ISSN 1028!334X, Doklady Earth Sciences, 2014, Vol. 454, Part 1, pp. 37–39. © Pleiades Publishing, Ltd., 2014.
Original Russian Text © O.G. Onishchenko, O.A. Pokhotelov, V. Fedun, 2014, published in Doklady Akademii Nauk, 2014, Vol. 454, No. 1, pp. 89–91.
GEOPHYSICS
Convection Cells of Internal Gravity Waves
in the Terrestrial Atmosphere
O. G. Onishchenkoa, b, O. A. Pokhotelova, and V. Fedunc
Presented by Academician O.A. Gliko November 29, 2012
Received December 7, 2012
DOI: 10.1134/S1028334X14010036
INTRODUCTION
2
– 1! + !!
1! dT
ω g = g  γ!!!!!!!!
!!!!!
 γH T dz 
Investigation of how the lithosphere interacts with
the neutral atmosphere and ionosphere is one of the
basic problems of the fundamental geophysics and
applied geophysical research. According to modern
ideas, one of the most well!known links between the
lithosphere and ionosphere is internal gravity waves
(IGWs). The IGW sources in the atmosphere are vol!
canic eruptions, typhoons, zonal winds in the atmo!
sphere, earthquakes, and some other processes.
Reaching high altitudes in the atmosphere, with a
decrease in density and increase in the amplitude of
disturbance, IGWs can disturb the ionosphere [1–4].
Satellite and ground!based electromagnetic sounding
of the neutral atmosphere and lower ionosphere indi!
cates the relationship between the catastrophic events
on the Earth’s surface and disturbances in the lower
weakly ionized ionosphere. Therefore, the problem
regarding prediction and observation of natural catas!
trophes from electromagnetic sounding is closely
related to the problem of IGW propagation. Genera!
tion of zonal structures by nonlinear IGWs [5, 6] and
eddy structures that control amplitude growth [7, 8]
were studied in the respective works under the assump!
tion of a stably stratified atmosphere and with the ver!
tical temperature gradient neglected.
(1)
d
is above zero. Here, g is the free fall acceleration; !!!! is
dz
the derivative on altitude; γ is the adiabatic parameter;
H is the local reduced scale of atmospheric altitude;
and T is the undisturbed temperature. In the region
where the temperature decreases with altitude and the
2
inverse inequality ω g < 0 is valid, strong vertical con!
2
vection transfer emerges. The condition ω g < 0 corre!
sponds to the absolute instability of IGWs, when the
wave frequency is a purely imaginary value. In the
near!surface layer of the real atmosphere, the condi!
2
tion ω g > 0 and, therefore, the condition of IGW
propagation can change, as well as the parameters of
nonlinear IGW eddy structures.
The aim of the present work is investigation of non!
linear IGW eddy structures (convection cells) in the
atmosphere with the finite vertical temperature gradi!
ents taken into consideration. In contrast to the previ!
ous studies, eddy structures will be examined both
under the assumed stably stratified atmosphere and
under an assumed unstably stratified one. The second
section will derive the system of nonlinear equations
for IGWs. The third section will be devoted to discus!
sion of the results.
The vertical temperature gradient is one of the
most important parameters determining the dynamics
of vertical transfer and convection stability of the real
atmosphere. The atmosphere is considered stably
stratified if the square of Brunt–Väisälä frequency
ωgaveraged over the altitude
MODEL DESCRIPTION OF INTERNAL GRAVITY
WAVES OF FINITE AMPLITUDE
IN AN ATMOSPHERE WITH NONUNIFORM
TEMPERATURE
a
Schmidt Joint Institute of Physics of the Earth,
Russian Academy of Sciences, Moscow, Russia
b Institute of Space Research,
Russian Academy of Sciences, Moscow, Russia
c Department of Automatic Control and Systems Engineering,
University of Sheffield, Sheffield, Great Britain
e!mail: [email protected]
Below we will focus on the dynamic characteristics
of the atmosphere with dissipation processes (viscos!
ity, thermal conductivity, heat transfer from outside,
friction) neglected. The initial equations are that of
motion
37
38
ONISHCHENKO et al.
ρ du
!!!!! + ∇p – ρg = 0
dt
(2)
1/γ
p
and that of potential temperature transfer Θ = !!!!!!
