- Email: [email protected]
Ionosphere F2-region under the influence of the evolutional atmospheric gravity waves in horizontal shear flow
\ PERGAMON
Journal of Atmospheric and Solar!Terrestrial Physics 50 "0888# 368Ð378
Ionosphere F1!region under the in~uence of the evolutional atmospheric gravity waves in horizontal shear ~ow G[G[ Didebulidze \ A[D[ Pataraya Town Department of Abastumani\ Astrophysical Observatory\ A[ Kazbegi av[ 1a\ 279959 Tbilisi\ Geor`ia Received 15 June 0887^ received in revised form 4 January 0888^ accepted 03 January 0888
Abstract The ambipolar di}usion equation for the height distribution of electron density in the ionospheric F1!layer is solved in the presence of neutral horizontal shear ~ow[ By using this nonstationary solution the reaction of the F1!region electron density on the evolution of atmospheric acousticÐgravity waves "AGW# is investigated[ The evolution of the AGW and the corresponding behaviour of the height distribution of the F1!region electron density are described by the characteristic time\ ta\ of transient development of shear waves in the horizontal shear ~ow[ For long times t × ta\ the gravity wave frequency tends to the isothermal BruntÐVaisala frequency\ which appears in the F1!layer as wavelike behaviour of hmF1 and NmF1 with periods close to 05Ð19 min\ when the scale height of the neutral gas is H 59 km[ The shear wave\ which is due to the presence of horizontal shear ~ow\ gives su.cient changes of the height pro_le of electron density for times of t ¾ ta[ Þ 0888 Elsevier Science Ltd[ All rights reserved[
0[ Introduction Many types of inhomogeneities of charged particles density in the ionospheric F!region are considered as a result of the neutrals| perturbation[ Up to now many authors "Hines\ 0859^ Khantadze and Gvelesiani\ 0868^ Hunsucker\ 0871^ Hocke and Schlegel\ 0885# have con! sidered travelling ionospheric disturbances "TID# as a result of atmospheric acousticÐgravity waves "AGW# propagation in the ionospheric F1!region[ The study of the gravity waveÐTID relationship improves the under! standing of the interaction between the thermosphere and ionosphere "Kirchengast\ 0885^ Hocke and Schlegel\ 0885#[ The in~uence of atmospheric gravity wave propagation on the behaviour of the ionospheric F1!layer for di}erent cases was considered by many authors[ In Hooke "0857#\ Testud and Francois "0860#\ Klostermeyer "0861a\ b#\ Morgan and Calderon "0867# and Shibata "0872#\ the undisturbed ionospheric parameters are assumed as inde! pendent of time and their changes due to the propagation of the gravity waves are considered as linear[ According to Porter and Tuan "0863# and Didebulidze and Pataraya
Corresponding author[ E!mail] gochaÝdtapha[kheta[ge
"0877# the height pro_le of electron density without grav! ity waves corresponds to the simple Chapman function with damping in time[ In these papers the behaviour of the height pro_le of electron density can be considered as a linear change of the background electron density[ Their investigation is based on the solution of the ambi! polar di}usion equation of the ionospheric F1!region using Green|s function\ when the gravity wave propa! gation is taken into account[ In the present paper we develop the solution of the ambipolar di}usion equation of the ionospheric F1! region by using Green|s function "Gliddon\ 0848^ Porter and Tuan\ 0863^ Didebulidze and Pataraya\ 0877#\ but with the presence of horizontal shear ~ow being taken into account[ Hereafter\ the horizontal shear ~ow cor! responds to the neutral meridional winds with zonal con! stant shear[ The in~uence of the neutral mean ~ow is essential in the coupled ionosphere:thermosphere models of Millward et al[ "0882a\ b#[ The mean ~owÐwave inter! action in~uences the formation of TIDs "Millward et al[\ 0882a\ b^ Hocke\ 0885# in the ionospheric F1!region[ The horizontal wind with constant vertical shear according to Farrell and Ioannou "0882# and Rogava and Mahajan "0886# sometimes is the source of the excitation of atmo! spheric waves with time varying wavenumber and periods[ The investigation of the evolution of the AGW
0253!5715:88:, ! see front matter Þ 0888 Elsevier Science Ltd[ All rights reserved PII] S 0 2 5 2 ! 5 7 1 5 " 8 8 # 9 9 9 9 3 ! 7
379
G[G[ Didebulidze\ A[D[ Pataraya : Journal of Atmospheric and Solar!Terrestrial Physics 50 "0888# 368Ð378
in the horizontal shear ~ow by the same method using modi_ed modal analysis "Pataraya and Pataraya\ 0886^ Didebulidze\ 0886#\ for the spectrum and ampli! _caiton:damping processes of these waves\ gives di}erent results[ According to Didebulidze "0886#\ in the hori! zontal shear ~ow together with acoustic and gravity waves appears the shear wave which is due to the presence of the shear ~ow[ This method of excitation of atmo! spheric gravity waves in the horizontal shear ~ow is not considered in the well!known coupled thermosphere: ionosphere models "Mayr et al[\ 0889^ Millward et al[\ 0882a\ b#[ The height pro_le of the electron density obtained in the present paper allows us to investigate the evolution of gravity and shear waves in the ionospheric F1!region[ In this solution the in~uence of the horizontal shear ~ow\ as well as the presence of the gravity and shear waves and their amplitude changes in time\ caused by the presence of the horizontal shear ~ow\ are taken into account[ The obtained night time pro_le of the electron density in the presence of horizontal shear ~ow shows the possibility of formation of the wave!like perturbation which cor! responds to the large scale TIDs with small period[ The resulting characteristic periods of the F1!layer par! ameters after a certain time for the evolution of the grav! ity waves are close to 05Ð19 min\ which are in agreement to the observations given in Oya et al[ "0871#\ Williams et al[ "0877# and Bowman "0881#[ The shear wave\ which is due to the presence of hori! zontal shear ~ow\ sometimes gives appreciable changes of the F1!layer[ We shall consider those cases when the evolution of the gravity and shear waves in the F1!region are di}erent[
1[ Ambipolar diffusion equation in the F1!region For the height distribution of electron density Ne"z\ t# in the ionospheric F1!region for the nonstationary case the ambipolar di}usion equation is used[ In this case the main processes of the formation and maintenance of the F1!layer are the balance between the di}usion of the oxygen ions\ O¦\ and the loss of electrons due to the ¦ dissociative recombination of ions O¦ 1 and NO [ This ambipolar di}usion equation in the case of the existence of the atmospheric gravity waves and the neutral mer! idional wind in the mid!latitude night ionosphere has the following form "Porter and Tuan\ 0863^ Didebulidze and Pataraya\ 0877#]
$
0
1Ne 1 1Ne Ne −Da cos1 u ¦ ¦ 1t 1z 1z 1H
1 %
¦"u¦u9# sin u cos uNe¦w cos1 uNe −L\
"0#
where L is the loss rate^ 1H is the scale height for ions^ the x!axis is oriented along the meridian to the North and the y!axis to the West^ x h−h9 is the di}erence between the actual and some initial height h9^ u is the angle between the magnetic _eld and the zenith\ the angle between the horizontal component of magnetic _eld and the x!direction is considered as the same^ Da is the ambi! polar di}usion coe.cient and for it the following well! known expression "Ferraro and O Ý zdog¼an\ 0847# will be used] Da cos1 u D9 exp
01
z \ H
"1#
and for the loss rate the following linear form "Ratcli}e et al[\ 0845# is used]
0 1
L bNe\ b b9 exp −
oz \ H
"2#
D9\ b9 are the coe.cients of the ambipolar di}usion and recombination\ respectively\ at height h h9^ o is the gas mixing coe.cient^ for 0 ¾ o ¾ 1\ the upper atmosphere changes from perfect mixing "o 0# to di}usive equi! librium "o 1#[ In eqn "0# u and w are the disturbed neutral gas vel! ocities in the meridional and vertical directions\ respec! tively[ u9 is the neutral meridional wind velocity which in the present case corresponds to the horizontal shear ~ow u9 ay\
"3#
where a is the shear of the horizontal "meridional# wind[ In eqn "0# the ionospheric is assumed to be isothermal and the condition of small changes of height pro_le of the electron density in the horizontal plane\ "0:Ne#"1Ne:1x\ y# ð "0:H#\ is taken into account[
2[ Equations of evolution of AGW in horizontal shear ~ow According to Didebulidze "0886# and Pataraya and Pataraya "0886# the amplitudes of the atmospheric AGW are changing in time[ In these papers the ampli! _cation:damping of the AGW amplitudes is due to the presence of the horizontal shear ~ow[ In the present paper the disturbed neutrals| velocity u and w correspond to the atmospheric gravity and shear waves[ These types of waves will be referred to as modi_ed gravity waves[ According to Didebulidze "0886# the perturbed values of pressure "p#\ density "r#\ and velocity "v v"u\ v\ w## components for the isothermal atmosphere in horizontal shear ~ow may be described by the following trans! formation]
G[G[ Didebulidze\ A[D[ Pataraya : Journal of Atmospheric and Solar!Terrestrial Physics 50 "0888# 368Ð378
$
"p\ r#"x\ y\ z\ t# "Pk\ Rk#"t# = exp if"t#−
$
%
z \ 1H
"u\ v\ w#"x\ y\ z\ t# "Uk\ Vk\ Wk#"t# = exp if"t#¦
"4a#
%
z \ 1H "4b#
f"t# kxx¦ky"t#y¦kzz\
"4c#
ky"t# k9−akxt[
"4d#
In these expressions the evolution of complex valued amplitudes\ qk"t# 0 "Pk\ Rk\ Uk\ Vk\ Wk#"t#\ according to the equations of motion\ continuity and adiabatic behav! iour\ when the dissipation is neglected\ is described by the set of the following linearized equations "Didebulidze\ 0886#] r99U?k −ikxPk−ar99Vk\
"5a#
r99V?k −ik"t#Pk\
"5b#
0
1
0 P −`Rk\ r99W?k − ikz− 1H k
0
"5c#
1
0 R?k −iðkxUk¦ky"t#VkŁ− ikz− Wk\ r99 1H P?k r99cs1
−iðkxUk¦ky"t#VkŁ−"ikz−E#Wk\
"5d#
"5e#
where the prime denotes derivatives in time^ k9 ky"t 9#^ k"t# kðkx\ ky"t#\ kzŁ is the time varying wavenumber^ v is the component of the perturbed vel! ocity in the y!direction^ g is the acceleration due to grav! ity^ H cs1:g` is the atmospheric "neutral gas# scale height^ cs ðg"p9:r9#Ł0:1 is the velocity of sound\ g is the ratio of speci_c heats^ p9 p99 exp ð−"z:H#Ł\ r9 r99 exp ð−"z:H#Ł are the unperturbed barometric height pro_le for the unperturbed pressure and density\ respectively^ p99 and r99 are the pressure and density at height z 9\ respectively^ E ð"1−g#:1gHŁ is the iso! thermal Eckart coe.cient^ cs\ H and E are assumed con! stant with height[ Equations "4# and "5# make up a math! ematically closed system of di}erential equations[ These linearized equations\ in the absence of shear ~ow\ describe the atmospheric AGW "Hines\ 0859#[ The evolution of amplitudes qk for times of Dt ð ta\ where ta 1
b b
k9 \ akx
"6#
according to Didebulidze "0886# can be described by use of the following transformation] qk"t# q"t# = exp ð−iV"t#Ł\
"7#
where Re "V?# corresponds to the characteristic fre!
