Study on the Momentum Flux Spectrum of Gravity Waves in the Tropical Western Pacific Based on Integrated Satellite Remote Sensing and In Situ Observations

Abstract

Gravity wave (GW) momentum flux spectra help to uncover the mechanisms through which GWs influence momentum transfer in the atmosphere and provide crucial insights into accurately characterizing atmospheric wave processes. This study examines the momentum flux spectra of GWs in the troposphere (2–14 km) and stratosphere (18–28 km) over Koror Island (7.2°N, 134.3°W) using radiosonde data from 2013–2018. Utilizing hodograph analysis and spectral methods, the characteristics of momentum flux spectra are discussed. Given that the zonal momentum flux spectra of low-level atmospheric GWs generally follow a Gaussian distribution, Gaussian fitting was applied to the spectral structures. This fitting further explores the seasonal variations of the zonal momentum flux spectra and the average spectral parameters for each month. Additionally, the GW energy is analyzed using SABER (Sounding of the Atmosphere using Broadband Emission Radiometry) satellite data and compared with the results of the momentum flux spectra from radiosonde data, revealing the close negative correlation between wave energy and wave momentum for stratospheric GW changing with time. The findings indicate that the Gaussian peak shifts more eastward in both the troposphere and stratosphere, primarily due to the absorption of eastward-propagating GWs by the winter tropospheric westerly jet and critical layer filtering. The full width at half maximum (FWHM) in the stratosphere is larger than in the troposphere, especially in June and July, as the spectrum broadens due to propagation effects, filtering, and interactions among waves. The central phase speed in the stratosphere exceeds that in the troposphere, reflecting the influences of Doppler effects and background wind absorption. The momentum flux in the stratosphere is lower than in the troposphere, which is attributed to jet absorption, partial reflection, or the dissipation of GWs.

1. Introduction

GWs are one of the fundamental oscillations generated in a stably stratified atmosphere. As they propagate upward, they dissipate energy and momentum into the background atmosphere, affecting both local and global atmospheric motion processes. With increasing altitude, the atmosphere becomes thinner, and the amplitude of GWs increases until they reach saturation or encounter instability, leading to breaking and dissipation. This process transfers energy and momentum from lower layers to higher altitudes, ultimately influencing the structure and state of atmospheric circulation. Consequently, GWs have a significant impact on the dynamic and thermal structures of the global atmosphere. Moreover, GWs play a crucial role in the formation of meridional circulation in the summer and winter hemispheres, and the development of the stratospheric polar vortex [1]. With advancements in detection technology, researchers have begun exploring potential links between GWs and natural disasters such as earthquakes, tsunamis, and typhoons [2,3], indicating that this field is becoming increasingly sophisticated and mature.

Atmospheric circulation is significantly influenced by GW activity on a global scale, making accurate GW parameterization essential for improving predictive accuracy. Understanding the sources of GWs is crucial for comprehensively understanding their activity processes. In the lower atmosphere, the primary factors influencing GW generation include topography, convection and wind shear, among others. In the middle and upper atmosphere, GW dissipation and wave–wave interactions become key factors in generating GWs [4]. The occurrence of small- and medium-scale GWs varies across different regions, prompting scholars to conduct in-depth studies on GW activity worldwide. Through theoretical analysis based on observational data, model results, and numerical simulations, significant progress has been made in understanding these phenomena.

So far, numerous observational techniques have been employed to study GWs, including satellite observations [5], radiosondes [6], rocket soundings [7], lidar [8], radar observations [9], aircraft measurements [10], and flat floating balloons [11]. Due to the limitations of observational technologies, each method acts as a filter across the full spectrum of GWs, capturing only a portion of the spectral characteristics. Satellite platforms provide a global perspective and can observe higher altitude ranges [12]. However, these methods are constrained by vertical resolution limitations, making it challenging to capture the small-scale structures of GWs. In contrast, radiosonde data, with their high vertical resolution and long data accumulation periods, are better suited to capture GWs with short vertical wavelengths and slow propagation speeds. This makes radiosonde data particularly useful for climatological analysis. Additionally, radiosondes are relatively low-cost and flexible to deploy and operate, making them suitable for various environments and scenarios. In recent years, the application of radiosondes in GW observations has seen significant progress.

Inertial GWs (IGWs) are low-frequency waves influenced by Earth’s rotation and gravity, with frequencies higher than the inertial frequency f. Although current general circulation models (GCMs) cannot adequately resolve small-scale GW issues in the momentum balance of the middle atmosphere due to resolution constraints, the parameterization of GW drag (GWD) remains uncertain. More observational data are crucial for improving the accuracy of GWD parameterizations in models. The energy carried by GWs comprises both potential and kinetic energy. According to the linear GW theory, the ratio of these two energy components is constant [4,13]. Studying the potential energy of GWs is one of the primary ways to understand GW activity. However, potential energy only represents the amount of energy carried by GWs and does not account for the momentum released. When investigating the breaking height of GWs and their effects on the stratosphere, the vertical flux of horizontal momentum, or momentum flux, is one of the most representative parameters of GWs. Therefore, analyzing the momentum flux of GWs is essential in the study of GW activities.

