**1. Introduction**

302 Acoustic Waves – From Microdevices to Helioseismology

Vargas, H.; Miranda, L.C.M. (1988) Photoacoustic and related photothermal techniques.

Viator, J.A.; Komadin, J.; Svaasand, L.O.; Aguilar, G.; Choi, B.; Nelson, J.S. (2004) A

Comparative Study of Photoacoustic and Reflectance Methods for Determination of Epidermal Melanin Content. *Journal of Investigative Dermatology*, v.122, pp.1432-

*Physics Reports*, v.161, n.2, pp.43-101.

1439.

Terrestrial atmosphere shows a high variability over a broad range of periodicities, which mostly consists of wave-like perturbations characterized by various spatial and temporal scales. The interest for short time variability in ionospheric attributes is related to the role that ionosphere plays in the Earth's environment and space weather. Acoustic-gravity waves (AGWs), waves in the period range from sub-seconds to several hours, are sources of most of the short-time ionospheric variability and play an important role in the dynamics and energetics of atmosphere and ionosphere systems. Many different mechanisms are likely to contribute to the acoustic-gravity wave generation: for instance, excitation at high latitudes induced by geomagnetic and consequent auroral activity, meteorological phenomena, excitation in situ by the solar terminator passages and by the occurrence of solar eclipses.

During solar eclipse, the lunar shadow creates a cool spot in the atmosphere that sweeps at supersonic speed across the Earth's atmosphere. The atmosphere strongly responds to the decrease in ionization flux and heating. The very sharp border between sunlit and eclipsed region, characterized by strong gradients in temperature and ionization flux, moves throughout the atmosphere and drives it into a non-equilibrium state. Acoustic-gravity waves contribute to the return to equilibrium. At thermospheric heights, the reduction in temperature causes a decrease in pressure over the totality footprint to which the neutral winds respond. Thermal cooling and downward transport of gases lead to neutral composition changes in the thermosphere that have significant influence on the resulting electron density distribution. Although the mechanisms are not well understood, several studies show direct evidence that solar eclipses induce wave-like oscillations in the acousticgravity wave domain.

Many different mechanisms are likely to contribute to wave generation and enhancement at ionospheric heights. Hence, it is difficult to clearly separate or differentiate each contributing agent and to decide which part of wave field belongs to the in situ generated and which part comes from distant regions. First experimental evidence of the existence of gravity waves in the ionosphere during solar eclipse was reported by Walker et al. (1991), where waves with periods of 30–33 min were observed on ionosonde sounding virtual heights.

### **1.1 Ionospheric sounding**

As the solar radiation penetrates Earth's atmosphere it forms pairs of charged particles. Under a normal day-time conditions the ionization solar flux increases immediately after

Acoustic–Gravity Waves in the Ionosphere During Solar Eclipse Events 305

γ is the ratio of specific heats at constant pressure and constant volume, g is the gravitational

(a)

(b) Fig. 1. Typical day-time (a) and night-time ionogram (b) measured by digisonde DPS 4 in the Observatory Pruhonice. On both plots, there is real hight electron concentration (solid

line with error bars) provided as obtained by the NHPC routine.

acceleration, and H is the scale height. For diatomic gas γ ~ 1.4.

(1)

c = γgH (2)

ω =<sup>a</sup> c 2H

where c is speed of sound

sunrise, reaches maximum around local noon and decreases again till sunset. Under such conditions concentration of charged particles significantly grows in the atmosphere and forms atmospheric plasma called ionosphere. Due to the composition of the neutral atmosphere together with the changing efficiency of the incoming solar radiation, ionosphere is stratified into the layers denoted D, E, F1 and F2. After sunset, electrons and ions recombine rapidly in the D, E and F1 layer. Due to slower recombination processes of atomic ions that dominate at heights approximately above 150km altitude, F2 layer remains present all the night. Special stratification Es, sporadic E layer, occurs sometimes at heights of E layer (Davies, 1990).

Ionosphere significantly affects propagation of the electromagnetic waves. According to a frequency of the wave with respect to a concentration of the ionospheric plasma, wave propagates through the medium or it is reflected. Electromagnetic waves with frequency lower than plasma frequency of the particular plasma parcel are reflected, which allows to estimate plasma frequency. Higher frequency waves propagate through plasma. An instrument called ionosonde (or digisonde) transmits electromagnetic wave of a defined frequency and detects it after reflection from the ionosphere. Typical ionosonde sounding range is 1 MHz – 20 MHz. For each sounding wave ionosonde records time of flight τ on the path transmitter - reflection point – receiver. Time of flight is simply converted into a

virtual height virtual .c <sup>h</sup> 2 <sup>τ</sup> <sup>=</sup> that corresponds to wave propagation in the vacuum (c stands

here for speed of light). Virtual height is equal or higher than the corresponding real height. The output of the measurement is height-frequency characteristics called an ionogram. Real height electron concentration profiles can further be inverted from ionograms using for instance programs POLAN (Titheridge, 1985) or NHPC (Huang and Reinish, 1996). Ionosphere represents inhomogeneous and anisotropic medium which leads to a wave splitting into an ordinary and extraordinary wave modes. Hence, two reflection traces are recorded by the ionosonde (as seen on ionograms in Figure 1). However, the extraordinary mode is not further used for electron concentration profile inversion.