370
quencies of atmospheric waves and Im "V?# are their amplitudes ampli_cation:damping rates\ respectively[ Using transformation "7# in the set of eqns "4# and "5# for the AGW gives the following characteristic frequencies "Didebulidze\ 0886#]
0
1
0 0 1 va\g "t# cs1 =k"t#=1¦ 1 3H1 2
X 0
0 1 0 3 −vb1cs1"kx1¦ky1"t##\ cs =k"t#=1¦ 3 3H1
1
"8#
where =k"t#=1 kx1¦ky1"t#¦kz1^ vb z"g−0#`:gH is the isothermal BruntÐVaisala frequency[ In eqn "8# high fre! quency\ va\ corresponds to acoustic waves and low fre! quency\ vg\ to gravity waves\ respectively[ The ampli! _caiton:damping rates "Im "V?## for acoustic\ vai\ and for gravity waves\ vgi\ respectively\ are] acs1kxky"t# vb1 0− [ vai\gi"t# 3 1 1ðva1"t#−vg1"t#Ł va\g "t#
$
%
"09#
Sign {−| at the beginning of the right!hand!side of eqn "09# corresponds to acoustic waves and {¦| to gravity waves\ respectively[ Values va\g"t#\ "8#\ and vai\gi"t#\ "09#\ are applied for describing the evolution of acoustic and gravity waves in the horizontal shear ~ow for times 1 1 Dt ð ta when condition va\g Ł vai\gi is valid[ The set of eqns "4# and "5# also describe the evolution of a shear wave\ which has zero frequency "Re "V?# 9# and the ampli_caiton:damping rate is determined by the following expression "Didebulidze\ 0886#] vshi"t#
akxky"t# kx1¦ky1"t#
[
"00#
According to the characteristic dispersion eqns "8#Ð"00# the acoustic wave frequency\ va"t#\ continuously increases for times of t × ta^ for the same times the gravity wave frequency\ vg"t#\ tends to the isothermal BruntÐ Vaisala frequency vb[ These equations also show that the increase of amplitudes of the acoustic and gravity waves appears at time t × ta[ Equation "8# shows us that the minimum frequency of the acoustic and gravity waves occurs at time t "ta:1#[ According to dispersion eqn "8# the meridional phase velocity of the gravity waves for large times of t Ł ta tends to the value of vb:kx[ According to eqn "00# the shear wave excitation is due to the presence of horizontal shear ~ow[ According to eqns "5#\ "7# and "00# the shear wave corresponds to the standing type "Re "V?# 9# incompressible "div v ¹ 9# vortical ""rot v#z const# perturbation\ excited in hori! zontal shear ~ow[ For condition "a:vB# ð 0\ the part of term ar99Vk in "5a# is similar to the e}ect of the Coriolis force for the traditional approach when the vertical com! ponent of this force is neglected[ In this case vortical perturbation is excited during the established balance
371
G[G[ Didebulidze\ A[D[ Pataraya : Journal of Atmospheric and Solar!Terrestrial Physics 50 "0888# 368Ð378
between the perturbation of the shear direction velocity of the rotation driving background ~ow and the pressure gradient[ In this case\ according to eqns "5#\ "7# and "00# the ampli_cation:damping of the amplitudes of the shear waves Uk and Vk "=Uk=\ =Vk= Ł =Wk=# are essential at time t ¾ ta[ The total spectral energy "½=qk=1 ½ exp "1 Im "V### for the shear waves according to eqn "00# has the transient development for times of t ¾ ta[ In the cases when for the horizontal velocity of the winds the following approxi! mation u9 U¦ay is possible\ the shear wave propagates in the meridional direction with velocity U const[ Henceforth\ these types of evolving gravity and shear waves will be referred to as modi_ed gravity waves[ In the following\ for long time interval Dt × ta\ in which the appearance of both gravity and shear waves is essential\ we consider the evolution of these waves numerically using the initial condition in accordance with eqns "7#Ð "00#[ In the following we consider the behaviour of the ionospheric F1!layer which corresponds to the evolution for modi_ed large wavelength\ lx\ ly Ł H\ gravity waves[ Here lx and ly are the wavelengths in the meridional and zonal direction\ respectively[ It is possible to describe these types of long period waves of the neutral gas in the F1!region by eqn "0#[ The molecular viscosity\ which is essential for the dissipation of gravity waves in the ionospheric F1!region "Hines\ 0859#\ for the large scale gravity waves\ lx\y Ł H\ is com! parably small "Hines\ 0857^ Richmond\ 0867# and in eqns "5# for simplicity we have neglected it[ According to Rich! mond "0867# and Mayr et al[ "0889# for gravity waves with a horizontal phase velocity about 649 m s−0 the temperature and wind perturbations at thermospheric heights are decaying slowly before reaching the equator[ So one may expect that gravity wave as well as the shear wave!like vortical perturbation\ which propagates with the wind at velocity U\ can possibly appear at the mid! latitude F!region[ In this case we consider the polar regions as the main source of the AGW generation of which the ionospheric manifestation is the large scale TIDs[ In the present consideration the presence of hori! zontal shear ~ow is possibly an additional source of grav! ity wave generation and it would be the reason for the large scale TIDs observed in the mid!latitude ionosphere[ In the following we choose the values of horizontal winds shear\ a\ in the ionospheric F1!region in accord! ance with the review of the satellite wind measurements given in Mayr et al[ "0889#[ According to Mayr et al[ "0889# the horizontal thermospheric wind measured in the polar regions is 799Ð1999 m s−0 and more on mag! netically disturbed days[ In some cases the change of this value of velocity for 1Ð4> of latitude is sometimes of the same order as the value[ Thus\ it is possible to choose the horizontal shear of the horizontal winds as a 9[9990Ð 9[990 s−0\ and even more[ The set of eqns "4# and "5# describes the evolving ampli! tudes of the coupled waves[ In this case coupling between
acoustic\ gravity and shear waves is due to the strati! _cation and the horizontal shear ~ow[ The values of ampli_cation:damping rates "09# and "00# are valid for the small time intervals of Dt ð ta in which the change of amplitudes of the considered waves due to the coupling between di}erent waves are negligibly small[ These considerations correspond to the cases when "a:vB# ð 0 for which the changes of amplitudes of shear and gravity waves due to their coupling in the horizontal shear ~ow is negligible[ For simplicity we have not con! sidered the cases of large shear\ "a:vB# − 0\ when it would be expected that the changes of amplitudes of all types of waves due to their energy exchange in the horizontal shear ~ow is essential[
3[ Evolution of gravity and shear waves in the F1!region In eqn "0# values u and w correspond to the real parts of the perturbed velocity of atmospheric waves which will be taken from eqn "4b#[ For the solution of eqn "0# we use a new variable z k = exp ð−"z:H#Ł and the following transformation]
$
Ne z0:3 exp −1Az¦
%0
0 V "z\ t#¦c"t# D9 n
1
Y9¦ s Yl \ l 0
"01a# where k
H 1 zu99 ¦3b9D9 D9
"01b#
u99 u9 sin u cos u\ u99
A
1 99
"01c# \
"01d#
3zu ¦3b9D9 Vn
g
z
z?
e− 1HVn9"z?\ t?# dz?\
"01e#
Vn9 Re "ðUk"t# sin u¦Wk"t# cos uŁ cos u eif#[
"01f #
The cases when the production rate of electrons in eqn "0# is given by Q f "z\ t#Ne\ can be easily included in the following formalism[ Here f "z\ t# is some function which depends on the source of production[ With eqns "7# in "0# for function Y9"z\ t#\ Y0"z\ t#\ Y1"z\ t#\ [ [ [ the set of following equations can be obtained] L
Y9 9\
"02a#
L
Yl Fl"Yl−0#\ l 0\ 1\ [ [ [ [
"02b#
Here operator L
has the form]
L
0 2 −1 11 0 1 \ −n9 z ¦Az−0− −n9 z 1t 05 3 1z1
$
%
"03#
372
G[G[ Didebulidze\ A[D[ Pataraya : Journal of Atmospheric and Solar!Terrestrial Physics 50 "0888# 368Ð378
where n9 "kD9:H1#[ Function Fl will be considered as the inhomogeneous part of eqns "02b# and has the fol! lowing form]
$ 01 % $ 0 1 $0 %
−
¦
dz
m
"z
m−0 1
% 1
0 1A− z−0 Yl−0 3
e−z#
"05#
where 0 7H
1
kD9 −u99 [ H
0 1
Y9 s Cm exp ð−lm"t−t9#Łz0:3 exp − m 9
G"z\ z?\ t−t?#
1Yl"z?\ t?# 1z? z?
%
1G"z\ z?\ t−t?# 1z?
dt?[
"08#
z? 9
Using solution "07# of the homogeneous part of eqns "02#\ the Green|s function can be obtained in the fol! lowing closed form "Whittaker and Watson\ 0841^ Lebedev\ 0852#]
G"z\ z?\ t−t?#
"zz?#0:3 "1pshðn0"t−t?#Ł#0:1
× exp
6
ch
6
"zz?#0:1 shðn0"t−t?#Ł
7
7
z¦z? u99 [ "t−t?#− 3H 1thðn0"t−t?#Ł
"19#
This Green|s function in the absence of the neutral wind velocity is the same as in Moschandreas and Tuan "0863# and Porter and Tuan "0863#[ The Green|s function\ "19#\ when the neutral wind velocity is constant\ has the same form as obtained in Didebulidze and Pataraya "0877#[ Note that the form of the Green|s function for the di}er! ent scale height of the ions "1H# must be other than given by eqn "19# "Lvova et al[\ 0861^ Makeev et al[\ 0862#[ Introduction of the function c"t# in transformation "01# allows us to solve eqns "02#[ The condition that Y9"z\ t# is the solution of the set of eqns "02# is equivalent to the equation]
"06#
The solution of eqn "02a# can be written in the following form]
g$
−Yl"z?\ t?# "04#
lm mn9¦1n0\
0 10
G"z\ z?\ t−t?#Yl"z?\ t9# dz? z?
t9
"Whittaker and Watson\ 0841^ Lebedev\ 0852# and their eigenvalues\ respectively\ are]
n0
G"z\ z?\ t−t?#Fl"z?\ t?# dz? dt?
