On the Frequency of Internal Gravity Waves in the Atmosphere: Comparing Theory with Observations

Abstract

This paper is devoted to the dynamics of the propagation of non-planetary scale internal gravity waves (IGWs) in the stratified atmosphere. We consider the system of equations describing internal gravity waves in three approximations: (1) the incompressible fluid approximation, (2) the anelastic gas (compressible fluid) approximation, and (3) a new approximation called the non-Boussinesq gas approximation. For each approximation, a different dispersion relation is given, from which it follows that the oscillation frequency of internal gravity waves depends on the direction of propagation, the horizontal and vertical components of the wave vector, the vertical gradient of the background temperature, and the background wind shear. In each of the three cases, the maximum frequency of internal gravity waves is different. Moreover, in the anelastic gas approximation, the maximum frequency is equal to the Brunt–Väisälä buoyancy frequency, and in the incompressible fluid approximation, it is larger than the Brunt–Väisälä frequency by a factor of $\sqrt{7}\cong 2.6$. In the model proposed in this paper, the value of the maximum frequency of internal gravity waves occupies an intermediate position between the above limits. The question arises: which of the above fluid representations adequately describe the dynamics of internal gravity waves? This paper compares the above theories with observational data and experiments.

1. Introduction

Many papers, reviews, and especially monographs [1,2,3,4,5,6,7,8,9,10] have been devoted to internal gravity waves (IGWs). The need to address this topic is due to the fact that gravity waves are the only means of relatively fast energy transfer from one point in the planetary atmosphere to another. It is now generally accepted [11] that IGWs play an important role in the large-scale atmospheric circulation, forming the vertical distribution of such atmospheric parameters as velocity and temperature.

IGWs (sometimes simply called internal waves) are waves within a continuous density stratified medium [4]. If the medium is a liquid, then they are waves in the ocean [7]. If the medium is air, then they are waves in the atmosphere, for example [11,12]. If the medium is plasma, they are waves in the ionosphere [13,14] or other astrophysical objects [15]. IGWs have been observed in the ocean since the late 18th century (for an interesting review of IGWs in the ocean, see [16,17], see also [7]). Observations of IGWs in the atmosphere are described in [5] and in reviews [17] (see references there), [8,11,18].

Note that the physics or mechanism of formation of IGWs (or buoyancy waves) is the same for all media, so the mathematical apparatus describing IGWs is the same. For IGWs to occur, a density disturbance is necessary in some volume of the medium, on which the buoyancy force will act, trying to return a parcel of liquid to the equilibrium position. The difference lies solely in the equation of state, which varies depending on the type of fluid. The equation of state is most developed for air, since the equation of state of an ideal gas is approximately fulfilled. But depending on the assumption of the equation of state, the final expression for the natural frequency of oscillation will also be different. Two approximations are usually distinguished: the first is the approximation of an incompressible fluid and the second is the approximation of a compressible fluid. Each approximation has its own expression for the maximum frequency of oscillation.

In some cases, the adiabatic equation is used as the equation of state for air in the compressible fluid approximation [5,6,11,13,19]. However, most studies use the incompressible fluid representation to describe IGWs [8,10].

Therefore, it is very important to know what determines the wave properties such as frequency, wavelength, and propagation velocity. To answer these questions, it is necessary to construct a mathematical model that adequately describes the wave propagation in a stratified atmosphere. Note that gravity waves, as well as any phenomena of geophysical hydrodynamics, are described by a system of nonlinear partial derivative equations, which has no solution in general form. To solve the above system of equations, certain assumptions are made in order to find an analytical solution. It is clear that the solutions obtained depend on the assumptions made.

Differences in the expression for the frequency of oscillations also arise depending on the temperature stratification we assume: isothermal atmosphere or linear stratification. This aspect is not fundamentally critical; while stratification does influence wave refraction, we will not address it in this paper.

For the existing different approximations in the description of IGWs, see [10,18]: the non-Boussinesq fluid approximation, the anelastic gas approximation, and the adiabatic approximation. In all these cases, the expression for the maximum frequency of IGW oscillations will be different.

In the compressible fluid approximation, the maximum frequency at which IGWs can oscillate is the Brunt–Väisäl buoyancy frequency. For an isothermal atmosphere, it corresponds to the minimum period of oscillation, which is 5.5 min for horizontally propagating waves. In the incompressible fluid approximation, IGWs oscillate with a maximum frequency greater than the Brunt–Väisäl frequency. For an isothermal atmosphere, it corresponds to the minimum period of oscillations equal to 3 min for horizontally propagating waves. Therefore, it seems important to compare the theory with observational data to clarify the adequacy of the model.

In a strictly vertical direction, IGWs do not propagate. However, as the angle of inclination of the wave vector to the horizon increases, the frequency decreases and the period increases. Therefore, to compare theory with experiment, we should be able to measure the maximum frequency of horizontally propagating IGW oscillations. However, horizontally propagating waves do not increase in amplitude as they propagate, so they cannot always be unambiguously identified experimentally. To determine which of the above models adequately describes wave propagation, it is necessary to know the source of the wave generation and the direction of wave propagation. However, the source of internal atmospheric waves is often unknown or impossible to determine [20]. Unfortunately, it is impossible to experimentally measure all parameters of IGWs simultaneously [21]. Observations allow us to determine the frequency of oscillation, the projection of the wave vector in the horizontal and vertical directions. However, it is currently impossible to simultaneously determine the source of the oscillation, its direction, its frequency, and its wavelength instrumentally [8].

In this paper, a new approximation, which we call the non-Boussinesq gas approximation, is proposed. It is shown that in the proposed approximation the maximum frequency is larger than the buoyancy frequency in the compressible fluid approximation but smaller than the buoyancy frequency in the incompressible fluid approximation.

On the one hand, the purpose of the present work is to analyze the system of equations describing the dynamics of IGWs in the stratified atmosphere in different approximations, allowing us to determine the maximum frequency of wave oscillations and to show that this frequency is different in each approximation. That is, we will obtain three expressions for the frequency of IGW oscillations (dispersion relations), which will include the vertical gradient of the background temperature and the vertical shear of the background wind. On the other hand, the aim of this paper is to compare the results of the theory with the observational data.

The existence of IGWs in fluids has been known for many years. However, their role in the upper atmosphere was not understood until the middle of the last century. Some of the first evidence for short-period fluctuations in the middle atmosphere came in the 1950s from observations of meteor trails and radio wave reflections from the ionosphere. The idea that these fluctuations are caused by upward propagation of internal waves was first proposed in [13]. This theory has since become almost universally accepted. A very comprehensive review of this early work is given in [22]. More recent studies are summarized in [11,23,24].

Along with the theory [13] came the idea that for waves propagating upward through the atmosphere, the amplitude increases exponentially due to decreasing air density. It became apparent that internal waves must become statically or dynamically unstable at some point in their exponential growth. Assuming that this instability leads to turbulent diffusion and wave dissipation, in [25] the turbulent eddy viscosity was calculated using the value of the Brunt–Väisälä frequency. In [25] and other works, estimates of eddy diffusion were obtained that were in agreement with estimates obtained from observations [22].

Therefore, knowledge of the expression for the frequency of oscillation of IGWs (dispersion relation) is also important for elucidating the mechanism of turbulence generation in the upper atmosphere. According to the available ideas based on observations, this mechanism is related to the breaking of IGWs propagating almost vertically upward when the wave amplitude increases with height. In this case, wave breaking occurs in a stable atmosphere. But wave collapse also occurs when the atmosphere is unstable, in which case the frequency of oscillation becomes an imaginary quantity, and this depends on the expression for the frequency itself. In the compressible fluid approximation, the atmosphere is unstable when the vertical temperature gradient is greater than the adiabatic gradient. In the incompressible fluid approximation, the atmosphere is unstable when the vertical temperature gradient is greater than the autoconvection gradient, which is more than three times the adiabatic gradient.