Figure 1 shows typical day-time and night-time ionograms recorded by a digisonde in the observatory Pruhonice. Together with the ionograms there are plots of the real height electron concentration derived by NHPC routine. On the day-time electron concentration profile, three ionospheric layers E, F1 and F2 are present while on the night-time profile there is only F2 layer detected by the ionosonde.

Sequences of ionograms are widely used for analyses of variability of atmospheric plasma ranging from detection of rapid changes with periods of minutes to the study of long-term trends.

### **2. Basic theory of AGWs in the Earth's atmosphere**

Most of the wave-like oscillations in the atmosphere can be described/parametrized using basic acoustic-gravity wave theory in the atmosphere. Details can be found, for instance, in works of Davies (1990), Bodo et al. (2001), Hargreaves (1982), Yeh & Liu (1974) among others. Here, we show brief derivation of the dispersion relation that any wave motion of the AGW type must satisfy. In a plane-stratified, isothermal atmosphere under gravity that is constant with height, two frequency domains exist in the atmosphere where atmospheric waves can propagate, acoustic and gravity wave. Atmosphere represents compressible gas that once compressed and then released would expand and oscillate about its equilibrium state. Its oscillation frequency is known as an acoustic cut-off frequency

$$\text{co}\_{\text{a}} = \frac{\text{c}}{2\text{H}} \tag{1}$$

where c is speed of sound

304 Acoustic Waves – From Microdevices to Helioseismology

sunrise, reaches maximum around local noon and decreases again till sunset. Under such conditions concentration of charged particles significantly grows in the atmosphere and forms atmospheric plasma called ionosphere. Due to the composition of the neutral atmosphere together with the changing efficiency of the incoming solar radiation, ionosphere is stratified into the layers denoted D, E, F1 and F2. After sunset, electrons and ions recombine rapidly in the D, E and F1 layer. Due to slower recombination processes of atomic ions that dominate at heights approximately above 150km altitude, F2 layer remains present all the night. Special

stratification Es, sporadic E layer, occurs sometimes at heights of E layer (Davies, 1990).

virtual height virtual

trends.

.c <sup>h</sup>

2

there is only F2 layer detected by the ionosonde.

**2. Basic theory of AGWs in the Earth's atmosphere** 

state. Its oscillation frequency is known as an acoustic cut-off frequency

mode is not further used for electron concentration profile inversion.

Ionosphere significantly affects propagation of the electromagnetic waves. According to a frequency of the wave with respect to a concentration of the ionospheric plasma, wave propagates through the medium or it is reflected. Electromagnetic waves with frequency lower than plasma frequency of the particular plasma parcel are reflected, which allows to estimate plasma frequency. Higher frequency waves propagate through plasma. An instrument called ionosonde (or digisonde) transmits electromagnetic wave of a defined frequency and detects it after reflection from the ionosphere. Typical ionosonde sounding range is 1 MHz – 20 MHz. For each sounding wave ionosonde records time of flight τ on the path transmitter - reflection point – receiver. Time of flight is simply converted into a

here for speed of light). Virtual height is equal or higher than the corresponding real height. The output of the measurement is height-frequency characteristics called an ionogram. Real height electron concentration profiles can further be inverted from ionograms using for instance programs POLAN (Titheridge, 1985) or NHPC (Huang and Reinish, 1996). Ionosphere represents inhomogeneous and anisotropic medium which leads to a wave splitting into an ordinary and extraordinary wave modes. Hence, two reflection traces are recorded by the ionosonde (as seen on ionograms in Figure 1). However, the extraordinary

Figure 1 shows typical day-time and night-time ionograms recorded by a digisonde in the observatory Pruhonice. Together with the ionograms there are plots of the real height electron concentration derived by NHPC routine. On the day-time electron concentration profile, three ionospheric layers E, F1 and F2 are present while on the night-time profile

Sequences of ionograms are widely used for analyses of variability of atmospheric plasma ranging from detection of rapid changes with periods of minutes to the study of long-term

