Vertical Wavelengths of Downward Phase Propagating Gravity Waves Determined by Vertical Fluctuation of Idealized Radiosonde Balloons

Tingting Qian aInstitute of Tibetan Plateau Meteorology, Chinese Academy of Meteorological Sciences, Beijing, China

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Junhong Wei bSchool of Atmospheric Sciences, Sun Yat-sen University, Zhuhai, China
cGuangdong Province Key Laboratory for Climate Change and Natural Disaster Studies, Sun Yat-sen University, Zhuhai, China
dSouthern Marine Science and Engineering Guangdong Laboratory (Zhuhai), Zhuhai, China

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Yongqiang Sun eNOAA/Geophysical Fluid Dynamics Laboratory, Princeton, New Jersey

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Yinghui Lu fSchool of Atmospheric Sciences, Nanjing University, Nanjing, China

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James H. Ruppert Jr. gSchool of Meteorology, University of Oklahoma, Norman, Oklahoma

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Abstract

This paper investigates the limitation in calculating the vertical wavelength of downward phase propagating gravity waves from the vertical fluctuation of idealized radiosonde balloons in a homogeneous background environment. The wave signals are artificially observed by an idealized weather balloon with a constant ascent rate. The apparent vertical wavelengths obtained from the moving radiosonde balloon are compared to the true vertical wavelength obtained from the dispersion relation, both in the no-wind case and in the constant-zonal-flow case. The node method and FFT method are employed to calculate the apparent vertical wavelength from the sounding profile. The difference between the node apparent vertical wavelength and the true vertical wavelength is attributed to the fact that the ascent rate of the balloon and the downward phase speed induce a strong Doppler-shifting bias on the apparent vertical wavelength from the observation records. The difference between the FFT apparent vertical wavelength and the true vertical wavelength includes both the Doppler-shifting bias and the mathematical bias. The extent to which the apparent vertical wavelength is reliable is discussed. The Coriolis parameter has negligible effects on the comparison between the true vertical wavelength and the apparent one.

Significance Statement

The purpose of this study is to discuss the Doppler-shifting bias induced by the ascent rate of radiosonde balloon when measuring the apparent vertical wavelengths of downward phase propagating gravity waves from the vertical fluctuation of idealized radiosonde balloons. This is an easily omitted problem. However, it can dramatically affect the gravity wave diagnosis when the ascent rate profile is treated as a quasi-instantaneous data. Further, such uncertainty could lead to remarkable errors in other derived wave propagating properties (e.g., phase velocity, which is the key input parameter in gravity wave parameterization).

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Tingting Qian, qiantt@cma.cn

Abstract

This paper investigates the limitation in calculating the vertical wavelength of downward phase propagating gravity waves from the vertical fluctuation of idealized radiosonde balloons in a homogeneous background environment. The wave signals are artificially observed by an idealized weather balloon with a constant ascent rate. The apparent vertical wavelengths obtained from the moving radiosonde balloon are compared to the true vertical wavelength obtained from the dispersion relation, both in the no-wind case and in the constant-zonal-flow case. The node method and FFT method are employed to calculate the apparent vertical wavelength from the sounding profile. The difference between the node apparent vertical wavelength and the true vertical wavelength is attributed to the fact that the ascent rate of the balloon and the downward phase speed induce a strong Doppler-shifting bias on the apparent vertical wavelength from the observation records. The difference between the FFT apparent vertical wavelength and the true vertical wavelength includes both the Doppler-shifting bias and the mathematical bias. The extent to which the apparent vertical wavelength is reliable is discussed. The Coriolis parameter has negligible effects on the comparison between the true vertical wavelength and the apparent one.

Significance Statement

The purpose of this study is to discuss the Doppler-shifting bias induced by the ascent rate of radiosonde balloon when measuring the apparent vertical wavelengths of downward phase propagating gravity waves from the vertical fluctuation of idealized radiosonde balloons. This is an easily omitted problem. However, it can dramatically affect the gravity wave diagnosis when the ascent rate profile is treated as a quasi-instantaneous data. Further, such uncertainty could lead to remarkable errors in other derived wave propagating properties (e.g., phase velocity, which is the key input parameter in gravity wave parameterization).

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Tingting Qian, qiantt@cma.cn

1. Introduction

Gravity wave dynamics and effects in the middle atmosphere (reviewed by Fritts and Alexander 2003) have been widely documented for several decades. The sources of gravity waves include topography (Smith 1980; Epifanio and Durran 2001; Epifanio and Rotunno 2005; Epifanio and Qian 2008), sea breezes (Rotunno 1983; Qian et al. 2009, 2012), convection (Alexander et al. 1995; Lane et al. 2001; Qian et al. 2020), jets (Zhang 2004; Wang and Zhang 2007; Wang et al. 2009; Wei and Zhang 2014; Wei et al. 2016), fronts (Snyder et al. 1993; Griffiths and Reeder 1996), and shear instability (Bühler et al. 1999; Bühler and McIntyre 1999). Among the gravity wave studies, the downward phase (upward energy) propagating gravity wave plays an important role in transferring the energy from the synoptic systems in the low level to the large-scale systems in the high level. Therefore, the downward phase (upward energy) propagating gravity wave achieved a considerable focus in the middle atmosphere.

Although several studies (Zhang et al. 2004, 2015; Dörnbrack et al. 2017; Vosper and Ross 2020) have pointed out the uncertainties in retrieving gravity wave characteristics from different measurements, a considerable number of observational studies have been performed to expand our knowledge of gravity wave characteristics. The measurement techniques include the remote sensing from space (Preusse et al. 2002; Ern et al. 2017; Jiang et al. 2019), radiosonde sounding observation (Zhang and Yi 2007; Geller and Gong 2010; Gong and Geller 2010), radar observation (Nakamura et al. 1993; Sato 1994), lidar observation (Whiteway and Carswell 1995; Hecht et al. 2018), rocket soundings observation (Eckermann et al. 1995), and aircraft observation (Zhang et al. 2015; Fritts et al. 2018). Among them, radiosonde observation is one of the most popular techniques to describe the vertically propagating gravity wave, with global coverage and at least twice-daily observation frequency. It is generally accepted that the radiosonde sounding can capture the low-frequency gravity waves.