Many works have been devoted to the experimental verification of dispersion relations corresponding to different approximations (compressible or incompressible fluid) [20,26,27,28,29,30]. The problem is that different observations show agreement with different models for describing IGWs. For example, experimental observations of perturbations caused by the collapse of IGWs show that the frequency of the oscillations is exactly the Brunt–Väisälä frequency [11,20,31], i.e., the observations are consistent with the results of the compressible fluid approximation. On the other hand, experimental measurements performed in [28,32,33] show a satisfactory agreement with the results of the theory written in the incompressible fluid approximation.

The reason for this ambiguity between theory (compressible and incompressible fluid approximations) and observations and experiments can be explained as follows. The most obvious and dominant sources of IGWs are three main sources: topography, convection, and wind shear [11]. Note that thermal convection alone causes an air parcel to oscillate at the Brunt–Väisälä frequency. However, although it has been known for decades that convection can excite gravity waves, there is still considerable controversy and ongoing research to understand this wave generation mechanism [20]. In general, waves generated by convection are not characterized by a single noticeable frequency as in the case of topographic waves. Instead, convection can generate waves over the entire range of wave frequencies. In particular, low-frequency waves can be observed in the middle atmosphere at large horizontal distances from the convection source, making them difficult to correlate with clouds or other indicators of convection. However, in the tropics, far from topography, the occurrence of inertial-gravity waves is attributed to convection as the source [20].

Thus, observations and experiments do not clearly indicate the correctness of one or the other approximation.

The structure of this paper is as follows. For each approximation, the dispersion relation is presented, taking into account the vertical gradient of the background temperature and the vertical shear of the background wind. Section 2 considers the incompressible fluid approximation, in which the equation of state is described by the constancy of the fluid density during motion. Section 3 considers the anelastic gas approximation, in which the equation of state is described by the constancy of the potential temperature during motion. Section 4 proposes a new non-Boussinesq gas approximation in which the equation of state is described by the heat conduction equation, i.e., by the change in internal energy during motion. Section 5 discusses the obtained dispersion relations and compares the results of the theory with observational and experimental data.

2. Incompressible Fluid Approximation

Almost everything we know about the nature of atmospheric gravity waves is derived from linear theory. From a computational point of view, simplified equations describing linear systems are much faster to solve than nonlinear ones. But perhaps more importantly, linear systems are more tractable and explicit than nonlinear systems.

For example, we can think of the background flow as an average flow that still varies vertically. The disturbances may be due to various causes, but we will not go into the nature of the disturbances here but will accept that for one reason or another there is a deviation in the density disturbance and the air pressure from the values taken in the static state. The sources of gravity waves can vary, but the main ones are topography, thermal convection, and background flow shear [11].

Observations of gravity waves by, for example, [34,35,36,37,38,39,40] show the presence of complex wave structures with time-varying amplitudes and usually with multiple frequencies. A single-frequency wave, i.e., a monochromatic wave that is constant in time, is never observed in nature. The application of linear analysis to observations of wave phenomena is often frustrating, as discussed in [41,42]. For example, wave amplitudes often vary with time; waves within the planetary boundary layer are often nonlinear due to the presence of the Earth’s surface. Despite these difficulties, however, linear theory is still useful and provides a first-order estimate of most wave phenomena. In addition, the simplicity of linear theory provides an understandable picture of wave processes and observations [8].

The generally accepted approach to the representation of gravity waves in the incompressible fluid approximation is most fully summarized in [8,10]. This approximation is called the non-Boussinesq fluid approximation according to the terminology of [10,18], since the air density in the undisturbed atmosphere is assumed to be a known function of height. We call this approach the incompressible fluid approach. Furthermore, in this approximation, the state of the disturbed atmosphere is defined by four parameters: $u$, $\mathit{w}$, $p^{\prime }$, and ${\mathrm{\rho }}^{\prime }$, i.e., horizontal and vertical velocity projections, pressure disturbance, and density disturbance. The temperature disturbance, i.e., the heat conduction equation, is not included in this system because the density disturbance is assumed to be proportional to the temperature disturbance (Boussinesq approximation).

We introduce the definition of internal frequency ${\mathsf{\omega }}_{\mathrm{i}}$ [8] as the frequency of the wave relative to the background flow $\overline{u}$, which is a function of the vertical coordinate $\mathit{z}$, i.e., the frequency of the wave measured by an observer moving with the flow at velocity $\overline{u}$; hence

$${\mathsf{\omega }}_{\mathrm{i}}=\mathsf{\omega }-\overline{u}k,$$

(1)

where $k$ is the horizontal component of the wave vector.

It follows that frequency ${\mathsf{\omega }}_{\mathrm{i}}$ is a function of altitude. Note that $\mathsf{\omega }$ is the frequency of a wave measured in a fixed coordinate system, for instance, with the help of a barograph placed on the surface of the earth. Therefore, frequency $\mathsf{\omega }$ is a constant quantity. In [43] ${\mathsf{\omega }}_{\mathrm{i}}$ is defined as the Doppler shift of the intrinsic frequency of the wave. Wind velocity $\overline{u}$ is the component of the background wind in the direction of wave propagation. It is typically a geostrophic wind, a westerly transport.

If we write (1) as

$$\mathsf{\omega }={\mathsf{\omega }}_{\mathrm{i}}+\overline{u}k,$$

(2)

it is clear that the observed frequency $\mathsf{\omega }$ is greater than the internal frequency ${\mathsf{\omega }}_{\mathrm{i}}$ if the wave propagates in the wind direction and less than ${\mathsf{\omega }}_{\mathrm{i}}$ if the wave propagates against the wind. From (2) for the observed horizontal phase velocity of the wave $c$ we obtain

$$c={\displaystyle \frac{\mathsf{\omega }}{k}}={\displaystyle \frac{{\mathsf{\omega }}_{\mathrm{i}}}{k}}+\overline{u}=c_{\mathrm{i}}+\overline{u},$$

(3)

where $c_{\mathrm{i}}={\mathsf{\omega }}_{\mathrm{i}}/k$ is the internal phase velocity of the wave in the direction of the $x$ axis.