Most of the wave-like oscillations in the atmosphere can be described/parametrized using basic acoustic-gravity wave theory in the atmosphere. Details can be found, for instance, in works of Davies (1990), Bodo et al. (2001), Hargreaves (1982), Yeh & Liu (1974) among others. Here, we show brief derivation of the dispersion relation that any wave motion of the AGW type must satisfy. In a plane-stratified, isothermal atmosphere under gravity that is constant with height, two frequency domains exist in the atmosphere where atmospheric waves can propagate, acoustic and gravity wave. Atmosphere represents compressible gas that once compressed and then released would expand and oscillate about its equilibrium

<sup>τ</sup> <sup>=</sup> that corresponds to wave propagation in the vacuum (c stands

$$\mathbf{c} = \sqrt{\mathbf{\hat{g}} \mathbf{H}} \tag{2}$$

γ is the ratio of specific heats at constant pressure and constant volume, g is the gravitational acceleration, and H is the scale height. For diatomic gas γ ~ 1.4.

Fig. 1. Typical day-time (a) and night-time ionogram (b) measured by digisonde DPS 4 in the Observatory Pruhonice. On both plots, there is real hight electron concentration (solid line with error bars) provided as obtained by the NHPC routine.

Acoustic–Gravity Waves in the Ionosphere During Solar Eclipse Events 307

ω = γ− ( ) 2 2 <sup>2</sup>

In the terrestrial atmosphere the buoyancy period depends on the height. The height variance of the acoustic cut-off and buoyancy frequencies in the isothermal atmosphere is

Fig. 2. Height dependence of acoustic cut-off period ta and Brunt-Vaisala period tB that represent limits dividing periods into acoustic and gravity wave domains. Period domain between acoustic cut-off and Brunt-Vaisala represents region where no AGW propagates. Wave motion in the atmosphere can be described using mass conservation (continuity

∂ρ → → <sup>+</sup> ρ∇+ ∇ <sup>ρ</sup> <sup>=</sup> <sup>∂</sup> .u u. 0

<sup>→</sup> <sup>∂</sup> → → <sup>ρ</sup> + ∇ = −∇ + <sup>ρ</sup> <sup>∂</sup>

 <sup>∂</sup> → → ∂ρ <sup>ρ</sup> +∇ = <sup>γ</sup> + ∇ ∂ ∂ <sup>p</sup> u. p p u. t t

are parameters of the atmosphere – density, pressure, ratio of specific

u. u p <sup>g</sup> <sup>t</sup>

t

u

where pressure gradients and gravity are the only forces causing the acceleration.

<sup>B</sup> 1 g /c (11)

tB

ta

(12)

(13)

(14)

In isothermal atmosphere ωB reduces to

equation), and equation of motion:

Oscillation takes place adiabatically

→

where ρ, p,γ and u

heats, and velocity.

shown in Figure 2.

Single element of fluid, parcel of the atmosphere, at height z with density ρ which is displaced in the vertical by Δz to a place where its density changes to ρ+Δρ, remains in pressure equilibrium with its surroundings. Displacement takes place adiabatically. This is valid when the motion is so slow that sound waves with speed

$$\mathbf{c} = \sqrt{\frac{\mathrm{d}\mathbf{p}}{\mathrm{d}\mathfrak{p}}} \tag{3}$$

where p stands for pressure can traverse the system faster than the time-scale of interest and the motion is so fast that the entropy is preserved. The parcel is no longer in equilibrium and starts to oscillate about its equilibrium height with buoyancy frequency.

The buoyancy force which acts on the parcel is balanced by inertial force (Newton's second law):

$$\rho \frac{\text{d}^2}{\text{d} \text{t}^2} (\Delta \mathbf{z}) = -\text{g} \Delta \rho \tag{4}$$

where Δρ is the difference between internal and external densities. Internal and external Δρ are derived as:

$$\left(\Delta\mathfrak{p}\right)\_{\text{internal}} = \Delta\mathfrak{p} / \mathfrak{c}^2 = -\frac{\text{g}\mathfrak{p}}{\mathfrak{c}^2} \Delta\mathfrak{x} \tag{5}$$

which is due to compressibility of the fluid within the membrane and

$$\left(\Delta\mathfrak{p}\right)\_{\text{external}} = -\frac{\text{d}\mathfrak{p}}{\text{d}\mathbf{z}}\Delta\mathbf{z} \tag{6}$$

is the change of background density at new position due to inhomogeneous nature of the atmosphere. Taking both the contributions of Δρ we get

$$\frac{\text{d}^2}{\text{d}\text{d}^2}(\Delta \mathbf{z}) = \left(\text{g}\,\frac{\text{d}}{\text{d}\text{z}}(\ln \mathfrak{p}) + \text{g}^2/\text{c}^2\right)\Delta \mathbf{z} \tag{7}$$

which can be recast into

$$\frac{\text{d}^2}{\text{d}\text{t}^2}(\Delta \mathbf{z}) + \text{co}\_\text{\text{\textquotedblleft}}^2 \Delta \mathbf{z} = 0 \tag{8}$$

where

$$\mathrm{d}\mathbf{o}\_{\mathrm{b}}^{2} = -\mathrm{g}\left(\frac{\mathrm{d}}{\mathrm{d}\mathbf{z}}(\ln\mathfrak{p}) + \mathrm{g}/\mathrm{c}^{2}\right) \tag{9}$$

