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
ω2=kxgγ-1. Also, the possibility of the
existence of a pseudo-mode related to it is indicated. The case of two
isothermal media differing in temperature at the interface is studied in
detail. It is shown that a non-divergent pseudo-mode with a horizontal scale
kx∼1/2H1 can be realized on the interface with dispersion
ω2=kxg. Dispersion relation ω2=kxgγ-1 at 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. The applicability of the obtained
results to clarify the properties of the f-mode observed on the Sun is
analyzed.
Introduction
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.
Evanescent modes in the isothermal atmosphere
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
Vx,Vz∼expazexpiωt-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.
Non-divergent and pseudo-non-divergent modes
Let us note the well-known hydrodynamics approximation of perturbations
incompressibility (see, e.g., Ladikov-Roev and Cheremnykh, 2010),
for which
divV=∂Vx∂x+∂Vz∂z=0.
In frames of this approximation, we
obtain the following equations from Eqs. () and ():
5∂2Vx∂t2=-g∂Vz∂x,6∂2Vz∂t2=g∂Vx∂x.
After substituting Eq. () into Eqs. () and (6), we find
that
-ω2Vx=ikxgVz,-ω2Vz=-ikxgVx.
This yields a dispersion equation for incompressible wave modes in the form
ω2=kxg.
Given the dispersion found, we obtain an expression for the polarization of
the incompressible modes:
Vz=iVx.
Further, from the condition (Eq. ) and
polarization (Eq. ) we get a=kx. Insofar as a is a real
value, then non-divergent (ND) wave mode has no periodic vertical solution
and is horizontally propagating.
Let us show that the dispersion relation (Eq. ) is also satisfied by
another wave mode. After using this relation in
Eqs. () and (), we get
9VxγHkx-1-iVz1-γaH=0,10iVx1-γHa-γ-Vz1+γHa2kx-γakx=0.
From Eqs. () and () follow
a2-aH+kxH1-kxH=0,
which implies that there are two solutions to this equation:
a=kx,a=1H-kx.
The first solution in Eq. () corresponds to the non-divergent (ND)
wave mode, and the second one we call pseudo-non-divergent mode (NDp). The
expression for polarization NDp is obtained from Eq. () and has the
form
Vx1γH-kx=-ikx-γ-1γHVz.
Also for this mode, the following equation holds:
divV=VzH1-2kxH1-γkxH,
which shows that for the NDp mode divV=0 only when kx=1/2H.
Anelastic and pseudo-anelastic modes
Let us show that Eqs. () and () indicate that another wave mode, not
previously studied, may exist. To do this, we introduce, according to Bannon (1996), the anelastic linear perturbations, which satisfy the condition
divρ0V=0.
In the isothermal atmosphere with barometric density distribution we have
∂ρ0∂z=-ρ0H;
therefore, for such anelastic perturbations, the following equation holds:
divV=VzH.
Substituting Eq. () into Eqs. () and (), we get
∂2Vx∂t2=gγ-1∂Vz∂x,∂2Vz∂t2=-gγ-1∂Vx∂x.
Thus, given Eq. (), this should be
14ω2Vx=ikxgγ-1Vz,15ω2Vz=-ikxγ-1gVx.
Then the dispersion equation for anelastic (AE) modes takes the form
ω2=kxgγ-1.
With the resulting dispersion, polarization follows from Eqs. () and ():
Vx=iVz.
Further, taking into account Eq. (), we obtain a=1H-kx.
Consequently, the AE mode also does not have a solution periodic vertically
and can only propagate horizontally.
After substituting the dispersion (Eq. ) into
Eqs. () and (), we get
18Vx1-γ+γHkx-iVz1-γaH=0,19iVx1-γ+γHa+Vz1-γ-γHa2kx+γakx=0,
whence we get a pair of values a identical to Eq. ().
Consequently, there is another wave solution that satisfies Eq. ():
we call it the pseudo-anelastic (AEp) mode. The first value in
Eq. () corresponds to the AEp wave mode, and the second to the AE
one.
Polarization of the AEp mode has the form
Vxkx-γ-1γH=-i1γH-kxVz
that follows from Eqs. () or ().
General properties of evanescent modes
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
1γH-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
a=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.
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 (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.
Dispersion dependencies ω=fkx:
(1) boundaries between acoustic and gravity regions for freely propagating
waves (dashed lines); (2) evanescent modes: ω=kxg (upper
solid curve) and ω=kxgγ-1 (lower
solid curve), ω=N (thin horizontal line), and ω=kxc (thin
sloping straight line).