This study combines satellite remote sensing and in situ observational techniques to investigate the momentum flux spectra of GWs in the tropical western Pacific region. Using radiosonde data from Koror Island Station (7.2°N, 134.3°W) in the western Pacific, collected between 2013 and 2018, spectral analysis is performed on the momentum flux of IGWs in the upper atmosphere. Additionally, GW activity over the corresponding area is captured using SABER satellite data. Section 2 introduces the radiosonde data from the Koror region and describes the methods for extracting quasi-monochromatic GWs and momentum flux. Section 3 details the background atmospheric conditions over the Koror region. In Section 4, the momentum flux spectra are analyzed through Gaussian fitting. Section 5 investigates the physical characteristics of the momentum flux spectra by examining the peak, FWHM, and central phase speed of the Gaussian fit. In Section 6, the seasonal variations of the zonal momentum flux in both the troposphere and stratosphere across four seasons—spring, summer, autumn, and winter—are analyzed. The fitting parameters for different months are calculated to understand the seasonal changes in the momentum flux spectra. Section 7 integrates the monthly distribution of potential energy in the troposphere and stratosphere from SABER satellite data with the previously obtained parameters for a comprehensive analysis. Finally, Section 8 provides a summary of the findings.

2. Data and Methods

2.1. Data

The data used in this study from the Koror Station were sourced from the Radio Sounding Replacement System (RRS) data provided by the U.S. National Oceanic and Atmospheric Administration (NOAA), the National Centers for Environmental Information (NCEI), and the Stratosphere–troposphere Processes And their Role in Climate (SPARC) project. A total of six years of data, from 2013 to 2018, with a resolution of 1 s, were obtained. The radiosondes were launched daily at 00:00 UTC and 12:00 UTC, with an average ascent rate of 5 m/s, and only nighttime data were selected. Using sensors and the Global Positioning System (GPS) carried by the Vaisala RS90 radiosonde, measurements of temperature, pressure, relative humidity, GPS altitude, and smoothed zonal and meridional winds were acquired. To mitigate the effects of non-uniform data intervals and eliminate errors caused by random balloon motions, the data were uniformly interpolated at 50 m intervals using cubic spline interpolation, corresponding to a 10 s time interval. During the balloon’s ascent, sensors on the radiosonde collect real-time data on pressure, temperature, and relative humidity, and accurate positional information is obtained via satellite navigation systems. This allows for the calculation of zonal and meridional winds. These high-resolution vertical profile data, analyzed using hodograph techniques, enable the extraction of several key parameters of IGWs, such as propagation direction, intrinsic frequency, phase speed, and vertical and horizontal wavelengths.

2.2. Methods

Considering the rapid changes in buoyancy frequency and temperature near the tropopause, GW parameters are extracted separately for the troposphere and stratosphere. In this study, the height intervals chosen for extracting GW parameters are 2–14 km and 18–28 km, representing the troposphere and stratosphere, respectively. The range below 2 km is not considered, to avoid errors caused by strong disturbances near the planetary boundary layer. Currently, the extraction of GW background wind and temperature profiles typically employs second-order [14], third-order [15], or fourth-order [16] polynomial fitting. Although the differences in GW parameters obtained using linear, second-order, third-order, or fourth-order polynomial fits are generally minor [17], the more pronounced variations in wind profiles in the tropical regions warrant the use of a fourth-order polynomial fit to extract the background profiles in this study.

2.2.1. Sine Curve Fitting

For the original profile data of temperature $T$, zonal wind $u$, and meridional wind $v$, the background profiles $\overline{T}$, $\overline{u}$, and $\overline{v}$ are obtained separately for the 2–14 km and 18–28 km altitude ranges using fourth-order polynomial fitting. The perturbation profiles $T^{\prime }$, $u^{\prime }$ and $v^{\prime }$ are then derived by subtracting the background profiles from the original profiles. These perturbation profiles contain waves of various scales and amplitudes. By applying Lomb–Scargle power spectral analysis, the waves can be categorized into different scales. The dominant wavelength is identified as the one with the maximum amplitude, from which the vertical wavelengths of temperature, zonal wind, and meridional wind can be determined. Once the vertical wavelength $\lambda$ is obtained, the amplitude and phase of the quasi-monochromatic IGWs can be extracted using the sinusoidal harmonic fitting method as follows:

$$U=A\mathit{sin}\left(\left(2\pi /\lambda z\right)Z+\phi \right)$$

(1)

In this context, $U=[u^{\prime },v^{\prime },T^{\prime }]$ represents the perturbation components of the zonal wind, meridional wind, and temperature. $A=[\tilde{u},\tilde{v},\tilde{T}]$ and $\phi =[{\phi }_u,{\phi }_v,{\phi }_T]$ denote the amplitude and phase of the fitted IGW, respectively.