If <sup>2</sup> ω<sup>B</sup> >0, the solution is oscillatory and the fluid parcel will oscillate with characteristic buoyancy frequency ωB called Brunt-Vaisala frequency. More convenient form used for atmosphere is following:

$$\text{Ca}\_{\text{u}}^{2} = (\gamma - 1) \text{g}^{2} \left/ \text{c}^{2} + \text{g} \,/\, \text{c}^{2} \, \text{d} \text{c}^{2} \, / \, \text{d} \, \text{z} \tag{10}$$

This approximation is valid in the atmosphere-ionosphere system of our interest.

In isothermal atmosphere ωB reduces to

306 Acoustic Waves – From Microdevices to Helioseismology

Single element of fluid, parcel of the atmosphere, at height z with density ρ which is displaced in the vertical by Δz to a place where its density changes to ρ+Δρ, remains in pressure equilibrium with its surroundings. Displacement takes place adiabatically. This is

> dp <sup>c</sup> <sup>d</sup> <sup>=</sup> <sup>ρ</sup>

where p stands for pressure can traverse the system faster than the time-scale of interest and the motion is so fast that the entropy is preserved. The parcel is no longer in equilibrium

The buoyancy force which acts on the parcel is balanced by inertial force (Newton's second

ρ ( ) Δ =− Δρ

<sup>z</sup> <sup>g</sup> dt

( ) <sup>ρ</sup> Δρ =Δ =− Δ <sup>2</sup> internal 2

( ) <sup>ρ</sup> Δρ =− Δ external

is the change of background density at new position due to inhomogeneous nature of the

( ) ( ) Δ = <sup>ρ</sup> + Δ

( ) Δ +ω Δ =

( ) ω =− ρ + 2 2

If <sup>2</sup> ω<sup>B</sup> >0, the solution is oscillatory and the fluid parcel will oscillate with characteristic

2 B <sup>d</sup> z z0

> d g ln g c dz

ω = γ− + ( ) 2 2 2 2 <sup>2</sup>

This approximation is valid in the atmosphere-ionosphere system of our interest.

2

2

dt

B

d d z g ln g c z dt dz

<sup>g</sup> p/c z c

> d z dz

> > 2 2

<sup>B</sup> 1 g c g /c dc /dz (10)

2 2 d

and starts to oscillate about its equilibrium height with buoyancy frequency.

where Δρ is the difference between internal and external densities.

which is due to compressibility of the fluid within the membrane and

atmosphere. Taking both the contributions of Δρ we get

buoyancy frequency ωB called Brunt-Vaisala frequency. More convenient form used for atmosphere is following:

2

2

Internal and external Δρ are derived as:

which can be recast into

where

(3)

(4)

(5)

(6)

(8)

(9)

(7)

valid when the motion is so slow that sound waves with speed

law):

$$
\alpha\_{\rm b}^2 = (\gamma - 1) \lg^2 / c^2 \tag{11}
$$

In the terrestrial atmosphere the buoyancy period depends on the height. The height variance of the acoustic cut-off and buoyancy frequencies in the isothermal atmosphere is shown in Figure 2.

Fig. 2. Height dependence of acoustic cut-off period ta and Brunt-Vaisala period tB that represent limits dividing periods into acoustic and gravity wave domains. Period domain between acoustic cut-off and Brunt-Vaisala represents region where no AGW propagates.

Wave motion in the atmosphere can be described using mass conservation (continuity equation), and equation of motion:

$$\frac{\partial \rho}{\partial t} + \rho \nabla \cdot \vec{\mathbf{u}} + \left(\vec{\mathbf{u}} \cdot \vec{\nabla}\right) \rho = 0 \tag{12}$$

$$\rho \left( \stackrel{\scriptstyle \rightarrow}{\partial \mathbf{\dot{u}}} + \stackrel{\scriptstyle \rightarrow}{\left( \stackrel{\scriptstyle \rightarrow}{\mathbf{u}}, \nabla \right)} \stackrel{\scriptstyle \rightarrow}{\mathbf{u}} \right) = -\nabla \mathbf{p} + \rho \mathbf{g} \tag{13}$$

where pressure gradients and gravity are the only forces causing the acceleration. Oscillation takes place adiabatically

$$\mathbf{p}\left(\frac{\partial \mathbf{p}}{\partial \mathbf{t}} + \stackrel{\rightarrow}{\mathbf{u}} . \nabla \mathbf{p}\right) = \gamma \mathbf{p} \left(\frac{\partial \mathbf{p}}{\partial \mathbf{t}} + \stackrel{\rightarrow}{\mathbf{u}} . \nabla\right) \tag{14}$$

where ρ, p,γ and u → are parameters of the atmosphere – density, pressure, ratio of specific heats, and velocity.