The system of linearized equations describing the propagation of IGWs in a stratified atmosphere in the incompressible fluid approximation reduces to a single equation for the stream function [10]:

$$-{\left({\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }t}}+\overline{u}{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }x}}\right)}^2\left({\nabla }^2\mathsf{\psi }-{\displaystyle \frac{1}{H_{\mathrm{\rho }}}}{\mathsf{\psi }}_{\mathit{z}}\right)+\left({\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }t}}+\overline{u}{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }x}}\right)\left({\overline{u}}^{″}-{\displaystyle \frac{1}{H_{\mathrm{\rho }}}}{\overline{u}}^{\prime }\right){\mathsf{\psi }}_x={\displaystyle \frac{g}{H_{\mathsf{\rho }}}}{\mathsf{\psi }}_{xx},$$

(4)

or for the vertical velocity component

$$-{\left({\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }t}}+\overline{u}{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }x}}\right)}^2\left({\nabla }^2\mathit{w}-{\displaystyle \frac{1}{H_{\mathrm{\rho }}}}{\displaystyle \frac{\mathsf{\partial }\mathit{w}}{\mathsf{\partial }\mathit{z}}}\right)+\left({\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }t}}+\overline{u}{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }x}}\right)\left({\overline{u}}^{″}-{\displaystyle \frac{1}{H_{\mathrm{\rho }}}}{\overline{u}}^{\prime }\right){\displaystyle \frac{\mathsf{\partial }\mathit{w}}{\mathsf{\partial }x}}={\displaystyle \frac{g}{H_{\mathsf{\rho }}}}{\displaystyle \frac{{\mathsf{\partial }}^2\mathit{w}}{\mathsf{\partial }{x}^2}},$$

(5)

Solution of the linearized system of equations describing the propagation of IGWs in the stratified atmosphere in the incompressible fluid approximation as a traveling wave

$$\mathit{w}=\mathit{W}\left(\mathit{z}\right)e^{i\left(kx-\mathsf{\omega }t\right)}$$

(6)

leads to equation

$${\displaystyle \frac{{\mathrm{d}}^2\mathit{W}}{\mathrm{d}{\mathit{z}}^2}}-{\displaystyle \frac{1}{H_{\mathrm{\rho }}}}{\displaystyle \frac{\mathrm{d}\mathit{W}}{\mathrm{d}\mathit{z}}}+\left[{\displaystyle \frac{N^2{k}^2}{{\left(\mathsf{\omega }-k\overline{u}\right)}^2}}+\left({\displaystyle \frac{{\mathrm{d}}^2\overline{u}}{\mathrm{d}{\mathit{z}}^2}}-{\displaystyle \frac{1}{H_{\mathrm{\rho }}}}{\displaystyle \frac{\mathrm{d}\overline{u}}{\mathrm{d}\mathit{z}}}\right){\displaystyle \frac{k}{\mathsf{\omega }-k\overline{u}}}-k^2\right]\mathit{W}=0.$$

(7)

Using the transformation

$$\mathit{W}=f\left(\mathit{z}\right)\tilde{\mathit{W}}$$

(8)

Equation (7) is reduced to the canonical form

$${\displaystyle \frac{{\mathrm{d}}^2\tilde{\mathit{W}}}{\mathrm{d}{\mathit{z}}^2}}+\left[{\displaystyle \frac{N^2}{{\left(c-\overline{u}\right)}^2}}+{\displaystyle \frac{{\overline{u}}^{″}}{c-\overline{u}}}-{\displaystyle \frac{1}{H_{\mathrm{\rho }}}}{\displaystyle \frac{{\overline{u}}^{\prime }}{c-\overline{u}}}-{\displaystyle \frac{1}{4H_{\mathrm{\rho }}^2}}-k^2\right]\tilde{\mathit{W}}=0,$$

(9)

where $f=\exp \left(\mathit{z}/2H_{\mathrm{\rho }}\right)$. This is the Taylor–Goldstein equation for the amplitude of the vertical velocity component [8,44,45]. Here $N\equiv \sqrt{g/H_{\mathrm{\rho }}}=\sqrt{\alpha \left({\gamma }_{\mathrm{A}}-\gamma \right)g}$ is the IGWs buoyancy frequency in the incompressible fluid approximation; $g$ is the free-fall acceleration; $H_{\mathrm{\rho }}=1/\alpha \left({\gamma }_{\mathrm{A}}-\gamma \right)$ is the scale of atmospheric height, which characterizes the vertical distribution of air density; ${\gamma }_{\mathrm{A}}=g/R_{\mathrm{d}}=34\mathrm{K}/\mathrm{km}$ is the so-called autoconvection gradient; $R_{\mathrm{d}}$ is the specific gas constant of dry air; and $\gamma =-\mathrm{d}\overline{T}/\mathrm{d}\mathit{z}$ is the vertical gradient of the background temperature in the atmospheric static state equal to $\gamma =6\mathrm{K}/\mathrm{km}$ for a standard atmosphere. The following approximation is also used in the derivation $1/\overline{T}\approx 1/T_0\equiv \alpha$, $T_0=273\mathrm{K}$.

The second term in (9) is related to the curvature of the background velocity profile, and the third term is related to the background velocity shear. The fourth term has no special name and is neglected when small amplitudes (or short waves) are assumed [8]. A different derivation of Equation (9) is given in [8].

Let us consider the wave propagation up to the critical level when $c=\overline{u}$ [8,18]. Moreover, we simplify Equation (9) by putting $\overline{u}\ll c$. Then we obtain

$${\displaystyle \frac{{\mathrm{d}}^2\tilde{\mathit{W}}}{\mathrm{d}{\mathit{z}}^2}}+\left({\displaystyle \frac{N^2}{c^2}}+{\displaystyle \frac{1}{c}}{\overline{u}}^{″}-{\displaystyle \frac{1}{c{H}_{\mathrm{\rho }}}}{\overline{u}}^{\prime }-{\displaystyle \frac{1}{4H_{\mathrm{\rho }}^2}}-k^2\right)\tilde{\mathit{W}}=0.$$

(10)

Equation (10) is still difficult to analyze, so let us assume that there is a linear shear of the background velocity, i.e., $\overline{u^{\prime }}=\mathrm{\Gamma }=\mathrm{const}$. Then Equation (10) will take the form

$${\displaystyle \frac{{\mathrm{d}}^2\tilde{\mathit{W}}}{\mathrm{d}{\mathit{z}}^2}}+\left({\displaystyle \frac{N^2}{c^2}}-{\displaystyle \frac{\mathrm{\Gamma }}{c{H}_{\mathrm{\rho }}}}-{\displaystyle \frac{1}{4H_{\mathrm{\rho }}^2}}-k^2\right)\tilde{\mathit{W}}=0.$$

(11)

The requirement that the solution be in the form of a wave leads to the expression for the vertical component of the wave vector:

$$m^2={\displaystyle \frac{N^2{k}^2}{{\mathsf{\omega }}^2}}-{\displaystyle \frac{\mathrm{\Gamma }k}{\mathsf{\omega }{H}_{\mathrm{\rho }}}}-{\displaystyle \frac{1}{4H_{\mathrm{\rho }}^2}}-k^2.$$

(12)

From (8), taking into account the expression $f=\exp \left(\mathit{z}/2H_{\mathrm{\rho }}\right)$, we can see that the amplitude of the vertical velocity projection increases with height. The increase in amplitude leads to wave breaking and instability in the middle and upper atmosphere.

Solving (12) with respect to $\mathsf{\omega }$, we obtain the dispersion relation

$${\mathsf{\omega }}_{1,2}=N{\displaystyle \frac{2k}{\frac{N\mathrm{\Gamma }}{g}\pm \sqrt{{\left(\frac{N\mathrm{\Gamma }}{g}\right)}^2+4\left(m^2+k^2+\frac{1}{4H_{\mathrm{\rho }}^2}\right)}}}.$$

(13)

Hence, at $\mathrm{\Gamma }=0$, we obtain the well-known dispersion relation for IGWs in the incompressible fluid approximation [8,10]:

$${\mathsf{\omega }}_{1,2}=\pm N{\displaystyle \frac{k}{\sqrt{m^2+k^2+\frac{1}{4H_{\mathrm{\rho }}^2}}}}.$$

(14)

Correspondingly, for the phase velocity we obtain

$$c_{1,2}={\displaystyle \frac{2N}{\frac{N\mathrm{\Gamma }}{g}\pm \sqrt{{\left(\frac{N\mathrm{\Gamma }}{g}\right)}^2+4\left(m^2+k^2+\frac{1}{4H_{\mathrm{\rho }}^2}\right)}}}.$$

(15)

The dispersion relation (13) (and (14)) expresses the property of IGW dispersion that the wave frequency depends not only on the wavelength but also on the direction of propagation. The phase velocity depends only on the wavelength and is independent of the direction.