2.2.2. Velocity Plot Analysis and Intrinsic Frequency

According to IGW theory, the disturbance amplitudes of the wind field along the major axis (parallel to the horizontal propagation direction of the wave) and the minor axis (perpendicular to the horizontal propagation direction of the wave) of the polarization ellipse are $\widehat{u}$ and $\widehat{v}$, respectively, and they satisfy the following polarization equation [4]:

$$\frac{\widehat{u}}{\widehat{v}}=i\frac{\widehat{\omega }}{f}$$

(2)

In the equations, $\widehat{\omega }$ is the intrinsic frequency, and $f$ is the inertial frequency, which can be derived from the Earth’s rotation rate $\Omega$ and the latitude $\beta$ of the observation point: $f=2\Omega sin\beta$. For the Koror Island region, $f$ is approximately 3.38 × 10⁻⁵ rad/s. The horizontal wind disturbances along the major axis $\widehat{u}$ and the minor axis $\widehat{v}$ of the polarization ellipse, as well as the azimuth angle $\theta$ (the angle measured clockwise from true north) corresponding to the major axis, can be determined using the following relationships:

$$2{\widehat{u}}^2={\tilde{u}}^2+{\tilde{v}}^2+{\left[{\left({\tilde{u}}^2-{\tilde{v}}^2\right)}^2+4F_{uv}^2\right]}^{\begin{array}{c}1\\ 2\end{array}}$$

(3)

$$2{\tilde{v}}^2={\tilde{u}}^2+{\tilde{v}}^2-{\left[{\left({\tilde{u}}^2-{\tilde{v}}^2\right)}^2+4F_{uv}^2\right]}^{\begin{array}{c}1\\ 2\end{array}}$$

(4)

$$\theta =\frac{1}{2}\left(\pi n+\mathit{arctan}\frac{2F_{uv}}{{\tilde{v}}^2-{\tilde{u}}^2}\right)$$

(5)

where $F_{uv}=\tilde{u}\tilde{v}\mathit{cos}\left({\phi }_u-{\phi }_v\right)$. When $\tilde{v}<\tilde{u}$, $n=1$; when $\tilde{v}>\tilde{u}$ and $F_{uv}>0$, $n=0$; when $\tilde{v}>\tilde{u}$ and $F_{uv}<0$, $n=2$. The direction of the horizontal propagation speed of the GW is along the major axis of the polarization ellipse. However, $\theta$ has a 180° ambiguity that needs to be resolved using the polarization characteristics of the GW. The horizontal wind disturbance formula is $u_h^{\prime }=u^{\prime }\mathit{sin}\theta +v^{\prime }\mathit{cos}\theta$a. When $\left(u_h^{\prime },T^{\prime }\right)$ rotates counterclockwise (or clockwise) with height, the actual propagation direction is opposite (or the same as) $\theta$. To determine the vertical propagation direction of the GW using the velocity disturbance components, if $\left(u^{\prime },v^{\prime }\right)$ rotates clockwise with height, it indicates upward energy propagation and downward phase propagation. Conversely, if $\left(u^{\prime },v^{\prime }\right)$ rotates counterclockwise with height, it indicates downward energy propagation and upward phase propagation.

2.2.3. Wave Energy and Momentum Flux

To quantitatively describe the intensity of IGW activity, wave kinetic energy $E_k$ and wave potential energy $E_p$ are introduced as follows [18]:

$$E_k=\frac{1}{2}\left(\overline{u^{\prime 2}}+\overline{v^{\prime 2}}\right)$$

(6)

$$E_p=\frac{g^2}{2N^2}\overline{{\widehat{T}}^{\prime 2}}$$

(7)

where ${\widehat{T}}^{\prime }=T^{\prime }/\overline{T}$ is the normalized temperature perturbation. The overline denotes an unweighted spatial average over the height interval, which is 2–14 km for the troposphere and 18–28 km for the stratosphere. The contribution of vertical wind disturbances is neglected here because, for IGWs, the perturbations in the vertical wind speed are much smaller than those in the horizontal direction.

Changes in the momentum flux of GWs can lead to changes in the background wind field. As a significant factor influencing large-scale mean flows, the vertical fluxes of zonal momentum ($\overline{u^{\prime }{w}^{\prime }}$) and meridional momentum ($\overline{v^{\prime }{w}^{\prime }}$) per unit mass can be expressed as [4]:

$$\overline{u^{\prime }{w}^{\prime }}=\frac{g\widehat{\omega }}{N^2}\overline{u^{\prime }{\left(\frac{T^{\prime }}{\overline{T}}\right)}_{+90}}\left[1-{\left(\frac{f}{\widehat{\omega }}\right)}^2\right]$$

(8)

$$\overline{v^{\prime }{w}^{\prime }}=\frac{g\widehat{\omega }}{N^2}\overline{v^{\prime }{\left(\frac{T^{\prime }}{\overline{T}}\right)}_{+90}}\left[1-{\left(\frac{f}{\widehat{\omega }}\right)}^2\right]$$

(9)

where ${\left(\frac{T^{\prime }}{\overline{T}}\right)}_{+90}$ represents the normalized temperature perturbation with a 90° phase shift applied using the Hilbert transform. The overline denotes an unweighted spatial average.