t
The homogeneous part of each of eqns "02# is the same and can be solved as the SturmÐLiouville equation "Tuan\ 0857#[ Thus\ using transformation "01# in "0# gives us one homogeneous equation "02a# and l inhomogeneous equations "02b#[ Transformation "01# allows us to obtain the linear expression with the modi_ed gravity waves| velocity in the inhomogeneous part of eqns "02b#\ and function c"t# will be obtained together with the solution of the eqns "02b#[ The separation of the variables "z and t# allows us to reduce eqn "02a# to the equation whose eigenfunctions are the generalized Laguerre polynomials dm
t9
¦
1Yl−0 [ 1z
Lm−0:1"z# ezz0:1
9
gg 9
Yl−0 1 0 c"t#¦ Vn"z\ t# z 1t D9 0:1
t
¦
Yl−0
Vn9"z\ t# k H z
g
o−0
b9 z Fl"Yl−0# 0− k k −
Yl"z\ t#
z −0:1 L "z#\ 1 m "07#
where t9 is some initial time[ The expression for Ne"z\ t# using eqns "01a# and "07# gives the solution to eqn "0#\ when the modi_ed gravity waves are absent and o 0 "Chamberlain\ 0850#[ The solution of the inhomogeneous equations "02b# by using Green|s function G"z\ z?\ t−t?# can be written "Tikhonov and Samarskiy\ 0866# as]
t
9
t9
gg
G"z\ z?\ t−t?#F0"z?\ t?# dz? dt? 9[
"10#
Thus\ eqn "10# means that the inhomogeneous solution Y0 9 and\ according to eqns "08#\ all Yl 9[ So solution "07# and solution c"t# of eqn "10# by use of transformation "01# gives us the solution of eqn "0#[ Hereafter\ for the gravity waves we use the isothermal model where relations "4# and "5# are valid "Hines\ 0859^ Didebulidze\ 0886#[ Some other models of the gravity waves\ when their amplitude dependence on height is di}erent from that in eqns "4#\ may be included in our formalism[ Expansion of Green|s function\ "19#\ in Laguerre poly! nomials gives]
373
G[G[ Didebulidze\ A[D[ Pataraya : Journal of Atmospheric and Solar!Terrestrial Physics 50 "0888# 368Ð378
n; 0:3 0 exp ð−ln"t−t?#Ł"zz?# G"n¦ # n 9 1
G"z\ z?\ t−t?# s
0
× exp −
1
z¦z? −0:1 Ln "z#Ln−0:1"z?# 1
"11#
where "12#
ln nn9¦1n0[
Terms in eqns "07# and "11# when m\ n 0\ 1\ [ [ [ damp more rapidly than the term when m\ n 9[ If we limit ourselves in eqns "05#Ð"07#\ "11# and "12# by the terms\ when m\ n 9 for the initial condition c"t t9# 9\ from "10# the solution for c"t# can be easily obtained and the height distribution of the F1!region electron density can be written as]
6
Ne N exp −l"t−t9#−
−
z a z − e− H 1H 1
$
%7
z 0 0 c0"y\ t# Vn0 e− 1H− Vd zpk
\
a k¦ Vd
b9 zpk
$
0 1 0 1%
k0−oG o¦
0 0 −G 0¦ 1 1
u99H \ D9
D9 \ 1H
Vn0
Ak z0¦Az−1 cos "f¦f0¦f1#\ Az
g $
c0
\
t? t e−z? Vn"z?\ t?#=t? t ¦1n0 9
9
g
t
"13c# "13d#
Vn0 ð 0\ Vd
"13e#
%
where Ak ="Uk sin u¦Wk cos u# cos u=\
"13g#
Az 1Hkz\
"13h#
$
%
Im "Uk sin u¦Wk cos u# \ Re "Uk sin u¦Wk cos u#
f1 arctan "Az#[
%
"13b#
"13f#
f0 arctan
0 1
where hm9 h9¦H = ln a is the maximum of the simple Chapman layer[ In the case of =Az= Ł 0 ð=kz= Ł "0:1H#Ł the behaviour of the main maximum "maximum close to hm9# of Ne"z\ t# can be approximately described by the expression "14# when ðin "14#Ł Vn9 Vn9"z H = ln a#[ According to eqns "13# and "14#\ when the conditions
"13i# "13j#
The value Vd\ "13d#\ corresponds to the characteristic velocity of the ambipolar di}usion of ions[ Solution for Ne"z\ t#\ "13#\ allows us to investigate analytically and numerically the evolution of the modi_ed gravity waves in the horizontal shear ~ow[ According to solution "13# and eqns "4# and "5# we may use the analytical as well as numerical results for describing the behaviour of the
Vn9 1zaVd
ð 0\
0¦
0 Vn9 1 Vn9 − \ 3a Vd 1zaVd
hmF1 hm9−1H = ln
Vn9"z?\ t?# dt? dz?\
t9
$X
"13a#
where l 1n0¦
electron density height pro_le\ Ne"z\ t#\ in the horizontal shear ~ow[ In the case of no shear "a 9 or u9 const# the height pro_le of electron density corresponds to the propagation of gravity waves for which the spectrum is given by Hines "0859# and in this case the values c0\ "13f#\ will be easily obtained[ Solution "13# without the modi_ed gravity waves "Vn0 9\ c0 9# corresponds to the simple Chapman function damping in time[ In the present case this func! tion describes the background F1!layer[ In "13a# the time decay coe.cient l\ "13b#\ and value a\ "13c#\ depend on the velocity "u9# of the horizontal winds[ Terms Vn0 and c0 in "13# correspond to the part of the in~uence of the evolution of the modi_ed gravity wave in horizontal shear ~ow[ According to solution "13# the height of the maximum of the F1!layer electron density for the case of kx 9 can be written as] "14#
"15#
are valid\ the height pro_le of Ne"z\ t# can be considered as the linear disturbed simple Chapman layer damping in time[ Hooke "0857#\ Testud and Francois "0860#\ Klo! stermeyer "0861a\ b#\ Morgan and Calderon "0867# and Shibata "0872# assumed that the basic undisturbed F1! layer is independent of time and that perturbations to its parameters due to the gravity waves are linear[ The val! idity of the inequalities "15# essentially depends on the value of Az\ "13h#[ For large vertical wavenumbers =kz= Ł "0:1H# "=Az= Ł 0#\ in many cases of gravity wave propagation the linear approach of the change of Ne"z\ t# would be true[ In the present cases\ according to eqn "13#\ the conditions "Vn0:Vd# − 0\ "Vn9:1zaVd# − 0 give more appreciable changes of Ne than the linear disturbed sim! ple Chapman layer[ These results are di}erent from those obtained by Huang and Li "0880# or Nekrasov et al[ "0884# where the nonlinear changes of the F!region ion density are due to the nonlinear gravity waves[ By use of solutions "13# and "14# it is possible to con! sider many di}erent cases for describing the evolution of the modi_ed gravity waves in the horizontal shear ~ow[ We shall limit ourselves to only two cases which cor! respond to the variation of hmF1"t# and its maximum electron density NmF1"t# for the case of the evolution of
G[G[ Didebulidze\ A[D[ Pataraya : Journal of Atmospheric and Solar!Terrestrial Physics 50 "0888# 368Ð378
the gravity and shear waves[ According to eqns "13#Ð "15# the in~uence of the modi_ed gravity waves on the behaviour of the F1!layer electron density is more e}ec! tive for the cases of kz 9\ which are partially dem! onstrated in Figs 0 and 1[ In these _gures the evolution of amplitudes Uk"t# and Wk"t# of the disturbed wave vel! ocity and the corresponding behaviour of the maxima of electron density\ NmF1"t#\ their heights\ hmF1"t#\ for the atmospheric gravity and shear waves\ are plotted\ respec! tively[ For all _gures the mid!latitude F1!region and the wave parameters are chosen as follows[ kx 4[1 = 09−5 m−0\ k9 1[6 = 09−5 m−0\ a 0[05 = 09−3 s−0\ t9 9\ H 5 = 093 m\ b9 1 = 09−3 s−0\ D9 8 = 094 m1 s−0\ N 095 cm−2\ h9 299 km[ For de_niteness we have chosen the condition sin u cos u ¹ −9[3 which cor! responds to the mid!latitude northern hemisphere[ For simplicity we choose the gas mixing coe.cient o 0[64\ which corresponds to di}usive equilibrium\ when the recombination of ions O¦ is mainly due to collisions with nitrogen\ N1\ molecules[ In Figs 0 and 1 for the cases "c# and "d# the dark lines correspond to the case of the absence of the modi_ed gravity waves "background F1! layer#[ According to Fig[ 0\ for large times\ t × ta "ta ¹ 049 min#\ the periods of the gravity waves are weakly changed and are nearly 05Ð19 min[ This case corresponds to the formation of large scale TIDs with small period and a phase velocity ð¹"vb:kx#Ł in the mer! idional direction of about 0999Ð0199 m s−0[ The period of nearly 06 min is in agreement with the observation of large scale TIDs by Williams et al[ "0877#\ Bowman "0881# and also with observations by Oya et al[ "0871#[ In these cases we try to choose parameters of the modi_ed gravity waves for which the changes of wave amplitudes caused by the coupling between acoustic\ gravity and shear wave modes would be small[ For Figs 0 and 1 the initial con! ditions are chosen in accordance to expressions "8#Ð"00# and to the condition =Uk=\ =Uk= Ł =Wk=[ With horizontal shear ~ow the changes of amplitude of the vertical com! ponent of the velocity perturbation\ Wk\ due to gravity waves\ occurs for times of t × ta "see Fig[ 0"b##[ Thus\ for the considered time intervals this amplitude becomes comparable to the values of amplitudes of the meridional perturbed velocity Uk "see Fig[ 0"a##[ The case when the horizontal wavelengths "lx\ ly# of gravity waves are comparable to the neutral gas scale height\ lx\y O"H#\ would lead to the formation of med! ium!scale TIDs\ due to the horizontal shear ~ow[ In this case in eqn "0# the terms describing the horizontal inhom! ogeneity of the height pro_le of electron density must be added "Shibata\ 0872# and dissipative processes in eqns "5# of the evolution of modi_ed atmospheric gravity waves would be taken into account "Richmond\ 0867^ Mayr et al[\ 0889#[ According to the cases demonstrated in Fig[ 1\ the evolution of a shear wave gives a di}erent change of the background F1!layer than in the case of the evolution of
374
the gravity waves[ For the shear wave\ su.cient changes of the F1!layer parameters hmF1"t# and NmF1"t# appear for times of t ¾ ta[ Similar types of changes of the iono! spheric F1!layer on magnetically disturbed days over times 1Ð4 h are known "Fatkullin\ 0867#[ In this case\ plotted in Fig[ 1\ the initial condition of the amplitude of the shear wave!like meridional velocity perturbation\ Uk\ is chosen to be opposite to the North direction[ The corresponding maximum of the F1!layer electron density\ hmF1 "Fig[ 1"d# thin line# according to "14# is located above that in the case of no perturbation "Fig[ 1"d# full line#[ Note that the cases when Da"z\ t# Da"z#¦D0"x\ y\ z\ t# "Da Ł D0# and b b"z#¦b0"x\ y\ z\ t# "b Ł b0# are easy to include in the above applied formalism[ For reasons of brevity we have not considered in more detail the small changes of electron density ð"0:Ne#"1Ne:1x\ y# ð "0:H#Ł\ the time decay l\ "13b#\ and of the value a\ "13c#\ in the horizontal plane[
