Acoustic-gravity waves (AGWs) in the Earth's atmosphere have been studied theoretically and experimentally for more than 60 years. The linear theory of AGW (Hines, 1960; Yeh and Liu, 1974; Francis, 1975) admits the existence in the atmosphere of a continuous spectrum of freely propagating waves, consisting of acoustic and gravity regions on the dispersion plane, as well as of evanescent modes, which can only propagate horizontally.

The freely propagating AGWs effectively transfer the energy and momentum between various atmospheric layers and thus play an important role in the dynamics and energy balance of the atmosphere. These waves are generated by various sources (both natural and technogenic ones), which are accompanied by a significant energy output into the atmosphere. Further, when the AGWs propagate upward, the energy conservation compensates for the decrease in the atmospheric density with the height by exponentially increasing amplitude. Therefore at a certain height the waves become nonlinear. Significant progress in the development of the nonlinear theory of AGWs was achieved by a number of authors, in particular, Belashov (1990), Nekrasov et al. (1995), Kaladze et al. (2008), Stenflo and Shukla (2009), and Huang et al. (2014). Numerical modeling of the freely propagating AGWs in the realistic viscous and heat-conducting atmosphere is an important area of modern studies of these waves (i.e., Cheremnykh et al., 2010; Vadas and Nicolls, 2012).

Satellite observations of AGWs in the Earth's polar thermosphere indicate a prevailing presence of waves with oscillation periods concentrated around the Brunt–Väisälä period and of horizontal scales of about 500–700 km (Johnson et al., 1995; Innis and Conde, 2002; Fedorenko et al., 2015). Azimuths of the propagation of these AGWs demonstrate the close connection with the directions of background winds in the thermosphere. Moreover, the amplitudes of the waves depend on the speed of headwind, but do not depend on height (Fedorenko and Kryuchkov, 2013; Fedorenko et al., 2018). These experimental results cannot be sufficiently explained by the theory of freely propagating AGWs. They may indicate waveguide or evanescent (along a horizontal surface) propagation of at least part of the observed waves.

As well as freely propagating AGWs, evanescent wave modes also play an important role in atmospheric dynamics of the Sun and planets. Evanescent waves propagate horizontally in an atmosphere, vertically stratified by gravity, subject to the presence of vertical gradients of parameters. The energy of these waves should decrease both up and down from the level at which they are generated. Therefore, evanescent waves are most effectively generated in areas of presence of significant vertical gradients of temperature and density or strong local currents. For example, in the solar atmosphere suitable conditions for realization of evanescent modes occur at the boundary between the chromosphere and corona. This follows from the analysis made by Jones (1969) for the so-called non-divergent modes of solar oscillations. In the Earth's atmosphere, such waves can be efficiently generated at sharp vertical temperature gradients, for example, at the base of the thermosphere or at the heights of the tropopause and mesopause. Also, evanescent wave modes can emerge in the presence of strong inhomogeneous winds, for example, in the region of the polar circulation of the thermosphere.

The study of evanescent waves traditionally gets less attention than the study of freely propagating AGWs. The most known of them are the horizontal Lamb wave and vertical oscillations with Brunt–Väisälä (BV) frequency (Beer, 1974; Waltercheid and Hecht, 2003). In hydrodynamics, physics of terrestrial and solar atmosphere, the surface gravity mode with dispersion ω2=kxg, is also well studied (Tolstoy, 1963; Jones, 1969). In particular, it was shown that it is the fundamental mode (f-mode) of oscillations in the solar atmosphere (Jones, 1969). Experimental f-mode observations are used to study flows, refinement of the solar radius, and other parameters of the Sun (Ghosh et al., 1995; Antia, 1998). In the Earth's atmosphere, evanescent waves are often observed at altitudes near the mesopause using ground-based instrumentation (Shimkhada et al., 2009).

In this paper, different types of evanescent acoustic-gravity modes characteristic of an isothermal atmosphere are investigated using a set of linearized hydrodynamic equations. In particular, the possibility of the existence of a new type of evanescent acoustic-gravity mode with the dispersion ω2=kxgγ-1 is proved in the assumption of anelasticity of the disturbance. Also, the possibility of realizing the evanescent modes in the model of a thin temperature gap is studied.

