observed, with periods, wavelengths, and phase velocities of 1.99/1.33 h, 1274/851 m, and 0.17 m/s, respectively. A relative amplitude of 3.85 corresponds to an atmospheric density perturbation spanning approximately 600 m vertically, centered around a 35 km atmospheric altitude. The phase velocity of 0.17 m/s is lower than the observed phase velocities of gravity waves generated by atmospheric activity, indicating that these fluctuations originate from gravity wave breaking or saturation dissipation [32].

3.2. Seasonal Variation

Through processing and analysis of the 2022 lidar observational data for the upper stratosphere (i.e., altitudes of 30–40 km), relatively clear time series of relative perturbations in atmospheric density exhibiting quasi-monochromatic characteristics were obtained from 100 days of nighttime cross-day data. Figure 6 presents the seasonal distribution and statistics of small-scale horizontal gravity wave parameters over these 100 days. In Figure 6a, from top to bottom, are the seasonal distributions of wavelength, frequency, and phase velocity, respectively. The black solid line, dashed line, and dot-dashed line in the figure represent the time–oscillation fitted curves for the fluctuation parameters, along with the 68% and 95% confidence intervals of the fitted curves. The wavelength error exhibits a linear relationship with the relative disturbance error. The error bars presented in Figure 5a were obtained under the condition of a 10% relative disturbance error.

The seasonal variations in parameters such as wavelength, frequency, and phase velocity of small-scale horizontal gravity waves uniformly exhibit SAO. Among these, peaks in wavelength and phase velocity variations occur in winter, followed by spring, with the weakest activity in summer. Period variations are smaller around the summer and winter solstices of that year and larger around the spring and autumn equinoxes. Due to the observation aperture being only 35 m, theoretically, the maximum observable horizontal wavelength does not exceed 5 km [33]. The wavelength parameter statistics indicate that those less than 1 km account for 60%, with the maximum wavelength not exceeding 3 km, and wavelengths greater than 1 km being more frequent in winter. The period parameter statistics show that those less than 3 h account for 75%, among which those greater than 2 h account for 58%. Periods greater than 6 h account for 3%, but their wavelengths are only in the order of hundreds of meters. The phase velocity statistics reveal an overall value less than 0.5 m/s, with those less than 0.1 m/s accounting for 58%, yet conforming to a linear relationship between wavelength and phase velocity, i.e., longer wavelengths correspond to faster phase velocities.

Lindzen’s linear theory provides mechanisms for instability phenomena such as gravity wave breaking and dissipation during propagation in the atmosphere [34]. It is generally considered that gravity wave sources with small phase velocity values are local. In summer, the stratosphere has easterly (east-to-west) winds (the summer easterly jet). These strong easterlies act as a critical level for any gravity wave and filter out most of the eastward-propagating component, so far fewer gravity waves survive to propagate upward. In winter, the stratospheric polar vortex creates strong westerly winds, allowing waves with westward or slow eastward phase speeds to propagate upward freely. A broad spectrum of gravity waves from the turbulent lower atmosphere can reach the middle atmosphere. This seasonal filtering is a key reason why the mesospheric circulation and temperature structure reverse dramatically between summer and winter, and why gravity wave effects show strong seasonal differences [2]. The fluctuation observation data in Figure 6 demonstrates that gravity wave activity in the winter stratosphere is stronger than in summer, aligning with theoretical predictions.

The characteristics of gravity waves formed by atmospheric activity include horizontal wavelength scales ranging from several kilometers to hundreds of kilometers, corresponding to phase velocities from several to hundreds of m/s. The horizontal fluctuations shown in Figure 6 exhibit features of short wavelengths, low phase velocities, and long periods. Gravity waves propagating upward from lower layers are influenced by the boundary layer, becoming confined to certain spaces, where they dissipate through nonlinear processes, or lose energy via saturation effects. The breaking and saturation dissipation mechanisms of gravity waves enable the distribution of momentum and energy from lower layers in the middle atmosphere, driving middle atmospheric circulation. When atmospheric gravity waves break or dissipate in the stratosphere, new wave sources can form through secondary wave generation processes. This phenomenon is driven by nonlinear dynamics associated with gravity wave breaking or dissipation. Such breaking involves nonlinear interactions that transfer energy from the primary wave to smaller-scale motions, including the generation of local perturbations in wind and temperature fields. As sources of secondary gravity waves, these waves typically have shorter horizontal and vertical wavelengths and different phase velocities compared to the primary waves [35,36]. Fritts et al. suggest that breaking gravity waves in the stratosphere generate secondary waves through nonlinear interactions, which are commonly observed as small-scale wave packets in radar and lidar data [37]. Breaking mountain waves with vertical wavelengths of 5–10 km may produce secondary waves with wavelengths of 1–5 km and phase velocities as low as 0.1–5 m/s.

4. Conclusions

Atmospheric circulation models must incorporate gravity wave effects to realistically reproduce and predict atmospheric activity, underscoring the importance of gravity waves. Research on gravity wave parameterization has revealed that small-scale horizontal wave activity serves as a crucial carrier for the redistribution of energy and momentum. Due to the illumination cross-section of middle atmospheric Rayleigh lidar, the photon counts recorded by the photodetector represent spatial sampling of horizontal spatiotemporal fluctuations by the laser illumination cross-section, meaning that amplitude variations in the time series contain spatial information about the fluctuations. Through analysis of the operating principles of middle atmospheric Rayleigh lidar, we provide a method for horizontal gravity wave observations using monostatic Rayleigh lidar and a data inversion workflow. The illumination cross-section, controlled by the laser divergence angle, determines the observational limit for horizontal scales. In theory, this method can be applied to observational studies of horizontal gravity waves with scales less than a dozen kilometers. The Beijing Yanqing lidar system, designed for middle atmospheric temperature observations, has an illumination cross-section aperture of only 35 m at a 35 km altitude, theoretically limiting the observable wavelength of horizontal gravity waves to no more than 5 km. This method was employed to process and analyze one year of Rayleigh lidar data. The results indicate that over the year, small-scale horizontal gravity waves exhibit SAO with weak activity in summer and strong activity in winter, consistent with general trends in middle atmospheric gravity wave activity. The observed small-scale horizontal gravity waves have wavelengths not exceeding 3 km, with those phase velocities generally not exceeding 0.5 m/s, and periods less than 3 h account for 75%. These fluctuation characteristics of short wavelengths, low phase velocities, and long periods suggest that the observed fluctuations are secondary waves generated by gravity wave breaking or dissipation. Atmospheric lidar designs are typically focused on probing higher altitudes, often selecting very small laser divergence angles to improve signal-to-noise ratios. If the divergence angle is designed to be 20 milliradians and a vertical spatial resolution of tens of meters is chosen, routine observations of horizontal gravity wave activity on scales less than 10 km can be achieved.