In recent decades, some studies pointed out that convectively generated gravity wave can be recorded by the fluctuations in the ascent rate of the radiosonde sounding (Lane et al. 2003; Geller and Gong 2010; Gong and Geller 2010). Lane et al. (2003) pointed out that the apparent vertical wavelength of convective gravity wave from the idealized radiosonde is different from the corresponding theoretical vertical wavelength when the background wind is different from the “source” background wind. They argued that the difference was caused by the tilted radiosonde trajectory deviating from the gravity wave propagation direction (paragraphs 3 and 4 in section 6 of Lane et al. 2003). Geller and Gong (2010) described that the vertical fluctuation energy (VE) is a good indicator of convectively forced high-frequency gravity waves. Gong and Geller (2010) further indicated that the apparent dominant vertical wavelength varies with the VE.

Unlike the gravity waves excited through flow over orography and/or jet imbalance, convectively generated gravity waves are characterized by a much wider range of phase speeds, frequencies, and vertical and horizontal wavenumbers (Song et al. 2003; Wei et al. 2016). The high-frequency waves, in particular, have shown a very close correspondence with deep convective clouds, of which the wave periods (minutes to hours) are on the same order of the radiosonde ascent time. Considering the relative speed between the ascent balloon and the vertically moving gravity wave phase line, a Doppler-shifting bias is introduced in gravity wave diagnosis even though the result meets the two criteria requirements previously published in literature (Lane et al. 2003; Gong and Geller 2010; details in section 2c). Furthermore, the vertical wavelength of gravity wave is an important quantity to be measured from the ascending radiosonde. Any significant uncertainty in its estimation could lead to remarkable errors in other derived wave propagating properties (e.g., phase velocity which is the key input parameter in gravity wave parameterization). Therefore, the current study seeks to explore how large the differences between the true vertical wavelength and the apparent vertical wavelength from the idealized radiosonde balloon are for different waves/wind conditions and whether the criteria previously published in literature give a good estimate of the limitations. Section 2 describes the basic wave equations, the idealized radiosonde assumption, three different methods in calculating the vertical wavelength, and the two criteria. Section 3 compares the vertical wavelength from different methods in the no-wind cases. The vertical wavelengths of gravity wave excited in the constant basic-state wind environment are compared with three methods in section 4. A discussion of Coriolis effects on measuring the inertia–gravity waves from sounding profile is presented in section 5. The conclusion is presented in section 6.

2. Basic wave equation and experiment design

The basic gravity wave assumption and experiment design are discussed in this section, with an emphasis on the introduction of three different methodologies in calculating the vertical wavelength. All comparisons for the remaining sections are based on the resulting vertical wavelengths.

a. Basic equations

In our simplified experiment, we consider an idealized downward phase (upward energy) propagating gravity wave with given horizontal wavelength and ground-based frequency in a uniform background state, in which the static stability N and the basic-state wind speed U are both constant. A hypothetical balloon with a specified ascent rate (5 m s−1 in our calculation) is released in the air. We assume that the artificial balloon takes 30 min to ascend 9 km, and it records the wave velocity every second (the vertical grid ∼5 m) during its ascent.

Considering the wave band selection of radiosonde mentioned in Alexander (1998), the observed vertical wavelength is limited by the balloon ascent height (9 km in this study), and the term associated with the atmospheric density scale height H is a negligible factor (Fritts and Alexander 2003) for the waves with the vertical wavelength less than 9 km (i.e., m2 ≫ 1/4H2, where m is the vertical wavenumber). Here we adopt the simplified wave equation without the term associated with the atmospheric density scale height H. The idealized plane wave is defined as
w(x,y,z,t)=w0exp[i(kx+ly+mzω^t)],
where w0 is the amplitude of vertical velocity, k = 2π/λx is the zonal wavenumber with λx the zonal wavelength, l = 2π/λy is the meridional wavenumber with λy the meridional wavelength, m = 2π/λz is the vertical wavenumber with λz the vertical wavelength, and the intrinsic frequency is ω^=ωUk, with ω = 2π/T being the ground-based wave frequency, T the wave period, and U the constant background wind. For simplicity, we assume l = 0. We also exclude the critical level and the vertically trapped waves. The background wind and the “source” background wind (Lane et al. 2003) are the same in our study, i.e., the background is constant from the surface to the top.
The simplified dispersion relation is given by
m2=N2ω^2ω^2f2k2.
Since the wave of interest is the downward phase propagating wave, we set k > 0, m < 0 (m2 > 0), and ω^>0 to restrict the properties of the reference wave. This implies that the gravity waves with constant phase lines tilting eastward with increasing height are employed. The phase line propagates eastward and downward, and the energy propagates upward in the example wave. The details are described in Holton (2004, his chapter 7.4).

In our experiment, we assume f = 0 in sections 3 and 4. The Earth rotation (Coriolis) effect is discussed in section 5. Here we focus on the upper troposphere and lower stratosphere (N = 0.02 s−1), which starts at a specified height of 15 km. The start height only adds a constant to the phase of the idealized plane wave and will not affect the conclusion. In terms of dimensional quantities, we set the atmosphere stability to be N = 0.02 s−1. The background wind is held to be constant with height. The background wind is selected to be −20, −5, 0, 5, and 20 m s−1. We assume that the wave of interest in the current study ranges from low ground-based frequency wave to high ground-based frequency wave. The low ground-based frequency varies from 5.0 × 10−5 s−1 (period of ∼35 h) to 1.0 × 10−3 s−1 (∼1.7 h). The high ground-based frequency is taken from 1.0 × 10−3 s−1 (∼1.7 h) to 7.0 × 10−3 s−1 (∼15 min) described as convectively generated gravity wave frequency in Lane et al. (2001, 2003). The horizontal wavelength ranges from 10 to 1000 km for low ground-based frequency waves and from 10 to 110 km for high ground-based frequency waves, in which the choice of horizontal wavelength is based on size of meso- and large-scale systems (section 1.3 in Holton 2004) and convection (Lane et al. 2001, 2003). The maximum horizontal phase speed is less than 30 m s−1 in all calculations. The maximum vertical phase speed of low (high)-frequency waves is less than 0.09 m s−1 (4 m s−1).

b. Methodology

The methodology used to calculate the vertical wavelength is illustrated in this section. The apparent vertical wavelength is extracted from the vertical profile of vertical velocity. A low ground-based frequency wave and a high ground-based frequency wave are taken as examples to explain the methodology.

Figure 1 shows the xz cross section of the vertical velocity of the example wave in the no-wind case. The vertical velocity distribution of the low ground-based frequency gravity wave with ω = 1.0 × 10−4 s−1 and λx = 900 km is shown in Fig. 1a. The vertical velocity distribution of the high ground-based frequency gravity wave with ω = 3.0 × 10−3 s−1 and λx = 40 km is shown in Fig. 1b.