From (14) and (15), provided that $1/\left(4H_{\mathrm{\rho }}^2\right)\ll \left(k^2,m^2\right)\ll {\left(N\mathrm{\Gamma }/g\right)}^2$, we find the maximum frequency and phase velocity of the wave:

$${\mathsf{\omega }}_{\max }={\displaystyle \frac{gk}{\mathrm{\Gamma }}},c_{\max }={\displaystyle \frac{g}{\mathrm{\Gamma }}}$$

(16)

If $m=0$, $\mathrm{\Gamma }=0$, and $k^2,m^2\gg 1/\left(4H_{\mathrm{\rho }}^2\right)$, then the maximum oscillation frequency of IGWs is equal to ${\mathsf{\omega }}_{\max }=N$, and the velocity is $c_{\max }=N/k$.

From expression (13), we can see that the frequency of IGWs in the incompressible fluid approximation depends on the atmospheric stratification $\gamma$, which is included in the expressions for $N$, the background velocity shear $\mathrm{\Gamma }$, the wavelength $\lambda$, and the propagation direction $\left(k,m\right)$.

3. Anelastic Gas Approximation Considering Background Flow

We consider a special case of the adiabatic approximation, the so-called anelastic gas approximation, which is related to the compressible fluid approximation. In the anelastic gas approximation, the system of equations describing the propagation of IGWs in a stratified atmosphere [18], taking into account the background flow, is written in the form

$$\left({\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }t}}+\overline{u}{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }x}}\right)u+{\displaystyle \frac{\mathsf{\partial }\overline{u}}{\mathsf{\partial }\mathit{z}}}\mathit{w}=-{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }x}}\left({\displaystyle \frac{p^{\prime }}{\overline{\mathrm{\rho }}}}\right),$$

(17)

$$\left({\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }t}}+\overline{u}{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }x}}\right)\mathit{w}=-{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }\mathit{z}}}\left({\displaystyle \frac{p^{\prime }}{\overline{\mathrm{\rho }}}}\right)+{\displaystyle \frac{1}{H_{\mathrm{\rho }}}}{\displaystyle \frac{p^{\prime }}{\overline{\mathrm{\rho }}}}+g{\displaystyle \frac{{\mathsf{\Theta }}^{\prime }}{\overline{\mathsf{\Theta }}}},$$

(18)

$$\left({\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }t}}+\overline{u}{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }x}}\right)\left({\displaystyle \frac{{\mathsf{\Theta }}^{\prime }}{\overline{\mathsf{\Theta }}}}\right)=-{\displaystyle \frac{N_{\mathrm{BV}}^2}{g}}\mathit{w},$$

(19)

$${\displaystyle \frac{\mathsf{\partial }u}{\mathsf{\partial }x}}+{\displaystyle \frac{\mathsf{\partial }\mathit{w}}{\mathsf{\partial }\mathit{z}}}={\displaystyle \frac{1}{H_{\mathit{\rho }}}}\mathit{w}.$$

(20)

Here

$$N_{\mathrm{BV}}^2={\displaystyle \frac{g}{\overline{\mathsf{\Theta }}}}{\displaystyle \frac{\mathrm{d}\overline{\mathsf{\Theta }}}{\mathrm{d}\mathit{z}}}={\displaystyle \frac{g}{H_{\mathit{\rho }}}}-{\displaystyle \frac{g^2}{c_{\mathrm{s}}^2}}=N^2-{\displaystyle \frac{g^2}{c_{\mathrm{s}}^2}}=\alpha \left({\gamma }_{\mathrm{a}}-\gamma \right)g$$

(21)

is the square of the Brunt–Väisälä buoyancy frequency; $c_{\mathrm{s}}$ the speed of sound; ${\gamma }_{\mathrm{a}}=g/c_p\approx 10\mathrm{K}/\mathrm{km}$ is the dry adiabatic vertical temperature gradient; and $c_p$ is the specific heat capacity of air at constant pressure.

Recall that in the incompressible fluid approximation the zero equality of the substantive derivative of the air density is assumed for the equation of state: $\mathrm{D}\mathit{\rho }/\mathrm{D}t=0$, i.e., the density of a moving air parcel remains constant during motion along the trajectory. In the compressible fluid (anelastic gas approximation) the zero equality of the substantive derivative of the potential temperature is taken as the equation of state: $\mathrm{D}\mathsf{\Theta }/\mathrm{D}t=0$. This leads to the fact that in the incompressible fluid approximation the velocity divergence is zero, and in the anelastic gas approximation the velocity divergence is not zero.

It is possible by analogy with the incompressible fluid approximation [10] (step by step) to reduce the system of Equations (17)–(20) to one equation for the vertical velocity component

$$-{\left({\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }t}}+\overline{u}{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }x}}\right)}^2\left({\nabla }^2-{\displaystyle \frac{2}{H_{\mathrm{\rho }}}}{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }\mathit{z}}}+{\displaystyle \frac{1}{H_{\mathrm{\rho }}^2}}\right)\mathit{w}+\left({\overline{u}}^{″}-{\displaystyle \frac{N_{\mathrm{BV}}^2}{g}}{\overline{u}}^{\prime }\right)\left({\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }t}}+\overline{u}{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }x}}\right){\displaystyle \frac{\mathsf{\partial }\mathit{w}}{\mathsf{\partial }x}}=N_{\mathrm{BV}}^2{\displaystyle \frac{{\mathsf{\partial }}^2\mathit{w}}{\mathsf{\partial }{x}^2}}.$$

(22)

In the absence of background flow, the equation takes the following form:

$$\left[{\displaystyle \frac{{\mathsf{\partial }}^2}{\mathsf{\partial }{t}^2}}\left({\displaystyle \frac{{\mathsf{\partial }}^2}{\mathsf{\partial }{x}^2}}+{\displaystyle \frac{{\mathsf{\partial }}^2}{\mathsf{\partial }{\mathit{z}}^2}}+{\displaystyle \frac{1}{H_{\mathrm{\rho }}^2}}\right)+N_{\mathrm{BV}}^2{\displaystyle \frac{{\mathsf{\partial }}^2}{\mathsf{\partial }{x}^2}}-{\displaystyle \frac{2}{H_{\mathrm{\rho }}}}{\displaystyle \frac{{\mathsf{\partial }}^2}{\mathsf{\partial }{t}^2}}{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }\mathit{z}}}\right]\mathit{w}=0.$$

(23)

The solution is sought in the form of a traveling wave (6). Substituting (6) into (22), we obtain

$${\displaystyle \frac{{\mathsf{\partial }}^2\mathit{W}}{\mathsf{\partial }{\mathit{z}}^2}}-{\displaystyle \frac{2}{H_{\mathrm{\rho }}}}{\displaystyle \frac{\mathsf{\partial }\mathit{W}}{\mathsf{\partial }\mathit{z}}}+\left[{\displaystyle \frac{N_{\mathrm{BV}}^2{k}^2}{{\left(\mathsf{\omega }-\overline{u}k\right)}^2}}+\left({\overline{u}}^{″}-{\displaystyle \frac{N_{\mathrm{BV}}^2}{g}}{\overline{u}}^{\prime }\right){\displaystyle \frac{k}{\mathsf{\omega }-\overline{u}k}}+{\displaystyle \frac{1}{H_{\mathrm{\rho }}^2}}-k^2\right]\mathit{W}=0.$$

