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Plasma density perturbations in the ionospheric E-region
__ __ i!B
&
ELSEVIER
27 November
1995
PHYSICS
LETTERS
A
Physics Letters A 208 (1995) 214-222
Plasma density perturbations in the ionospheric E-region A.V. Gurevich ‘, N.D. Borisov 2, A.N. Karashtin 3, K.P. Zybin Max-Planck-Institute fur Aeronomie, 37191 Katlenburg-Lindau,
’
Germany
Received 14 August 1995; accepted for publication 22 September Communicated by V.M. Agranovich
1995
Abstract The theory of gradient drift instability in a height inhomogeneous the instability increments is demonstrated.
ionospheric
E-region is revised. A strong decrease of
1. Introduction Plasma density perturbations are usually observed in the ionospheric E-layer by radio wave scattering and in situ measurements on rockets [l]. Recently, it was supposed [2], that the low frequency part of these perturbations is induced in an ionospheric plasma by the turbulent motion of the neutral atmosphere. A qualitative agreement of this theory with the observations on board of rockets is demonstrated [3]. The theory was developed in Ref. [2] for a homogeneous plasma only. The plasma density in the real ionosphere is height inhomogeneous, N = N(z), and this inhomogeneity plays an important role first of all due to the gradient-drift instability, which can occur in an inhomogeneous plasma [1,4]. This instability can excite and essentially amplify the plasma density perturbations. It should be noted, that in the theory of gradient-drift instability the following relation is usually supposed to be fulfilled,
dN x=z’
N
where L is constant (see Refs. [1,4]). That allowed one to consider the process phenomenologically in the same way as in a homogeneous plasma, what simplifies the problem radically. But in an inhomogeneous medium boundary conditions usually play an essential role. This means that the theory of gradient-drift instability developed phenomenologically like in an unbounded homogeneous medium cannot be considered as fully constructed.
’ On leave from P.N. Lebedev Physical Institute, 117924 Moscow, Russian Federation. 2 On leave from Institute of Terrestrial Magnetism, Ionosphere and Radio Wave Propagation, 142092 Troitsk, Moscow Region, Russian Federation. 3 On leave from Radiophysical Research Institute (NIRFI), Nizhny Novgorog, Russian Federation. Elsevier Science B.V. SSDI 0375-9601(95)00738-5
A.V. Gureuich et al. / Physics Letters A 208 (1995) 214-222
215
In the present paper, we will consider the general theory describing the development of density perturbations in an inhomogeneous ionospheric plasma. We show that realistic boundary conditions strongly affect the excitation criterium and the increments of the gradient-drift instability.
2. Basic equations We will start with the linearized continuity equations for the electron, n,, and ion, ni, density perturbation equations taking into account plasma quasi-neutrality, i.e. n, = ni = n (n is the plasma density perturbation), an n z + - + div j,,i = 0.
(1)
TN
Here rN is the electron life time determined by the balance between ionization and recombination processes, j,,i the corresponding electron and ion flux perturbations. According to Ref. [5] the particle flux perturbations are determined by the particle drift V&, the perturbation density gradient Vn and the perturbation electric field that will be assumed electrostatic - V4, 1 j,.i = V,,in + --Of ee,i
ivy - ~~ iVn. ’
Here e,i are the electron and ion charges, CQ and be,i the tensors of the kinetic coefficients (conductivity and diffusion) for electrons and ions. Both tensors are coupled under the E-layer conditions by the Einstein relation
El,
A simplified quasi-hydrodynamic treatment results in the following expressions for the kinetic coefficients, eziN
eiiN
-1 ak.i 11-
ueil meivei’ I
.
=
(
m,,i
vei
o&
+ 2~:~’
(3)
are the electron and ion cyclotron frequencies, B, the external magnetic field where wee i = e,,iB,/m,,ic strength, c’the velocity of light, m, i the electron and ion masses. An exact kinetic theory correction (see Ref. [5]) affects the numerical coefficients in expressions (3) but does not change essentially neither their dependence on the plasma density N, nor the relations between the cyclotron and collision frequencies v&wB,,~. The unperturbed particle drifts are determined by the external electric field E, as 1 V,,i =
--+$E,. %,iN
In a homogeneous plasma a solution of Eqs. Cl), (2) can be found in the form of a plain wave exp ( - i ot + i k r) that gives the dispersion relation l
w=k
I * V, - iy,
(5)
216
A. V. Gurevich et al. /Physics
Letters A 208 (1995) 214-222
where
k:_u,,,+ k; a, k,2ui,,+ k;
*=
ui
L
I
(8)
’
The dispersion relations (5)-(S) density perturbations [ 1,2].