Fig. 1.
Fig. 1.

The xz cross section of the vertical velocity (color shading, contour interval = 0.2 m s−1; red shading is positive; blue shading is negative) distribution for the (a) gravity wave with low ground-based frequency 0.0001 s−1 and horizontal wavelength 900 km and (b) gravity wave with high ground-based frequency 0.003 s−1 and horizontal wavelength 40 km in the no-wind case.

Citation: Journal of Atmospheric and Oceanic Technology 39, 9; 10.1175/JTECH-D-21-0137.1

With given ground-based frequency and horizontal wavelength, three different vertical wavelength results calculated from three different methods are compared in our study: the true vertical wavelength, the node apparent vertical wavelength, and the fast Fourier transform (FFT) apparent vertical wavelength. The true vertical wavelength is obtained directly from the dispersion relation [Eq. (2)] with the given horizontal wavelength and ground-based frequency. Figures 2a and 2c depict the vertical velocity variation with height for the low ground-based frequency wave (Fig. 1a) and the high ground-based frequency wave (Fig. 1b) at x = 0 km and t = 0 s. True vertical wavelength for low-frequency wave and high-frequency wave is ∼4.5 and ∼6.0 km, respectively.

Fig. 2.
Fig. 2.

The vertical profile of vertical velocity for the example gravity wave. The true vertical profile of vertical velocity for (a) the wave with low ground-based frequency 0.0001 s−1 and horizontal wavelength 900 km and (b) the wave with high ground-based frequency 0.003 s−1 and horizontal wavelength 40 km at x = 0 km and t = 0 s. The corresponding observed (apparent) vertical profile for (c) the wave shown in Fig. 1a and (d) the wave shown in Fig. 1b.

Citation: Journal of Atmospheric and Oceanic Technology 39, 9; 10.1175/JTECH-D-21-0137.1

The corresponding apparent (observed) vertical velocity variations with height (Figs. 2b,d) are derived by substituting the balloon ascent path into Eq. (1). The balloon path is defined as
x(t)=x(tΔt)+UΔt+δssinα,
z(t)=z(tΔt)+WΔt+δscosα,
where Δt = 1 s, the balloon displaced a distance δs = cos[(Ncos α)t] along the phase line, α is the angle between the tilted phase line and the vertical. The first term represents the position of balloon at time t − Δt. The second term is the movement in horizontal distance or ascent height of the balloon in Δt. The third term is the balloon oscillation associated with the parcel fluctuation (section 7.4.1 in Holton 2004).

The node apparent vertical wavelength is obtained directly from the vertical velocity profile observed from the idealized radiosonde balloon, i.e., the vertical distance between the two nearest peaks (with the same sign) in the observed profile (Figs. 2b,d). The difference between the node apparent vertical wavelength and the true one is totally caused by the Doppler-shifting bias associated with ascent rate of balloon.

Since the FFT method is one of the common tools used in extracting the dominant wave from the observed sounding profile, the apparent vertical wavelength obtained with FFT method is also exhibited in the current study. The FFT apparent vertical wavelength is calculated from the power spectrum of the observed vertical velocity profile (Figs. 2b,d) by using the one-dimensional FFT method, then the solution of vertical wavelength is assumed to be the spectral peak. This power spectrum method has been applied by the past studies to diagnose gravity wave characteristics from radiosondes (Wang et al. 2005; Gong and Geller 2010). The difference between the FFT apparent vertical wavelength and the true one is caused by both the Doppler-shifting bias associated with the ascent rate of balloon and the bias associated with FFT method.

It is obvious that the vertical wavelength observed from the ascent balloon (Figs. 2b,d) is different from the true one (Figs. 2a,c), although this effect is much smaller for the low ground-based frequency wave. The true vertical wavelength for the low ground-based frequency wave is ∼4500 m, the node apparent vertical wavelength is ∼4444 m, and the FFT apparent vertical wavelength is 4505 m. The ratio of the node/FFT apparent vertical wavelength and the true vertical wavelength for the low-frequency gravity wave is ∼0.99/∼1.001. The true vertical wavelength for the high ground-based frequency wave is ∼6069 m, the node apparent vertical wavelength is ∼3927 m and the FFT apparent vertical wavelength is ∼4481 m. The ratio of the node/FFT apparent vertical wavelength and the true vertical wavelength for this high-frequency gravity wave is ∼0.647/∼0.738. This shortening with node method is induced by the Doppler-shifting bias associated with the ascent rate of balloon and the descent rate of phase line for the downward phase propagating waves. The difference with FFT method is related to both the Doppler-shifting bias and the FFT bias.

c. Criteria

As mentioned in the previous studies (Lane et al. 2003; Gong and Geller 2010), two conditions need to be met so that a radiosonde sounding can be treated as an instantaneous vertical profile when diagnosing gravity waves:
|λZUλxW¯|1
and
|ωmW¯|1,
where W¯ is the mean ascent rate of the balloon. The first criterion requires that the horizontal wavelength (λx) is much longer than the horizontal trajectory λZU/W¯ of the radiosonde, so that the radiosonde sounding profile can be treated as a quasi-instantaneous vertical profile. The second criterion requires that the ascent rate of the balloon (W¯) is much larger than the ground-based vertical phase velocity of gravity wave (ω/m). In reality, the above-mentioned two criteria are calculated based on the observationally diagnosed vertical wavelength. This research will examine how reliable these two criteria are, and quantify its validity in the vertical wavelength diagnosis.

Note that the smaller criterion threshold can limit the apparent vertical wavelength to a smaller bias. The cost of smaller criterion threshold is the exclusion of some reasonable observation result with small bias. Since most of the vertical fluctuations of radiosonde are used to extract the high ground-based frequency waves in the observation study, and the criterion threshold 0.1 works well with them, the criterion threshold value 0.1 is employed in the whole work.

3. The no-wind cases

As mentioned in section 1, a considerable amount of past research uses radiosondes to explore the features of low-frequency waves (Guest et al. 2000; Wang et al. 2005) and high-frequency waves (Lane et al. 2003; Gong and Geller 2010). The vertical wavelength is one of the most common estimates of gravity wave features obtained from the observed sounding profile. In this section, we examine the apparent vertical wavelength of the internal gravity waves obtained from the hypothetical radiosonde in the no-wind atmosphere, with comparisons to the true vertical wavelength obtained from the dispersion relation.

a. Low ground-based frequency gravity waves

The frequency range of the low ground-based frequency gravity wave at the equator is set to be 5.0 × 10−5–1.0 × 10−3 s−1, which is larger than f (0 s−1) and less than the buoyancy frequency N (0.02 s−1) in the upper troposphere and lower stratosphere. Due to the limitation of radiosonde balloon ascent height between the upper troposphere and the lower stratosphere, we set the longest vertical wavelength of the observed gravity wave to be 9 km.