Acoustic–Gravity Waves in the Ionosphere During Solar Eclipse Events 309

The motions of the air parcels are, in general, ellipses in the plane of propagation and have components transverse to the direction of wave propagation. The ratio of the horizontal

<sup>ξ</sup> <sup>ω</sup> <sup>ω</sup> <sup>ω</sup> = −− ζ ω ωω <sup>−</sup> <sup>ω</sup>

On the frequencies just above the acoustic cutoff the air motion is essentially vertical. With acoustic waves on high frequencies the motion is radial as in sound waves. The motion is circular with horizontal propagation at a frequency <sup>a</sup> ω γ 2 / . Gravity wave propagation is

sin , <sup>−</sup> <sup>ω</sup> φ =π− <sup>ω</sup>

max

The sense of rotation of the air for gravity waves is opposite than for acoustic waves. As Φ approaches its asymptotic values the air motion becomes linear and transverse to the direction of propagation. Air parcel rotation is clockwise in case of acoustic waves while anticlockwise in case of gravity waves. Energy vector lies in the same quadrant as direction of propagation of acoustic waves. Energy flows up when phase travels down and vice versa in case of gravity waves propagation. This is important property since it accounts for the observed downward phase propagation when the source is below the level at which a

The horizontal ux and vertical uz components of the packet velocity are obtained from

ω −ω <sup>=</sup> ω ω −ω −

c k

u

u

<sup>ω</sup> <sup>=</sup> ω ω −ω −

Due to coupling between neutral and charged components the initial wave-like oscillation in the neutral atmosphere induces wave-like perturbation in the ionosphere. Perturbation in the ion production is the most effective when solar ionizing rays are nearly in alignment with the initial wave front. Perturbations in the neutral atmosphere may cause perturbations in chemical processes. Presence of AGW influences the ionisation rate through changes in the local neutral density and temperature, and through changes in the ionisation radiation

Acoustic-gravity waves are always present in the Earth's atmosphere. AGWs arise from many natural sources like convection, topography, wind shear, moving solar terminator, earthquakes, tsunami, etc. Increase in wave-like activity is associated also with human

( ) ( )

2 22 x g x 2 2 22 a 2 2 z z 2 2 22 a

2 ck c k

( )

2 ck

1

g

ck <sup>i</sup>

2 x

ck <sup>1</sup>

<sup>−</sup> <sup>ω</sup> φ = <sup>ω</sup> 1

g

min

ck

<sup>x</sup> <sup>2</sup> <sup>2</sup> a g z

(22)

sin (23)

(24)

ζis:

displacement

ξ

limited to angles between

disturbance is observed.

absorption (Hooke, 1970).

**3. AGW in the ionospheric plasma** 

disperse relation:

to its vertical displacement

Applying the perturbation approach we are searching for wave-like solutions for the perturbation quantities. Further simplification comes from the assumption that the background state is of constant temperature T in which p0/ρ0 must be a constant.

$$\mathbf{p}\_0 \nmid \mathbf{p}\_0 = \mathbf{c}^z \nmid \mathbf{y} \tag{15}$$

Then the system (12), (13) and (14) reduces to:

$$\frac{\partial \mathbf{p'}}{\partial \mathbf{t}} + \mathbf{p}\_0 \vec{\nabla} . \mathbf{u'} - \mathbf{p}\_0 \mathbf{u}\_x \;/\,\mathrm{H} = 0\tag{16}$$

$$
\rho\_0 \frac{\partial \stackrel{\rightarrow}{\mathbf{u}}}{\partial \mathbf{t}} + \nabla \mathbf{p} \text{--} \rho \text{'} \mathbf{g} = 0 \tag{17}
$$

$$\mathfrak{p}\_0 \left( \frac{\partial \mathfrak{p'}}{\partial \mathfrak{t}} - \mathfrak{p}\_0 \mathfrak{u}\_{\boldsymbol{\omega}}^{\cdot} / \mathcal{H} \right) = \mathfrak{M}\_0 \left( \frac{\partial \mathfrak{p'}}{\partial \mathfrak{t}} - \mathfrak{p}\_0 \mathfrak{u}\_{\boldsymbol{\omega}}^{\cdot} / \mathcal{H} \right) \tag{18}$$

where index 0 denotes stationary (non fluctuating) component and the apostrophe denotes perturbation. These are the basic governing equations for the gravity waves. For a nontrivial solution the following prescription of the dispersion relation must be satisfied:

$$\left(\alpha^4 - \alpha^2 \alpha\_\text{a}^2 - \mathbf{k}\_\text{x}^2 \mathbf{c}^2 \left(\alpha \mathbf{o}^2 - \alpha\_\text{o}^2\right) - \mathbf{c}^2 \alpha^2 \mathbf{k}\_\text{x}^2 = 0\tag{19}$$