Consider an unbounded ideal isothermal atmosphere, stratified in a field of gravity. Linear perturbations in such a medium satisfy a set of four first-order hydrodynamic equations (Hines, 1960). These equations are convenient to bring to a set of two second-order equations for the perturbations of the horizontal Vx and vertical Vz particle velocities (Tolstoy, 1963): 1ρ0∂2Vx∂t2=-ρ0g∂Vz∂x+∂∂xρ0c2∂Vx∂x+∂Vz∂z,2ρ0∂2Vz∂t2=ρ0g∂Vx∂x+∂∂zρ0c2∂Vx∂x+∂Vz∂z, where ρ0, γ, and g denote background atmosphere density, ratio of specific heats, and acceleration of gravity, respectively; c=γgH is the sound speed, H=-ρ0/dρ0/dz=kT/mg is the density-scale height, T is the temperature, k is the Boltzmann constant, and m is the molecular mass of the atmospheric gas.

Solutions to Eqs. () and () are searched for in the form 3Vx,Vz∼exp⁡azexp⁡iωt-kxx, where ω and kx are cyclic frequency and horizontal component of the wave vector, respectively; parameter a sets the vertical scale of the change in the amplitude of velocities, Vx and Vz, with the height, z. For brevity, we will refer to a as the stratification of the corresponding mode.

Equations () and () admit, similarly to Hines (1960), on the existence on the “frequency–wave number” plot of regions of freely propagating gravity and acoustic waves, in which a=12H±ikz and kz is the vertical component of the wave vector. Also, from Eqs. () and () we get the solutions in the form of evanescent wave modes with real a and propagating horizontally (Waltercheid and Hecht, 2003). Solutions in the form of evanescent modes are usually obtained by imposing additional conditions on the perturbation properties.

Let us prove that the different types of evanescent modes characteristic of an isothermal atmosphere are related. We substitute Eq. () into Eqs. () and () without additional conditions that were imposed in Sect. 2 when deriving ND and AE modes. As a result, we get 20Vza-gc2-ikxVx1-ω2kx2c2=0,21Vxa-N2g-ikxVzN2ω2-1=0, where N2=gHγ-1γ is the square of the Brunt–Väisälä frequency.

From Eqs. () and () we obtain the dispersion equation 221γH-aa-γ-1γH=kx2N2ω2-11-ω2kx2c2. Expressions ω2=N2 and ω2=kx2c2 are well-known dispersions of Brunt–Väisälä oscillations with a=1/γH and Lamb waves (L) with a=γ-1/γH; in addition to these known modes, dispersion (Eq. ) also admits the existence of additional solutions in the form of BV pseudo-modes (BVp) with ω2=N2, a=γ-1/γH and Lamb pseudo-modes (Lp) with ω2=kx2c2, a=1/γH (Beer, 1974; Waltercheid and Hecht, 2003).

Then represent Eq. () in the form of a quadratic equation with respect to a: a2-aH+ω2c2-kx2+kx2g2ω2c2γ-1=0. The solution to this equation is 23a=12H±kxg-ω2ω2-kxgγ-1ω2c2+kx-12H2, from which it follows that for modes with dispersions ω2=kxgγ-1 and ω2=kxg there are two possible values: a=kx and a=1H-kx. The first value corresponds to modes ND and AEp, and the second to NDp and AE.

Thus, each evanescent mode can be associated with a pseudo-mode which satisfies the same dispersion relation but differs in polarization and dependence of the amplitude from the height, i.e., in its stratification. Table 1 presents the properties of different evanescent modes characteristic of the isothermal atmosphere: BV oscillations, Lamb waves, non-divergent and anelastic modes, along with associated pseudo-modes: BVp, Lp, NDp, AEp. Table 1 shows that for all pseudo-modes, the polarization changes depending on the value of kx. Wave modes AE and ND at kx=1/2H completely coincide with AEp and NDp, respectively.

The location of the dispersion curves for anelastic and non-divergent modes relative to gravity and acoustic regions in the (ω,kx) plane is shown in Fig. 1. The ω2=kxgγ-1 mode touches the gravity region of freely propagating AGWs at the same value kx=1/2H at which the ω2=kxg curve touches the acoustic region (see Fig. 1). In this case, the dispersion curves of AE and ND modes are symmetric relative to the “characteristic” curve (see Beer, 1974), which separates the AGW acoustic region from the AGW gravity region. In fact, the characteristic curve is the geometric mean of the dispersion curves of AE and ND modes with ω2=kx2g2γ-1=Nkxc.