An analogous calculation for the vertical wavelength of low ground-based frequency waves is shown in Fig. 3. The true vertical wavelength, the node apparent vertical wavelength, and the FFT apparent vertical wavelength are obtained individually through three different methods. As expected from the dispersion relation, the vertical wavelength increases with both horizontal wavelength and the wave frequency (Fig. 3a). The node apparent vertical wavelengths (Fig. 3b) agree well with the truth for relatively shorter vertical wavelengths, but are slightly shortened.

Fig. 3.
Fig. 3.

The distribution of (a) true vertical wavelength (color shading, contour interval = 1 km), (b) node apparent vertical wavelength (color shading, contour interval = 1 km), and (c) FFT apparent vertical wavelength (color shading, contour interval = 1 km) of the low ground-based frequency waves varying with ground-based frequency between 5.0 × 10−5 and 1.0 × 10−3 s−1 and horizontal wavelength in the zero background wind cases. (a) The corresponding second criterion value 0.1 (i.e., |ω/mW¯|) is included with black solid line calculated from true wave result, red dotted line calculated from node apparent result, and blue dashed line calculated from FFT apparent result. (b) The second criterion value less than 1 (black solid lines, contour interval = 0.1) calculated with the node apparent vertical wavelength. (c) The second criterion value less than 1 (black solid lines, contour interval = 0.1) calculated with the FFT apparent vertical wavelength.

Citation: Journal of Atmospheric and Oceanic Technology 39, 9; 10.1175/JTECH-D-21-0137.1

Figure 4 shows the ratio of the apparent wavelength to the true wavelength, which is used to quantify the deviation of observation results. The deviation of node apparent vertical wavelength from the true wavelength is less than 20% (Fig. 4a), which is totally caused by the Doppler-shifting bias associated ascent radiosonde balloon.

Fig. 4.
Fig. 4.

The ratio (color shading, contour interval = 0.1) of (a) node and (b) FFT apparent vertical wavelength to true wavelength for the low ground-based frequency gravity waves. The distribution of the second criterion value 0.1 (i.e., |ω/mW¯|, black lines) calculated from (a) node apparent result and (b) FFT apparent result.

Citation: Journal of Atmospheric and Oceanic Technology 39, 9; 10.1175/JTECH-D-21-0137.1

Due to the discontinuity characteristics of the FFT algorithm, the FFT method introduces the discrete constraint to the results, and it does not fully resolve the distribution of relatively long vertical wavelength (Fig. 3c). The deviation of FFT apparent vertical wavelength from the true one is less than 50% (Fig. 4b), which includes the bias caused by both the Doppler-shifting and the FFT algorithm. The vertical wavelength solution for the low-frequency gravity wave is therefore sensitive to the methodology, especially for the wave with vertical wavelength greater than half of the ascent height.

The extent to which the criteria can guarantee the reliability result is examined. In this section for U = 0 cases, we only consider the second condition. Figures 3 and 4 also show the corresponding second criterion value less than 0.1, which is considered to meet Eq. (6). The criterion can confirm the validity of node apparent vertical wavelength with bias less than 10% (Fig. 4a). For the FFT apparent results, the criterion value less than 0.1 does not accurately indicate the significant bias of the longer vertical wavelength with horizontal wavelength > 350 km and frequency between ∼1.2 × 10−4 and ∼3.7 × 10−4 s−1 (Figs. 3c and 4b). Even with the small criterion, the maximum bias for FFT apparent vertical wavelength can reach ∼50%.

b. High ground-based frequency gravity waves

Convection is regarded as an important source of high-frequency downward phase (upward energy) propagating gravity waves. Besides the two frequencies of convectively generated gravity wave in Lane et al. (2003), here we investigate the gravity waves with the ground-based frequency ranging from 1.0 × 10−3 to 7.0 × 10−3 s−1 and the true vertical wavelengths less than 9 km.

Figure 5 shows the vertical wavelengths of the idealized high ground-based frequency waves obtained with the three different methods. The observational diagnosis of the vertical wavelength from radiosonde (Figs. 5b,c) strongly deviates from the truth (Fig. 5a), i.e., the vertical wavelength longer than 2 km is dramatically shortened in the observational results. This is because the downward phase speed of the high ground-based frequency wave is much faster than the phase speed of low ground-based frequency wave. In the same altitude, the ascending balloon sees more downward-moving phase lines of high ground-based frequency wave than it sees the phase lines of low ground-based frequency wave, which leads to an artificially shorter vertical wavelength than the true result.

Fig. 5.
Fig. 5.

As in Fig. 3, but for waves with high ground-based frequency between 1.0 × 10−3 and 7.0 × 10−3 s−1 in the zero background wind cases. The contour interval of the second criterion value is 0.2.

Citation: Journal of Atmospheric and Oceanic Technology 39, 9; 10.1175/JTECH-D-21-0137.1

The discrete feature of the FFT method further introduces another bias to the apparent vertical wavelength calculation (Fig. 5c). The longer the vertical wavelength is, the more obvious the bias is.

The ratio of the apparent wavelength to the true wavelength for the high ground-based frequency wave is shown in Fig. 6. The maximum deviation of node apparent vertical wavelength from the true wavelength increased to 70% (Fig. 6a), which is totally caused by the Doppler-shifting bias associated ascent radiosonde balloon. The maximum deviation of FFT apparent vertical wavelength from the true wavelength is also 70% (Fig. 6b), which includes both the Doppler-shifting bias and the FFT algorithm effect.

Fig. 6.
Fig. 6.

As in Fig. 4, but for waves with high ground-based frequency between 1.0 × 10−3 and 7.0 × 10−3 s−1 in the cases without background wind.

Citation: Journal of Atmospheric and Oceanic Technology 39, 9; 10.1175/JTECH-D-21-0137.1

The second criterion for the high ground-based frequency wave is depicted in both Figs. 5 and 6. The requirement of small criterion (<0.1) can accurately indicate the reliability of the vertical wavelength of the waves with weakly high ground-based frequency and short (horizontal and vertical) wavelength. The apparent vertical wavelength (both with node method and FFT method) satisfying the criterion requirement is conditionally less than 3 km (Fig. 5). The corresponding deviation of node apparent result from the true result is less than 10% (Fig. 6a). The deviation of FFT apparent result from the true result is less than 20% with the criterion requirement.

