ISSN 0884-5913, Kinematics and Physics of Celestial Bodies, 2007, Vol. 23, No. 2, pp. 73–76. © Allerton Press, Inc., 2007. Original Russian Text © Yu.V. Kyz’yurov, 2007, published in Kinematika i Fizika Nebesnykh Tel, 2007, Vol. 23, No. 2, pp.107–111.
SPACE PHYSICS
Diurnal Variations in the Spectrum of Lower-Ionosphere Plasma Irregularities Yu. V. Kyz’yurov Main Astronomical Observatory, National Academy of Sciences of Ukraine, vul. Akademika Zabolotnoho 27, Kyiv, 03680 Ukraine Received October 10, 2006
Abstract—We examined the variations caused in the spectrum of lower-ionosphere plasma irregularities by diurnal changes of ionospheric parameters. We demonstrate that at a height of 95 km, where the irregularities arise due to the turbulence of the neutral atmosphere, the expected level of plasma irregularity fluctuations at a constant turbulent mixing intensity is smaller in the daytime than at night. The one-dimensional spectrum which can be measured in experiments also changes, namely, the spectrum slope is greater at night. PACS numbers: 94.20.Vv DOI: 10.3103/S0884591307020055
INTRODUCTION The structure of the ionosphere is known to undergo regular temporal changes: in the course of a day, a season, an 11-year solar activity cycle [1, 3]. In the lower ionosphere, for example, the mean electron density N0, the characteristic scale LN of vertical density gradient, and the temperature T at night are smaller than in the daytime [1, 3, 6]. The ionosphere below the turbopause is characterized by a developed chaotic nonuniform structure which is associated in large measure with turbulent motions of atmospheric gas [1–3]. The density irregularity depends to a large extent on turbulent velocity and height gradient of plasma density [2, 3]. That is the reason why the chaotic nonuniform structure of the ionosphere is expected to change even at a constant turbulent mixing intensity. It is well known [4] that the atmospheric turbulence at ionosphere heights is brought about by the breaking of the internal gravity waves and tides coming from the lower atmosphere as well as by the nonlinear interaction of planetary waves and tides. In this study we analyze the influence of the changes experienced by the lower-ionosphere parameters in the course of a day on the plasma irregularities produced by the turbulence of the neutral atmosphere. INPUT EQUATIONS AND RELATIONS Inasmuch as the plasma density N is small in comparison to the neutral gas density Nn, the plasma in the lower ionosphere is regarded as a passive admixture and the effect of charged particles on gas motion can be ignored, the gas velocity u being considered a given parameter. When we take into consideration that the characteristic time t t of turbulent motions is vastly greater than the characteristic time t s between the collisions of charged particles with neutrals (collisions between charged particles can be ignored in the height range we consider here), that the typical spatial scales far exceed the free path length, and that the hydrodynamic motion velocities vs are lower than the thermal velocities vTs , the entrainment of charged particles by neutrals can be described by the set of equations [2, 5, 8] dN s dt + Ñ( N svs ) = 0,
(1)
2 t -s 1 ( v s - u ) = qs E ms + W s ( v s ´ b) - v Ts N s-1 ÑN s .
(2)
Here subscript s denotes electrons (s = e) or ions (s = i), E is electric field, qe = -qi are electron and ion charges, ms is charged particle mass, b is a unit vector aligned with the external magnetic field, and Ws is charged particle gyrofrequency. In the ionosphere below the turbopause the inequality ti W i < 1 as well as the quasi-neutrality and isothermality conditions N e = N i = N and Te = Ti = Tn = T are readily met. The gas at these heights can be 73
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regarded as noncompressible (Ñu = 0), and its velocity u can be represented as sums of the mean u0 = áuñ and the deviation u1 from the mean, u1 being the turbulent gas velocity field (as a rule, u1 < u0). The density N also can be divided into the mean part N0 = áNñ and the fluctuating part N1 < N0. As shown in [5, 8], the formation of plasma irregularities in a turbulent gas flow can be described by the set of equations (1) and (2), and the spatiotemporal spectrum Y(k, w) of the fluctuations dN = N1/N0 can thus be obtained. In this case the characteristic dimensions of turbulent motions are limited to the inertial scale interval, where the turbulence is uniform and isotropic and the spatiotemporal spectral tensor of the random field u1 is of the form F ab ( k, w ) = (d ab - k a k b k 2 ) ´ [4p 2 (1 + w 2 t 2t )] -1 t tC 1e 2/ 3 k -11/ 3 ,