3. Equations
for an inhomogeneous
Substitution i
describe
the so-called
drift mode, which plays the main role in the E-layer
plasma
of (2) into (1) gives
+ t
+ div V,,i + ( Ve*i* V) - div( 8c,iV)
n+ 1
div( &,iV4)
= 0.
e.* Let us consider a one-dimensional inhomogeneous plasma supposing that its parameters depend only on the altitude (vertical). The magnetic field B, is assumed to have an arbitrary angle (Y with respect to the vertical. Both the electron and ion unperturbed drift velocities Ve,i are supposed to be in the horizontal plane - as usual we assume that a small background electric field along the magnetic field B, compensates all vertical drifts due to the high conductivity along B,. Under the typical conditions of the E-layer of the ionosphere the electrons of the ions changes with the altitude are always strongly magnetized ws, * vem while the magnetization according to the altitude dependence of the ion collisional frequency. In the main part of the current carrying region of the E-layer vi Z+ osi. We will use a nonortogonal, non-normalized coordinate system with the z-axis along the magnetic field. The y-axis is chosen in the plane of the magnetic meridian while the x-axis is perpendicular to both of them to create the right trio. The ort length along the z-axis is chosen to be l/cos (Y so that the unit length along the z-axis corresponds to the unit length along the vertical. The other ort lengths are chosen to be unity. The metric tensor of this coordinate system is g,, =
1 0
0 1
0 tan (Y
.
(10) 0 tan (Y l/cos2cu i I Looking for the solution of Eq. (9) proportional to exp( - i w t + k, x + k, y) we obtain that the combination of Eqs. (9) for electrons and ions leads to an ordinary fourth order equation for C#J(or n). Let us first neglect the mass transport - which means neglecting the diffusion terms, and formulating a purely electrostatic problem. In this case the system of equations (9) for electrons and ions can be reduced to the second order equation
[(o-k,
.vJ,“]$-((W-k, .vi)~+(o-k, .$yg
*vJq_=+(W-kl
-[(u-k, (w-k,
.V,)u,, *Vi)k,
+(w-k,
dR,e
*T+(w--k,
.V,)k,
dR,i
s---g
1
+=O.
(11)
A. V. Gurevich et al. /Physics Letters A 208 (1995) 214-222
217
Here a,,;* = u,,~,, cos2 a + Q I sin’ (Y, k; = k; + k;/cos2
a,
k,A
= k,A” + k,AY,
R,,,i = ak,iH sin( a) x - ak,iI tan( (Y)y.
It should be mentioned that k,A is not a scalar product of vectors, but only a notation for simplicity. The plasma inhomogeneity in (11) is described mainly by the second term connected with the dependence of the ion collision frequency vi(z) on z and the fourth term with N(z) could be responsible for the gradient-drift instability. Using expressions (3) for the kinetic coefficients one can obtain from (11) the second order equation for CD= [(u-k,
+i)q”+(w-kl
exp( - $1 dz)
- V,)giz”]
(12)
in the form (13) where
(14) (15) Here we divided the effective potential in the Schrijdinger equation (13) into two parts: V, that depends on the altitude through the ion collision frequency only and U, that includes the plasma density inhomogeneity. If the plasma density does not depend on the altitude U, = 0. We used here also the nondimensional parameters
E=-$
fi= k,b,-o
/E, mi vi
CLW
k,
v,, = ~ ’
k,
*ve .
(16)
In the E-region of the ionosphere the ion collision frequency is determined by the collisions with neutrals and is proportional to the neutral gas density. The latter can be sufficiently well approximated by an exponential function of the altitude, therefore, we will suppose 5 = exp<-z/L). Here L is the characteristic scale of the neutral atmosphere, supposed to be constant. Using 5 as an independent variable one can easily obtain from equation (13)
$-(V,+V,)p=O, ?P=\/5 CD.
(17)
The renormalized potentials V,,, are
(
v, = A2 1 V, = ipk,L
0
t2--t
tan CY--P~ i ’
sin (Y 1 dN
(19)
Here A = pk, L/cos CT.The analyses show, that even. at VI = 0 Eq. (17) has four singular points in the complex plane 5: two single zeros and two single poles of the potential V,.
218
A. V. Gureuich et al. /Physics
Letters A 208 (1995) 214-222
4. WKB solution We will analyze Eq. (7) in the Wentzel-Kramers-Brillouin (17) in the form
(WKB) approximation. Looking for a solution of
/d5(&+4,+...
)) and supposing that $Q GK+,, and d&/d& * 4i we obtain 4f=V,,
40=
kg-9
~1 = _
1 1 --+_--____++-_ d4l VI 24, d5 2 4,
1
1 dV, VI 4V, d.$-2&’
(21)
Therefore, the solution of (17) in the WKB approximation can be written as
(22) Let us first consider the topology of the complex 5 surface for Eq. (17) with potential (18). The potential is determined at two sheets of the complex surface which are linked at two cuts that we will choose between the zeros and poles of the potential. Therefore we have a toroidal surface on which the solutions of (17) should be considered. Let us suppose that the solution is well behaved (tending to zero) at 5 = 0 corresponding to z --$ +m, what means that there are no sources for the E-region plasma density perturbations. In that case we can select a contour C around 5 -+ 0 without any singular points inside. If the solution is traced along such a contour it should become the same after the full path. This is the usual WKB consideration that leads to the condition for the existence of a well behaved solution,
Suppose also that the solution is regular at 5 + CQ,which means natural transition conditions to the neutral atmosphere below the ionosphere. Expanding the selected contour to infinity through the corn lex surface we will get separate contours around cuts between zeros and poles. The square root branches of PV, in the integral (23) should be in appropriate conjunction with each other because this is the only way to transform the contour to 6 + ~0.After such an expansion only the phase shift 2inT is allowed to be added to the solution, where n is an integer (in this case the solution will be the same). Taking into account that passing around a null and a pole gives the same value but an opposite sign we transform the condition (23) to
(24) where the same branches of E should be chosen in both integrals. Integrating along straight lines between zeros and poles using the parametrization ,$= 5, + t( tb - &a>,where 5, and tb are the beginning and the end of the appropriate contour, one can get the following condition for the existence of a solution which behaves well at z + + CO(23), (rO-rm),,‘dt\/T(I+-&)
=$,
(25)
Here the following notations are used, B=-
2r, r. - r, ’
ro=
tarA+4(1+~)0,
and arithmetic values of all square roots are assumed.