!
ρ
dΘ! = 0.
!!!!!
dt
(3)
d
∂
Here, ρ is the mass density; p is the pressure; !!!! = !!!! +
dt ∂t
u · ∇ is the Euler (convection) time derivative on time;
u is the velocity of matter; g = –g ẑ is the acceleration
due to gravity; ẑ is the unit vector of the local Carte!
sian coordinate system (x, y, z), directed along the ver!
tical. To describe the two!dimensional motion of non!
compressible gas in terms of gravity waves, let us assume
a velocity of matter u = (u, 0, w), where u = – ∂ψ
!!!!!! , and
∂z
∂ψ
w = !!!!!! , ψ(t, x, z) is the stream function.
∂x
We assume that disturbances are small, ρ = ρ0(z) +
ρ̃ (t, x, z) and p = p0(z) + p̃ (t, x, z), where ρ0 is undis!
turbed density, p0 is undisturbed pressure, and ρ̃ and p̃
are the disturbances (| ρ̃ | ! ρ0 and | p̃ | ! p0). Under
these assumptions, from Eqs. (2) and (3) we obtain
∂!  ∇ 2 ψ d ln ρ 0 ∂ψ + ( , 2 ψ )
!!!
+ !!!!!!!!!!! !!!!!!
J ψ ∇
∂t 
dz ∂z 
∂χ! + !!!
1! J ( ρ̃, p̃ )
= – !!!!
∂x ρ 20
(4)
2
2
2
1
χ
ψ
E = ! ρ 0 ( ∇ψ ) + !!!!!!!2 + !!!!!2 dx dz.
2
4L ρ ω g
(8)
(5)
2
∂
∂
gρ̃
!!!!!!2 + !!!!!!2 ; and χ = !!!!! .
ρ0
∂x
∂z
In the undisturbed atmosphere, pressure decreases
exponentially with altitude, p0(z – z0) =
2
2 ∂ψ̂
∂χ̂
(7)
!!!!! – ω g !!!!!! + J ( ψ̂, χ̂ ) = 0.
∂t
∂x
The closed system of equations (6) and (7) can be used
for numerical modelling of the dynamics of nonlinear
IGWs in the atmosphere with a finite vertical temper!
ature gradient. In terms of the isothermal atmosphere
(LT " H), the system of equations (6) and (7) coin!
cides with the respective system of equations studied in
[5–8]. From Eqs. (4) and (5) we can deduce the law of
∂E! = 0, where
wave energy conservation, !!!!
∂t
It is seen from Eq. (8) that, in an instably stratified
atmosphere, energy can take on negative values.
∂a ∂b
∂a ∂b
Here J(a, b) = !!!!! !!!!! – !!!!! !!!!! is the Jacobian; ∇2 =
∂x ∂z
∂z ∂x
2
z–z
and χ = χ̂ (t, x, z)exp !!!!!!!!!!0 , we rearrange Eqs. (4)
2L ρ
and (5) as follows:
2
∂ 2
ψ̂
∂χ̂
!!!!  ∇ ψ̂ – !!!!!!!2  + J ( ψ̂, ∇ ψ̂ ) = – !!!!!,
(6)


∂t
∂x
4L ρ
∫
and
2 ∂ψ
∂χ
!!!!! – ω g !!!!!! + { ψ, ρ̃ } = 0.
∂t
∂x
p
inversion layer LT < 0. Using the ideal gas law, !!!!!! =
ρT
const, we obtain the equation of undisturbed density
z–z
ρ0(z – z0) = ρ0(z0)exp – !!!!!!!!!!0 , where Lρ is the charac!
Lρ
teristic scale of the vertical density gradient. From the
ideal gas law p(ρT)–1 = const, we also obtain the ratio
z–z
1
1
1
!!!! = !!! – !!!!! . Substituting ψ = ψ̂ (t, x, z)exp !!!!!!!!!!0
Lρ H LT
2L ρ
z–z
c
p0(z0)exp – !!!!!!!!!!0 , where H = !!!!s is the scale of alti!