3. Background Conditions

Based on the climatic characteristics of the Koror region, the year is divided into four seasons: spring (March, April, and May), summer (June, July, and August), autumn (September, October, and November), and winter (December, January, and February). For this study, the selected height range covers the profile from 2 km to 28 km. Before delving into the specifics of GW activity, it is essential to have a clear outline of the background atmospheric conditions in this region, as the generation and propagation of GWs are closely related to the state of the background atmosphere. The monthly averages of the zonal wind, meridional wind, temperature, and buoyancy frequency profiles are computed and interpolated over the six-year period to represent the background atmospheric conditions, as illustrated in Figure 1.

In the lower troposphere (0 to 5 km altitude), the zonal wind primarily exhibits easterly flow, particularly weakening significantly during the summer, while stronger westerlies prevail in winter and spring. In the mid to upper troposphere, easterlies dominate, but during spring and summer, they tend to become weak easterlies. In the stratosphere, there is an alternating pattern of easterlies and westerlies, with easterlies being more predominant. Above 18 km in the stratosphere, there is a noticeable quasi-biennial oscillation (QBO) in the zonal wind. However, this oscillation was interrupted in the winter of 2015/16, where the westerly phase at 20 km was replaced by easterlies, and the westerly phase at 27 km strengthened and moved upward. This interruption is associated with the El Niño event of the same year, which impacted the upper atmospheric circulation systems in the western Pacific region [19]. The slowly propagating IGWs in the stratosphere were absorbed, dissipating momentum into the lower stratosphere and generating additional enhanced westward forcing [20]. In contrast, meridional winds are generally weaker across seasons but significantly intensify in the altitude range of 12 to 15 km, exhibiting clear interannual variations. During winter, the maximum northward wind speeds in this range can reach 7 to 9 m/s.

In the lower and middle troposphere, the temperature exhibits a more distinct stratified structure, with little variation across months. However, near the tropopause and the lower stratosphere, temperature changes become more pronounced. The equatorial cold point tropopause typically lies at approximately 16 to 18 km [21]. Near the tropopause, there is a noticeable “cold pool” where temperatures drop below −80 degrees Celsius, corresponding to the minimum temperatures in the stratosphere. The highest tropopause temperatures generally occur in summer, being 4 to 8 °C higher than those at the same altitude in other seasons. The stratospheric temperature shows a clear annual cycle, indicating that even at the same latitude, there are local climatic differences between different regions. The monthly average buoyancy frequency decreases with increasing altitude, reaching a distinct minimum between 10 and 15 km, with frequencies ranging between 0.5 and 0.7 rad/s. This is related to the static instability caused by strong convective activity. Near the tropopause, the buoyancy frequency increases rapidly, with a peak in the lower stratosphere ranging between 2.5 and 2.8 rad/s. Additionally, the rapid increase in buoyancy frequency in the mid-levels shows clear interannual oscillations, reflecting interannual variations in the tropopause, with the highest tropopause in winter and the lowest in summer.

4. Momentum Flux Spectrum Analysis

In the process of calculating the momentum flux spectrum $F\left(c\right)$ (where $c$ represents the horizontal phase speed of GWs), the phase speed is first divided into bins with a width of 2 m/s. Then, the average momentum flux of GWs within the $i$-th phase speed interval $c_i$ is calculated. This results in the momentum flux spectrum as follows:

$$F(c_i)=\frac{{\sum }_{n=1}^{N_3}{F}_n\left(c_i\right)}{N_U}$$

(10)

In the formula, $F\left(c_i\right)$ represents the momentum flux of the $n$-th GW in the $i$-th phase speed interval. $N_3$ denotes the number of GWs observed within the $i$-th phase speed interval, and $N_U$ is the total number of upward-propagating GWs observed at the radiosonde station. Generally, only upward-propagating GWs significantly impact the dynamical and thermodynamical structure of the middle and upper atmosphere. Therefore, the analysis focuses solely on the momentum flux spectrum of upward-propagating GWs.

The blue fine solid line histogram in Figure 2 illustrates the upward-propagating GW momentum flux spectra in the troposphere and stratosphere at Koror Station. Taking the tropospheric zonal momentum flux spectrum as an example, the phase speed distribution is roughly divided into two regions: the point where the momentum flux is zero separates the positive peak (influenced by eastward winds, referred to as eastward) and the negative peak (influenced by westward winds, referred to as westward). The spectrum width for the eastward peak ranges between −7 and 18 m/s, while the westward peak ranges between −23 and −7 m/s. The peak of the eastward spectrum is located between −2 and −1 m/s, with a peak value of 1.87 × 10⁻⁴ m²/s². The peak of the westward spectrum is located between −15 and −14 m/s, with a peak value of −1.10 × 10⁻⁴ m²/s². The distribution of the momentum flux spectra resembles a Gaussian distribution and exhibits an asymmetry in horizontal propagation. The tropospheric meridional momentum flux spectrum and the stratospheric zonal and meridional momentum flux spectra at this station display similar characteristics. Next, we will attempt to further explore the fundamental properties of the momentum flux spectra using Gaussian fitting.