r_={q
(26)
A.V. Gurevich et al. /Physics
For large I 0 1compared
219
Letters A 208 (1995) 214-222
to tan*8 (4 I 0 I B- tan*01 one can obtain
/tan*8 + 41.(.,12
B=2v,_
r. - r, = r-0-2@, fi
10
(27)
’
We see that 1B ( -=x 1. In this limit the integral in (25) can be evaluated B+...,
~‘d~/~=ln&-1-i-p
Substitution of (271, (28) into (25) results in the conditions determined by
as a series on B with the first term
IBI el.
(28)
Im a = 0 and Re 0 > 0 with the eigenvalues
of 0
(29) In the opposite case of small I 0 I (4 10 I x tan*61 BZ-
20 ro - rcc= ltan
tan*0 0
’
(30)
9
and I B I x=-1. In this limit the integral in (25) can be evaluated
Substitution n=
of (30), (31) into (2.5) results in the condition
as a series on Be1 with the first term
for the eigenvalues,
IZI tan 8 I h .
(32)
Expresssions (29) and (32) together with (23) describe the eigenmodes of the inhomogeneous plasma with a small plasma density gradient dN/dz --) 0. It should be mentioned that these eigenmodes are always stable.
5. Gradient-drift instability example Let us take now the plasma density gradient into consideration.
It changes the WKB condition
(24) to
(33) Using the same parametrization
in the integrals
as for (25) one can obtain from (33)
(ro-r.)[l;ld~j/~+/:dl~~~~(r)]=l.
(34)
Suppose the plasma density layer to be N = No exp( - j35). Here we take B +z 1 and under this condition a perturbation to V,. For such layer parameters the potential VI becomes constant VI = ipk,
sin (Y pL= + ip2 = ie,,, +
l$’
VI is
A.V. Gureuich et al. /Physics
220
and the second integral in (34) can be evaluated
z
2,
Letters A 208 (1995) 214-222
in two limits on 1B 1,
IBI ~1.
(35)
Using (34) and (35) we obtain the dispersion relations for the gradient-drift instability in the cases of large and small 1f2 I. In both cases the corrections to the real parts of 0 can be neglected due to /3 -=x 1. For large 4 ( 0 I B tan2 8 the expression for Re 0 is determined by the same relation (29) as in the absence of plasma density gradient and for Im 0 we obtain
(36) Here 0 = Re 0. In the case of small 4 10 I -=Ktan28 the real part of 0 is determined for homogeneous plasma density and r=Im
0=
-20~~.
also by expression
(32)
(37)
According to (361, (37) the waves are unstable. Let us estimate the value of the growth rate of the gradient-drift instability for 8 = 0. Using (36), (29) and (16) we estimate the maximum growth rate as $=Im
o=
V,, k, sin cx r N k, ’
(38)
where L, = L/p is the scale of the plasma density inhomogeneity. Let us consider now a simple model of a plasma layer from x = 0 to x = x,,. Suppose the density gradient along x, k, = k and the drift velocity uY= u. Introducing the potential in the form CF, a +( x) exp[i( wt - b)l 7 we can rewrite Eq. (11) in a simple form, 1-t i
1
div(oiV4)+div(&V$)=0.
To resolve Eq. (39) it is necessary to formulate the conditions at the boundaries of the plasma layer x = 0 and x = x0. The natural boundary conditions for this equation are zero normal current at the boundaries, j, = 0,
at x=0,
x=x0.
There conditions are equivalent layer is given by -ik where & = ~~(1 - ku/w). The boundary conditions d+ =ikOK+, dx ai
(40) to the well behaved conditions
N’ VH ----+,@,=O, &ii+1 N
(23) in an unbounded
plasma. Eq. (39) in the
(41)
(40) for Eq. (411 are if x=0,
x0.
(42)
We consider a simple profile of the plasma density, N = N, exp( x/L,).
(43)
A. V. Gureuich et al. /Physics Letters A 208 (1995) 214-222
Introducing
relation (43) in (41) we have a differential
g’.+$+f+iS+-kz&=O,
6=
N
221
equation with constant coefficients,
ken (44)
L,(GiiUL).