4. Background wind effects

In this section, limitation of extracting the vertical wavelength from the observational profile is examined in a uniform background flow. To compare the methods under the simplified condition, we exclude the critical level and the vertically trapped waves. Only waves with k > 0, m < 0, and ω^>0 are considered.

a. Low ground-based frequency waves in nonzero wind condition

The estimates of the vertical wavelength for low ground-based frequency wave in nonzero wind condition are examined in this subsection. Figure 7 shows the vertical wavelength variation with ground-based frequency and horizontal wavelength in different background flows. Figure 8 depicts the ratio of the apparent vertical wavelength to the true vertical wavelength, and the overlays are the corresponding criteria |λZU/λxW¯| and |ω/mW¯|. Both criteria less than 0.1 represent the results satisfying Eqs. (5) and (6).

Fig. 7.
Fig. 7.

The distribution of (a1)–(d1) true vertical wavelength (color shading, contour interval = 1 km), (a2)–(d2) node apparent vertical wavelength (color shading, contour interval = 1 km), and (a3)–(d3) FFT apparent vertical wavelength (color shading, contour interval = 1 km) of the low ground-based frequency waves varying with wave ground-based frequency and horizontal wavelength. The background winds U are (a1)–(a3) −20, (b1)–(b3) −5, (c1)–(c3) 5, and (d1)–(d3) 20 m s−1. The corresponding first criterion (i.e., |λZU/λxW¯|; black solid lines, contour interval = 0.2, starting from 0.1) and the second criterion (i.e., |ω/mW¯|; red solid lines, contour interval = 0.2, starting from 0.1) are overlaid.

Citation: Journal of Atmospheric and Oceanic Technology 39, 9; 10.1175/JTECH-D-21-0137.1

Fig. 8.
Fig. 8.

The ratio (color shading, contour interval = 0.1) of (a)–(d) the node apparent vertical wavelength to the true wavelength and (e)–(h) the FFT apparent vertical wavelength to the true wavelength varying with ground-based frequency and horizontal wavelength for the low ground-based frequency waves with the background winds U = (a),(e) −20, (b),(f) −5, (c),(g) 5, and (d),(h) 20 m s−1. The distribution of the corresponding first criterion value |λZU/λxW¯|=0.1 (black solid line) and the second criterion value |ω/mW¯|=0.1 (red solid line) calculated from (a)–(d) node apparent result and (e)–(h) FFT apparent result.

Citation: Journal of Atmospheric and Oceanic Technology 39, 9; 10.1175/JTECH-D-21-0137.1

Adding the background wind U = −20 m s−1 in the regime of the given ground-based low frequency and the given horizontal wavelength, the apparent vertical wavelength of the downward phase propagating waves (Figs. 7a2,a3) obviously deviates from the true result (Fig. 7a1). According to the two criteria, only waves with the ground-based frequency less 2 × 10−4 s−1, horizontal wavelength longer than ∼200 km, and vertical wavelength 7–8 km (Figs. 7a1–a3) satisfy the two criteria requirement (≤0.1). With criteria less than 0.1, the bias of the apparent vertical wavelength from the true vertical wavelength is less than 30% with node method (Fig. 8a) and 50% with FFT method (Fig. 8e).

Adding a weak negative background wind (Figs. 7b1–b3) or a positive background wind (Figs. 7c1–c3,d1–d3), the true solutions and the node apparent solutions of vertical wavelength (less than half observation altitude) mostly satisfy the two criteria and the deviation is less than 10% (Figs. 8c,d). For the results with FFT method, the deviation of the apparent vertical wavelength from the true result can reach 90%. The criteria cannot completely exclude the evident bias (more than 20%) of the long vertical wavelength for the waves with long horizontal wavelength (Figs. 8f–h).

In the analyzed range of ground-based frequency and horizontal wavelength, for the downward phase propagating waves with low ground-based frequency, waves excited in the strong negative background wind cannot be accurately extracted from the sounding profile. For the low ground-based frequency waves excited in a weak negative background wind or a positive background wind, the criteria can limit the bias of node apparent vertical wavelength to be less than 10%, but the criteria do not work well with the FFT apparent vertical wavelength.

b. High ground-based frequency waves in nonzero wind condition

Figure 9 shows the vertical wavelength comparison of the true solution, node solution, and FFT solution for the high ground-based frequency wave in the background flow. When the strong negative background wind is −20 m s−1, the horizontal wavelengths of downward phase propagating gravity waves are confined to 10–50 km (Fig. 9a1) in the regime of the given ground-based frequency and the given horizontal wavelength. The apparent vertical wavelength is dramatically shortened (Figs. 9a2,a3). The maximum deviation of the apparent vertical wavelength from the true vertical wavelength can reach ∼90% (Figs. 10a,e). The apparent vertical wavelength does not satisfy the criteria (5) and (6) simultaneously.

Fig. 9.
Fig. 9.

As in Fig. 7, but for waves with high ground-based frequency between 1.0 × 10−3 and 7.0 × 10−3 s−1 in the cases with background wind.

Citation: Journal of Atmospheric and Oceanic Technology 39, 9; 10.1175/JTECH-D-21-0137.1

Fig. 10.
Fig. 10.

As in Fig. 8, but for waves with high ground-based frequency between 1.0 × 10−3 and 7.0 × 10−3 s−1 in the cases with background wind.

Citation: Journal of Atmospheric and Oceanic Technology 39, 9; 10.1175/JTECH-D-21-0137.1

When the negative background flow decreases to −5 m s−1 (Figs. 9b1–b3), the apparent vertical wavelengths are underestimated (Figs. 9b2,b3). The apparent vertical wavelengths meet the two criteria when horizontal wavelength is between 20 and 45 km and the ground-based frequency is less than 2 × 10−3 s−1 (Figs. 10b,f). The bias of the apparent vertical wavelength with criteria less than 0.1 is less than 10% for node method and 20% for FFT method.

When the background wind is positive (U = 5 or 20 m s−1, Figs. 9c1–c3,9d1–d3), the vertical wavelength less than 2 km can be well reproduced by the apparent vertical wavelength with node method and FFT method. The criteria can restrict the error less than 10% for both the node method and the FFT methods (Figs. 10c,g,d,h).