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.
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.
LambLamb's pseudo-BVBV pseudo-Non-divergentPseudo-non-AnelasticPseudo-anelasticwave (L)mode (Lp)oscillationsmode (BVp)mode (ND)divergentmode (AE)mode (AEp)(BV)mode (NDp)BVpBVLpLLpLLpLNDpNDNDpNDBVpBVBVpBVAEpAEAEpAEThe energy of evanescent modes in an isothermal atmosphere
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. () and (). However, in an infinitely
extended medium, the necessary condition for the existence of evanescent
modes is the absence of unlimited growth of oscillation energy above and
below the height level at which they are generated. It is easy to verify
that in an isothermal infinite atmosphere, none of the modes listed in Table 1 satisfy this condition.
Suppose further that an evanescent wave is generated at a certain altitude
level z=0. The kinetic energy density E∼ρzVx2+Vz2 of waves should decrease both up and down
from the level z=0. When z→+∞ the energy density E∼exp2a-1Hz→0, if a<1/2H, and E→∞, if
a>1/2H. When z→-∞ the energy density E→0, if a>1/2H, and
E→∞, if a<1/2H. Based on these considerations, it is not
difficult to understand how the energy density varies with height for
different types of evanescent modes in an infinite isothermal atmosphere (see
Table 3). Therefore, for the realization of such modes, it is necessary to
have boundaries in the medium at which the condition for reducing energy in
both directions from this boundary can be satisfied.
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 a=1/2H holds for these
modes, then their energy does not vary with height in an isothermal
atmosphere. For an infinite atmosphere, this solution does not seem to be
physical, but it can make sense for a real atmosphere of finite height. As
follows from Eq. (), for the ND and AE modes, as well as their
pseudo-modes, the condition a=1/2H performed at the point kx=1/2H.
Also, at this point, the ND mode is identical to the NDp mode, and the AE
mode completely coincides with AEp. In addition, when kx=1/2H these
evanescent modes adjoin the border of regions of freely propagating AGWs (see
Fig. 1).
Consider some features of the energy balance for the evanescent modes. It
follows from Eq. () that
Vz2a-gc22=kx2Vx21-ω2kx2c22.
Combining Eqs. () and () gives the relation
ρ0Vx21-ω2kx2c2a-N2g25=ρ0Vz2N2ω2-1gc2-a.
The average density of the kinetic energy of the perturbations is
Ek=14ρ0Vx2+Vz2,
and of the potential energy it is Ep=14ρ0Vx2ω2kx2c2+Vz2N2ω2 (Yeh and Liu, 1974; Fedorenko, 2010). Therefore, from
Eq. () it follows that for the evanescent modes Ek≠Ep. At the same time, for freely propagating AGWs, the equality
Ek=Ep is always fulfilled (Yeh and Liu, 1974). At the
point a=1/2H where evanescent modes on the plane (ω,kx) in
Fig. 1 are adjacent to areas of freely propagating AGWs, the equality
a-N2g=gc2-a holds. Taking this circumstance into
account, from Eq. () we obtain
ρ04Vx2+Vz2=ρ04Vx2ω2kx2c2+Vz2N2ω2;
that is, at this point Ek=Ep.
Evanescent modes at the interface of isothermal media
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∼expa1zexpiωt-kxx for the lower half-plane and in the form
Vx,Vz∼expa2zexpiωt-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∼exp2a2-1H2z→0 if a2<1/2H2. In the lower half-space (z<0),
when z→-∞, the energy density E1∼exp2a1-1H1z→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 ():
ρ1c12gkx2-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
gkx2-ω2a1ω2-c12kx2=gkx2-ω2a2ω2-c22kx2.
Dispersion dependencies ω=fkx at the
boundary of the discontinuity for different values of the parameter d.
General dependence (a), long-wave part in more detail (b).
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
ω2=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
ω2=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).
Dispersion dependencies of the ω=fkx
type at the temperature discontinuity boundary for d=2(a),
d=3(b), d=5(c), and d=20(d). The dashed
curves represent the boundaries of the areas with free propagation of AGWs in
the upper and lower half-spaces, respectively.
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.
Characteristic scales of ND and AE evanescent modes on the discontinuity
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.