From disperse relation, it is evident that between buoyancy frequency and acoustic cut-off frequencies one cannot have both kx and kz real. Figure 2 shows two period domains with border limits of acoustic cut-off period and buoyancy period.

An attenuation or growth in the wave amplitude must occur in either the vertical or the horizontal directions. We suppose that there is no variation in amplitude in horizontal directions so that kx is purely real and kz has an imaginary component. At frequencies exceeding acoustic cut-off ωa, expression (19) becomes simple and the waves may be termed as ACOUSTIC WAVES. At frequencies smaller than Brunt-Vaisala frequency where gravity plays an important role, the waves are called GRAVITY or INTERNAL GRAVITY WAVES. Brunt-Vaisala frequency and acoustic cut-off frequency divide the frequency spectrum into two domains in which ω<sup>g</sup> forms the high frequency limit for one class ω<ωg normally called internal gravity waves and ω<sup>a</sup> is the low frequency limit for another class ω>ωa called the acoustic waves. A gap in the frequency spectrum exists between ω<sup>g</sup> and ωa where no internal waves can propagate.

Important approximations can be obtained under the assumption k 1 2H <sup>z</sup> >> and ω << ω<sup>g</sup> then:

$$\mathbf{k}\_{\mathbf{x}}^{2} = \left(\alpha\_{\mathcal{E}}^{2} \;/\; \mathbf{o}^{2}\right) \mathbf{k}\_{\mathbf{x}}^{2} \tag{20}$$

These approximations apply to much of the observed gravity waves. From (20) we see that the angle of ascent of the phase α is:

$$\mathbf{t}\mathbf{g}\mathbf{\alpha} = \mathbf{k}\_{\mathbf{z}} \;/\; \mathbf{k}\_{\mathbf{x}} = \mathbf{o}\_{\mathbf{g}} \;/\; \mathbf{o} = \mathbf{\tau} \;/\; \mathbf{\tau}\_{\mathbf{g}}\tag{21}$$

Applying the perturbation approach we are searching for wave-like solutions for the perturbation quantities. Further simplification comes from the assumption that the

<sup>+</sup> <sup>ρ</sup> ∇ −ρ <sup>=</sup> <sup>∂</sup> 0 0z

<sup>ρ</sup> +∇ − <sup>ρ</sup> <sup>=</sup> <sup>∂</sup> <sup>0</sup> <sup>u</sup> p' 'g <sup>0</sup>

 <sup>∂</sup> ∂ρ ρ − = γ −ρ ∂ ∂ ' '

where index 0 denotes stationary (non fluctuating) component and the apostrophe denotes perturbation. These are the basic governing equations for the gravity waves. For a nontrivial solution the following prescription of the dispersion relation must be satisfied:

ω −ω ω − ω −ω − ω = ( ) 4 2 2 22 2 2 2 2 2

From disperse relation, it is evident that between buoyancy frequency and acoustic cut-off frequencies one cannot have both kx and kz real. Figure 2 shows two period domains with

An attenuation or growth in the wave amplitude must occur in either the vertical or the horizontal directions. We suppose that there is no variation in amplitude in horizontal directions so that kx is purely real and kz has an imaginary component. At frequencies exceeding acoustic cut-off ωa, expression (19) becomes simple and the waves may be termed as ACOUSTIC WAVES. At frequencies smaller than Brunt-Vaisala frequency where gravity plays an important role, the waves are called GRAVITY or INTERNAL GRAVITY WAVES. Brunt-Vaisala frequency and acoustic cut-off frequency divide the frequency spectrum into two domains in which ω<sup>g</sup> forms the high frequency limit for one class ω<ωg normally called internal gravity waves and ω<sup>a</sup> is the low frequency limit for another class ω>ωa called the acoustic waves. A gap in the frequency spectrum exists between ω<sup>g</sup> and ωa where no

Important approximations can be obtained under the assumption k 1 2H <sup>z</sup> >> and

These approximations apply to much of the observed gravity waves. From (20) we see that

0 0 z 0 0z p' ' p u /H p u /H t t

.u' u /H 0 <sup>t</sup>

ρ = γ <sup>2</sup> p/ c/ 0 0 (15)

ax g z kc c k 0 (19)

=ω ω ( ) 2 2 22 k /k <sup>z</sup> <sup>g</sup> <sup>x</sup> (20)

α= =ω ω=τ τ z x g g tg k /k / / (21)

(16)

(17)

(18)

background state is of constant temperature T in which p0/ρ0 must be a constant.