In the given ground-based frequency and horizontal wavelength, for the downward phase propagating waves with high ground-based frequency, few waves excited in the negative background wind can be accurately extracted from the sounding profile. The apparent vertical wavelength of high ground-based frequency waves excited in a positive background wind can be extracted from the sounding profile with bias less than 10%.

5. Discussion of Coriolis effects on extracting gravity wave

Most radiosonde observations take place outside the equatorial region. Under the Coriolis forcing effect, the horizontal velocity vectors rotate anticyclonically with height in Northern Hemisphere for waves with downward phase propagation [e.g., waves with k > 0, m < 0, l = 0, ω^>0, and f > 0, described in chapter 7.5 of Holton (2004)]. The Coriolis effects on low ground-based frequency waves are relatively large. It is therefore worthy to examine if such effects can change the Doppler shifting effects on extracting gravity wave from sounding profile.

Figure 11 shows the three vertical wavelengths obtained with different methods for the downward phase propagating waves at latitudes 30° and 45°N. For the waves with same ground-based frequency and horizontal wavelength, increasing Coriolis parameter f can slightly elongate the corresponding vertical wavelength (Figs. 3a and 11a1,b1), and exclude the lowest frequency waves from inertia–gravity wave (bottom part of each panel in Fig. 11). However, the Coriolis parameter f has negligible effects on comparison result between the true solution (Figs. 11a1,b1) and apparent solutions (Figs. 11a2,a3,b2,b3).

Fig. 11.
Fig. 11.

The distribution of (a1),(b1) true vertical wavelength (color shading, contour interval = 1 km), (a2),(b2) node apparent vertical wavelength (color shading, contour interval = 1 km), and (a3),(b3) FFT apparent vertical wavelength (color shading, contour interval = 1 km) of the low-frequency wave varying with ground-based frequency and horizontal wavelength and background wind U = 0 m s−1 at latitudes (a1)–(a3) 30° and (b1)–(b3) 45°N.

Citation: Journal of Atmospheric and Oceanic Technology 39, 9; 10.1175/JTECH-D-21-0137.1

6. Conclusions

A large body of past research has utilized the motion of weather balloons to estimate characteristics of atmospheric gravity waves. However, there has been relatively little study of the Doppler-shifting biases on the apparent vertical wavelength induced by the balloon’s ascent rate. Here we explore this issue using idealized downward phase propagating waves and the artificial radiosonde balloons. Three different methods are used to diagnose the vertical wavelength of gravity waves with given ground-based frequency and horizontal wavelength. Then, two criteria typically invoked to assess the validity of gravity wave diagnoses from vertical radiosonde sounding profiles are computed. The results from three methods are compared for the low ground-based frequency waves (with ground-based frequency between 5.0 × 10−5 and 1.0 × 10−3 s−1) and high ground-based frequency waves (with ground-based frequency between 1.0 × 10−3 and 7.0 × 10−3 s−1), and in both no-wind cases and background wind cases.

In the no-wind cases, the node method can reasonably capture the vertical wavelength of the low ground-based frequency gravity wave from the radiosonde observation. The second criterion value 0.1 limits the bias less than 10%. The FFT method exhibits the inaccurate apparent results for the waves with long vertical wavelength. The criterion cannot distinguish the significantly deformed long apparent vertical wavelength of the low ground-based frequency waves with long horizontal wavelength in the no-wind condition. The maximum difference between the FFT apparent vertical wavelength and the true vertical wavelength of low ground-based frequency waves can reach 50%.

In the no-wind cases, the apparent vertical wavelength of high ground-based frequency gravity waves from observations is much shorter than the true value, i.e., the vertical wavelength ≥ 3 km is dramatically shortened in the observational results. The strict criterion effectively qualifies the validity of apparent vertical wavelength calculation. The bias between the true vertical wavelength and apparent wavelength is 10% with the node method and 20% with the FFT method.

The apparent vertical wavelengths of gravity wave excited in the uniform background wind are also explored. For gravity waves with low ground-based frequency in a large negative background wind (−20 m s−1) and criteria less than 0.1, the maximum bias of the apparent vertical wavelength can reach 30% with node method and 50% with FFT method. When the background wind is a small negative value or positive values, the criteria can control the bias of node apparent vertical wavelength to be less than 10%. The criteria cannot completely exclude the evident deviation (more than 20%) of the long vertical wavelength for the waves with FFT method.

For the high ground-based frequency gravity waves in nonzero wind condition, few downward phase propagating waves in the negative background wind cases can be extracted from the sounding profile. For the high ground-based frequency waves excited in the positive background wind, the criteria can limit the deviation of the apparent vertical wavelength from the true vertical wavelength to be less than 10% with node method and FFT method.

In summary, the current research demonstrates the limitation in measuring vertical wavelengths of downward phase propagating gravity waves from the vertical fluctuation of radiosonde balloons. For the waves with high ground-based frequency, the criteria work well to limit the deviation of apparent vertical wavelength from the true wavelength by 10% with node method and 20% with FFT method. For the waves with low ground-based frequency, the criteria cannot readily distinguish the long vertical wavelength deformation with both the node method and the FFT method. The Coriolis parameter f is found to have negligible effects on vertical wavelength comparison between the true solution and apparent solutions.

Finally, it should be emphasized that we do not expect our simple calculation to provide an accurate solution to gravity wave extracted from the vertical profile of ascent rate. Most notably, the calculation misses the complex gravity wave envelope, varying background conditions, and varying ascent rate of hypothetical balloon. However, it is reasonable to expect that this result provides some insight into limitation of measuring vertical wavelength of gravity waves from the vertical fluctuation of idealized radiosonde balloon. At the very least, the current study discussed an easily omitted problem when extracting vertical wavelength from the vertical fluctuation of idealized radiosonde balloon, which has not been highlighted in the previous studies. Further work is also needed to obtain a suitable solution to accurately diagnose gravity wave characteristics from radiosonde observations.

Acknowledgments.

This research was funded by the CN National Research and Development Projects, Grant 2019YFC1505705; the Basic Research Fund of the Chinese Academy of Meteorological Sciences, Grant 2019Z008; the U.S. National Science Foundation, Grants AGS-1305798 and 1712290; the National Natural Science Foundation of China (Grant 42075005); Guangdong Province Key Laboratory for Climate Change and Natural Disaster Studies (Grant 2020B1212060025); the Innovation Group Project of Southern Marine Science and Engineering Guangdong Laboratory (Zhuhai) (Grant 311021001).

Data availability statement.