From Fig. 1 we see that the dispersion curves of different evanescent modes have intersections at separate points. A Lamb dispersion curve with ω2=kx2c2 intersects the BV curve with ω2=N2 at the point kx=N/c. However, these modes cannot interact with each other by reason of different polarizations and values of a. At the same time, the pairs Lp-BV and L-BVp completely coincide in these properties and are indistinguishable at the intersection points.

Dispersion curves ω2=kxg and ω2=kxgγ-1 intersect with the Lamb curve and the BV curve at points kx=1/γH, kx=γ-1/γH. In addition, the ND mode curve intersects with the Lamb curve at the same value kx at which the AE mode curve intersects with the BV curve (see Fig. 1). ND and AE modes cannot interact with the Lamb mode and BV oscillations due to different polarizations (Table 1). Pseudo-modes NDp and AEp, at the points of intersection with the Lamb wave and the BV oscillations, have the same polarization and values of a. Similarly, ND and AE are indistinguishable at the points of intersection with Lp and BVp. Table 2 shows all evanescent modes that coincide with each other at the points of intersection of the dispersion curves, and between which interaction is possible. The cases of ND and AE mode curves intersection with curves (a=1/2H), which separate the area of freely propagating AGWs from the evanescent area, are not presented in Table 2.

Let us consider the possibility of realization of evanescent modes in the atmosphere at a thin interface between two isothermal half-spaces of infinite extent, which differ in temperature T. Let the boundary be localized at some altitude level z=0. In the lower half-space (z<0) we have T=T1, while in the upper half-space (z>0) we have T=T2 and it is assumed that T2>T1. Note that a similar model was considered by Rosental and Gough (1994). We will search for solutions to Eqs. () and () in the form of Vx,Vz∼exp⁡a1zexp⁡iωt-kxx for the lower half-plane and in the form Vx,Vz∼exp⁡a2zexp⁡iωt-kxx for the upper half-plane. Substituting these dependencies into Eqs. () and () yields 27a1=12H1±14H12-ω2c12+kx2-kx2N12ω21/2,28a2=12H2±14H22-ω2c22+kx2-kx2N22ω21/2.

Here indices 1 and 2 denote the values in the lower and upper half-spaces, respectively.

The density of the kinetic energy of evanescent waves should decrease from the level z=0 both up and down. This condition limits the possible values of a1 and a2. In the upper half-space (z>0), when z→+∞, the energy density E2∼exp⁡2a2-1H2z→0 if a2<1/2H2. In the lower half-space (z<0), when z→-∞, the energy density E1∼exp⁡2a1-1H1z→0 if a1>1/2H1. Therefore, it is necessary to take in Eq. () for a1 the solution with a “+” sign and in Eq. (28) for a2 with a “-” sign, so that the energy decreases on both sides of the interface.

It is also necessary to consider that the possible values of a1 and a2 must satisfy the boundary condition (Tolstoy, 1963; Rosental and Gough, 1994), arising from Eqs. () and (): 29ρ1c12gkx2-ω2a1ω2-c12kx2z=-0=ρ2c22gkx2-ω2a2ω2-c22kx2z=+0, where ρ1 and ρ2 are the densities on both sides of the boundary. The procedure for deriving equality (Eq. ) is exactly the same as in the papers by Cheremnykh et al. (2018a, b). When obtaining Eq. () we require continuity of the vertical velocity component (kinematic condition) and perturbed pressure (dynamic condition). In the barometric atmosphere we have ρc2=γp0, where p0 is the equilibrium pressure, which must be continuous across the interface. Therefore, when γ1=γ2, Eq. () can be written as 30gkx2-ω2a1ω2-c12kx2=gkx2-ω2a2ω2-c22kx2.

Dispersion dependencies of ω=fkx calculated numerically by means of Eq. () are shown in Fig. 2a for different values of the parameter d=H2/H1. On each of these curves, the condition for decreasing energy up and down from the interface is satisfied. The long-wavelength part of the spectrum, where the most interesting features appear, is shown in more detail in Fig. 2b. Also shown in these figures are the dispersion curves ω=kxg and ω=kxgγ-1 for the ND and AE wave modes. The discontinuities of the ω=fkx curves, as well as their cut-off for smaller kx values, are due to requirements a1>1/2H1 and a2<1/2H2. Some features of the behavior of ω=fkx will be discussed below.