∂ρ → →

→ ∂

t

'

border limits of acoustic cut-off period and buoyancy period.

internal waves can propagate.

the angle of ascent of the phase α is:

ω << ω<sup>g</sup> then:

Then the system (12), (13) and (14) reduces to:

The motions of the air parcels are, in general, ellipses in the plane of propagation and have components transverse to the direction of wave propagation. The ratio of the horizontal displacement ξ to its vertical displacement ζis:

$$\frac{\xi}{\xi} = \frac{\frac{\text{ck}\_{\text{x}}}{\text{co}}}{\left(\frac{\text{ck}\_{\text{x}}}{\text{co}}\right)^{2} - 1} \left(\frac{\text{ck}\_{\text{x}}}{\text{co}} - \text{i}\sqrt{\left(\frac{\text{co}\_{\text{a}}}{\text{co}}\right)^{2} - \left(\frac{\text{co}\_{\text{g}}}{\text{co}}\right)^{2}}\right) \tag{22}$$

On the frequencies just above the acoustic cutoff the air motion is essentially vertical. With acoustic waves on high frequencies the motion is radial as in sound waves. The motion is circular with horizontal propagation at a frequency <sup>a</sup> ω γ 2 / . Gravity wave propagation is limited to angles between

$$\boldsymbol{\phi}\_{\text{min}} = \sin^{-1}\left(\frac{\boldsymbol{\alpha}}{\boldsymbol{\alpha}\_{\boldsymbol{\ell}}}\right) \; \; \boldsymbol{\phi}\_{\text{max}} = \pi - \sin^{-1}\left(\frac{\boldsymbol{\alpha}}{\boldsymbol{\alpha}\_{\boldsymbol{\ell}}}\right) \tag{23}$$

The sense of rotation of the air for gravity waves is opposite than for acoustic waves. As Φ approaches its asymptotic values the air motion becomes linear and transverse to the direction of propagation. Air parcel rotation is clockwise in case of acoustic waves while anticlockwise in case of gravity waves. Energy vector lies in the same quadrant as direction of propagation of acoustic waves. Energy flows up when phase travels down and vice versa in case of gravity waves propagation. This is important property since it accounts for the observed downward phase propagation when the source is below the level at which a disturbance is observed.

The horizontal ux and vertical uz components of the packet velocity are obtained from disperse relation:

$$\mathbf{u}\_{\times} = \frac{\mathbf{c}^2 \mathbf{k}\_{\times} \left(\mathbf{o}\mathbf{o}^2 - \mathbf{o}\mathbf{o}\_{\times}^2\right)}{\cos\left(2\mathbf{o}\mathbf{o}^2 - \mathbf{o}\mathbf{o}\_{\mathbf{a}}^2 - \mathbf{c}^2 \mathbf{k}^2\right)}\tag{24}$$

$$\mathbf{u}\_{\times} = \frac{\mathbf{c}^2 \mathbf{k}\_{\times} \mathbf{o}\mathbf{o}^2}{\cos\left(2\mathbf{o}^2 - \mathbf{o}\mathbf{o}\_{\mathbf{a}}^2 - \mathbf{c}^2 \mathbf{k}^2\right)}$$

Due to coupling between neutral and charged components the initial wave-like oscillation in the neutral atmosphere induces wave-like perturbation in the ionosphere. Perturbation in the ion production is the most effective when solar ionizing rays are nearly in alignment with the initial wave front. Perturbations in the neutral atmosphere may cause perturbations in chemical processes. Presence of AGW influences the ionisation rate through changes in the local neutral density and temperature, and through changes in the ionisation radiation absorption (Hooke, 1970).