For this idealized work, data were not used, nor created for this research.

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  • Alexander, M. J., 1998: Interpretations of observed climatological patterns in stratospheric gravity wave variance. J. Geophys. Res., 103, 86278640, https://doi.org/10.1029/97JD03325.

    • Search Google Scholar
    • Export Citation
  • Alexander, M. J., J. R. Holton, and D. R. Durran, 1995: The gravity wave response above deep convection in a squall line simulation. J. Atmos. Sci., 52, 22122226, https://doi.org/10.1175/1520-0469(1995)052<2212:TGWRAD>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Bühler, O., and M. E. McIntyre, 1999: On shear-generated gravity waves that reach the mesosphere. Part II: Wave propagation. J. Atmos. Sci., 56, 37643773, https://doi.org/10.1175/1520-0469(1999)056<3764:OSGGWT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Bühler, O., M. E. McIntyre, and J. F. Scinocca, 1999: On shear-generated gravity waves that reach the mesosphere. Part I: Wave generation. J. Atmos. Sci., 56, 37493763, https://doi.org/10.1175/1520-0469(1999)056<3749:OSGGWT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Dörnbrack, A., S. Gisinger, and B. Kaifler, 2017: On the interpretation of gravity wave measurements by ground-based lidars. Atmosphere, 8, 49, https://doi.org/10.3390/atmos8030049.

    • Search Google Scholar
    • Export Citation
  • Eckermann, S. D., I. Hirota, and W. K. Hocking, 1995: Gravity wave and equatorial wave morphology of the stratosphere derived from long-term rocket soundings. Quart. J. Roy. Meteor. Soc., 121, 149186, https://doi.org/10.1002/qj.49712152108.

    • Search Google Scholar
    • Export Citation
  • Epifanio, C. C., and D. R. Durran, 2001: Three-dimensional effects in high-drag-state flows over long ridges. J. Atmos. Sci., 58, 10511065, https://doi.org/10.1175/1520-0469(2001)058<1051:TDEIHD>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Epifanio, C. C., and R. Rotunno, 2005: The dynamics of orographic wake formation in flows with upstream blocking. J. Atmos. Sci., 62, 31273150, https://doi.org/10.1175/JAS3523.1.

    • Search Google Scholar
    • Export Citation
  • Epifanio, C. C., and T. Qian, 2008: Wave–turbulence interactions in a breaking mountain wave. J. Atmos. Sci., 65, 31393158, https://doi.org/10.1175/2008JAS2517.1.

    • Search Google Scholar
    • Export Citation
  • Ern, M., L. Hoffmann, and P. Preusse, 2017: Directional gravity wave momentum fluxes in the stratosphere derived from high-resolution AIRS temperature data. Geophys. Res. Lett., 44, 475485, https://doi.org/10.1002/2016GL072007.

    • Search Google Scholar
    • Export Citation
  • Fritts, D. C., and M. J. Alexander, 2003: Gravity wave dynamics and effects in the middle atmosphere. Rev. Geophys., 41, 1003, https://doi.org/10.1029/2001RG000106.

    • Search Google Scholar
    • Export Citation
  • Fritts, D. C., and Coauthors, 2018: Large-amplitude mountain waves in the mesosphere accompanying weak cross-mountain flow during DEEPWAVE research flight RF22. J. Geophys. Res. Atmos., 123, 999210 022, https://doi.org/10.1029/2017JD028250.

    • Search Google Scholar
    • Export Citation
  • Geller, M. A., and J. Gong, 2010: Gravity wave kinetic, potential, and vertical fluctuation energies as indicators of different frequency gravity waves. J. Geophys. Res., 115, D11111, https://doi.org/10.1029/2009JD012266.

    • Search Google Scholar
    • Export Citation
  • Gong, J., and M. A. Geller, 2010: Vertical fluctuation energy in United States high vertical resolution radiosonde data as an indicator of convective gravity wave sources. J. Geophys. Res., 115, D11110, https://doi.org/10.1029/2009JD012265.

    • Search Google Scholar
    • Export Citation
  • Griffiths, M., and M. J. Reeder, 1996: Stratospheric inertia–gravity waves generated in a numerical model of frontogenesis. I: Model solutions. Quart. J. Roy. Meteor. Soc., 122, 11531174, https://doi.org/10.1002/qj.49712253307.

    • Search Google Scholar
    • Export Citation
  • Guest, F. M., M. J. Reeder, C. J. Marks, and D. J. Karoly, 2000: Inertia–gravity waves observed in the lower stratosphere over Macquarie Island. J. Atmos. Sci., 57, 737752, https://doi.org/10.1175/1520-0469(2000)057<0737:IGWOIT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Hecht, J. H., and Coauthors, 2018: Observations of the breakdown of mountain waves over the Andes Lidar Observatory at Cerro Pachon on 8/9 July 2012. J. Geophys. Res. Atmos., 123, 276299, https://doi.org/10.1002/2017JD027303.

    • Search Google Scholar
    • Export Citation
  • Holton, J. R., 2004: An Introduction to Dynamic Meteorology. 4th ed. Elsevier, 535 pp.

  • Jiang, Q., J. D. Doyle, S. D. Eckermann, and B. P. Williams, 2019: Stratospheric trailing gravity waves from New Zealand. J. Atmos. Sci., 76, 15651586, https://doi.org/10.1175/JAS-D-18-0290.1.

    • Search Google Scholar
    • Export Citation
  • Lane, T. P., M. J. Reeder, and T. L. Clark, 2001: Numerical modeling of gravity wave generation by deep tropical convection. J. Atmos. Sci., 58, 12491274, https://doi.org/10.1175/1520-0469(2001)058<1249:NMOGWG>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Lane, T. P., M. J. Reeder, and F. M. Guest, 2003: Convectively generated gravity waves observed from radiosonde data taken during MCTEX. Quart. J. Roy. Meteor. Soc., 129, 17311740, https://doi.org/10.1256/qj.02.196.

    • Search Google Scholar
    • Export Citation
  • Nakamura, T., T. Tsuda, S. Fukao, S. Kato, and R. A. Vincent, 1993: Comparison of the mesospheric gravity waves observed with the MU radar (35°N) and the Adelaide MF radar (35°S). Geophys. Res. Lett., 20, 803806, https://doi.org/10.1029/93GL00365.

