The objects of research in this work are evanescent wave
modes in a gravitationally stratified atmosphere and their associated
pseudo-modes. Whereas the former, according to the dispersion relation,
rapidly decrease with distance from a certain surface, the latter, having the
same dispersion law, differ from the first by the form of polarization and
the nature of decrease from the surface. Within a linear hydrodynamic model,
the propagation features of evanescent wave modes in an isothermal atmosphere
are studied. Research is carried out for different assumptions about the
properties of the disturbances. In this way, a new wave mode – anelastic
evanescent wave mode – was discovered that satisfies the dispersion relation

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

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

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

Solutions to Eqs. (

Equations (

Let us note the well-known hydrodynamics approximation of perturbations
incompressibility (see, e.g., Ladikov-Roev and Cheremnykh, 2010),
for which

After substituting Eq. (

Let us show that the dispersion relation (Eq.

Let us show that Eqs. (

After substituting the dispersion (Eq.

Polarization of the AEp mode has the form

Let us prove that the different types of evanescent modes characteristic of
an isothermal atmosphere are related. We substitute Eq. (

From Eqs. (

Then represent Eq. (

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

Properties of different evanescent acoustic-gravity modes.

The location of the dispersion curves for anelastic and non-divergent modes
relative to gravity and acoustic regions in the (

Dispersion dependencies

From Fig. 1 we see that the dispersion curves of different evanescent modes
have intersections at separate points. A Lamb dispersion curve with

Dispersion curves

The coincidence of the evanescent mode properties at the intersection points of the dispersion curves. Note: the bottom rows show the modes that are indistinguishable from the corresponding mode of the top row at the point of intersection of the dispersion curves.

In Sects. 2 and 3, we considered a model of an unbounded isothermal
stratified atmosphere to determine which types of evanescent modes can
satisfy the initial system of Eqs. (

Suppose further that an evanescent wave is generated at a certain altitude
level

The change in energy density of evanescent modes with height in an infinite isothermal atmosphere.

The presence of boundaries is not the only condition that can limit the
energy of the evanescent mode. If the equality

Consider some features of the energy balance for the evanescent modes. It
follows from Eq. (

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

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

It is also necessary to consider that the possible values of

Dispersion dependencies

Dispersion dependencies of

As shown by Miles and Roberts (1992), the dispersion Eq. (

If

In the long-wave limit, i.e., at

The evanescent modes' frequencies lie on the

Dispersion dependencies of the

Note that the dispersion curves

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

For some values of

As was shown in Sect. 2, for the dispersion relations

For a dispersion of the form

Horizontal scales

Take the stratification of the NDp modes in the form of

When combining the stratifications for ND modes as

Thus, consideration of the possible values of

For the AE stratification of the form

It should be noted that for the dispersion of the form

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

With the

For the Earth's atmosphere, the maximum possible value of

In other layers of the Earth's atmosphere we have

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

For AE and ND evanescent modes, the value of

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

It was demonstrated that on the interface of two isothermal media with

It is important to note that according to our analysis in the framework of
the temperature discontinuity model, (1) the

No data sets were used in this article.

This article has been prepared by the authors with equal contributions.

The authors declare that they have no conflict of interest.

This article is part of the special issue “Solar magnetism from interior to corona and beyond”. It is a result of Dynamic Sun II: Solar Magnetism from Interior to Corona, Siem Reap, Angkor Wat, Cambodia, 12–16 February 2018.

This publication is based on work supported in part by Integrated Scientific Programmes of the National Academy of Science of Ukraine on Space Research and Plasma Physics.

This paper was edited by Sergiy Shelyag and reviewed by Tamaz Kaladze and one anonymous referee.