### **3. AGW in the ionospheric plasma**

Acoustic-gravity waves are always present in the Earth's atmosphere. AGWs arise from many natural sources like convection, topography, wind shear, moving solar terminator, earthquakes, tsunami, etc. Increase in wave-like activity is associated also with human

Acoustic–Gravity Waves in the Ionosphere During Solar Eclipse Events 311

plasma variations. Despite intensive research many questions in the problem of the

Studies by Fritts and Luo (1993) suggest that perturbations generated by the eclipse induced ozone heating interruption may propagate upwards into the thermosphere–ionosphere system where they have an important influence. Temperature fluctuations and electron density changes propagate as a wave, away from the totality path, cf. Muller-Wodarg et al. (1998). By means of vertical ionospheric sounding, Liu et al. (1998) detected waves excited during solar eclipse event at F1 layer heights and their generation and/or enhancement attributed to changes of temperatures and variations of the height of the transition level for the loss coefficient and the height of the peak of electron production. Studies reported by Farges et al. (2001) suggest a longitudinal diversity of the disturbances with respect to prenoon and postnoon phases. Xinmiao et al. (2010) reported synchronous oscillations in the Es and F layer during the recovery phase of the solar eclipse. Ivanov et al. (1998) found that during solar eclipse with maximum obscuration of about 70% the F-region electron density decreased by 6-8% compared to a control day and detected travelling ionospheric disturbances. Additionally, they detected strong variations in the difference group delays with a period about 40 minutes associated with the start and end of the eclipse. Oscillations in the ionosphere, similar to gravity waves, were observed following some solar eclipse events (Chimonas and Hines, 1970; Cheng et al., 1992; Liu et al., 1998; Sauli et al., 2006). Investigation of the latitudinal dependence of NmF2 (the maximum electron density of the F2 layer) indicated that the strongest response was at middle latitudes (Le et al., 2009). The response of the sporadic-E (Es) layer also differed in each solar eclipse event. A remarkable decrease in Es layer ionization was observed during the eclipse of 20 July 1963 (Davis et al., 1964). Enhancement of Es layer ionization has also been reported and it has been suggested that it is related to internal gravity waves generated in the atmosphere during the solar

During the solar eclipse, on the time scale shorter than day-night change, the ionosphere reconfigures itself into a state similar to that of night situation. Photochemical ionization falls heavily almost to a night-time level. With the decreasing solar flux, atmospheric temperature falls in the moon shadow creating a cool spot with well defined border. Then the increasing solar flux starts ionization processes and warms the atmosphere again to

Such changes in the ionization cause variation in the reflection heights, decrease/increase in electron concentration at all ionospheric heights, decrease/increase in the total electron content, rising/falling of the layer height. Such effects are characteristic for the processes during sunrise/sunset in the ionosphere. However, supersonic movement of the eclipsed region represents a key difference from the regular solar terminator motion at sunrise and sunset times. These changes in the neutral atmosphere and ionosphere induced by solar eclipse force the evolution of the ionospheric plasma toward a new equilibrium state. The return to equilibrium is likely accompanied by the eclipse induced wave motions excited in the atmosphere. Any moving discontinuity of gas parameters such as temperature, pressure etc. will generate transit-like waves. In the upper ionosphere, waves can be generated by a strong horizontal electron pressure gradient. Possible mechanisms contributing to the wave generation in the region of solar terminator are in detail discussed by Somsikov & Ganguly

generation and propagation remain to be understood.

eclipse (Datta, 1972).

daytime level.

(1995).

**4.2 Processes induced by solar eclipse** 

activity including coordinated experiments or unwilling accidents. AGWs influence on the upper atmosphere is not yet understood enough. They produce a great amount of variability and contribute to the background conditions in a specific parcel of the atmosphere. Gravity waves propagating from lower laying atmosphere have been long regarded as a very important source of the energy and momentum transfer in the upper atmosphere (Hines, 1960). The breaking of the upward propagating waves affects wind system, generates turbulence and heats the atmospheric gas.

Waves that reach upper atmosphere produce travelling atmospheric disturbances (TAD) or travelling ionospheric disturbances (TID) and even form the ionospheric inhomogenities which grow and finally break into the plasma instabilities observed by radar techniques that might cause scintillation of the communication signals propagating through the ionosphere. From the observation it is evident that the thermosphere is continuously swept by the acoustic-gravity waves. Statistically, the waves show a moderate preference for southward travel, with this preference being reduced or shifted to southeastward travel during disturbed times (Oliver et al., 1997). Experimental studies show that AGW activity in the ionosphere slightly increases during dawn and dusk periods of the day (Galushko et al., 1998; Somsikov & Ganguly, 1995; Sauli et al., 2005 among others). Influence of infrasonic waves generated by ground experimental sources on the ionosphere was reported for instance by Rapoport et al. (2004).

Solar eclipse represents well defined source of the AGW in the atmosphere and ionosphere systems. During solar eclipse event, solar ionization flux decreases producing well-defined cool spot in the atmosphere that moves through the Earth's atmosphere. Moving source in the atmosphere can emit both acoustic and gravity waves. Supersonic motion of the source forms wave field with bow wave. Both acoustic and gravity waves can be radiated in association with supersonic motion in the atmosphere. When the source is moving within atmosphere with subsonic velocity only gravity waves can be emitted (Kato et al., 1977).