Studies on lower atmospheric GWs using radiosonde data have shown that the characteristics of tropospheric GWs are determined by the properties of their sources [4]. Modeling the global lower atmospheric GW momentum flux spectra with a simple function, such as a Gaussian function, based on radiosonde data, could significantly benefit GCMs. We apply the least squares method to perform Gaussian fitting on the momentum flux spectra. The red curve in Figure 2 represents the Gaussian fitting results for the momentum flux spectra in both the troposphere and stratosphere. The total root mean square error (RMSE) of the eight fitted curves is relatively small, at 2.3 × 10⁻⁵ m²/s², indicating that using Gaussian functions to model the momentum flux spectra of GWs in the troposphere and stratosphere is reasonable. Observing the figure, it can be seen that for the zonal momentum flux, the Gaussian fit matches the momentum flux spectra closely. However, for the meridional momentum flux, as shown in Figure 2d, the average relative error for the northward spectrum reaches 4400%. On one hand, the large error in the meridional spectrum results in poor fitting; on the other hand, the peak values of the meridional spectrum are much smaller than those of the zonal spectrum (by an order of magnitude), indicating that the momentum carried by GWs is primarily propagated in the zonal direction. Therefore, subsequent research will focus on discussing the phase speed spectra of zonal momentum flux.

5. Gaussian Parameter Analysis

Given that the Gaussian function can effectively fit the observed momentum flux spectra, we can analyze three key parameters of the Gaussian fit—peak value, FWHM, and Gaussian central phase speed—to study the momentum flux spectra and their physical characteristics. The peak value is the extremum of the Gaussian fit spectrum, the Gaussian central phase speed corresponds to the phase speed at the spectrum peak, and the FWHM indicates the wavelength range at half the peak height.

Table 1. Gaussian fitting parameters of tropospheric momentum flux spectrum at Koror Station.

Direction	Peak Value/(10−4 m2/s2)	Central Phase Speed/(m/s)	FWHM/(m/s)
Eastward	1.26	−0.42	14.55
Westward	−0.84	−14.29	10.15
Northward	−0.02	39.75	39.41
Southward	−0.59	−14.75	55.82

presents the Gaussian fitting parameters for the tropospheric momentum flux spectra. In the zonal direction, the peak value for the eastward direction is 1.26 × 10⁻⁴ m²/s², which is larger in absolute value than that for the westward direction. This is because the site experiences a strong westerly jet stream in the troposphere during winter, where critical layer filtering causes the absorption of westward-propagating GWs. The FWHM for the eastward direction is 14.55 m/s, slightly larger than that for the westward direction, indicating that the difference in monochromaticity between waves propagating eastward and westward is minimal. The peak value and FWHM can characterize the total momentum flux; the larger these two parameters, the greater the total momentum flux. Therefore, in Figure 2, the total westward momentum flux in the troposphere is greater than the eastward flux, primarily due to the absorption of eastward-propagating GWs by the winter westerly jet stream. The Gaussian central phase speed for westward-propagating waves is −14.29 m/s, and the difference between this value and the demarcation point of −7 m/s is larger than the difference for the eastward direction, which is −0.42 m/s. This can be attributed to the Doppler effect of the jet stream, where the background westerly wind reduces the phase speed of eastward-propagating waves and increases the phase speed of westward-propagating waves. In the meridional direction, the peak value and FWHM for the southward direction are larger than those for the northward direction, indicating that the total momentum flux for southward-propagating waves is greater than that for northward-propagating waves.

Table 2. Gaussian fitting parameters of the stratospheric momentum flux spectrum at Koror Station.

Direction	Peak Value/(10−4 m2/s2)	Central Phase Speed/(m/s)	FWHM/(m/s)
Eastward	2.05	6.41	36.73
Westward	−0.13	−28.63	16.02
Northward	0.10	32.79	54.37
Southward	−0.16	−34.61	63.32

presents the Gaussian fitting parameters for the stratosphere. The peak value and the difference between the central phase speed and the demarcation point are significantly larger for the eastward direction compared to the westward direction. For the meridional direction, the absolute values of the peak for the southward and northward directions are approximately equal, but the difference in central phase speed from the demarcation point is slightly larger for the southward direction than for the northward direction. These characteristics in the stratosphere are similar to those in the troposphere because the sources of upward-propagating GWs in the stratosphere are primarily located in the troposphere. Thus, their spectral characteristics largely retain the properties of upward-propagating waves from the troposphere. The FWHM in the stratosphere is larger than that in the troposphere, indicating that the sources in the troposphere are more singular and the waves exhibit stronger monochromaticity. This is because most upward-propagating waves in the stratosphere are excited in the troposphere, and as the waves propagate from the troposphere to the stratosphere, effects such as propagation, filtering, and interactions among waves lead to a broadening of the spectrum. The central phase speed is also greater in the stratosphere. This shift in phase speed from the troposphere to the stratosphere is due to the Doppler effect: when GWs propagate against the wind, the Doppler effect increases the phase speed; conversely, when propagating with the wind, the low-frequency components of the waves are absorbed by the background wind, allowing only the higher-frequency components to reach the stratosphere.