Due to the linear character of Eq. (44) one can look for its solution in the form 4=
C, exp( h,x)
+ C, exp( A,x).
(45)
Taking into account boundary conditions (42) at x = 0 and x =x0 requirement of resolution of the linear system (441, (45), (42),
l-!!+a,= w
we find the dispersion
relation
from the
o;,
ui
(46) ’
~~+ikLNa,[l+(a,/a,)2]
From relation (47), one can find the increment
y. In the limit u _L< ai -K a,,
k.L, > 1, it has the form
YE;.
(47) N
Increment (47) corresponds well to the one in (381, obtained from the general solution in an unbounded inhomogeneous plasma. The position of the effective boundaries under real ionospheric conditions could be chosen as follows. From the homogeneous treatment it is evident that the phase velocity of the waves (6) changes significantly while the parameter Q (8) becomes larger than unity. This occurs in the lower part of the E-region at heights of 90-100 km when the electrons are already strongly magnetized. This level can be taken as the lower boundary of the ionospheric plasma layer where electrostatic perturbations connected to plasma particle drift could be excited. The upper boundary can be chosen at altitudes z > 115 km where the ions become magnetized and the electric current connected with the difference between the electron and ion drift velocities becomes rather small. These boundaries allow one to formulate a model problem for the eigenvalues of the gradient-drift instability in the plasma layer with appropriate boundary conditions.
6. Discussion Let us compare now with the previous theory (see, for example, Refs. [1,4]) where the plasma density inhomogeneity was taken into account only phenomenologically by including the local gradient terms into the homogeneous equations. That resulted in modifying the decrement (7) y = y + -yD, where 1
k,a,, k,2ui,,+k;qL
?‘O= - (I++)~
k,
*(y-vi)
(
(48)
1+
Here N/N = LN ’ is the relative plasma density gradient in the direction perpendicular to the magnetic field B, and k, is the perpendicular projection of the wave vector with respect to the plasma density gradient direction. For waves satisfying the condition k,N > 0 it is easy to see that yD < 0 and so an additional term (48) can result in an instability which is usually called “gradient-drift instability”. Assuming k, + 0, I) -+ 0 and k I in the direction of the density gradient one can obtain from (48) the well-known estimation of the gradient-drift instability growth rate, ‘i YD=
“0
(49)
---LY
BI
N
where u. = V,, is determined according to k I u. = k, widely used in the literature (see Refs. [1,2,4]).
* (V, - Vi). This phenomenological
growth
rate is
A. V. Gureuich et al. /Physics
222
Letters A 208 (1995) 214-222
A comparison of the direct (38) and phenomenological estimations (49) shows an essential difference between them. The growth rate (38) is smaller than (49) by a factor wBi/vi that is always supposed to be small in the phenomenological calculations. The major difference between the direct and phenomenological estimations appears in the boundary conditions. The phenomenological estimations do not assume any boundary conditions. However, for an inhomogeneous plasma, the boundary conditions for the absence of current below as well as above the current sheet, which is the ionospheric E-region, should be fulfilled. Such an approach is made for the direct calculations which resulted in estimations (38). The role of the boundary conditions can be clearly demonstrated for Eq. (44). Let us neglect the boundary conditions (40). In this case we can look for a solution of Eq. (41) in the form #aexp(iKx).
(50)
Eq. (50) leads to the dispersion relation which gives an increment geometry this increment has the form (49), "gH
-=y=
L,u,
‘i
of the gradient-drift
instability.
In our
vO
(51)
wL3iLN.
But neglecting the boundary conditions means (as one can see from (421, (50)) that we allow the current pass through the boundaries of our plasma layer, J=~[(u~+cT~)K-cT~~]
J to
exp(iKx).
This current is the main source of instability amplifies its increase.
in the plasma. As one can see, comparing
(51) with (47), it strongly
7. Conclusions So we see that the correct formulation and solution of the problem of the plasma density perturbations development in an inhomogeneous E-layer leads to a strong decrease of the increments of the gradient-drift instability in comparison with the usually used phenomenological theory. The growth rate (38) is smaller than (49) by a factor wsi/vi which is always small at heights z < 115 km. Note that perturbation decrements, connected with diffusive processes were not considered here. One can suppose that due to the diffusion gradient the drift instability would be attenuated even more and could completely disappear for a wide region of parameters.
Acknowledgement The authors are grateful to T. Hagfors for valuable discussions. M8WOOO and R8EOOO from the International Science Foundation.
This research was partly supported by grants
References [l] [2] [3] [4] [5]
M.C. Kelley, The earth’s ionosphere: plasma physics and electrodynamics (Academic Press, New York, 1989). A.V. Gurevich, N.D. Borisov and K.P. Zybin, submitted to J. Geophys. Res. A.V. Gurevich, K. Rinnert and K. Shlegel, submitted to J. Geophys. Res. R. Sudan, Unified theory of type I and type II irregularities in equatorial electrojet, J. Geophys. Res. 88 (1983) 4853. A.V. Gurevich, Nonlinear phenomena in the ionosphere (Springer, Berlin, 1978).