4[ Conclusion We have studied the reaction of the ionospheric F1! region to the evolution of gravity and shear waves in the horizontal shear ~ow when the production of electrons is not taken into account[ A solution of the ambipolar di}usion eqn "0#\ using Green|s function when the neutral horizontal shear ~ow is taken into account\ is developed[ It has been found that the height pro_le of electron density\ Ne"z\ t#\ allows us to investigate the evolution of the gravity and shear waves in horizontal shear ~ow[ The evolution of the meridional and vertical components of the velocity of the gravity and shear waves and their in~uence on the behaviour of the F1!region electron den! sity are demonstrated[ Appreciable changes of the F1! layer parameters NmF1"t# and hmF1"t# appear for the shear waves for times of t ¾ ta[ For atmospheric gravity waves\ for times of t × ta\ hmF1"t# is changed and the frequency of the oscillations tends to the isothermal BruntÐVaisala frequency[ For the scale height of the F1! region neutral gas H 59 km the characteristic periods of variation of hmF1"t# and NmF1"t# due to the evolution of the gravity wave in the horizontal shear ~ow at large times t × ta are close to 05Ð19 min[ It has been shown that the changes of the F1!layer electron density due to the evolution of gravity and shear waves can be described as a linear change of the simple Chapman layer\ with damping in time[
Acknowledgement This work was supported\ in part\ by a Grant awarded by the Georgian Academy of Science[
375
G[G[ Didebulidze\ A[D[ Pataraya : Journal of Atmospheric and Solar!Terrestrial Physics 50 "0888# 368Ð378
Fig[ 0[ Evolution of the velocity amplitudes "a# Uk"t#\ "b# Wk"t# of modi_ed atmospheric gravity waves\ and the behaviour of "c# NmF1"t# and "d# hmF1"t# of the F1!layer[
G[G[ Didebulidze\ A[D[ Pataraya : Journal of Atmospheric and Solar!Terrestrial Physics 50 "0888# 368Ð378
376
Fig[ 1[ Evolution of the velocity amplitudes "a# Uk"T#\ "b# Wk"T# of atmospheric shear waves\ and the behaviour of "c# NmF1"t# and "d# hmF1"t# of the F1!layer[
377
G[G[ Didebulidze\ A[D[ Pataraya : Journal of Atmospheric and Solar!Terrestrial Physics 50 "0888# 368Ð378
References Bowman\ G[G[\ 0881[ Some aspects of large!scale travelling ionospheric disturbances[ Planetary and Space Science 39\ 718Ð734[ Chamberlain\ J[W[\ 0850[ Physics of the Aurora and Airglow[ Academic Press\ New York[ Didebulidze\ G[G[\ 0886[ Ampli_cation:damping processes of atmospheric acoustic!gravity waves in horizontal winds with linear shear[ Physics Letters A 124\ 54Ð69[ Didebulidze\ G[G[\ Pataraya\ A[D[\ 0877[ Reaction of the iono! spheric F!layer on oblique propagation of the inner gravi! tational waves in the presence of a local wind[ Bulletin of Abastumani Astrophysical Observatory 55\ 112Ð121[ Farrell\ B[F[\ Ionannou\ P[J[\ 0882[ Transient development of perturbation in strati_ed shear ~ow[ Journal of Atmospheric Science 49\ 1190Ð1103[ Fatkullin\ M[N[\ 0867[ Ionospheric disturbances[ Science and Technical Results[ Geomagnetism and Upper Layers of Atmosphere 3\ 5Ð096 "in Russian#[ zdog¼an\ I[\ 0847[ The e}ect of di}usion on Ferraro\ V[C[A[\ O vertical distribution of ionization in a quite F!region[ Journal of Atmospheric and Terrestrial Physics 01\ 039Ð038[ Gliddon\ J[E[C[\ 0848[ Use of Green|s function in the solution of ionospheric di}usion problems[ Quarterly Journal of Mech! anics and Applied Mathematics 01\ 236Ð242[ Hines\ C[O[\ 0859[ Internal atmospheric gravity waves at iono! spheric heights[ Canadian Journal of Physics 27\ 0330Ð0370[ Hines\ C[O[\ 0857[ An e}ect of molecular dissipation in upper atmospheric gravity waves[ Journal of Atmospheric and Ter! restrial Physics 29\ 734Ð738[ Hocke\ K[\ Schlegel\ K[\ 0885[ A review of atmospheric gravity waves and traveling ionospheric disturbances] 0871Ð0884[ Annales Geophysicae 01\ 806Ð839[ Hooke\ W[H[\ 0857[ Ionospheric irregularities produced by internal atmospheric gravity waves[ Journal of Atmospheric and Terrestrial Physics 29\ 684Ð712[ Huang\ C[S[\ Li\ 0880[ Weak nonlinear theory of the ionospheric response to atmospheric gravity waves in the F!region[ Jour! nal of Atmospheric and Terrestrial Physics 42\ 892Ð897[ Hunsucker\ R[D[\ 0871[ Atmospheric gravity waves generated in the high!latitude ionosphere] a review[ Reviews of Geophysics and Space Physics 19\ 182Ð204[ Khantadze\ A[G[\ Gvelesiani\ A[I[\ 0868[ On the Theory of Di}usion in Ionosphere F!region[ Moskva\ Nauka "in Russian#[ Kirchengast\ G[\ 0885[ Elucidation of the physics of the gravity waveÐTID relationship with the aid of theoretical simulations[ Journal of Geophysical Research 090\ 02\242Ð02\257[ Klostermeyer\ J[\ 0861a[ Numerical calculation of gravity wave propagation in a realistic thermosphere[ Journal of Atmo! spheric and Terrestrial Physics 23\ 654Ð663[ Klostermeyer\ J[\ 0861b[ Comparison between observed and numerically calculated atmospheric gravity waves in the F! region[ Journal of Atmospheric and Terrestrial Physics 23\ 0282Ð0390[ Lebedev\ N[N[\ 0852[ Special Functions and Their Application\ Moskva "in Russian#[ Lvova\ A[A[\ Poliakov\ V[M[\ Ribin\ V[V[\ 0861[ Peculiarities of nonstationary solutions of the equation of di}usion in the gravitational _eld[ Izvestia Vuzov Radiophysica 04\ 739Ð740 "in Russian#[
Makeev\ V[V[\ Poliakov\ V[M[\ Ribin\ V[V[\ 0862[ Non! stationary solution of di}usion equation in gravitational _eld with consideration of disappearance of particles and their transfer[ Izvestia Vuzov Radiophysica 05\ 0559Ð0569 "in Rus! sian#[ Mayr\ H[G[\ Harris\ I[\ Herraro\ F[A[\ Spencer\ N[W[\ Varosi\ F[\ Pesnell\ W[D[\ 0889[ Thermospheric gravity waves] obser! vations and interpretation using the transfer function model "TFM#[ Space Science Reviews 43\ 186Ð264[ Millward\ G[H[\ Mo}ett\ R[J[\ Quegan\ S[\ Fuller!Rowell\ T[J[\ 0882b[ E}ects of an atmospheric gravity wave on the mid! latitude ionospheric F layer[ Journal of Geophysical Research 87\ 08\062Ð08\068[ Millward\ G[H[\ Quegan\ S[\ Mo}ett\ R[J[\ Fuller!Rowell\ T[J[\ Rees\ D[\ 0882a[ A modelling study of the coupled ionospheric and thermospheric response to an enhanced high!latitude elec! tric _eld event[ Planetary and Space Science 30\ 34Ð45[ Morgan\ M[G[\ Calderon\ 0867[ Testing Hook|s perturbation formula for the production of TID|s by gravity waves[ Journal of Geophysical Research 72\ 4626Ð4639[ Moschandreas\ D[J[\ Tuan\ T[F[\ 0863[ On the behavior of the polar ionosphere under the in~uence of incident electron ~ux[ Journal of Atmospheric and Terrestrial Physics 25\ 176Ð292[ Nekrasov\ A[K[\ Shalimov\ S[L[\ Shukla\ P[K[\ Sten~o\ L[\ 0884[ Nonlinear disturbances in the ionosphere due to acoustic grav! ity waves[ Journal of Atmospheric and Terrestrial Physics 46\ 626Ð630[ Oya\ H[\ Takahashi\ T[\ Morioka\ A[\ Miyaoka\ H[\ 0871[ Wavy patterns of ionospheric electron density pro_les triggered by TID!observation results of the electron density by Taiyo Sat! ellite[ Journal of Geomagnetism and Geoelectricity 23\ 498Ð 414[ Pataraya\ A[D[\ Pataraya\ T[A[\ 0886[ An excitement of the gravitational and acoustic waves in the solar atmosphere caused by inhomogeneous ~ow[ Astronomical and Astro! physical Transactions 02\ 010Ð015[ Porter\ H[S[\ Tuan\ T[F[\ 0863[ On the behaviour of the F!layer under the in~uence of gravity waves[ Journal of Atmospheric and Terrestrial Physics 25\ 024Ð046[ Ratcli}e\ J[A[\ Schmerling\ E[R[\ Setty\ C[S[G[K[\ Thomas\ J[O[\ 0845[ The rates of production and loss of electrons in the F region of the ionosphere[ Philosophical Transactions 137\ 510Ð531[ Richmond\ A[D[\ 0867[ Gravity wave generation\ propagation\ and dissipation in the thermosphere[ Journal of Geophysical Research 72\ 3020Ð3034[ Rogava\ A[D[\ Mahajan\ S[M[\ 0886[ Coupling of sound and internal waves in shear ~ows[ Physics Reviews E 44\ 0074Ð 0077[ Shibata\ T[\ 0872[ A numerical calculation of the ionospheric response to atmospheric gravity waves in the F1!region[ Jour! nal of Atmospheric and Terrestrial Physics 34\ 686Ð798[ Testud\ J[\ Francois\ P[\ 0860[ Importance of di}usion processes in the interaction between neutral waves and ionization[ Jour! nal of Atmospheric and Terrestrial Physics 22\ 654Ð663[ Tikhonov\ A[N[\ Samarskiy\ A[A[\ 0866[ Equations of Math! ematical Physics[ Moskva\ Nauka "in Russian#[ Tuan\ T[F[\ 0857[ On the theory of the ionospheric F layer[ Journal of Physics B "Proceedings of Physical Society# 0\ 855Ð 861[ Whittaker\ E[T[\ Watson\ G[N[\ 0841[ A Course of Modern Analysis[ Cambridge University Press[
G[G[ Didebulidze\ A[D[ Pataraya : Journal of Atmospheric and Solar!Terrestrial Physics 50 "0888# 368Ð378 Williams\ P[J[S[\ Crowley\ G[\ Schlegel\ K[\ Virdi\ T[S[\ McCrea\ I[\ Watkins\ G[\ Wade\ N[\ Hargreaves\ J[K[\ Lachlan!Cope\ T[\ Muller\ H[\ Baldwin\ J[E[\ Warner\ P[\ Van Eyken\ A[P[\ Hapgood\ M[A[\ Rodger\ A[S[\ 0877[ The generation and
378
propagation of atmospheric gravity waves observed during the Worldwide AcousticÐGravity wave Study "WAGS#[ Jour! nal of Atmospheric and Terrestrial Physics 49\ 212Ð226[
Journal of Atmospheric and Solar!Terrestrial Physics 50 "0888# 368Ð378
Ionosphere F1!region under the in~uence of the evolutional atmospheric gravity waves in horizontal shear ~ow G[G[ Didebulidze \ A[D[ Pataraya Town Department of Abastumani\ Astrophysical Observatory\ A[ Kazbegi av[ 1a\ 279959 Tbilisi\ Geor`ia Received 15 June 0887^ received in revised form 4 January 0888^ accepted 03 January 0888