Dispersion of the form ω2=kxg
For a dispersion of the form ω2=kxg, we first analyze the
stratification of the ND mode with a1=kx, a2=kx. In order
for the energy of this mode to decay in both directions from the
discontinuity, the following inequalities 1/2H1<kx<1/2H2 must
be satisfied, i.e., H1>H2. Therefore, the ND mode can be realized
at the discontinuity if the ambient temperature in the upper region is less
and the density is greater than they are in the lower region. This situation
corresponds to the unstable state of the atmosphere (see Roberts, 1991).
Horizontal scales kxH1, on which the modes with the
dispersions ω2=kxg(a) and ω2=kxgγ-1(b) are realized, depending on d=H2/H1. (See text for details.)
Take the stratification of the NDp modes in the form of a1=1H1-kx, a2=1H2-kx. The energy in
this case decreases both ways from the discontinuity, if 1/2H2<kx<1/2H1, i.e., when H2>H1. This condition corresponds to the
stable state and the case under consideration. For the NDp mode from the
dispersion Eq. () we get
H21γH2-kx2kx-1H1=H11γH1-kx2kx-1H2,34kx≠1/γH1,kx≠1/γH2.
From Eq. () it follows that
kx=d+14dH11±1-8dγd2-1.
Figure 4a shows values of kx for which the dispersion curve ω2=kxg intersects with the calculated dispersion curve ω=fkx depending on the parameter d. The upper solid
curve in this figure corresponds to the solution (Eq. ) with the
sign “+” before the radical and shows the points of intersection with the
shorter wavelength branch. The lower dashed curve corresponds to the solution
with a sign “-” and represents the points of intersection with the
long-wavelength branch. For the upper curve kx→1/2H1 when d→∞. For d<2.5, there are no intersections of the curve ω=fkx calculated numerically from Eq. () with
the curve for the dispersion ω2=kxg.
When combining the stratifications for ND modes as a1=kx and for
NDp modes as a2=1H2-kx, Eq. () yields the only
possible value of kx=1/2H2. For a combination of stratifications
a1=1H1-kx (NDp), a2=kx (ND) we get kx=1/2H1. Both of these cases do not satisfy the condition of energy
decrease with height.
Thus, consideration of the possible values of a1 and a2 leads to
the conclusion that on the interface of two isothermal media with H2>H1, the NDp mode can only be
implemented with a dispersion ω2=kxg and a specific scale
kx∼1/2H1.
Dispersion of the form ω2=kxgγ-1
For the AE stratification of the form a1=1H1-kx,
a2=1H2-kx and for the AEp stratification of the form
a1=kx, a2=kx, from the dispersion Eq. ()
follows the identity H1=H2. Therefore, such modes are not realized
at a temperature discontinuity. Apparently, to study the conditions of
realization of AE and AEp modes, it is necessary to consider atmospheric
models in which height profile Hz is continuous.
It should be noted that for the dispersion of the form ω2=kxgγ-1, cases of combined mode stratifications are
possible, satisfying the condition of decreasing energy on both sides of the
boundary. So, for the combination of stratifications a1=kx (AEp),
a2=1H2-kx (AE) from Eq. (), we obtain the
relation
H2kx2-γ=H1γkx-γ-1γH1,
whence kx=γ-1H1γ-2-γd. In this case, the inequality d<γ/2-γ must be satisfied. When γ=5/3 we get the following
restriction: d<5. Given this limitation and condition kx>1/2dH1, we find that a mode with a dispersion of ω2=kxgγ-1 and stratification of AE type for the upper
half-space and of AEp type for the lower half-space can propagate at the
boundary in the range 1<d<5 and for kx>1/2H1. For the
stratifications a1=1H1-kx (AE), a2=kx (AEp)
from Eq. () we obtain the relation
H2γkx-γ-1γH2=H1kx2-γ.
It implies the ratio kx=γ-1H1γd-2-γ, in which the parameter d can take any values
with d>1, and the horizontal wave number is limited by the inequality
kx<1/2H1. Features of the behavior of the ω2=kxgγ-1 mode at the discontinuity, depending on the
scale kx, are shown in Fig. 4b.
Discussion
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∼expkxz 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∼exp1H-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.
Main results
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.
Data availability
No data sets were used in this article.
Author contributions
This article has been prepared by the authors with equal contributions.
Competing interests
The authors declare that they have no conflict of
interest.
Special issue statement
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.
Acknowledgements
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.
Review statement
This paper was edited by Sergiy Shelyag and reviewed by Tamaz Kaladze and one anonymous referee.
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