As shown by Miles and Roberts (1992), the dispersion Eq. () can be rewritten to a polynomial form suitable for analysis: ω8-2c12d+1kx2ω6+c14d+12kx4+2γ-1kx2g231×ω4-2γ-1c12d+1kx4g2ω2-c14d-12kx6g2=0. Non-physical solutions (Miles and Roberts, 1992) arising from quadratic expressions under the radicals were excluded from consideration while obtaining Eq. () (see Eqs.  and 28). Solutions of Eq. () can be analyzed by studying their asymptotic behavior.

If kx2c12≫ω2, then from Eq. (), we get ω4-2N12d+1ω2-d-12d+12kx2g2≈0. It follows from this expression that 32ω2=1d+1N12+N14+d-12kx2g2. Equation () contains an interesting dependence of the frequency on the parameter d. In the limit d→∞, the dispersion ω2≈kxg of the ND (NDp) mode, independent of the properties of both environments, follows from Eq. (). With d→1 and using Eq. (), we obtain the dispersion of the BV (BVp) mode with the parameters of the lower medium, that is, ω2≈N12. The indicated asymptotic features are visible on the curves shown in Fig. 2 below.

In the long-wave limit, i.e., at kx→0, from Eq. () it follows that 2γ-1ω4-2γ-1c12d+1kx2ω2-c14d-12kx4≈0. Hence we find that 33ω2=c12kx22γ-1γ-1d+1+γ2d+12-4d2γ-1. For the considered small kx, for different values of d, from Eq. () we obtain the family of Lamb-type acoustic modes (see Fig. 2b). For large values of d, using Eq. (), we obtain the expression ω2≈c12kx2d=c22kx2; i.e., the oscillation frequency is determined by the characteristics of the medium in the upper half-space.

The evanescent modes' frequencies lie on the ω,kx plane between the acoustic and gravity regions of freely propagating AGWs determined for upper and lower media separately (see Fig. 1). It is necessary to take into account when considering evanescent modes at the boundary of two isothermal media with different temperatures that the evanescent regions are different in the upper and lower half-planes. On the ω,kx plane, these regions are shifted more relative to each other the more the value of d is. At the same time, the wave modes at the interface of the media should remain evanescent in both media, and their dispersions should be enclosed within the overlap region of two evanescent regions. The cut-off curves for evanescent regions in the media under consideration are obtained in the case of the null expressions under the radicals in Eqs. () and (28). Gaps on the ω=fkx dispersion curves are due to the evanescent areas of the two media not matching (see Fig. 3).

Note that the dispersion curves ω=fkx for values d≤4 are mostly inside both evanescent regions (see Fig. 3a, b), except for the longest waves. When d≥4, the dispersion curve ω=fkx breaks into two separate branches (see Fig. 3c, d). The long-wave branch is acoustic, and another branch with kx≥0.4H1 is surface gravity by its physical nature.

In an unlimited isothermal medium, evanescent modes are separate “pure” solutions of hydrodynamic equations. At the interface between two isothermal media with different temperatures, dispersion of the evanescent modes has a combined character, comprising different types of “pure” modes, depending on the value of the parameter d and spectral properties ωkx.

For some values of d, the curves of the dispersion Eq. () approach fairly closely the curves ω2=kxg and ω2=kxgγ-1, and also intersect them at different points. These intersection points correspond to the specific value of kx at which the dispersions of the ND and AE modes are realized, in the model under consideration, in a “pure” form. Let us now examine these cases in more detail. For this purpose, we substitute the dispersion relations ω2=kxg and ω2=kxgγ-1 directly into Eqs. () and (28), and then into the boundary condition (Eq. ).

As was shown in Sect. 2, for the dispersion relations ω2=kxg and ω2=kxgγ-1, the values of each of the parameters a1 and a2 are the same for both relations and are determined by Eq. (). Consider the valid values of a1 and a2 for these dispersions with regard to the requirement of energy decay in both directions from the interface a1>1/2H1 and a2<1/2H2.

Let us dwell on some of the results in terms of their use for the analysis of experimental data.