Abstract The ambipolar di}usion equation for the height distribution of electron density in the ionospheric F1!layer is solved in the presence of neutral horizontal shear ~ow[ By using this nonstationary solution the reaction of the F1!region electron density on the evolution of atmospheric acousticÐgravity waves "AGW# is investigated[ The evolution of the AGW and the corresponding behaviour of the height distribution of the F1!region electron density are described by the characteristic time\ ta\ of transient development of shear waves in the horizontal shear ~ow[ For long times t × ta\ the gravity wave frequency tends to the isothermal BruntÐVaisala frequency\ which appears in the F1!layer as wavelike behaviour of hmF1 and NmF1 with periods close to 05Ð19 min\ when the scale height of the neutral gas is H 59 km[ The shear wave\ which is due to the presence of horizontal shear ~ow\ gives su.cient changes of the height pro_le of electron density for times of t ¾ ta[ Þ 0888 Elsevier Science Ltd[ All rights reserved[
0[ Introduction Many types of inhomogeneities of charged particles density in the ionospheric F!region are considered as a result of the neutrals| perturbation[ Up to now many authors "Hines\ 0859^ Khantadze and Gvelesiani\ 0868^ Hunsucker\ 0871^ Hocke and Schlegel\ 0885# have con! sidered travelling ionospheric disturbances "TID# as a result of atmospheric acousticÐgravity waves "AGW# propagation in the ionospheric F1!region[ The study of the gravity waveÐTID relationship improves the under! standing of the interaction between the thermosphere and ionosphere "Kirchengast\ 0885^ Hocke and Schlegel\ 0885#[ The in~uence of atmospheric gravity wave propagation on the behaviour of the ionospheric F1!layer for di}erent cases was considered by many authors[ In Hooke "0857#\ Testud and Francois "0860#\ Klostermeyer "0861a\ b#\ Morgan and Calderon "0867# and Shibata "0872#\ the undisturbed ionospheric parameters are assumed as inde! pendent of time and their changes due to the propagation of the gravity waves are considered as linear[ According to Porter and Tuan "0863# and Didebulidze and Pataraya
Corresponding author[ E!mail] gochaÝdtapha[kheta[ge
"0877# the height pro_le of electron density without grav! ity waves corresponds to the simple Chapman function with damping in time[ In these papers the behaviour of the height pro_le of electron density can be considered as a linear change of the background electron density[ Their investigation is based on the solution of the ambi! polar di}usion equation of the ionospheric F1!region using Green|s function\ when the gravity wave propa! gation is taken into account[ In the present paper we develop the solution of the ambipolar di}usion equation of the ionospheric F1! region by using Green|s function "Gliddon\ 0848^ Porter and Tuan\ 0863^ Didebulidze and Pataraya\ 0877#\ but with the presence of horizontal shear ~ow being taken into account[ Hereafter\ the horizontal shear ~ow cor! responds to the neutral meridional winds with zonal con! stant shear[ The in~uence of the neutral mean ~ow is essential in the coupled ionosphere:thermosphere models of Millward et al[ "0882a\ b#[ The mean ~owÐwave inter! action in~uences the formation of TIDs "Millward et al[\ 0882a\ b^ Hocke\ 0885# in the ionospheric F1!region[ The horizontal wind with constant vertical shear according to Farrell and Ioannou "0882# and Rogava and Mahajan "0886# sometimes is the source of the excitation of atmo! spheric waves with time varying wavenumber and periods[ The investigation of the evolution of the AGW
0253!5715:88:, ! see front matter Þ 0888 Elsevier Science Ltd[ All rights reserved PII] S 0 2 5 2 ! 5 7 1 5 " 8 8 # 9 9 9 9 3 ! 7
379
G[G[ Didebulidze\ A[D[ Pataraya : Journal of Atmospheric and Solar!Terrestrial Physics 50 "0888# 368Ð378
in the horizontal shear ~ow by the same method using modi_ed modal analysis "Pataraya and Pataraya\ 0886^ Didebulidze\ 0886#\ for the spectrum and ampli! _caiton:damping processes of these waves\ gives di}erent results[ According to Didebulidze "0886#\ in the hori! zontal shear ~ow together with acoustic and gravity waves appears the shear wave which is due to the presence of the shear ~ow[ This method of excitation of atmo! spheric gravity waves in the horizontal shear ~ow is not considered in the well!known coupled thermosphere: ionosphere models "Mayr et al[\ 0889^ Millward et al[\ 0882a\ b#[ The height pro_le of the electron density obtained in the present paper allows us to investigate the evolution of gravity and shear waves in the ionospheric F1!region[ In this solution the in~uence of the horizontal shear ~ow\ as well as the presence of the gravity and shear waves and their amplitude changes in time\ caused by the presence of the horizontal shear ~ow\ are taken into account[ The obtained night time pro_le of the electron density in the presence of horizontal shear ~ow shows the possibility of formation of the wave!like perturbation which cor! responds to the large scale TIDs with small period[ The resulting characteristic periods of the F1!layer par! ameters after a certain time for the evolution of the grav! ity waves are close to 05Ð19 min\ which are in agreement to the observations given in Oya et al[ "0871#\ Williams et al[ "0877# and Bowman "0881#[ The shear wave\ which is due to the presence of hori! zontal shear ~ow\ sometimes gives appreciable changes of the F1!layer[ We shall consider those cases when the evolution of the gravity and shear waves in the F1!region are di}erent[
1[ Ambipolar diffusion equation in the F1!region For the height distribution of electron density Ne"z\ t# in the ionospheric F1!region for the nonstationary case the ambipolar di}usion equation is used[ In this case the main processes of the formation and maintenance of the F1!layer are the balance between the di}usion of the oxygen ions\ O¦\ and the loss of electrons due to the ¦ dissociative recombination of ions O¦ 1 and NO [ This ambipolar di}usion equation in the case of the existence of the atmospheric gravity waves and the neutral mer! idional wind in the mid!latitude night ionosphere has the following form "Porter and Tuan\ 0863^ Didebulidze and Pataraya\ 0877#]
$
0
1Ne 1 1Ne Ne −Da cos1 u ¦ ¦ 1t 1z 1z 1H
1 %
¦"u¦u9# sin u cos uNe¦w cos1 uNe −L\
"0#
where L is the loss rate^ 1H is the scale height for ions^ the x!axis is oriented along the meridian to the North and the y!axis to the West^ x h−h9 is the di}erence between the actual and some initial height h9^ u is the angle between the magnetic _eld and the zenith\ the angle between the horizontal component of magnetic _eld and the x!direction is considered as the same^ Da is the ambi! polar di}usion coe.cient and for it the following well! known expression "Ferraro and O Ý zdog¼an\ 0847# will be used] Da cos1 u D9 exp
01
z \ H
"1#
and for the loss rate the following linear form "Ratcli}e et al[\ 0845# is used]
0 1
L bNe\ b b9 exp −
oz \ H
"2#
D9\ b9 are the coe.cients of the ambipolar di}usion and recombination\ respectively\ at height h h9^ o is the gas mixing coe.cient^ for 0 ¾ o ¾ 1\ the upper atmosphere changes from perfect mixing "o 0# to di}usive equi! librium "o 1#[ In eqn "0# u and w are the disturbed neutral gas vel! ocities in the meridional and vertical directions\ respec! tively[ u9 is the neutral meridional wind velocity which in the present case corresponds to the horizontal shear ~ow u9 ay\
"3#
where a is the shear of the horizontal "meridional# wind[ In eqn "0# the ionospheric is assumed to be isothermal and the condition of small changes of height pro_le of the electron density in the horizontal plane\ "0:Ne#"1Ne:1x\ y# ð "0:H#\ is taken into account[
2[ Equations of evolution of AGW in horizontal shear ~ow According to Didebulidze "0886# and Pataraya and Pataraya "0886# the amplitudes of the atmospheric AGW are changing in time[ In these papers the ampli! _cation:damping of the AGW amplitudes is due to the presence of the horizontal shear ~ow[ In the present paper the disturbed neutrals| velocity u and w correspond to the atmospheric gravity and shear waves[ These types of waves will be referred to as modi_ed gravity waves[ According to Didebulidze "0886# the perturbed values of pressure "p#\ density "r#\ and velocity "v v"u\ v\ w## components for the isothermal atmosphere in horizontal shear ~ow may be described by the following trans! formation]
G[G[ Didebulidze\ A[D[ Pataraya : Journal of Atmospheric and Solar!Terrestrial Physics 50 "0888# 368Ð378
$
"p\ r#"x\ y\ z\ t# "Pk\ Rk#"t# = exp if"t#−
$
%
z \ 1H
"u\ v\ w#"x\ y\ z\ t# "Uk\ Vk\ Wk#"t# = exp if"t#¦
"4a#
%
z \ 1H "4b#
f"t# kxx¦ky"t#y¦kzz\
"4c#
ky"t# k9−akxt[
"4d#
In these expressions the evolution of complex valued amplitudes\ qk"t# 0 "Pk\ Rk\ Uk\ Vk\ Wk#"t#\ according to the equations of motion\ continuity and adiabatic behav! iour\ when the dissipation is neglected\ is described by the set of the following linearized equations "Didebulidze\ 0886#] r99U?k −ikxPk−ar99Vk\
"5a#
r99V?k −ik"t#Pk\
"5b#
0
1
0 P −`Rk\ r99W?k − ikz− 1H k
0
"5c#
1
0 R?k −iðkxUk¦ky"t#VkŁ− ikz− Wk\ r99 1H P?k r99cs1
−iðkxUk¦ky"t#VkŁ−"ikz−E#Wk\
"5d#
"5e#
where the prime denotes derivatives in time^ k9 ky"t 9#^ k"t# kðkx\ ky"t#\ kzŁ is the time varying wavenumber^ v is the component of the perturbed vel! ocity in the y!direction^ g is the acceleration due to grav! ity^ H cs1:g` is the atmospheric "neutral gas# scale height^ cs ðg"p9:r9#Ł0:1 is the velocity of sound\ g is the ratio of speci_c heats^ p9 p99 exp ð−"z:H#Ł\ r9 r99 exp ð−"z:H#Ł are the unperturbed barometric height pro_le for the unperturbed pressure and density\ respectively^ p99 and r99 are the pressure and density at height z 9\ respectively^ E ð"1−g#:1gHŁ is the iso! thermal Eckart coe.cient^ cs\ H and E are assumed con! stant with height[ Equations "4# and "5# make up a math! ematically closed system of di}erential equations[ These linearized equations\ in the absence of shear ~ow\ describe the atmospheric AGW "Hines\ 0859#[ The evolution of amplitudes qk for times of Dt ð ta\ where ta 1
b b
k9 \ akx
"6#
according to Didebulidze "0886# can be described by use of the following transformation] qk"t# q"t# = exp ð−iV"t#Ł\
"7#
where Re "V?# corresponds to the characteristic fre!