&
ELSEVIER
27 November
1995
PHYSICS
LETTERS
A
Physics Letters A 208 (1995) 214-222
Plasma density perturbations in the ionospheric E-region A.V. Gurevich ‘, N.D. Borisov 2, A.N. Karashtin 3, K.P. Zybin Max-Planck-Institute fur Aeronomie, 37191 Katlenburg-Lindau,
’
Germany
Received 14 August 1995; accepted for publication 22 September Communicated by V.M. Agranovich
1995
Abstract The theory of gradient drift instability in a height inhomogeneous the instability increments is demonstrated.
ionospheric
E-region is revised. A strong decrease of
1. Introduction Plasma density perturbations are usually observed in the ionospheric E-layer by radio wave scattering and in situ measurements on rockets [l]. Recently, it was supposed [2], that the low frequency part of these perturbations is induced in an ionospheric plasma by the turbulent motion of the neutral atmosphere. A qualitative agreement of this theory with the observations on board of rockets is demonstrated [3]. The theory was developed in Ref. [2] for a homogeneous plasma only. The plasma density in the real ionosphere is height inhomogeneous, N = N(z), and this inhomogeneity plays an important role first of all due to the gradient-drift instability, which can occur in an inhomogeneous plasma [1,4]. This instability can excite and essentially amplify the plasma density perturbations. It should be noted, that in the theory of gradient-drift instability the following relation is usually supposed to be fulfilled,
dN x=z’
N
where L is constant (see Refs. [1,4]). That allowed one to consider the process phenomenologically in the same way as in a homogeneous plasma, what simplifies the problem radically. But in an inhomogeneous medium boundary conditions usually play an essential role. This means that the theory of gradient-drift instability developed phenomenologically like in an unbounded homogeneous medium cannot be considered as fully constructed.
’ On leave from P.N. Lebedev Physical Institute, 117924 Moscow, Russian Federation. 2 On leave from Institute of Terrestrial Magnetism, Ionosphere and Radio Wave Propagation, 142092 Troitsk, Moscow Region, Russian Federation. 3 On leave from Radiophysical Research Institute (NIRFI), Nizhny Novgorog, Russian Federation. Elsevier Science B.V. SSDI 0375-9601(95)00738-5
A.V. Gureuich et al. / Physics Letters A 208 (1995) 214-222
215
In the present paper, we will consider the general theory describing the development of density perturbations in an inhomogeneous ionospheric plasma. We show that realistic boundary conditions strongly affect the excitation criterium and the increments of the gradient-drift instability.
2. Basic equations We will start with the linearized continuity equations for the electron, n,, and ion, ni, density perturbation equations taking into account plasma quasi-neutrality, i.e. n, = ni = n (n is the plasma density perturbation), an n z + - + div j,,i = 0.
(1)
TN
Here rN is the electron life time determined by the balance between ionization and recombination processes, j,,i the corresponding electron and ion flux perturbations. According to Ref. [5] the particle flux perturbations are determined by the particle drift V&, the perturbation density gradient Vn and the perturbation electric field that will be assumed electrostatic - V4, 1 j,.i = V,,in + --Of ee,i
ivy - ~~ iVn. ’
Here e,i are the electron and ion charges, CQ and be,i the tensors of the kinetic coefficients (conductivity and diffusion) for electrons and ions. Both tensors are coupled under the E-layer conditions by the Einstein relation
El,
A simplified quasi-hydrodynamic treatment results in the following expressions for the kinetic coefficients, eziN
eiiN
-1 ak.i 11-
ueil meivei’ I
.
=
(
m,,i
vei
o&
+ 2~:~’
(3)
are the electron and ion cyclotron frequencies, B, the external magnetic field where wee i = e,,iB,/m,,ic strength, c’the velocity of light, m, i the electron and ion masses. An exact kinetic theory correction (see Ref. [5]) affects the numerical coefficients in expressions (3) but does not change essentially neither their dependence on the plasma density N, nor the relations between the cyclotron and collision frequencies v&wB,,~. The unperturbed particle drifts are determined by the external electric field E, as 1 V,,i =
--+$E,. %,iN
In a homogeneous plasma a solution of Eqs. Cl), (2) can be found in the form of a plain wave exp ( - i ot + i k r) that gives the dispersion relation l
w=k
I * V, - iy,
(5)
216
A. V. Gurevich et al. /Physics
Letters A 208 (1995) 214-222
where
k:_u,,,+ k; a, k,2ui,,+ k;
*=
ui
L
I
(8)
’
The dispersion relations (5)-(S) density perturbations [ 1,2].