(24)

In the absence of background flow, Equation (24) takes the following form:

$${\displaystyle \frac{{\mathsf{\partial }}^2\mathit{W}}{\mathsf{\partial }{\mathit{z}}^2}}-{\displaystyle \frac{2}{H_{\mathrm{\rho }}}}{\displaystyle \frac{\mathsf{\partial }\mathit{W}}{\mathsf{\partial }\mathit{z}}}+\left[{\displaystyle \frac{1}{H_{\mathrm{\rho }}^2}}+\left({\displaystyle \frac{N_{\mathrm{BV}}^2}{{\mathsf{\omega }}^2}}-1\right)k^2\right]\cdot \mathit{W}=0.$$

(25)

Let us reduce Equation (24) to the canonical form using transformation (8) at $f=\exp \left(\mathit{z}/H_{\mathrm{\rho }}\right)$. Substituting (8) into (24), we obtain the equation of Taylor–Goldstein type in the anelastic gas approximation

$${\displaystyle \frac{{\mathsf{\partial }}^2\tilde{\mathit{W}}}{\mathsf{\partial }{\mathit{z}}^2}}+\left[{\displaystyle \frac{N_{\mathrm{BV}}^2{k}^2}{{\left(\mathsf{\omega }-\overline{u}k\right)}^2}}+\left({\overline{u}}^{″}-{\displaystyle \frac{N_{\mathrm{BV}}^2}{g}}{\overline{u}}^{\prime }\right){\displaystyle \frac{k}{\mathsf{\omega }-\overline{u}k}}-k^2\right]\tilde{\mathit{W}}=0.$$

(26)

Comparing (26) with (9), we see that the buoyancy frequency $N$ has been replaced by the Brunt–Väisälä buoyancy frequency $N_{\mathrm{BV}}$, and the term $1/4H_{\mathrm{\rho }}^2$ is missing. In the absence of background flow, we have

$${\displaystyle \frac{{\mathsf{\partial }}^2\tilde{\mathit{W}}}{\mathsf{\partial }{\mathit{z}}^2}}+\left({\displaystyle \frac{N_{\mathrm{BV}}^2}{{\mathsf{\omega }}^2}}-1\right)k^2\tilde{\mathit{W}}=0.$$

(27)

Equation (26) is also difficult to analyze, so let us assume, as in the case of an incompressible fluid, that there is a linear shift of the background velocity, i.e., $\overline{u^{\prime }}=\mathrm{\Gamma }=\mathrm{const}$ and $\overline{u}\ll c$. Then Equation (26) takes the following form:

$${\displaystyle \frac{{\mathsf{\partial }}^2\tilde{\mathit{W}}}{\mathsf{\partial }{\mathit{z}}^2}}+\left({\displaystyle \frac{N_{\mathrm{BV}}^2{k}^2}{{\mathsf{\omega }}^2}}-{\displaystyle \frac{N_{\mathrm{BV}}^2\mathrm{\Gamma }}{g}}{\displaystyle \frac{k}{\mathsf{\omega }}}-k^2\right)\tilde{\mathit{W}}=0.$$

(28)

The requirement that the solution be in the form of a wave leads to the expression for the vertical component of the wave vector

$$m^2={\displaystyle \frac{N_{\mathrm{BV}}^2{k}^2}{{\mathsf{\omega }}^2}}-{\displaystyle \frac{N_{\mathrm{BV}}^2\mathrm{\Gamma }}{g}}{\displaystyle \frac{k}{\mathsf{\omega }}}-k^2.$$

(29)

Comparing the expressions for $f=\exp \left(\mathit{z}/2H_{\mathrm{\rho }}\right)$ in the incompressible fluid approximation with the expression for $f=\exp \left(\mathit{z}/H_{\mathrm{\rho }}\right)$ in the anelastic gas approximation, we see that the wave amplitude grows faster with height in the anelastic gas approximation.

Solving (29) with respect to $\mathsf{\omega }$, we obtain the dispersion relation

$${\mathsf{\omega }}_{1,2}=N_{\mathrm{BV}}{\displaystyle \frac{2k}{\frac{N_{\mathrm{BV}}\mathrm{\Gamma }}{g}\pm \sqrt{{\left(\frac{N_{\mathrm{BV}}\mathrm{\Gamma }}{g}\right)}^2+4\left(k^2+m^2\right)}}}.$$

(30)

Hence, when $\mathrm{\Gamma }=0$, we obtain the well-known dispersion relation for IGWs in the anelastic gas approximation [10,18]:

$${\mathsf{\omega }}_{1,2}=\pm N_{\mathrm{BV}}{\displaystyle \frac{k}{\sqrt{k^2+m^2}}}.$$

(31)

Correspondingly, for the phase velocity we obtain

$$c_{1,2}={\displaystyle \frac{2N_{\mathrm{BV}}}{\frac{N_{\mathrm{BV}}\mathrm{\Gamma }}{g}\pm \sqrt{{\left(\frac{N_{\mathrm{BV}}\mathrm{\Gamma }}{g}\right)}^2+4\left(k^2+m^2\right)}}}.$$

(32)

From (30) and (32), provided that $\left(k^2,m^2\right)\ll {\left(N_{\mathrm{BV}}\mathrm{\Gamma }/g\right)}^2$, we find that the maximum frequency and phase velocity of the wave take the form (16), i.e., the same as in the incompressible fluid approximation. Therefore, experimental verification of Formula (16) does not allow us to determine which of the two models adequately describes the propagation of IGWs. If $m=0$ and $\mathrm{\Gamma }=0$, then the maximum frequency of IGW oscillations is equal to ${\mathsf{\omega }}_{\max }=N_{\mathrm{BV}}$, and the velocity is $c_{\max }=N_{\mathrm{BV}}/k$.

Comparing the dispersion relations for incompressible gas (13) and anelastic gas (30), we notice that they differ not only by replacing $N$ by $N_{\mathrm{BV}}$ but also by the presence of a term $1/4H_{\mathrm{\rho }}^2$ in the denominator of (13), which affects both the magnitude of the frequency and the direction of propagation of IGWs.

Thus, in both cases, both in the incompressible fluid approximation and in the anelastic gas approximation, the buoyancy force is due to density disturbance, i.e., a change in its value relative to the static (background) value. In the incompressible fluid approximation, the density remains constant during motion. This causes the density disturbance to be opposite to the density change in the surrounding atmosphere (static state). The background change in air density is considered to be a known function with a characteristic scale of atmospheric height $H_{\mathrm{\rho }}$. In other words, there is an equilibrium position (altitude) at which the density of the moving parcel coincides with the ambient air density. Since in this approximation the density of the moving parcel remains constant, when moving upward relative to the equilibrium position, the density of the parcel becomes greater than the density of the ambient air. And when moving downward relative to the equilibrium position, the density of the moving parcel is less than the density of the surrounding air. Thus, when the parcel deviates from the equilibrium position, a restoring force is generated, which leads to oscillations, which, in turn, generate waves.

In the anelastic gas approximation, the potential temperature remains constant during motion. The buoyancy force is due to perturbations of the potential temperature. Here there is also an equilibrium position at which the potential temperature of the moving particle coincides with the potential temperature of the environment. In a stable atmosphere $\left(\gamma <{\gamma }_{\mathrm{a}}\right)$, the potential temperature increases with altitude [6,8,10]. So, when deviating upward from the equilibrium position, the disturbance of the potential temperature is negative, and the buoyancy force is directed downward. Conversely, when deviating downward from the equilibrium position, the potential temperature disturbance is positive, and the buoyancy force is directed upward. This is how oscillations arise that generate waves [46].