370
quencies of atmospheric waves and Im "V?# are their amplitudes ampli_cation:damping rates\ respectively[ Using transformation "7# in the set of eqns "4# and "5# for the AGW gives the following characteristic frequencies "Didebulidze\ 0886#]
0
1
0 0 1 va\g "t# cs1 =k"t#=1¦ 1 3H1 2
X 0
0 1 0 3 −vb1cs1"kx1¦ky1"t##\ cs =k"t#=1¦ 3 3H1
1
"8#
where =k"t#=1 kx1¦ky1"t#¦kz1^ vb z"g−0#`:gH is the isothermal BruntÐVaisala frequency[ In eqn "8# high fre! quency\ va\ corresponds to acoustic waves and low fre! quency\ vg\ to gravity waves\ respectively[ The ampli! _caiton:damping rates "Im "V?## for acoustic\ vai\ and for gravity waves\ vgi\ respectively\ are] acs1kxky"t# vb1 0− [ vai\gi"t# 3 1 1ðva1"t#−vg1"t#Ł va\g "t#
$
%
"09#
Sign {−| at the beginning of the right!hand!side of eqn "09# corresponds to acoustic waves and {¦| to gravity waves\ respectively[ Values va\g"t#\ "8#\ and vai\gi"t#\ "09#\ are applied for describing the evolution of acoustic and gravity waves in the horizontal shear ~ow for times 1 1 Dt ð ta when condition va\g Ł vai\gi is valid[ The set of eqns "4# and "5# also describe the evolution of a shear wave\ which has zero frequency "Re "V?# 9# and the ampli_caiton:damping rate is determined by the following expression "Didebulidze\ 0886#] vshi"t#
akxky"t# kx1¦ky1"t#
[
"00#
According to the characteristic dispersion eqns "8#Ð"00# the acoustic wave frequency\ va"t#\ continuously increases for times of t × ta^ for the same times the gravity wave frequency\ vg"t#\ tends to the isothermal BruntÐ Vaisala frequency vb[ These equations also show that the increase of amplitudes of the acoustic and gravity waves appears at time t × ta[ Equation "8# shows us that the minimum frequency of the acoustic and gravity waves occurs at time t "ta:1#[ According to dispersion eqn "8# the meridional phase velocity of the gravity waves for large times of t Ł ta tends to the value of vb:kx[ According to eqn "00# the shear wave excitation is due to the presence of horizontal shear ~ow[ According to eqns "5#\ "7# and "00# the shear wave corresponds to the standing type "Re "V?# 9# incompressible "div v ¹ 9# vortical ""rot v#z const# perturbation\ excited in hori! zontal shear ~ow[ For condition "a:vB# ð 0\ the part of term ar99Vk in "5a# is similar to the e}ect of the Coriolis force for the traditional approach when the vertical com! ponent of this force is neglected[ In this case vortical perturbation is excited during the established balance
371
G[G[ Didebulidze\ A[D[ Pataraya : Journal of Atmospheric and Solar!Terrestrial Physics 50 "0888# 368Ð378
between the perturbation of the shear direction velocity of the rotation driving background ~ow and the pressure gradient[ In this case\ according to eqns "5#\ "7# and "00# the ampli_cation:damping of the amplitudes of the shear waves Uk and Vk "=Uk=\ =Vk= Ł =Wk=# are essential at time t ¾ ta[ The total spectral energy "½=qk=1 ½ exp "1 Im "V### for the shear waves according to eqn "00# has the transient development for times of t ¾ ta[ In the cases when for the horizontal velocity of the winds the following approxi! mation u9 U¦ay is possible\ the shear wave propagates in the meridional direction with velocity U const[ Henceforth\ these types of evolving gravity and shear waves will be referred to as modi_ed gravity waves[ In the following\ for long time interval Dt × ta\ in which the appearance of both gravity and shear waves is essential\ we consider the evolution of these waves numerically using the initial condition in accordance with eqns "7#Ð "00#[ In the following we consider the behaviour of the ionospheric F1!layer which corresponds to the evolution for modi_ed large wavelength\ lx\ ly Ł H\ gravity waves[ Here lx and ly are the wavelengths in the meridional and zonal direction\ respectively[ It is possible to describe these types of long period waves of the neutral gas in the F1!region by eqn "0#[ The molecular viscosity\ which is essential for the dissipation of gravity waves in the ionospheric F1!region "Hines\ 0859#\ for the large scale gravity waves\ lx\y Ł H\ is com! parably small "Hines\ 0857^ Richmond\ 0867# and in eqns "5# for simplicity we have neglected it[ According to Rich! mond "0867# and Mayr et al[ "0889# for gravity waves with a horizontal phase velocity about 649 m s−0 the temperature and wind perturbations at thermospheric heights are decaying slowly before reaching the equator[ So one may expect that gravity wave as well as the shear wave!like vortical perturbation\ which propagates with the wind at velocity U\ can possibly appear at the mid! latitude F!region[ In this case we consider the polar regions as the main source of the AGW generation of which the ionospheric manifestation is the large scale TIDs[ In the present consideration the presence of hori! zontal shear ~ow is possibly an additional source of grav! ity wave generation and it would be the reason for the large scale TIDs observed in the mid!latitude ionosphere[ In the following we choose the values of horizontal winds shear\ a\ in the ionospheric F1!region in accord! ance with the review of the satellite wind measurements given in Mayr et al[ "0889#[ According to Mayr et al[ "0889# the horizontal thermospheric wind measured in the polar regions is 799Ð1999 m s−0 and more on mag! netically disturbed days[ In some cases the change of this value of velocity for 1Ð4> of latitude is sometimes of the same order as the value[ Thus\ it is possible to choose the horizontal shear of the horizontal winds as a 9[9990Ð 9[990 s−0\ and even more[ The set of eqns "4# and "5# describes the evolving ampli! tudes of the coupled waves[ In this case coupling between
acoustic\ gravity and shear waves is due to the strati! _cation and the horizontal shear ~ow[ The values of ampli_cation:damping rates "09# and "00# are valid for the small time intervals of Dt ð ta in which the change of amplitudes of the considered waves due to the coupling between di}erent waves are negligibly small[ These considerations correspond to the cases when "a:vB# ð 0 for which the changes of amplitudes of shear and gravity waves due to their coupling in the horizontal shear ~ow is negligible[ For simplicity we have not con! sidered the cases of large shear\ "a:vB# − 0\ when it would be expected that the changes of amplitudes of all types of waves due to their energy exchange in the horizontal shear ~ow is essential[
3[ Evolution of gravity and shear waves in the F1!region In eqn "0# values u and w correspond to the real parts of the perturbed velocity of atmospheric waves which will be taken from eqn "4b#[ For the solution of eqn "0# we use a new variable z k = exp ð−"z:H#Ł and the following transformation]
$
Ne z0:3 exp −1Az¦
%0
0 V "z\ t#¦c"t# D9 n
1
Y9¦ s Yl \ l 0
"01a# where k
H 1 zu99 ¦3b9D9 D9
"01b#
u99 u9 sin u cos u\ u99
A
1 99
"01c# \
"01d#
3zu ¦3b9D9 Vn
g
z
z?
e− 1HVn9"z?\ t?# dz?\
"01e#
Vn9 Re "ðUk"t# sin u¦Wk"t# cos uŁ cos u eif#[
"01f #
The cases when the production rate of electrons in eqn "0# is given by Q f "z\ t#Ne\ can be easily included in the following formalism[ Here f "z\ t# is some function which depends on the source of production[ With eqns "7# in "0# for function Y9"z\ t#\ Y0"z\ t#\ Y1"z\ t#\ [ [ [ the set of following equations can be obtained] L
Y9 9\
"02a#
L
Yl Fl"Yl−0#\ l 0\ 1\ [ [ [ [
"02b#
Here operator L
has the form]
L
0 2 −1 11 0 1 \ −n9 z ¦Az−0− −n9 z 1t 05 3 1z1
$
%
"03#
372
G[G[ Didebulidze\ A[D[ Pataraya : Journal of Atmospheric and Solar!Terrestrial Physics 50 "0888# 368Ð378
where n9 "kD9:H1#[ Function Fl will be considered as the inhomogeneous part of eqns "02b# and has the fol! lowing form]
$ 01 % $ 0 1 $0 %
−
¦
dz
m
"z
m−0 1
% 1
0 1A− z−0 Yl−0 3
e−z#
"05#
where 0 7H
1
kD9 −u99 [ H
0 1
Y9 s Cm exp ð−lm"t−t9#Łz0:3 exp − m 9
G"z\ z?\ t−t?#
1Yl"z?\ t?# 1z? z?
%
1G"z\ z?\ t−t?# 1z?
dt?[
"08#
z? 9
Using solution "07# of the homogeneous part of eqns "02#\ the Green|s function can be obtained in the fol! lowing closed form "Whittaker and Watson\ 0841^ Lebedev\ 0852#]
G"z\ z?\ t−t?#
"zz?#0:3 "1pshðn0"t−t?#Ł#0:1
× exp
6
ch
6
"zz?#0:1 shðn0"t−t?#Ł
7
7
z¦z? u99 [ "t−t?#− 3H 1thðn0"t−t?#Ł
"19#
This Green|s function in the absence of the neutral wind velocity is the same as in Moschandreas and Tuan "0863# and Porter and Tuan "0863#[ The Green|s function\ "19#\ when the neutral wind velocity is constant\ has the same form as obtained in Didebulidze and Pataraya "0877#[ Note that the form of the Green|s function for the di}er! ent scale height of the ions "1H# must be other than given by eqn "19# "Lvova et al[\ 0861^ Makeev et al[\ 0862#[ Introduction of the function c"t# in transformation "01# allows us to solve eqns "02#[ The condition that Y9"z\ t# is the solution of the set of eqns "02# is equivalent to the equation]
"06#
The solution of eqn "02a# can be written in the following form]
g$
−Yl"z?\ t?# "04#
lm mn9¦1n0\
0 10
G"z\ z?\ t−t?#Yl"z?\ t9# dz? z?
t9
"Whittaker and Watson\ 0841^ Lebedev\ 0852# and their eigenvalues\ respectively\ are]
n0
G"z\ z?\ t−t?#Fl"z?\ t?# dz? dt?