3. Equations
for an inhomogeneous
Substitution i
describe
the so-called
drift mode, which plays the main role in the E-layer
plasma
of (2) into (1) gives
+ t
+ div V,,i + ( Ve*i* V) - div( 8c,iV)
n+ 1
div( &,iV4)
= 0.
e.* Let us consider a one-dimensional inhomogeneous plasma supposing that its parameters depend only on the altitude (vertical). The magnetic field B, is assumed to have an arbitrary angle (Y with respect to the vertical. Both the electron and ion unperturbed drift velocities Ve,i are supposed to be in the horizontal plane - as usual we assume that a small background electric field along the magnetic field B, compensates all vertical drifts due to the high conductivity along B,. Under the typical conditions of the E-layer of the ionosphere the electrons of the ions changes with the altitude are always strongly magnetized ws, * vem while the magnetization according to the altitude dependence of the ion collisional frequency. In the main part of the current carrying region of the E-layer vi Z+ osi. We will use a nonortogonal, non-normalized coordinate system with the z-axis along the magnetic field. The y-axis is chosen in the plane of the magnetic meridian while the x-axis is perpendicular to both of them to create the right trio. The ort length along the z-axis is chosen to be l/cos (Y so that the unit length along the z-axis corresponds to the unit length along the vertical. The other ort lengths are chosen to be unity. The metric tensor of this coordinate system is g,, =
1 0
0 1
0 tan (Y
.
(10) 0 tan (Y l/cos2cu i I Looking for the solution of Eq. (9) proportional to exp( - i w t + k, x + k, y) we obtain that the combination of Eqs. (9) for electrons and ions leads to an ordinary fourth order equation for C#J(or n). Let us first neglect the mass transport - which means neglecting the diffusion terms, and formulating a purely electrostatic problem. In this case the system of equations (9) for electrons and ions can be reduced to the second order equation
[(o-k,
.vJ,“]$-((W-k, .vi)~+(o-k, .$yg
*vJq_=+(W-kl
-[(u-k, (w-k,
.V,)u,, *Vi)k,
+(w-k,
dR,e
*T+(w--k,
.V,)k,
dR,i
s---g
1
+=O.
(11)
A. V. Gurevich et al. /Physics Letters A 208 (1995) 214-222
217
Here a,,;* = u,,~,, cos2 a + Q I sin’ (Y, k; = k; + k;/cos2
a,
k,A
= k,A” + k,AY,
R,,,i = ak,iH sin( a) x - ak,iI tan( (Y)y.
It should be mentioned that k,A is not a scalar product of vectors, but only a notation for simplicity. The plasma inhomogeneity in (11) is described mainly by the second term connected with the dependence of the ion collision frequency vi(z) on z and the fourth term with N(z) could be responsible for the gradient-drift instability. Using expressions (3) for the kinetic coefficients one can obtain from (11) the second order equation for CD= [(u-k,
+i)q”+(w-kl
exp( - $1 dz)
- V,)giz”]
(12)
in the form (13) where
(14) (15) Here we divided the effective potential in the Schrijdinger equation (13) into two parts: V, that depends on the altitude through the ion collision frequency only and U, that includes the plasma density inhomogeneity. If the plasma density does not depend on the altitude U, = 0. We used here also the nondimensional parameters
E=-$
fi= k,b,-o
/E, mi vi
CLW
k,
v,, = ~ ’
k,
*ve .
(16)
In the E-region of the ionosphere the ion collision frequency is determined by the collisions with neutrals and is proportional to the neutral gas density. The latter can be sufficiently well approximated by an exponential function of the altitude, therefore, we will suppose 5 = exp<-z/L). Here L is the characteristic scale of the neutral atmosphere, supposed to be constant. Using 5 as an independent variable one can easily obtain from equation (13)
$-(V,+V,)p=O, ?P=\/5 CD.
(17)
The renormalized potentials V,,, are
(
v, = A2 1 V, = ipk,L
0
t2--t
tan CY--P~ i ’
sin (Y 1 dN
(19)
Here A = pk, L/cos CT.The analyses show, that even. at VI = 0 Eq. (17) has four singular points in the complex plane 5: two single zeros and two single poles of the potential V,.
218
A. V. Gureuich et al. /Physics
Letters A 208 (1995) 214-222
4. WKB solution We will analyze Eq. (7) in the Wentzel-Kramers-Brillouin (17) in the form
(WKB) approximation. Looking for a solution of
/d5(&+4,+...
)) and supposing that $Q GK+,, and d&/d& * 4i we obtain 4f=V,,
40=
kg-9
~1 = _
1 1 --+_--____++-_ d4l VI 24, d5 2 4,
1
1 dV, VI 4V, d.$-2&’
(21)
Therefore, the solution of (17) in the WKB approximation can be written as
(22) Let us first consider the topology of the complex 5 surface for Eq. (17) with potential (18). The potential is determined at two sheets of the complex surface which are linked at two cuts that we will choose between the zeros and poles of the potential. Therefore we have a toroidal surface on which the solutions of (17) should be considered. Let us suppose that the solution is well behaved (tending to zero) at 5 = 0 corresponding to z --$ +m, what means that there are no sources for the E-region plasma density perturbations. In that case we can select a contour C around 5 -+ 0 without any singular points inside. If the solution is traced along such a contour it should become the same after the full path. This is the usual WKB consideration that leads to the condition for the existence of a well behaved solution,
Suppose also that the solution is regular at 5 + CQ,which means natural transition conditions to the neutral atmosphere below the ionosphere. Expanding the selected contour to infinity through the corn lex surface we will get separate contours around cuts between zeros and poles. The square root branches of PV, in the integral (23) should be in appropriate conjunction with each other because this is the only way to transform the contour to 6 + ~0.After such an expansion only the phase shift 2inT is allowed to be added to the solution, where n is an integer (in this case the solution will be the same). Taking into account that passing around a null and a pole gives the same value but an opposite sign we transform the condition (23) to
(24) where the same branches of E should be chosen in both integrals. Integrating along straight lines between zeros and poles using the parametrization ,$= 5, + t( tb - &a>,where 5, and tb are the beginning and the end of the appropriate contour, one can get the following condition for the existence of a solution which behaves well at z + + CO(23), (rO-rm),,‘dt\/T(I+-&)
=$,
(25)
Here the following notations are used, B=-
2r, r. - r, ’
ro=
tarA+4(1+~)0,
and arithmetic values of all square roots are assumed.