4. Non-Boussinesq Gas Approximation Considering Background Flow

In this paper, we consider a new approximation that differs from the well-known approximations discussed above in that the state of the system is described by the heat conduction Equation (35), which appears to be more general than density constancy or potential temperature constancy. We call this approximation the non-Boussinesq gas approximation. Thus, the system of equations describing the propagation of IGWs in a stratified atmosphere in the presence of a background flow in the non-Boussinesq gas approximation has the following form:

$$\left({\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }t}}+\overline{u}{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }x}}\right)u+{\displaystyle \frac{\mathsf{\partial }\overline{u}}{\mathsf{\partial }\mathit{z}}}\mathit{w}=-{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }x}}\left({\displaystyle \frac{p^{\prime }}{\overline{\mathrm{\rho }}}}\right),$$

(33)

$$\left({\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }t}}+\overline{u}{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }x}}\right)\mathit{w}=-{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }\mathit{z}}}\left({\displaystyle \frac{p^{\prime }}{\overline{\mathrm{\rho }}}}\right)+{\displaystyle \frac{1}{H_{\mathrm{\rho }}}}\left({\displaystyle \frac{p^{\prime }}{\overline{\mathrm{\rho }}}}\right)+g\alpha \mathsf{\theta },$$

(34)

$$\left({\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }t}}+\overline{u}{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }x}}\right)\alpha \mathsf{\theta }=\alpha \gamma \mathit{w}-{\displaystyle \frac{R_{\mathrm{d}}}{c_V}}\left(\nabla ,v\right),$$

(35)

$${\displaystyle \frac{\mathsf{\partial }u}{\mathsf{\partial }x}}+{\displaystyle \frac{\mathsf{\partial }\mathit{w}}{\mathsf{\partial }\mathit{z}}}={\displaystyle \frac{1}{H_{\mathrm{\rho }}}}\mathit{w}.$$

(36)

The heat conduction Equation (35) describes the time evolution of the internal energy of an air parcel during motion. Taking into account the continuity Equation (36), the heat conduction equation is written in the form

$$\left({\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }t}}+\overline{u}{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }x}}\right)\alpha \mathsf{\theta }=-{\displaystyle \frac{c_p}{c_V}}{\displaystyle \frac{N_{\mathrm{BV}}^2}{g}}\mathit{w}.$$

(37)

A comparison of this system of equations with the system of equations for inelastic gas shows that they transform into each other when the perturbation of the potential temperature is replaced by a perturbation of the temperature itself $\left({\mathsf{\Theta }}^{\prime }/\overline{\mathsf{\Theta }}\right)\to \alpha \mathsf{\theta }$ in Equations (18) and (19). Additionally, in Equation (19), a multiplier in the form of the adiabatic exponent appears on the right-hand side.

The similarity of the two systems allows us to immediately write down the equation for the vertical component of velocity:

$$-{\left({\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }t}}+\overline{u}{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }x}}\right)}^2\left({\nabla }^2-{\displaystyle \frac{2}{H_{\mathrm{\rho }}}}{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }\mathit{z}}}+{\displaystyle \frac{1}{H_{\mathrm{\rho }}^2}}\right)\mathit{w}+\left({\overline{u}}^{″}-{\displaystyle \frac{\mathsf{\kappa }{N}_{\mathrm{BV}}^2}{g}}{\overline{u}}^{\prime }\right)\left({\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }t}}+\overline{u}{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }x}}\right){\displaystyle \frac{\mathsf{\partial }\mathit{w}}{\mathsf{\partial }x}}=\mathsf{\kappa }{N}_{\mathrm{BV}}^2{\displaystyle \frac{{\mathsf{\partial }}^2\mathit{w}}{\mathsf{\partial }{x}^2}}$$

(38)

Here, $\mathsf{\kappa }=c_p/c_V$ is the adiabatic exponent. Accordingly, in the absence of background flow, the equation will take the following form:

$${\displaystyle \frac{{\mathsf{\partial }}^2}{\mathsf{\partial }{t}^2}}\left({\nabla }^2-{\displaystyle \frac{2}{H_{\mathrm{\rho }}}}{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }\mathit{z}}}+{\displaystyle \frac{1}{H_{\mathrm{\rho }}^2}}\right)\mathit{w}+\mathsf{\kappa }{N}_{\mathrm{BV}}^2{\displaystyle \frac{{\mathsf{\partial }}^2\mathit{w}}{\mathsf{\partial }{x}^2}}=0.$$

(39)

The equation of the Taylor–Goldstein type in the non-Boussinesq gas approximation will be written similarly:

$${\displaystyle \frac{{\mathsf{\partial }}^2\tilde{\mathit{W}}}{\mathsf{\partial }{\mathit{z}}^2}}+\left[{\displaystyle \frac{\mathsf{\kappa }{N}_{\mathrm{BV}}^2{k}^2}{{\left(\mathsf{\omega }-\overline{u}k\right)}^2}}+\left({\overline{u}}^{″}-{\displaystyle \frac{\mathsf{\kappa }{N}_{\mathrm{BV}}^2}{g}}{\overline{u}}^{\prime }\right){\displaystyle \frac{k}{\mathsf{\omega }-\overline{u}k}}-k^2\right]\tilde{\mathit{W}}=0.$$

(40)

In this approximation, the wave amplitude also grows with height, as in the previous two cases. The dispersion relation will be written in the following form:

$${\mathsf{\omega }}_{1,2}=\sqrt{\mathsf{\kappa }}{N}_{\mathrm{BV}}{\displaystyle \frac{2k}{\frac{\sqrt{\mathsf{\kappa }}{N}_{\mathrm{BV}}\mathrm{\Gamma }}{g}\pm \sqrt{{\left(\frac{\sqrt{\mathsf{\kappa }}{N}_{\mathrm{BV}}\mathrm{\Gamma }}{g}\right)}^2+4\left(k^2+m^2\right)}}}.$$

(41)

Hence it can be seen that at $m=0$ and $\mathrm{\Gamma }=0$, the maximum frequency of IGWs is ${\mathsf{\omega }}_{\max }=\sqrt{\mathsf{\kappa }}{N}_{\mathrm{BV}}$, and the velocity is $c_{\max }=\sqrt{\mathsf{\kappa }}{N}_{\mathrm{BV}}/k$. Under the condition $4\left(k^2+m^2\right)\ll {\left(\sqrt{\mathsf{\kappa }}{N}_{\mathrm{BV}}\mathrm{\Gamma }/g\right)}^2$ for the maximum frequency and phase velocity, we again obtain the Formula (16). Thus, we see that the difference between the models is most significant for short wavelengths.

5. Discussion

Three dispersion relations for three approximations are presented above, respectively. It is the dispersion relations that are subject to experimental verification.

Table 1. The main characteristics of the three considered approximations of the IGWs.