(3)
k 0 < k < kn , t t = ( nk 2 + e 1 3 k 2 3 ) -1 being the characteristic lifetime of a turbulent eddy of size k–1, C1 a dimensionless constant of the order of unity, e the mean dissipation rate of turbulent pulsation energy in a unit gas mass, k0 and k n = ( n 3 e ) -1 4 the wave numbers characterizing the outer and inner scales of turbulence, respectively, and n the kinematic gas viscosity. When deriving Y( k, w ), we took into consideration only the internal electric field E, which ensures the collective drift of charged particles; the characteristic dimension of irregularities was assumed to be less than LN , and the relationship between the disturbances on different scales in the generation of plasma irregularities by atmospheric turbulence was taken into account through the coefficient of turbulent diffusion DT : Y( k, w ) = [4p 2 (1 + w 2 t 2k )(1 + w 2 t 2t )] -1 t t t 2k Q( k ),
(4)
L-N1 < k < k d . Here t k = (D A k 2 + DT k 2 ) -1 = (D A k 2 + e 1 3 k 2 3 ) -1 , D A is the coefficient of ambipolar diffusion, the wave number k d = (D A3 e ) -1 4 defines the spatial scale of the fluctuations dN for which the coefficients DA and DT become equal, Q( k ) = [( n ´ k ) 2 (LN k ) 2 + ( b ´ k ) 2 ( t i W i ) 2] C 1e 2 3 k -11 3, and n = LN N 0-1ÑN 0 is a unit vector aligned with the gradient of N0. We integrate (4) with respect to w and obtain the spatial spectrum of fluctuations dN : SN ( k ) =
¥
ò Y( k, w) dw = [4p(1 + t t
t k )] -1 t t t k Q( k ).
(5)
-¥
In the integration we took into consideration that DA is not equal to n, although these quantities are of the same order of magnitude (this was not done in [5, 8]). Expression (5) allows us to estimate the root-mean-square fluctuation in the interval {k1, k2}: k2
dN
2
= ò SN ( k ) d k = ò S 0( k )dk = F (( k 2 k d ) 4 3 ) - F (( k 1 k d ) 4 3 ),
(6)
L-N2 k -3 + t 2i W 2i k -1 , [1 + ( k k d ) 4 3 ][2 + ( k k d ) 4 3 + ( k k n ) 4 3 ]
(7)
k1
where S0 (k ) =
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3 F ( x) = L-N2 k d-2 x - 3 2 [( 3 + Pr ) - 2 3] 8 +
+
3 L-N2 k d-2 [arctan x1 2 - ((1 + Pr ) 2) 5 2 arctan( x (1 + Pr ) 2)1 2 ] 2 1 - Pr
3 t 2i W 2i {2ln[ x ( 1 + x)] - (1 + Pr )ln[x(1 + Pr ) ( 2 + x(1 + Pr ))]}. 8 1 - Pr
Here Pr = n DA is the diffusion Prandtl number. Typically we have Pr » 1 in the lower ionosphere, but Pr ¹ 1. In the experimental investigations of ionospheric irregularities, the one-dimensional spectrum along some preferred direction is commonly measured. When such a measurement is made along the z-axis, the spectrum S 1 ( k z ) = ò S N ( k ) dk x dk y is obtained [3]. The expression for it is derived with the use of expression (5) and the cylindrical coordinate system (k ^ , j, k z ): kz
2p
0
0
S 1 ( k z ) = ò k ^ dk ^ ò S N ( k ) dj kz
1 = ò [L-N2 f ( k ^ , k z , q 1 ) + t 2i W 2i k 2 f ( k ^ , k z , q 2 )] P( k )k -7 k ^ dk ^ , 4 0
(8)
where k z2 = k d2 - k z2 ,
k 2 = k ^2 + k z2 ,
f ( k ^ , k z , q ) = k ^2 + k ^2 cos 2 q + 2k z2 sin 2 q, P( k ) = [(1 + ( k k d ) 4 3 )( 2 + ( k k d ) 4 3 + ( k k n ) 4 3 )] -1 , q 1 is the angle of the direction z along which the spectrum is measured with the direction n, and q 2 is the angle of z with b. IRREGULARITY SPECTRUM VARIABILITY IN THE COURSE OF A DAY We consider the diurnal variations of the plasma irregularities produced in the ionosphere at a height of 95 km by atmospheric turbulence. At this height we have N0 = 3.9 ´ 1010 m-3 , T = 220 K, LN = 8 km in the daytime and N0 = 4.3 ´ 10 8 m-3 , T = 204 K, LN = 4 km at night [3, 6]. We assume that the mean masses of ions mi » 31 amu and neutral particles mn » 29 amu as well as the quantities t i = 6.25 ´ 10 -5 s, W i = 155 s–1 ( t i W i » 0.0097 ), and e = 0.1 m2s–3 [1, 7] remain unchanged. With such a set of parameters, the level of electron density fluctuations ( N 1 N 0 ) 2