r_={q
(26)
A.V. Gurevich et al. /Physics
For large I 0 1compared
219
Letters A 208 (1995) 214-222
to tan*8 (4 I 0 I B- tan*01 one can obtain
/tan*8 + 41.(.,12
B=2v,_
r. - r, = r-0-2@, fi
10
(27)
’
We see that 1B ( -=x 1. In this limit the integral in (25) can be evaluated B+...,
~‘d~/~=ln&-1-i-p
Substitution of (271, (28) into (25) results in the conditions determined by
as a series on B with the first term
IBI el.
(28)
Im a = 0 and Re 0 > 0 with the eigenvalues
of 0
(29) In the opposite case of small I 0 I (4 10 I x tan*61 BZ-
20 ro - rcc= ltan
tan*0 0
’
(30)
9
and I B I x=-1. In this limit the integral in (25) can be evaluated
Substitution n=
of (30), (31) into (2.5) results in the condition
as a series on Be1 with the first term
for the eigenvalues,
IZI tan 8 I h .
(32)
Expresssions (29) and (32) together with (23) describe the eigenmodes of the inhomogeneous plasma with a small plasma density gradient dN/dz --) 0. It should be mentioned that these eigenmodes are always stable.
5. Gradient-drift instability example Let us take now the plasma density gradient into consideration.
It changes the WKB condition
(24) to
(33) Using the same parametrization
in the integrals
as for (25) one can obtain from (33)
(ro-r.)[l;ld~j/~+/:dl~~~~(r)]=l.
(34)
Suppose the plasma density layer to be N = No exp( - j35). Here we take B +z 1 and under this condition a perturbation to V,. For such layer parameters the potential VI becomes constant VI = ipk,
sin (Y pL= + ip2 = ie,,, +
l$’
VI is
A.V. Gureuich et al. /Physics
220
and the second integral in (34) can be evaluated
z
2,
Letters A 208 (1995) 214-222
in two limits on 1B 1,
IBI ~1.
(35)
Using (34) and (35) we obtain the dispersion relations for the gradient-drift instability in the cases of large and small 1f2 I. In both cases the corrections to the real parts of 0 can be neglected due to /3 -=x 1. For large 4 ( 0 I B tan2 8 the expression for Re 0 is determined by the same relation (29) as in the absence of plasma density gradient and for Im 0 we obtain
(36) Here 0 = Re 0. In the case of small 4 10 I -=Ktan28 the real part of 0 is determined for homogeneous plasma density and r=Im
0=
-20~~.
also by expression
(32)
(37)
According to (361, (37) the waves are unstable. Let us estimate the value of the growth rate of the gradient-drift instability for 8 = 0. Using (36), (29) and (16) we estimate the maximum growth rate as $=Im
o=
V,, k, sin cx r N k, ’
(38)
where L, = L/p is the scale of the plasma density inhomogeneity. Let us consider now a simple model of a plasma layer from x = 0 to x = x,,. Suppose the density gradient along x, k, = k and the drift velocity uY= u. Introducing the potential in the form CF, a +( x) exp[i( wt - b)l 7 we can rewrite Eq. (11) in a simple form, 1-t i
1
div(oiV4)+div(&V$)=0.
To resolve Eq. (39) it is necessary to formulate the conditions at the boundaries of the plasma layer x = 0 and x = x0. The natural boundary conditions for this equation are zero normal current at the boundaries, j, = 0,
at x=0,
x=x0.
There conditions are equivalent layer is given by -ik where & = ~~(1 - ku/w). The boundary conditions d+ =ikOK+, dx ai
(40) to the well behaved conditions
N’ VH ----+,@,=O, &ii+1 N
(23) in an unbounded
plasma. Eq. (39) in the
(41)
(40) for Eq. (411 are if x=0,
x0.
(42)
We consider a simple profile of the plasma density, N = N, exp( x/L,).
(43)
A. V. Gureuich et al. /Physics Letters A 208 (1995) 214-222
Introducing
relation (43) in (41) we have a differential
g’.+$+f+iS+-kz&=O,
6=
N
221
equation with constant coefficients,
ken (44)
L,(GiiUL).