Since the peak value and FWHM can characterize the total momentum flux, a comparison of Table 1 and Table 2 reveals that in all four directions, the peak values in the lower stratosphere are smaller than those in the troposphere, while the FWHM in the lower stratosphere is slightly larger than in the troposphere. This indicates that the total momentum flux in the lower stratosphere is smaller than in the troposphere. On one hand, the jet stream reflects and absorbs some of the upward-propagating GWs, reducing the number of waves that continue to carry momentum upward. On the other hand, before reaching the stratosphere, GWs often undergo breaking or dissipation through wave–mean flow and wave–wave interactions, depositing part of their momentum into the background atmosphere of the troposphere.

6. Seasonal Variation in Momentum Flux Spectra

6.1. Seasonally Averaged Gaussian Fit Spectra

Previous studies have shown that atmospheric GWs exhibit significant seasonal variations [22,23]. This study examines the seasonal variations in the GW spectra. Given that the zonal fit is better than the meridional fit and that the zonal momentum flux of GWs over the tropics is much higher than the meridional flux, this section focuses on the seasonal variations of the zonal momentum flux. The zonal momentum fluxes in both the troposphere and stratosphere are calculated for each season using six years of data.

Figure 3 shows the vertical profiles of the seasonal average zonal wind for the four seasons: spring, summer, autumn, and winter. In the troposphere (2–14 km), there are weak westerlies in the lower levels during the summer and autumn, while the stratosphere (18–28 km) is predominantly characterized by easterlies. During spring, autumn, and winter, there is a more easterly wind pattern near the tropopause, around 20 km. In summer, autumn, and winter, there are more westerly winds around 15 km. In the stratosphere, as the altitude increases, the easterlies become increasingly strong.

Figure 4 displays the distributions of the zonal momentum flux spectra and their Gaussian fits in the troposphere for the four seasons: spring, summer, autumn, and winter. The peak values of the eastward momentum flux are 1.06 × 10⁻⁴, 1.08 × 10⁻⁴, 1.44 × 10⁻⁴, and 1.23 × 10⁻⁴ m²/s², respectively, for spring, summer, autumn, and winter. The peak values for the westward momentum flux are −4.57 × 10⁻⁵, −5.59 × 10⁻⁵, −5.43 × 10⁻⁵, and −7.57 × 10⁻⁵ m²/s² for the same seasons. The eastward peak in autumn is slightly higher than in the other seasons, while the westward peak in winter is somewhat larger than in the other three seasons. Across all four seasons, the eastward peak values are higher than the westward peak values, and the overall spectral structure is skewed westward. This skewing is due to the prevalent easterlies in the troposphere (as shown in Figure 3). For the eastward momentum spectrum, the peak energy is near zero phase speed, indicating that the GWs excited by the tropospheric sources during this period primarily have low phase speeds and carry momentum as they propagate upward.

Figure 5 shows the Gaussian-fitted spectra of the zonal momentum flux in the stratosphere. The peak values of the eastward momentum flux are 2.47 × 10⁻⁴, 2.29 × 10⁻⁴, 1.99 × 10⁻⁴, and 1.69 × 10⁻⁴ m²/s² for spring, summer, autumn, and winter, respectively. The peak values for the westward momentum flux are −8.45 × 10⁻⁶, −7.12 × 10⁻⁶, −7.79 × 10⁻⁶, and −7.19 × 10⁻⁶ m²/s² for the same seasons. It can be observed that the eastward and westward peak values in spring are slightly higher than those in the other three seasons. The westward peak values are significantly lower than the eastward peak values, which is due to the presence of an easterly jet at the tropopause that filters out a large number of westward-propagating GWs during their upward propagation. The eastward peak values are generally higher in the stratosphere compared to the troposphere, which is attributed to the increase in GW amplitude as they propagate upward, carrying more momentum and energy. The eastward momentum flux spectrum corresponds to a broader range of GW phase speeds (−19.5 to 33 m/s), indicating a more extensive distribution of GWs as they interact with the background wind field during their upward propagation.

Comparing the troposphere and stratosphere, the overall spectral shape shifts to the left, indicating that the eastward–westward demarcation point moves leftward, which represents the strengthening of the background easterlies. On the other hand, the total westward momentum flux in the stratosphere is smaller than that in the troposphere, while the total eastward momentum flux in the stratosphere is greater than that in the troposphere. This is due to the increased background easterlies, which facilitate the upward propagation of the eastward GW momentum flux.