	Incompressible Fluid	Compressible Fluid (Anelastic Gas)	Non-Boussinesq Gas
Maximum frequency	$N=\sqrt{\alpha \left({\gamma }_{\mathrm{A}}-\gamma \right)}$	$N_{\mathrm{BV}}=\sqrt{\alpha \left({\gamma }_{\mathrm{a}}-\gamma \right)g}$	$\sqrt{\mathsf{\kappa }}{N}_{\mathrm{BV}}$
Amplitude growth	$\exp \left(\mathit{z}/2H_{\mathrm{\rho }}\right)$	$\exp \left(\mathit{z}/H_{\mathrm{\rho }}\right)$	$\exp \left(\mathit{z}/H_{\mathrm{\rho }}\right)$
Direction of propagation and dispersion relation equation	${\displaystyle \frac{{\mathsf{\omega }}_{1,2}}{N}}$	${\displaystyle \frac{{\mathsf{\omega }}_{1,2}}{N_{\mathrm{BV}}}}$	${\displaystyle \frac{{\mathsf{\omega }}_{1,2}}{\sqrt{\mathsf{\kappa }}{N}_{\mathrm{BV}}}}$
	(13)	(30)	(41)
Buoyancy force	$-g{\displaystyle \frac{{\mathrm{\rho }}^{\prime }}{\overline{\mathrm{\rho }}}}$	$g{\displaystyle \frac{{\mathsf{\Theta }}^{\prime }}{\overline{\mathsf{\Theta }}}}$	$g\alpha \mathsf{\theta }$
Equation of state	$\mathrm{D}\mathit{\rho }/\mathrm{D}t=0$	$\mathrm{D}\mathsf{\Theta }/\mathrm{D}t=0$	$\mathrm{D}T/\mathrm{D}t=-{\displaystyle \frac{R_{\mathrm{d}}\overline{T}}{c_V}}\left(\nabla ,v\right)$

below summarizes the main characteristics of the above approximations.

In the incompressible fluid approximation, the disturbed atmosphere behaves as a fluid of constant density, i.e., the density increases with respect to the surroundings exactly as much as the density decreases in the static state and vice versa. This leads to oscillations relative to the equilibrium position where the densities of the air parcel and the background atmosphere coincide.

In the anelastic gas approximation, the potential temperature remains constant in the disturbed state. Since for the background state the potential temperature increases with height, the disturbance of the potential temperature decreases by exactly the same amount. This also leads to oscillations of the air parcel relative to the equilibrium position where the potential temperature of the disturbed atmosphere coincides with the potential temperature of the background atmosphere.

Note that in the static state of the atmosphere, air density and potential temperature are distributed with height according to a different law. It can be shown that for an incompressible fluid the equilibrium position is lower and the oscillation amplitude is smaller than in the case of an anelastic gas, for which the equilibrium position is higher and the oscillation amplitude is larger. This could also serve as an experimental test of the adequacy of the above models.

Thus, we see that in both cases, both in the anelastic gas approximation and in the incompressible fluid approximation, the oscillations are due to the deviation of the density of the moving particle relative to the unperturbed environment. In the anelastic gas approximation, the density changes due to adiabatic expansion or contraction, thus creating a buoyancy force. In the incompressible fluid approximation, the density of the moving parcel does not change [4], but the density of the surrounding atmosphere decreases exponentially, which also results in a buoyancy force.

In the non-Boussinesq gas approximation, we follow the change in internal energy, i.e., the temperature of the gas, during motion. The temperature dynamics are described by the heat conduction equation. The buoyancy is due to the temperature disturbance. In this case, the IGWs oscillate with a frequency greater than the Brunt–Väisälä buoyancy frequency, namely with frequency $\sqrt{\mathsf{\kappa }}{N}_{\mathrm{BV}}$.

Thus, we see that the expression for the frequency of IGWs depends on the chosen mathematical model for their description. The differences lie in the equation by which the buoyancy dynamics are described. For an anelastic gas, the buoyancy dynamics are described by Equation (19) for the potential temperature disturbance. For an incompressible fluid, the buoyancy force dynamics are described by the equation for density disturbance, which is similar to Equation (19) if we make a substitution ${\mathsf{\Theta }}^{\prime }/\overline{\mathsf{\Theta }}\to -{\mathit{\rho }}^{\prime }/\overline{\mathit{\rho }}$, $N_{\mathrm{BV}}^2\to N^2$. For a non-Boussinesq gas, the dynamics of the buoyancy force are described by the heat conduction Equation (37).

Thus, the three mechanisms of buoyancy generation excite oscillatory motions that generate IGWs. In adiabatic oscillations, the maximum oscillation frequency is equal to the Brunt–Väisäl frequency $N_{\mathrm{BV}}$. In constant density oscillations, the maximum frequency of oscillation is greater than $N_{\mathrm{BV}}$ and equal to $N$. In the non-Boussinesq gas approximation, the maximum frequency of oscillation takes an intermediate value and is equal to $\sqrt{\mathsf{\kappa }}{N}_{\mathrm{BV}}$. In adiabatic oscillations (anelastic gas approximation), the equilibrium position is higher than in constant density oscillations (incompressible fluid approximation). The non-Boussinesq gas approximation differs from the anelastic gas approximation by the frequency of oscillations.

For the experimental verification of the theory of IGWs, it is important that the amplitude of the oscillations is larger for adiabatic oscillations than for constant density oscillations. Thus, when describing IGWs in the incompressible fluid approximation, we obtain the maximum frequency $N$, when describing IGWs in the anelastic gas approximation, we obtain the maximum frequency $N_{\mathrm{BV}}$, and in the non-Boussinesq gas approximation, we obtain $\sqrt{\mathsf{\kappa }}{N}_{\mathrm{BV}}$. An experimental technique could be based on this fact to determine which of the above models adequately describes the propagation of IGWs. However, such experimental results are not currently available.

Wave breaking, which is associated with amplitude growth in near-vertical propagation, occurs in a stable atmosphere and is described in the anelastic gas approximation, the incompressible fluid approximation, and the non-Boussinesq gas approximation. It is due to the phenomenon of wave breaking that it is proposed to explain the occurrence of turbulence and other inhomogeneities in the upper atmosphere, see e.g., [11,13,47]. In particular, according to [47], wave breaking appears to occur at an altitude of about 85 km. In other words, the breaking of IGWs leads to turbulence. However, based on the observational data given in [47], temperature gradients $\gamma >{\gamma }_{\mathrm{A}}$ in the upper atmosphere are not observed at the altitudes considered in [47]. Wave breaking occurs in a stable atmosphere. Therefore, the only breaking mechanism is the growth of the wave amplitude from the wave source located in the lower troposphere up to the indicated heights. This consideration does not take into account that the vertical temperature gradient has a kink at the transition from the troposphere to the stratosphere.

Gravity wave instability structures and turbulence have been observed experimentally in [27]. These data indicate that the maximum frequency of the oscillations of IGWs is the Brunt–Väisälä frequency $N_{\mathrm{BV}}$, not the frequency $N$ given by the theory in the incompressible fluid approximation, and not the frequency $\sqrt{\mathsf{\kappa }}{N}_{\mathrm{BV}}$ given by the theory in the non-Boussinesq gas approximation.

Determining the conditions under which instabilities and internal gravity wave breaking occur is also important for understanding how airglows and similar structures [31] are formed in the upper atmosphere. Short-period, quasi-monochromatic disturbances are ubiquitous in mesopause airglow layers, as evidenced by numerous airglow imaging observations (see, e.g., [48,49,50,51]). Most short-period disturbances fall into two categories: bands and ripples [52]. Band structures are often wave train fronts with horizontal wavelengths of tens of kilometers. Ripples are small-scale structures with a horizontal separation of less than 15 km. The observed structures are thought to be atmospheric gravity waves originating in the lower atmosphere, and ripples are thought to be generated in situ by convective or dynamical instabilities (see, e.g., [53,54] and review [55]).