With the f-mode observed on the Sun, one should identify the mode that we classify as the ND mode, for which ω2=kxg, Vz∼exp⁡kxz and divV=0 (Roberts, 1991). In the framework of the considered temperature discontinuity model, it was shown that with T1<T2 (corresponding to the chromosphere–corona interface) the condition for decreasing amplitude with height to both sides of the interface is satisfied only by the NDp mode with ω2=kxg, Vz∼exp⁡1H-kxz and divV≠0. When the ratio d→∞ (i.e., H2/H1→∞), the NDp mode with kx→1/2H1 asymptotically approaches the ND mode. On the interface between the chromosphere and the solar corona, d is large but of finite magnitude: d∼50 (Jones, 1969; Athay, 1976). Therefore, the condition of the presence of a free surface, which is required for the realization of the ND mode, is fulfilled only approximately. Therefore, in the framework of the temperature discontinuity model, the f-mode observed on the Sun should not be associated with the non-divergent ND mode, but with non-divergent pseudo-mode NDp.

For the Earth's atmosphere, the maximum possible value of d is observed at the interface between the thermosphere with T2∼800–1500 K (depending on solar activity) and the underlying atmosphere with T1∼300 K. When d=5, the dispersion (Eq. ) asymptotically tends to ω2=kxgγ-1 with kx→∞. Therefore, it can be expected that evanescent modes in this case will be close to ω2=kxgγ-1.

In other layers of the Earth's atmosphere we have d≤1.3 (Jursa, 1985). As follows from Eq. (), for small values of d≤1.3 and for the wavelengths in the interval kx∼0.5-1.5H1, the relation ω2→N2 is satisfied (see Fig. 2). Therefore, it can be expected that at small positive temperature gradients in the atmosphere, waves with a frequency close to the frequency of Brent–Väisälä should prevail. These conclusions experimentally confirm (Shimkhada et al., 2009) the results of observations of short-period evanescent waves with small wavelengths at altitudes near the mesopause.

In the paper, different types of evanescent acoustic-gravity modes characteristic of an isothermal atmosphere are investigated. A new mode was derived in the form of anelastic acoustic-gravity wave mode with the dispersion equation ω2=kxgγ-1. The main properties of the AE mode are presented in Table 1 in comparison with other known evanescent modes. It is shown that for both anelastic and non-divergent modes there are pseudo-modes that satisfy the same dispersions but have different polarization and the dependence of the amplitude of the disturbances on the height.

For AE and ND evanescent modes, the value of kx→1/2H sets a special scale (wavelength) at which these modes are identical to their pseudo-modes AEp and NDp. In addition, at the same point they are adjacent to the boundaries of the continuous spectrum (AE mode to the gravity region and ND mode to the acoustic region, respectively).

The features of the evanescent modes' realization at the interface of two isothermal media are considered. It is shown that in this case, dispersions of evanescent modes are combined, merging the features of different types of modes characteristic of an unbounded isothermal atmosphere. This effect is most pronounced in the following asymptotic cases: (1) when d→∞, we obtain the dispersion for the ND (NDp) mode in the form ω2≈kxg; (2) when d→1, for scales kx∼H1, a mode with ω2≈N12 is realized; (3) for kx→0, a Lamb wave with a dispersion relation of the form ω2≈c22kx2 is obtained, which depends only on the parameters of the medium in the upper half-space.

It was demonstrated that on the interface of two isothermal media with T2>T1, the NDp mode with the dispersion ω2=kxg and the selected scale kx∼1/2H1 is realized. At the same time, the ND mode does not satisfy the condition of decreasing energy on each side of the interface. Dispersion ω2=kxgγ-1 on the interface of two media is satisfied by the wave mode, which has different types of amplitude versus height dependencies at different horizontal scales kx. When kx>1/2H1, the height dependence of AE amplitude for z>0 and AEp amplitude for z<0 satisfy the condition of decreasing energy from the interface. By contrast, when kx<1/2H1, this condition is satisfied by AEp amplitude for z>0 and AE amplitude for z<0.

It is important to note that according to our analysis in the framework of the temperature discontinuity model, (1) the f-mode observed on the Sun should not be associated with the non-divergent (ω2=kxg, divV=0) mode, but with its non-divergent pseudo-mode (ω2=kxg, divV≠0). (2) At the interface between the Earth's thermosphere and the underlying atmosphere it can be expected that evanescent modes with short wavelengths will be close to the new mode (ω2=kxgγ-1). (3) Oscillations with a frequency close to the frequency of Brent–Väisälä should prevail at altitudes near the Earth's mesopause.