Due to the linear character of Eq. (44) one can look for its solution in the form 4=
C, exp( h,x)
+ C, exp( A,x).
(45)
Taking into account boundary conditions (42) at x = 0 and x =x0 requirement of resolution of the linear system (441, (45), (42),
l-!!+a,= w
we find the dispersion
relation
from the
o;,
ui
(46) ’
~~+ikLNa,[l+(a,/a,)2]
From relation (47), one can find the increment
y. In the limit u _L< ai -K a,,
k.L, > 1, it has the form
YE;.
(47) N
Increment (47) corresponds well to the one in (381, obtained from the general solution in an unbounded inhomogeneous plasma. The position of the effective boundaries under real ionospheric conditions could be chosen as follows. From the homogeneous treatment it is evident that the phase velocity of the waves (6) changes significantly while the parameter Q (8) becomes larger than unity. This occurs in the lower part of the E-region at heights of 90-100 km when the electrons are already strongly magnetized. This level can be taken as the lower boundary of the ionospheric plasma layer where electrostatic perturbations connected to plasma particle drift could be excited. The upper boundary can be chosen at altitudes z > 115 km where the ions become magnetized and the electric current connected with the difference between the electron and ion drift velocities becomes rather small. These boundaries allow one to formulate a model problem for the eigenvalues of the gradient-drift instability in the plasma layer with appropriate boundary conditions.
6. Discussion Let us compare now with the previous theory (see, for example, Refs. [1,4]) where the plasma density inhomogeneity was taken into account only phenomenologically by including the local gradient terms into the homogeneous equations. That resulted in modifying the decrement (7) y = y + -yD, where 1
k,a,, k,2ui,,+k;qL
?‘O= - (I++)~
k,
*(y-vi)
(
(48)
1+
Here N/N = LN ’ is the relative plasma density gradient in the direction perpendicular to the magnetic field B, and k, is the perpendicular projection of the wave vector with respect to the plasma density gradient direction. For waves satisfying the condition k,N > 0 it is easy to see that yD < 0 and so an additional term (48) can result in an instability which is usually called “gradient-drift instability”. Assuming k, + 0, I) -+ 0 and k I in the direction of the density gradient one can obtain from (48) the well-known estimation of the gradient-drift instability growth rate, ‘i YD=
“0
(49)
---LY
BI
N
where u. = V,, is determined according to k I u. = k, widely used in the literature (see Refs. [1,2,4]).
* (V, - Vi). This phenomenological
growth
rate is
A. V. Gureuich et al. /Physics
222
Letters A 208 (1995) 214-222
A comparison of the direct (38) and phenomenological estimations (49) shows an essential difference between them. The growth rate (38) is smaller than (49) by a factor wBi/vi that is always supposed to be small in the phenomenological calculations. The major difference between the direct and phenomenological estimations appears in the boundary conditions. The phenomenological estimations do not assume any boundary conditions. However, for an inhomogeneous plasma, the boundary conditions for the absence of current below as well as above the current sheet, which is the ionospheric E-region, should be fulfilled. Such an approach is made for the direct calculations which resulted in estimations (38). The role of the boundary conditions can be clearly demonstrated for Eq. (44). Let us neglect the boundary conditions (40). In this case we can look for a solution of Eq. (41) in the form #aexp(iKx).
(50)
Eq. (50) leads to the dispersion relation which gives an increment geometry this increment has the form (49), "gH
-=y=
L,u,
‘i
of the gradient-drift
instability.
In our
vO
(51)
wL3iLN.
But neglecting the boundary conditions means (as one can see from (421, (50)) that we allow the current pass through the boundaries of our plasma layer, J=~[(u~+cT~)K-cT~~]
J to
exp(iKx).
This current is the main source of instability amplifies its increase.
in the plasma. As one can see, comparing
(51) with (47), it strongly
7. Conclusions So we see that the correct formulation and solution of the problem of the plasma density perturbations development in an inhomogeneous E-layer leads to a strong decrease of the increments of the gradient-drift instability in comparison with the usually used phenomenological theory. The growth rate (38) is smaller than (49) by a factor wsi/vi which is always small at heights z < 115 km. Note that perturbation decrements, connected with diffusive processes were not considered here. One can suppose that due to the diffusion gradient the drift instability would be attenuated even more and could completely disappear for a wide region of parameters.
Acknowledgement The authors are grateful to T. Hagfors for valuable discussions. M8WOOO and R8EOOO from the International Science Foundation.
This research was partly supported by grants
References [l] [2] [3] [4] [5]
M.C. Kelley, The earth’s ionosphere: plasma physics and electrodynamics (Academic Press, New York, 1989). A.V. Gurevich, N.D. Borisov and K.P. Zybin, submitted to J. Geophys. Res. A.V. Gurevich, K. Rinnert and K. Shlegel, submitted to J. Geophys. Res. R. Sudan, Unified theory of type I and type II irregularities in equatorial electrojet, J. Geophys. Res. 88 (1983) 4853. A.V. Gurevich, Nonlinear phenomena in the ionosphere (Springer, Berlin, 1978).