6.2. Gaussian Fitting Parameters for Each Month

To avoid the impact of having too few samples for each individual month on the effectiveness of Gaussian fitting, the momentum flux data over six years were analyzed and calculated for each month separately. Figure 6 illustrates the seasonal variations of the Gaussian fitting parameters for the troposphere. The maximum values for the eastward and westward Gaussian peak occur in August and November, respectively. Apart from November and December, the eastward peak values are consistently higher than the westward ones, with the largest difference appearing in August. This pattern indicates the absorption of eastward-propagating GWs by the background winds in seasons other than summer. There is no significant anisotropy between the meridional and zonal directions. Considering the FWHM, the minimum values for the eastward and westward directions are observed in June and July, respectively, which are the periods when the background zonal wind is weakest. This suggests that during periods of stronger background winds, the interaction between GWs and the background wind not only causes Doppler frequency shifts but may also introduce more spectral components, leading to a broadening of the wave spectrum [24]. The maximum value for the eastward peak occurs in April, and the spectrum width is also broader, indicating a richer spectrum of eastward-propagating waves in April. Regarding the central phase speed, the central phase speed of the westward momentum spectrum is consistently greater than that of the eastward spectrum. This is due to the Doppler effect of the background westerlies, which reduces the phase speed of eastward-propagating waves and increases the phase speed of westward-propagating waves. In the meridional direction, the central phase speeds for the southward and northward directions show frequent variations, with the maximum value for the northward direction and the minimum value for the southward direction both occurring in February. The southward peak is larger from March to April, with a full width at half maximum greater than the zonal direction. This is due to the interaction between gravity waves and the background wind field, where stronger north winds enhance the propagation efficiency and momentum flux of gravity waves, consistent with the results in Figure 1b.

Figure 7 shows the seasonal variations of the Gaussian parameters in the stratosphere. For the Gaussian peak values, the maximum values for the eastward and westward directions occur in June and August, respectively. The peak value for the eastward direction is larger, reaching up to 3.77 × 10⁻⁴ m²/s², and the eastward values are consistently higher than the westward ones throughout the year. Regarding the FWHM, the minimum values for the eastward and westward directions occur in December and May, respectively. The eastward FWHM shows little variation throughout the year, while the Gaussian peak values are higher from June to September compared to other months. Since both the peak value and FWHM jointly determine the total momentum flux, the total momentum flux is greater during the summer months, which is consistent with the results shown in Figure 5. For the Gaussian central phase speed, the westward values are consistently higher than the eastward values throughout the year. Additionally, the southward values exceed the northward values from June to August and from October to December. Apart from the eastward direction, the peak values in the stratosphere are significantly lower than those in the troposphere for all months, while the differences in FWHM between the two altitude ranges are relatively small. This indicates that the total momentum flux in the stratosphere is smaller than that in the troposphere. The significant differences in total momentum flux between the troposphere and the stratosphere are due to two main factors: first, as previously mentioned, the jet stream absorbs or reflects a substantial portion of the upward-propagating GWs; second, as the waves propagate to the region near the tropopause (10–18 km), some of their momentum is deposited into the background atmosphere due to dissipation [25].

7. Potential Energy Analysis of Satellite Data

7.1. Data and Methodology

SABER is an instrument aboard NASA’s TIMED (Thermosphere Ionosphere Mesosphere Energetics and Dynamics) satellite. SABER measures infrared radiation emitted by various atmospheric gases to obtain vertical distribution data of temperature and composition, thereby providing high-resolution vertical temperature profiles ranging from the mesosphere to the thermosphere (15–105 km). These data are crucial for analyzing the vertical distribution and activity of GWs [26,27].

To extract GW potential energy from SABER satellite data, the following observational variables must be collected: temperature, longitude, latitude, altitude, and time. Missing or anomalous data should be removed, and profiles with the lowest point below 20 km should be filtered out to ensure data completeness and accuracy. In the 20–30 km range, interpolation is performed vertically at 1 km intervals. Latitude is selected within the 0–10° range, and interpolation along the meridian is carried out at intervals of 0:5:360. Harmonic fitting is applied with 0–6 waves, considered as large fluctuations along the meridian, and these are removed. High-pass filtering with a cutoff wavelength of 15 km is then applied, treating the remaining fluctuations as GW oscillations [28]. Temperature data are processed by stratifying it into different altitude layers and calculating the mean temperature for each layer, which serves as the background temperature field. The mean state temperature is subtracted from the raw temperature data to obtain temperature perturbations. Using GW theoretical formulas, the potential energy of GWs at each altitude layer is calculated. The formula is as follows:

$$E_p=\frac{1}{2}{(\frac{g}{N})}^2{\overline{\left(T^{\prime }\right)}}^2$$

(11)

In the formula, $E_p$ represents the GW potential energy, $g$ is the gravitational acceleration, $N$ is the buoyancy frequency, and $\overline{\left(T^{\prime }\right)}$ is the root mean square (RMS) value of temperature perturbations.

7.2. Analysis of Results

Figure 8 shows the monthly distribution of the stratospheric potential energy obtained from SABER satellite data. From 2013 to 2015 and from 2016 to 2018, a clear QBO in the stratospheric potential energy can be observed. However, this phenomenon was interrupted between 2015 and 2016, likely associated with the El Niño event that occurred during that period [19,20]. During the QBO cycle, alternating layers of easterlies and westerlies appear in the stratosphere, with each phase lasting approximately 1 to 2 years. The easterly phase refers to periods when the easterly regions in the stratosphere are stronger, while the westerly phase refers to periods when the westerly regions are more dominant. In the first half of 2014, the QBO was in the westerly phase, resulting in increased energy. In the second half of 2014, the QBO transitioned to the easterly phase, leading to a decrease in energy.