12
in the irregularities (which are smaller than 500 m in size;
Some characteristics of the ionosphere and plasma irregularities at a height of 95 km Time
n, m2s–1
DA , m2s–1
LB = ti Wi LN , m
k n-2 , m
k d-1 , m
ádN 2ñ1 2, %
p
Day
3.9
7.3
77.6
4.94
7.9
3.3
1.94
Night
3.6
6.8
38.8
4.67
7.47
6.3
2.26
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k1 = 0.002 m–1, k2 = kd) is 3.3% in the daytime and 6.3% at night, according to formula (6) (the quantity kd changes over the course of a day, see the table). The shape of the spectrum is calculated for the mid-latitude ionosphere (magnetic inclination 45°); in this case the direction of measurement coincides with the direction of the N0 gradient, i.e., sin2 q 1 = 0, cos 2 q 1 = 1, and sin 2 q 2 = cos 2 q 2 = 1/2. Then, instead of (8), we have kz
1 S 1 ( k z ) = ò [4L-N2 k ^2 + t i2 W i2 k 2 ( 3k ^2 + 2k z2 )]P( k )k -7 k ^ dk ^ . 8 0
(9)
The figure displays the calculation results for the one-dimensional spectrum S 1 ( k z ) at a height of 95 km under the daytime and night conditions. If S 1( k z ) is approximated by the simple power function k z- p , the exponent p is approximately equal to 1.94 in the daytime and 2.26 at night, that is, the slope of the spectrum increases at night. CONCLUSION When the turbulent mixing intensity in the lower ionosphere remains unchanged, the level of the turbulence-produced plasma 1/ 2 fluctuations dN 2 should be lower in the daytime (3.3%) than at night (6.3%) for irregularities smaller than 500 m. The shape of the one-dimensional spectrum S 1 ( k z ), which can be measured in experiments, also changes. When the spectrum is approximated by the The spectrum of plasma irregularities in the ionosphere calculated by forpower function k z- p , the exponent p is about 1.94 in the daytime and mula (9): (1) in the daytime, (2) at 2.26 at night; the spectrum is more inclined under night conditions. night; (3) relation k z-5 3. The increase in the fluctuation level and in the spectrum steepness at night is accountable to a decrease of LN , the characteristic scale of electron density gradient. Two scale intervals can be distinguished in the spatial spectrum of irregularities [8]. In the long-wave interval, where the characteristic size of irregularities is greater than L B = t i W i LN , the irregularities arise primarily due to the density gradient breaking resulting from turbulent mixing, while in the other interval, for smaller irregularities, of greater importance is the interaction of the charged particles entrained by turbulent gas flow with the magnetic field [8]. The turbulent pulsations of atmospheric gas are more intense and the slope of the spatial spectrum of irregularities is greater in the long-wave interval, and this scale interval becomes wider as LN diminishes at constant t i W i . REFERENCES 1. Al’pert, Ya.L., Rasprostranenie electromagnitnykh voln i ionosfera (Propagation of Electromagnetic Waves and the Ionosphere), Moscow: Nauka, 1972. 2. Gershman, B.N., Dinamika ionosfernoi plazmy (Ionospheric Plasma Dynamics), Moscow: Nauka, 1974. 3. Gershman, B.N., Erukhimov, L.M., and Yashin, Yu.Ya., Volnovye yavleniya v ionosfere i kosmicheskoi plazme (Wave Phenomena in the Ionosphere and in Cosmic Plasmas), Moscow: Nauka, 1984. 4. Kazimirovskii, E.S. and Kokourov, V.D., Meteorological Effects in the Ionosphere (Review), Geomagnetizm i Aeronomiya, 1995, vol. 35, no. 3, pp. 1–23. 5. Kyz’yurov, Yu.V., Effective Cross-Section of the Scattering from the Sporadic-E Irregularities Induced by Atmospheric Turbulence, Kinematika i Fizika Nebes. Tel, 2005, vol. 21, no. 3, pp.163–171. 6. Martynenko, S.I. and Chernogor, L.F., On Nonlinear Effects in Partial Reflections of Radio Waves in the Ionosphere, Geomagnetizm i Aeronomiya, 1976, vol. 16, no. 4, pp. 658–665. 7. Gurevich, A.V., Borisov, N.D., and Zybin, K.P., Ionospheric Turbulence Induced in the Lower Part of the {E} region by the Turbulence of the Neutral Atmosphere, J. Geophys. Res., 1997, vol. 102, no. 1, pp. 379–388. 8. Kyzyurov, Yu.V., On the Spectrum of Mid-Latitude Sporadic-E Irregularities, Ann. Geophys., 2000, vol. 18, no 10, pp. 283–292.
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