    • Search Google Scholar
    • Export Citation
  • Preusse, P., A. Dörnbrack, S. D. Eckermann, M. Riese, B. Schaeler, J. T. Bacmeister, D. Broutman, and K. U. Grossmann, 2002: Space-based measurements of stratospheric mountain waves by CRISTA 1. Sensitivity, analysis method, and a case study. J. Geophys. Res., 107, 8178, https://doi.org/10.1029/2001JD000699.

    • Search Google Scholar
    • Export Citation
  • Qian, T., C. C. Epifanio, and F. Zhang, 2009: Linear theory calculations for the sea breeze in a background wind: The equatorial case. J. Atmos. Sci., 66, 17491763, https://doi.org/10.1175/2008JAS2851.1.

    • Search Google Scholar
    • Export Citation
  • Qian, T., C. C. Epifanio, and F. Zhang, 2012: Topographic effects on the tropical land and sea breeze. J. Atmos. Sci., 69, 130149, https://doi.org/10.1175/JAS-D-11-011.1.

    • Search Google Scholar
    • Export Citation
  • Qian, T., F. Zhang, J. Wei, J. He, and Y. Lu, 2020: Diurnal characteristics of gravity waves over the Tibetan Plateau in 2015 summer using 10-km downscaled simulations from WRF-EnKF regional reanalysis. Atmosphere, 11, 631, https://doi.org/10.3390/atmos11060631.

    • Search Google Scholar
    • Export Citation
  • Rotunno, R., 1983: On the linear theory of the land and sea breeze. J. Atmos. Sci., 40, 19992009, https://doi.org/10.1175/1520-0469(1983)040<1999:OTLTOT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Sato, K., 1994: A statistical study of the structure, saturation and sources of inertio-gravity waves in the lower stratosphere observed with the MU radar. J. Atmos. Terr. Phys., 56, 755774, https://doi.org/10.1016/0021-9169(94)90131-7.

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  • Fig. 1.

    The xz cross section of the vertical velocity (color shading, contour interval = 0.2 m s−1; red shading is positive; blue shading is negative) distribution for the (a) gravity wave with low ground-based frequency 0.0001 s−1 and horizontal wavelength 900 km and (b) gravity wave with high ground-based frequency 0.003 s−1 and horizontal wavelength 40 km in the no-wind case.

  • Fig. 2.

    The vertical profile of vertical velocity for the example gravity wave. The true vertical profile of vertical velocity for (a) the wave with low ground-based frequency 0.0001 s−1 and horizontal wavelength 900 km and (b) the wave with high ground-based frequency 0.003 s−1 and horizontal wavelength 40 km at x = 0 km and t = 0 s. The corresponding observed (apparent) vertical profile for (c) the wave shown in Fig. 1a and (d) the wave shown in Fig. 1b.

  • Fig. 3.

    The distribution of (a) true vertical wavelength (color shading, contour interval = 1 km), (b) node apparent vertical wavelength (color shading, contour interval = 1 km), and (c) FFT apparent vertical wavelength (color shading, contour interval = 1 km) of the low ground-based frequency waves varying with ground-based frequency between 5.0 × 10−5 and 1.0 × 10−3 s−1 and horizontal wavelength in the zero background wind cases. (a) The corresponding second criterion value 0.1 (i.e., |ω/mW¯|) is included with black solid line calculated from true wave result, red dotted line calculated from node apparent result, and blue dashed line calculated from FFT apparent result. (b) The second criterion value less than 1 (black solid lines, contour interval = 0.1) calculated with the node apparent vertical wavelength. (c) The second criterion value less than 1 (black solid lines, contour interval = 0.1) calculated with the FFT apparent vertical wavelength.

  • Fig. 4.

    The ratio (color shading, contour interval = 0.1) of (a) node and (b) FFT apparent vertical wavelength to true wavelength for the low ground-based frequency gravity waves. The distribution of the second criterion value 0.1 (i.e., |ω/mW¯|, black lines) calculated from (a) node apparent result and (b) FFT apparent result.

  • Fig. 5.

    As in Fig. 3, but for waves with high ground-based frequency between 1.0 × 10−3 and 7.0 × 10−3 s−1 in the zero background wind cases. The contour interval of the second criterion value is 0.2.

  • Fig. 6.

    As in Fig. 4, but for waves with high ground-based frequency between 1.0 × 10−3 and 7.0 × 10−3 s−1 in the cases without background wind.

  • Fig. 7.

    The distribution of (a1)–(d1) true vertical wavelength (color shading, contour interval = 1 km), (a2)–(d2) node apparent vertical wavelength (color shading, contour interval = 1 km), and (a3)–(d3) FFT apparent vertical wavelength (color shading, contour interval = 1 km) of the low ground-based frequency waves varying with wave ground-based frequency and horizontal wavelength. The background winds U are (a1)–(a3) −20, (b1)–(b3) −5, (c1)–(c3) 5, and (d1)–(d3) 20 m s−1. The corresponding first criterion (i.e., |λZU/λxW¯|; black solid lines, contour interval = 0.2, starting from 0.1) and the second criterion (i.e., |ω/mW¯|; red solid lines, contour interval = 0.2, starting from 0.1) are overlaid.

  • Fig. 8.

    The ratio (color shading, contour interval = 0.1) of (a)–(d) the node apparent vertical wavelength to the true wavelength and (e)–(h) the FFT apparent vertical wavelength to the true wavelength varying with ground-based frequency and horizontal wavelength for the low ground-based frequency waves with the background winds U = (a),(e) −20, (b),(f) −5, (c),(g) 5, and (d),(h) 20 m s−1. The distribution of the corresponding first criterion value |λZU/λxW¯|=0.1 (black solid line) and the second criterion value |ω/mW¯|=0.1 (red solid line) calculated from (a)–(d) node apparent result and (e)–(h) FFT apparent result.

  • Fig. 9.

    As in Fig. 7, but for waves with high ground-based frequency between 1.0 × 10−3 and 7.0 × 10−3 s−1 in the cases with background wind.

  • Fig. 10.

    As in Fig. 8, but for waves with high ground-based frequency between 1.0 × 10−3 and 7.0 × 10−3 s−1 in the cases with background wind.

  • Fig. 11.

    The distribution of (a1),(b1) true vertical wavelength (color shading, contour interval = 1 km), (a2),(b2) node apparent vertical wavelength (color shading, contour interval = 1 km), and (a3),(b3) FFT apparent vertical wavelength (color shading, contour interval = 1 km) of the low-frequency wave varying with ground-based frequency and horizontal wavelength and background wind U = 0 m s−1 at latitudes (a1)–(a3) 30° and (b1)–(b3) 45°N.

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