Depending on how they are generated, the ripples exhibit different positions of their phase fronts relative to the band of accompanying waves and the background wind shear. Ripples oriented perpendicular to the high-frequency shortwave atmospheric brightness bands have been observed [56,57]. These ripples were associated with the breaking of low-frequency and long-wavelength IGWs detected in simultaneous lidar observations. Comparison of the observations with model simulations showed that the ripples were caused by convective instabilities [56,58]. Very different ripple structures were observed in [59], where the ripples were shown to be parallel to short-period wave bands. It was hypothesized that these ripples represent Kelvin–Helmholtz waves resulting from shear instability. These studies indicate the importance of simultaneous measurements of temperature and wind profiles in studying the occurrence of ripples [60].

To study disturbances in the upper atmosphere, a method based on the resonant scattering of test (low-frequency) radio waves on artificial inhomogeneities created when the ionosphere is heated by high-frequency radio waves is used in [28]. To analyze the experimental data, the mathematical model of IGWs in the incompressible fluid approximation was considered in [28]. In their experiment, waves with oscillation periods of 5 min or more were observed, corresponding to a vertical temperature gradient close to the autoconvection gradient ${\gamma }_{\mathrm{A}}$. Therefore, the authors concluded that the description of IGWs in the incompressible fluid approximation (Equation (14)) is in satisfactory agreement with the observational data.

It is now generally accepted that IGWs are important driving forces of the middle and upper atmospheric circulation. For example, in [61], the influence of IGWs on the destruction of the Antarctic polar vortex was investigated. However, the insufficient number of observations and the incomplete mathematical description of the waves limit our understanding of their contribution to atmospheric dynamics.

Thus, from the analysis presented in this paper, it follows that in the anelastic gas approximation (compressible fluid), the maximum frequency of IGW oscillations is equal to the Brunt–Väisäl buoyancy frequency $N_{\mathrm{BV}}=\sqrt{\alpha \left({\gamma }_{\mathrm{a}}-\gamma \right)g}$, whose maximum value corresponds to an isothermal atmosphere $\gamma =0$, i.e., ${\left(N_{\mathrm{BV}}\right)}_{\max }=\sqrt{\alpha {\gamma }_{\mathrm{a}}g}$. In the incompressible fluid approximation, the maximum frequency is equal to $N=\sqrt{\alpha \left({\gamma }_{\mathrm{A}}-\gamma \right)g}$, whose maximum value also corresponds to an isothermal atmosphere and is equal to $N_{\max }=\sqrt{\alpha {\gamma }_{\mathrm{A}}g}$. In our proposed non-Boussinesq gas approximation, the maximum frequency of IGWs in an isothermal atmosphere is $\kappa \sqrt{\alpha {\gamma }_{\mathrm{a}}g}$. These three values are essentially different, and at the present level of measurement technology, this difference should be revealed experimentally. However, as we have seen above, the observational and experimental data give a wide range of frequency values and do not allow us to determine unambiguously which of the considered models adequately describes the propagation of IGWs.

Most authors, based on observational data, conclude that the maximum frequency of IGW oscillations is equal to the Brunt–Väisälä frequency, which results from the description of waves in the compressible fluid approximation. However, as can be seen from Formula (30), the frequency value is influenced not only by the propagation direction $\left(k,m\right)$ and not only by the vertical temperature stratification $\gamma$, but also by the wind shear $\mathrm{\Gamma }$, which was not taken into account, i.e., the authors draw this conclusion using Formula (31).

In [28], instabilities in the upper atmosphere were artificially created and their characteristics were determined. The experiments conducted in [28] to artificially create inhomogeneities in the upper atmosphere and measure their characteristics showed satisfactory agreement of the experimental measurements with the results of the IGWs theory in the incompressible fluid approximation. The authors draw their conclusion using Formula (14), which is obtained from Formula (13) when $\mathrm{\Gamma }=0$, i.e., in the absence of wind shear.

Thus, the choice of an adequate mathematical model describing IGWs, as well as methods of their observation and experimental measurement and verification of the dispersion relations, remain open.

6. Conclusions

This paper discusses the fact that, in the framework of the non-Boussinesq fluid approximation (incompressible stratified fluid), the maximum frequency of oscillations of internal gravitational waves in the stratified atmosphere is equal to $N=\sqrt{g/H_{\mathrm{\rho }}}=\sqrt{\alpha g\left({\gamma }_{\mathrm{A}}-\gamma \right)}$. This frequency is different from the Brunt–Väisälä frequency $N_{\mathrm{BV}}=\sqrt{\alpha g\left({\gamma }_{\mathrm{a}}-\gamma \right)}$, which is related to the frequency of oscillation of an air parcel when it is adiabatically deflected from its equilibrium position. The Brunt–Väisälä oscillation frequency is related to the adiabatic motion of the air, while the frequency $N$ is related to internal oscillations in which the density of the air parcel remains constant. These oscillations differ not only quantitatively but also in physical meaning. Both of these frequencies are due to the buoyancy force, but the meaning is different. In Brunt–Väisälä oscillations, an air parcel that is warmer than its surroundings rises adiabatically to a position of equilibrium where its potential temperature is equal to the potential temperature of the surrounding air. The air parcel then continues its adiabatic upward motion, becoming colder than the surrounding medium. This creates a downward buoyancy force (caused by the potential temperature disturbance). This is how the oscillations of the air parcel occur in a stratified atmosphere. The mechanism of oscillation of IGWs in the incompressible fluid approximation is quite different. At equilibrium, the density of the air parcel coincides with the density of the surrounding atmosphere, which is assumed to be undisturbed. As the air parcel rises, its density remains constant and the density of the surrounding atmosphere decreases, i.e., the air parcel becomes heavier, resulting in a negative buoyancy force (caused by the density disturbance) that attempts to return the air parcel to equilibrium. Similarly, if the air parcel falls below the equilibrium position of constant density, it becomes lighter than the surrounding air and a positive buoyancy force occurs, also returning it to equilibrium. But the frequency of these oscillations is different from the Brunt–Väisälä frequency.

Similar to the previous cases, in our proposed non-Boussinesq gas approximation there is an equilibrium position where the temperature of the air parcel coincides with the background temperature of the ambient air. Any deviation from the equilibrium position will create a buoyancy force due to the temperature disturbance. This buoyancy force causes oscillations with a frequency higher than the frequency in the compressible fluid approximation $N_{\mathrm{BV}}$, but lower than the frequency in the incompressible fluid approximation $N$. Therefore, the maximum frequency of the IGWs oscillations in our approximation will be greater than the Brunt–Väisälä frequency $N_{\mathrm{BV}}$, but less than the $N$ frequency.

The question of the oscillation frequency of IGWs is important for elucidating the mechanism of their breaking and the formation of turbulence and other disturbances in the upper atmosphere. A number of instrumental observations [11] of structures formed as a result of the breaking of IGWs indicate that they correspond to the Brunt–Väisälä frequency. However, as shown in this paper, to make the correct conclusion, it is necessary to use Formula (30), which also takes into account the wind shear, instead of Formula (31), which is traditionally used.

The experiments carried out in [28,32,33] on the artificial creation of inhomogeneities in the upper atmosphere and the measurement of their properties showed a satisfactory agreement of the experimental measurements with the results of the IGW theory in the incompressible fluid approximation. However, this conclusion follows from Formula (14), which does not consider wind shear, instead of Formula (13). Our proposed model gives the value of the maximum frequency between these two bounds.

This paper discusses possible ways to empirically validate one or another IGW model. It is noted that at present the choice of an adequate mathematical approximation to describe internal gravitational waves, as well as the methods of their observation and experimental measurement, remain open.