t
The homogeneous part of each of eqns "02# is the same and can be solved as the SturmÐLiouville equation "Tuan\ 0857#[ Thus\ using transformation "01# in "0# gives us one homogeneous equation "02a# and l inhomogeneous equations "02b#[ Transformation "01# allows us to obtain the linear expression with the modi_ed gravity waves| velocity in the inhomogeneous part of eqns "02b#\ and function c"t# will be obtained together with the solution of the eqns "02b#[ The separation of the variables "z and t# allows us to reduce eqn "02a# to the equation whose eigenfunctions are the generalized Laguerre polynomials dm
t9
¦
1Yl−0 [ 1z
Lm−0:1"z# ezz0:1
9
gg 9
Yl−0 1 0 c"t#¦ Vn"z\ t# z 1t D9 0:1
t
¦
Yl−0
Vn9"z\ t# k H z
g
o−0
b9 z Fl"Yl−0# 0− k k −
Yl"z\ t#
z −0:1 L "z#\ 1 m "07#
where t9 is some initial time[ The expression for Ne"z\ t# using eqns "01a# and "07# gives the solution to eqn "0#\ when the modi_ed gravity waves are absent and o 0 "Chamberlain\ 0850#[ The solution of the inhomogeneous equations "02b# by using Green|s function G"z\ z?\ t−t?# can be written "Tikhonov and Samarskiy\ 0866# as]
t
9
t9
gg
G"z\ z?\ t−t?#F0"z?\ t?# dz? dt? 9[
"10#
Thus\ eqn "10# means that the inhomogeneous solution Y0 9 and\ according to eqns "08#\ all Yl 9[ So solution "07# and solution c"t# of eqn "10# by use of transformation "01# gives us the solution of eqn "0#[ Hereafter\ for the gravity waves we use the isothermal model where relations "4# and "5# are valid "Hines\ 0859^ Didebulidze\ 0886#[ Some other models of the gravity waves\ when their amplitude dependence on height is di}erent from that in eqns "4#\ may be included in our formalism[ Expansion of Green|s function\ "19#\ in Laguerre poly! nomials gives]
373
G[G[ Didebulidze\ A[D[ Pataraya : Journal of Atmospheric and Solar!Terrestrial Physics 50 "0888# 368Ð378
n; 0:3 0 exp ð−ln"t−t?#Ł"zz?# G"n¦ # n 9 1
G"z\ z?\ t−t?# s
0
× exp −
1
z¦z? −0:1 Ln "z#Ln−0:1"z?# 1
"11#
where "12#
ln nn9¦1n0[
Terms in eqns "07# and "11# when m\ n 0\ 1\ [ [ [ damp more rapidly than the term when m\ n 9[ If we limit ourselves in eqns "05#Ð"07#\ "11# and "12# by the terms\ when m\ n 9 for the initial condition c"t t9# 9\ from "10# the solution for c"t# can be easily obtained and the height distribution of the F1!region electron density can be written as]
6
Ne N exp −l"t−t9#−
−
z a z − e− H 1H 1
$
%7
z 0 0 c0"y\ t# Vn0 e− 1H− Vd zpk
\
a k¦ Vd
b9 zpk
$
0 1 0 1%
k0−oG o¦
0 0 −G 0¦ 1 1
u99H \ D9
D9 \ 1H
Vn0
Ak z0¦Az−1 cos "f¦f0¦f1#\ Az
g $
c0
\
t? t e−z? Vn"z?\ t?#=t? t ¦1n0 9
9
g
t
"13c# "13d#
Vn0 ð 0\ Vd
"13e#
%
where Ak ="Uk sin u¦Wk cos u# cos u=\
"13g#
Az 1Hkz\
"13h#
$
%
Im "Uk sin u¦Wk cos u# \ Re "Uk sin u¦Wk cos u#
f1 arctan "Az#[
%
"13b#
"13f#
f0 arctan
0 1
where hm9 h9¦H = ln a is the maximum of the simple Chapman layer[ In the case of =Az= Ł 0 ð=kz= Ł "0:1H#Ł the behaviour of the main maximum "maximum close to hm9# of Ne"z\ t# can be approximately described by the expression "14# when ðin "14#Ł Vn9 Vn9"z H = ln a#[ According to eqns "13# and "14#\ when the conditions
"13i# "13j#
The value Vd\ "13d#\ corresponds to the characteristic velocity of the ambipolar di}usion of ions[ Solution for Ne"z\ t#\ "13#\ allows us to investigate analytically and numerically the evolution of the modi_ed gravity waves in the horizontal shear ~ow[ According to solution "13# and eqns "4# and "5# we may use the analytical as well as numerical results for describing the behaviour of the
Vn9 1zaVd
ð 0\
0¦
0 Vn9 1 Vn9 − \ 3a Vd 1zaVd
hmF1 hm9−1H = ln
Vn9"z?\ t?# dt? dz?\
t9
$X
"13a#
where l 1n0¦
electron density height pro_le\ Ne"z\ t#\ in the horizontal shear ~ow[ In the case of no shear "a 9 or u9 const# the height pro_le of electron density corresponds to the propagation of gravity waves for which the spectrum is given by Hines "0859# and in this case the values c0\ "13f#\ will be easily obtained[ Solution "13# without the modi_ed gravity waves "Vn0 9\ c0 9# corresponds to the simple Chapman function damping in time[ In the present case this func! tion describes the background F1!layer[ In "13a# the time decay coe.cient l\ "13b#\ and value a\ "13c#\ depend on the velocity "u9# of the horizontal winds[ Terms Vn0 and c0 in "13# correspond to the part of the in~uence of the evolution of the modi_ed gravity wave in horizontal shear ~ow[ According to solution "13# the height of the maximum of the F1!layer electron density for the case of kx 9 can be written as] "14#
"15#
are valid\ the height pro_le of Ne"z\ t# can be considered as the linear disturbed simple Chapman layer damping in time[ Hooke "0857#\ Testud and Francois "0860#\ Klo! stermeyer "0861a\ b#\ Morgan and Calderon "0867# and Shibata "0872# assumed that the basic undisturbed F1! layer is independent of time and that perturbations to its parameters due to the gravity waves are linear[ The val! idity of the inequalities "15# essentially depends on the value of Az\ "13h#[ For large vertical wavenumbers =kz= Ł "0:1H# "=Az= Ł 0#\ in many cases of gravity wave propagation the linear approach of the change of Ne"z\ t# would be true[ In the present cases\ according to eqn "13#\ the conditions "Vn0:Vd# − 0\ "Vn9:1zaVd# − 0 give more appreciable changes of Ne than the linear disturbed sim! ple Chapman layer[ These results are di}erent from those obtained by Huang and Li "0880# or Nekrasov et al[ "0884# where the nonlinear changes of the F!region ion density are due to the nonlinear gravity waves[ By use of solutions "13# and "14# it is possible to con! sider many di}erent cases for describing the evolution of the modi_ed gravity waves in the horizontal shear ~ow[ We shall limit ourselves to only two cases which cor! respond to the variation of hmF1"t# and its maximum electron density NmF1"t# for the case of the evolution of
G[G[ Didebulidze\ A[D[ Pataraya : Journal of Atmospheric and Solar!Terrestrial Physics 50 "0888# 368Ð378
the gravity and shear waves[ According to eqns "13#Ð "15# the in~uence of the modi_ed gravity waves on the behaviour of the F1!layer electron density is more e}ec! tive for the cases of kz 9\ which are partially dem! onstrated in Figs 0 and 1[ In these _gures the evolution of amplitudes Uk"t# and Wk"t# of the disturbed wave vel! ocity and the corresponding behaviour of the maxima of electron density\ NmF1"t#\ their heights\ hmF1"t#\ for the atmospheric gravity and shear waves\ are plotted\ respec! tively[ For all _gures the mid!latitude F1!region and the wave parameters are chosen as follows[ kx 4[1 = 09−5 m−0\ k9 1[6 = 09−5 m−0\ a 0[05 = 09−3 s−0\ t9 9\ H 5 = 093 m\ b9 1 = 09−3 s−0\ D9 8 = 094 m1 s−0\ N 095 cm−2\ h9 299 km[ For de_niteness we have chosen the condition sin u cos u ¹ −9[3 which cor! responds to the mid!latitude northern hemisphere[ For simplicity we choose the gas mixing coe.cient o 0[64\ which corresponds to di}usive equilibrium\ when the recombination of ions O¦ is mainly due to collisions with nitrogen\ N1\ molecules[ In Figs 0 and 1 for the cases "c# and "d# the dark lines correspond to the case of the absence of the modi_ed gravity waves "background F1! layer#[ According to Fig[ 0\ for large times\ t × ta "ta ¹ 049 min#\ the periods of the gravity waves are weakly changed and are nearly 05Ð19 min[ This case corresponds to the formation of large scale TIDs with small period and a phase velocity ð¹"vb:kx#Ł in the mer! idional direction of about 0999Ð0199 m s−0[ The period of nearly 06 min is in agreement with the observation of large scale TIDs by Williams et al[ "0877#\ Bowman "0881# and also with observations by Oya et al[ "0871#[ In these cases we try to choose parameters of the modi_ed gravity waves for which the changes of wave amplitudes caused by the coupling between acoustic\ gravity and shear wave modes would be small[ For Figs 0 and 1 the initial con! ditions are chosen in accordance to expressions "8#Ð"00# and to the condition =Uk=\ =Uk= Ł =Wk=[ With horizontal shear ~ow the changes of amplitude of the vertical com! ponent of the velocity perturbation\ Wk\ due to gravity waves\ occurs for times of t × ta "see Fig[ 0"b##[ Thus\ for the considered time intervals this amplitude becomes comparable to the values of amplitudes of the meridional perturbed velocity Uk "see Fig[ 0"a##[ The case when the horizontal wavelengths "lx\ ly# of gravity waves are comparable to the neutral gas scale height\ lx\y O"H#\ would lead to the formation of med! ium!scale TIDs\ due to the horizontal shear ~ow[ In this case in eqn "0# the terms describing the horizontal inhom! ogeneity of the height pro_le of electron density must be added "Shibata\ 0872# and dissipative processes in eqns "5# of the evolution of modi_ed atmospheric gravity waves would be taken into account "Richmond\ 0867^ Mayr et al[\ 0889#[ According to the cases demonstrated in Fig[ 1\ the evolution of a shear wave gives a di}erent change of the background F1!layer than in the case of the evolution of
374
the gravity waves[ For the shear wave\ su.cient changes of the F1!layer parameters hmF1"t# and NmF1"t# appear for times of t ¾ ta[ Similar types of changes of the iono! spheric F1!layer on magnetically disturbed days over times 1Ð4 h are known "Fatkullin\ 0867#[ In this case\ plotted in Fig[ 1\ the initial condition of the amplitude of the shear wave!like meridional velocity perturbation\ Uk\ is chosen to be opposite to the North direction[ The corresponding maximum of the F1!layer electron density\ hmF1 "Fig[ 1"d# thin line# according to "14# is located above that in the case of no perturbation "Fig[ 1"d# full line#[ Note that the cases when Da"z\ t# Da"z#¦D0"x\ y\ z\ t# "Da Ł D0# and b b"z#¦b0"x\ y\ z\ t# "b Ł b0# are easy to include in the above applied formalism[ For reasons of brevity we have not considered in more detail the small changes of electron density ð"0:Ne#"1Ne:1x\ y# ð "0:H#Ł\ the time decay l\ "13b#\ and of the value a\ "13c#\ in the horizontal plane[