Figure 9 shows the Gaussian fitting parameters for the zonal momentum flux in the stratosphere, calculated for each season over six years of data analysis. Since both the peak value and the FWHM jointly determine the total momentum flux, the central phase speed of the Gaussian fit is not discussed further. As seen in Figure 9, the eastward peak values are consistently higher than the westward ones, with the highest value occurring in the summer of 2016, reaching 22.9 × 10⁻⁴ m²/s². Except for the autumn of 2014, the spring of 2015, the summer of 2016, and the winter of 2017, the FWHM for the eastward direction is larger than that for the westward direction, consistent with the results shown in Figure 5. As GWs propagate upward, potential energy is converted into kinetic energy, and the momentum flux describes the vertical transfer of momentum, typically induced by the vertical gradient of the wind field. To some extent, a decrease in potential energy is associated with an increase in momentum flux. The trends in the distribution of potential energy shown in Figure 8 are approximately opposite to the Gaussian-fitted peak parameters of zonal momentum flux in the stratosphere shown in Figure 9. For example, in the summer and autumn of 2014, the GW potential energy observed by the SABER satellite reached its highest levels, while the corresponding peak values of the GW momentum flux spectrum obtained from the radiosonde data at the same heights were the lowest, accompanied by the weakest total momentum flux. In the spring of 2016, the eastward fitted peak value of momentum flux was at its highest, with a relatively large total momentum flux, while the potential energy in the summer was at a lower level, due to strong westerly jet streams accompanying the upward propagation of GWs. By comparing these results, we find that the satellite data and the radiosonde data provide mutually corroborating insights into the characteristics of stratospheric GW activity in the tropical upper atmosphere. This further confirms the reliability and validity of the results.

8. Conclusions

Based on radiosonde data from Koror Island Station in the western Pacific region collected from 2013 to 2018, and supplemented by SABER satellite data, this study utilizes hodograph analysis and spectral analysis to examine the momentum flux spectra of IGWs in the upper atmosphere. The momentum flux spectra are discussed through Gaussian fitting, focusing on the analysis of the peak value, FWHM, and Gaussian central phase speed of the fitting function. The primary focus is on the characteristics and seasonal variations of the zonal momentum flux spectra.

The analysis of the Gaussian fitting parameters and their seasonal variations in the momentum flux spectra reveals the following: (1) In the troposphere, the Gaussian peak values for eastward-propagating waves are higher than those for westward-propagating waves, with this difference being more pronounced in the stratosphere. This is due to a strong easterly jet stream in the troposphere during winter, which leads to the absorption of westward-propagating GWs through critical layer filtering. (2) The FWHM in the stratosphere is larger than in the troposphere. The minimum FWHM values in the troposphere for the zonal direction occur in June and July, which are periods when the background zonal wind is weakest. This broader distribution in the stratosphere results from the fact that most upward-propagating waves originate in the troposphere. As these waves propagate from the troposphere to the stratosphere, propagation effects, filtering, and interactions among waves cause the spectrum to broaden, resulting in a wider range of distribution characteristics. (3) The central phase speed in the stratosphere is higher than in the troposphere due to the Doppler effect, which increases the phase speed of GWs propagating against the wind and absorbs low-frequency components when propagating with the wind, allowing only higher-frequency components to reach the stratosphere. (4) The peak energy of the eastward momentum spectrum in the troposphere is near zero phase speed, indicating that during this period, the primary excitation source generates GWs with low phase speeds that carry momentum upward. (5) Except for the eastward direction, the peak values in the stratosphere are significantly lower than those in the troposphere across all months. The differences in FWHM between the two altitude ranges are relatively small, indicating that the total momentum flux in the stratosphere is smaller than in the troposphere. This is due to the fact that the jet stream absorbs or reflects a substantial portion of the upward-propagating GWs, and some of the GWs dissipate near the tropopause, depositing their momentum into the background atmosphere. The analysis of the momentum flux spectra indicates that in the lower atmosphere of the tropical western Pacific region, the tropospheric jet stream is a key factor in determining the characteristics of wave activity and dynamics.

The temporal distribution of the stratospheric potential energy obtained from SABER satellite data, combined with the Gaussian fitting parameters of the zonal momentum flux spectra in the stratosphere, reveals the following: (1) The stratospheric potential energy exhibits a clear QBO phenomenon, which was interrupted between 2015 and 2016, likely related to the El Niño event during that period. During the westerly phase of the QBO, the potential energy increases; during the easterly phase, the potential energy decreases. (2) As GWs propagate upward, potential energy is converted into kinetic energy. To some extent, a decrease in potential energy is associated with an increase in momentum flux, with the trend in potential energy distribution being approximately opposite to the trend in the eastward peak parameter. The GW potential energy characteristics observed by the SABER satellite correspond well with the GW momentum flux characteristics observed by radiosonde data. This correspondence from different observational methods demonstrates the inverse relationship between GW potential energy and momentum flux, highlighting the processes of internal energy and momentum transfer and transport.

In conclusion, the analysis of the IGW momentum flux spectra over the tropical western Pacific region, using radiosonde data from Koror Island, significantly supplements existing research in low-latitude tropical areas. This study demonstrates the potential of leveraging long-term radiosonde data and the widespread distribution of observation sites to gradually establish and refine an observation-based parameterization model for GW source spectra. Such developments will contribute to the advancement of more observationally consistent GCMs.