γg
H
γp 1/2
tudes and cs =  !!!!!!0
is the sound speed. Assume
 ρ0 
that the undisturbed temperature changes with alti!
z–z
tude as follows T(z – z0) = T(z0)exp – !!!!!!!!!!0 , where
LT
LT is the characteristic scale of the vertical tempera!
ture gradient. In the atmospheric layer where temper!
ature decreases with altitude LT > 0, whereas in the
CONVECTION CELLS
To search for stationary eddy solutions in the coor!
dinate system moving at speed v along the axis x, let us
introduce the new variable η = x – vt. Resulting from
this substitution, the system of two equations (6) and
(7) can be reduced to one equation
2
J 1 ( ∇ ψ̂ – Λψ̂, ψ̂ – vz ) = 0,
(9)
∂a ∂b ∂a ∂b
where J1(a, b) = !!!!! !!!!! – !!!!! !!!!! and
∂η ∂z
∂z ∂η
2
ωg
1 ! – !!!!
Λ = !!!!!!
!2 .
(10)
2
4L ρ v
An equation similar to (9) has been studied in detail by
analytical and numerical methods; for example, it was
shown in [9] that the equation has a solution in the
form of dipole eddies. In the external zone of an eddy,
the following condition is satisfied:
2
∇ ψ̂ – Λψ̂ = 0.
DOKLADY EARTH SCIENCES
Vol. 454
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Part 1
2014
CONVECTION CELLS OF INTERNAL GRAVITY WAVES
Isolated singular eddies exist in the atmosphere a pos!
sessing finite vertical temperature gradient only in the
case when Λ > 0, where the Brunt–Väisälä frequency
ωg is defined by Eq. (1). Relation (10) connects the
characteristic size of an eddy (a convection cell) Λ–1/2
with the characteristic parameters of the atmosphere,
2
ω g , scale Lρ, and the horizontal velocity of eddy v.
From the condition Λ > 0, we obtain
2
2
H 2 4c γ – 1 H
v  1 – !!!!! > !!!!!!s  !!!!!!!!! – !!!!! .

L T
γ  γ
L T
(12)
Neglecting the vertical temperature gradient,
H
!!!!! ! 1, and assuming the effective value at γ = 1.4, we
LT
obtain (in accordance with [7, 8] the condition v >
γp 1/2
0.9cs, where cs =  !!!!!!0 is the sound speed. Note that
 ρ0 
eddies moving at the velocity of eddy structures must
have an indefinitely large horizontal scale, Λ  1/2 → ∞.
Additionally, the supersonic motion of structures
induces chock waves; hence, such a motion is unlikely
to occur in terms of the approximation from [7, 8]. By
the same reason, the existence of eddy IGW structures
is problematic in the stratosphere where temperature
grows with altitude (LT < 0).
It is seen from inequality (12) that the existence of
eddies moving at a speed significantly slower than
sound is possible in the thermosphere layers where the
vertical temperature gradient’s order is similar to that
Hγ
of the adiabatic gradient LT ! !!!!!!!!! . In the atmosphere
γ–1
stratified under nonequilibrium conditions, if LT >
Hγ
!!!!!!!!! , the condition described by Eq. (10) is always
γ–1
satisfied and there is no limitation to the horizontal
velocity of eddy motion.
CONCLUSIONS
We have studied the IGW eddy structures in the
atmosphere having a finite vertical temperature gradi!
DOKLADY EARTH SCIENCES
Vol. 454
Part 1
2014
39
ent. It has been shown that eddies can exist in the ther!
mosphere layers where the order of the vertical tem!
perature gradient is similar to that of the dry adiabatic
gradient. The condition of minimal speed of eddies in
such an atmosphere has been obtained. It has been
shown that the horizontal speed of these eddies is sig!
nificantly lower than the sound speed. Additionally, in
contrast to the previous studies, we investigated IGW
eddies in terms of an instably stratified atmosphere,
2
ω g < 0. It has been shown that these atmospheric
(thermospheric) layers have no lower cut!off value of
the speed of eddy motion.
ACKNOWLEDGMENTS
This work was supported by the Russian Founda!
tion for Basic Research (project no. 11!0500920),
International Space Science Institute, Bern, and Pro!
gram 22 of the Presidium of Russian Academy of Sci!
ence.
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Translated by N. Astafiev