4[ Conclusion We have studied the reaction of the ionospheric F1! region to the evolution of gravity and shear waves in the horizontal shear ~ow when the production of electrons is not taken into account[ A solution of the ambipolar di}usion eqn "0#\ using Green|s function when the neutral horizontal shear ~ow is taken into account\ is developed[ It has been found that the height pro_le of electron density\ Ne"z\ t#\ allows us to investigate the evolution of the gravity and shear waves in horizontal shear ~ow[ The evolution of the meridional and vertical components of the velocity of the gravity and shear waves and their in~uence on the behaviour of the F1!region electron den! sity are demonstrated[ Appreciable changes of the F1! layer parameters NmF1"t# and hmF1"t# appear for the shear waves for times of t ¾ ta[ For atmospheric gravity waves\ for times of t × ta\ hmF1"t# is changed and the frequency of the oscillations tends to the isothermal BruntÐVaisala frequency[ For the scale height of the F1! region neutral gas H 59 km the characteristic periods of variation of hmF1"t# and NmF1"t# due to the evolution of the gravity wave in the horizontal shear ~ow at large times t × ta are close to 05Ð19 min[ It has been shown that the changes of the F1!layer electron density due to the evolution of gravity and shear waves can be described as a linear change of the simple Chapman layer\ with damping in time[
Acknowledgement This work was supported\ in part\ by a Grant awarded by the Georgian Academy of Science[
375
G[G[ Didebulidze\ A[D[ Pataraya : Journal of Atmospheric and Solar!Terrestrial Physics 50 "0888# 368Ð378
Fig[ 0[ Evolution of the velocity amplitudes "a# Uk"t#\ "b# Wk"t# of modi_ed atmospheric gravity waves\ and the behaviour of "c# NmF1"t# and "d# hmF1"t# of the F1!layer[
G[G[ Didebulidze\ A[D[ Pataraya : Journal of Atmospheric and Solar!Terrestrial Physics 50 "0888# 368Ð378
376
Fig[ 1[ Evolution of the velocity amplitudes "a# Uk"T#\ "b# Wk"T# of atmospheric shear waves\ and the behaviour of "c# NmF1"t# and "d# hmF1"t# of the F1!layer[
377
G[G[ Didebulidze\ A[D[ Pataraya : Journal of Atmospheric and Solar!Terrestrial Physics 50 "0888# 368Ð378
References Bowman\ G[G[\ 0881[ Some aspects of large!scale travelling ionospheric disturbances[ Planetary and Space Science 39\ 718Ð734[ Chamberlain\ J[W[\ 0850[ Physics of the Aurora and Airglow[ Academic Press\ New York[ Didebulidze\ G[G[\ 0886[ Ampli_cation:damping processes of atmospheric acoustic!gravity waves in horizontal winds with linear shear[ Physics Letters A 124\ 54Ð69[ Didebulidze\ G[G[\ Pataraya\ A[D[\ 0877[ Reaction of the iono! spheric F!layer on oblique propagation of the inner gravi! tational waves in the presence of a local wind[ Bulletin of Abastumani Astrophysical Observatory 55\ 112Ð121[ Farrell\ B[F[\ Ionannou\ P[J[\ 0882[ Transient development of perturbation in strati_ed shear ~ow[ Journal of Atmospheric Science 49\ 1190Ð1103[ Fatkullin\ M[N[\ 0867[ Ionospheric disturbances[ Science and Technical Results[ Geomagnetism and Upper Layers of Atmosphere 3\ 5Ð096 "in Russian#[ zdog¼an\ I[\ 0847[ The e}ect of di}usion on Ferraro\ V[C[A[\ O vertical distribution of ionization in a quite F!region[ Journal of Atmospheric and Terrestrial Physics 01\ 039Ð038[ Gliddon\ J[E[C[\ 0848[ Use of Green|s function in the solution of ionospheric di}usion problems[ Quarterly Journal of Mech! anics and Applied Mathematics 01\ 236Ð242[ Hines\ C[O[\ 0859[ Internal atmospheric gravity waves at iono! spheric heights[ Canadian Journal of Physics 27\ 0330Ð0370[ Hines\ C[O[\ 0857[ An e}ect of molecular dissipation in upper atmospheric gravity waves[ Journal of Atmospheric and Ter! restrial Physics 29\ 734Ð738[ Hocke\ K[\ Schlegel\ K[\ 0885[ A review of atmospheric gravity waves and traveling ionospheric disturbances] 0871Ð0884[ Annales Geophysicae 01\ 806Ð839[ Hooke\ W[H[\ 0857[ Ionospheric irregularities produced by internal atmospheric gravity waves[ Journal of Atmospheric and Terrestrial Physics 29\ 684Ð712[ Huang\ C[S[\ Li\ 0880[ Weak nonlinear theory of the ionospheric response to atmospheric gravity waves in the F!region[ Jour! nal of Atmospheric and Terrestrial Physics 42\ 892Ð897[ Hunsucker\ R[D[\ 0871[ Atmospheric gravity waves generated in the high!latitude ionosphere] a review[ Reviews of Geophysics and Space Physics 19\ 182Ð204[ Khantadze\ A[G[\ Gvelesiani\ A[I[\ 0868[ On the Theory of Di}usion in Ionosphere F!region[ Moskva\ Nauka "in Russian#[ Kirchengast\ G[\ 0885[ Elucidation of the physics of the gravity waveÐTID relationship with the aid of theoretical simulations[ Journal of Geophysical Research 090\ 02\242Ð02\257[ Klostermeyer\ J[\ 0861a[ Numerical calculation of gravity wave propagation in a realistic thermosphere[ Journal of Atmo! spheric and Terrestrial Physics 23\ 654Ð663[ Klostermeyer\ J[\ 0861b[ Comparison between observed and numerically calculated atmospheric gravity waves in the F! region[ Journal of Atmospheric and Terrestrial Physics 23\ 0282Ð0390[ Lebedev\ N[N[\ 0852[ Special Functions and Their Application\ Moskva "in Russian#[ Lvova\ A[A[\ Poliakov\ V[M[\ Ribin\ V[V[\ 0861[ Peculiarities of nonstationary solutions of the equation of di}usion in the gravitational _eld[ Izvestia Vuzov Radiophysica 04\ 739Ð740 "in Russian#[
Makeev\ V[V[\ Poliakov\ V[M[\ Ribin\ V[V[\ 0862[ Non! stationary solution of di}usion equation in gravitational _eld with consideration of disappearance of particles and their transfer[ Izvestia Vuzov Radiophysica 05\ 0559Ð0569 "in Rus! sian#[ Mayr\ H[G[\ Harris\ I[\ Herraro\ F[A[\ Spencer\ N[W[\ Varosi\ F[\ Pesnell\ W[D[\ 0889[ Thermospheric gravity waves] obser! vations and interpretation using the transfer function model "TFM#[ Space Science Reviews 43\ 186Ð264[ Millward\ G[H[\ Mo}ett\ R[J[\ Quegan\ S[\ Fuller!Rowell\ T[J[\ 0882b[ E}ects of an atmospheric gravity wave on the mid! latitude ionospheric F layer[ Journal of Geophysical Research 87\ 08\062Ð08\068[ Millward\ G[H[\ Quegan\ S[\ Mo}ett\ R[J[\ Fuller!Rowell\ T[J[\ Rees\ D[\ 0882a[ A modelling study of the coupled ionospheric and thermospheric response to an enhanced high!latitude elec! tric _eld event[ Planetary and Space Science 30\ 34Ð45[ Morgan\ M[G[\ Calderon\ 0867[ Testing Hook|s perturbation formula for the production of TID|s by gravity waves[ Journal of Geophysical Research 72\ 4626Ð4639[ Moschandreas\ D[J[\ Tuan\ T[F[\ 0863[ On the behavior of the polar ionosphere under the in~uence of incident electron ~ux[ Journal of Atmospheric and Terrestrial Physics 25\ 176Ð292[ Nekrasov\ A[K[\ Shalimov\ S[L[\ Shukla\ P[K[\ Sten~o\ L[\ 0884[ Nonlinear disturbances in the ionosphere due to acoustic grav! ity waves[ Journal of Atmospheric and Terrestrial Physics 46\ 626Ð630[ Oya\ H[\ Takahashi\ T[\ Morioka\ A[\ Miyaoka\ H[\ 0871[ Wavy patterns of ionospheric electron density pro_les triggered by TID!observation results of the electron density by Taiyo Sat! ellite[ Journal of Geomagnetism and Geoelectricity 23\ 498Ð 414[ Pataraya\ A[D[\ Pataraya\ T[A[\ 0886[ An excitement of the gravitational and acoustic waves in the solar atmosphere caused by inhomogeneous ~ow[ Astronomical and Astro! physical Transactions 02\ 010Ð015[ Porter\ H[S[\ Tuan\ T[F[\ 0863[ On the behaviour of the F!layer under the in~uence of gravity waves[ Journal of Atmospheric and Terrestrial Physics 25\ 024Ð046[ Ratcli}e\ J[A[\ Schmerling\ E[R[\ Setty\ C[S[G[K[\ Thomas\ J[O[\ 0845[ The rates of production and loss of electrons in the F region of the ionosphere[ Philosophical Transactions 137\ 510Ð531[ Richmond\ A[D[\ 0867[ Gravity wave generation\ propagation\ and dissipation in the thermosphere[ Journal of Geophysical Research 72\ 3020Ð3034[ Rogava\ A[D[\ Mahajan\ S[M[\ 0886[ Coupling of sound and internal waves in shear ~ows[ Physics Reviews E 44\ 0074Ð 0077[ Shibata\ T[\ 0872[ A numerical calculation of the ionospheric response to atmospheric gravity waves in the F1!region[ Jour! nal of Atmospheric and Terrestrial Physics 34\ 686Ð798[ Testud\ J[\ Francois\ P[\ 0860[ Importance of di}usion processes in the interaction between neutral waves and ionization[ Jour! nal of Atmospheric and Terrestrial Physics 22\ 654Ð663[ Tikhonov\ A[N[\ Samarskiy\ A[A[\ 0866[ Equations of Math! ematical Physics[ Moskva\ Nauka "in Russian#[ Tuan\ T[F[\ 0857[ On the theory of the ionospheric F layer[ Journal of Physics B "Proceedings of Physical Society# 0\ 855Ð 861[ Whittaker\ E[T[\ Watson\ G[N[\ 0841[ A Course of Modern Analysis[ Cambridge University Press[
G[G[ Didebulidze\ A[D[ Pataraya : Journal of Atmospheric and Solar!Terrestrial Physics 50 "0888# 368Ð378 Williams\ P[J[S[\ Crowley\ G[\ Schlegel\ K[\ Virdi\ T[S[\ McCrea\ I[\ Watkins\ G[\ Wade\ N[\ Hargreaves\ J[K[\ Lachlan!Cope\ T[\ Muller\ H[\ Baldwin\ J[E[\ Warner\ P[\ Van Eyken\ A[P[\ Hapgood\ M[A[\ Rodger\ A[S[\ 0877[ The generation and
378
propagation of atmospheric gravity waves observed during the Worldwide AcousticÐGravity wave Study "WAGS#[ Jour! nal of Atmospheric and Terrestrial Physics 49\ 212Ð226[