Radiophysics and Quantum Electronics, Vol. 55, No. 6, November, 2012 (Russian Original Vol. 55, No. 6, June, 2012)
MAGNETOGRAVITY WAVES IN THE IONOSPHERE UNDER CONDITIONS OF FINITE CONDUCTIVITY N. A. Barkhatov,1,2,3 ∗ O. M. Barkhatova,1,2 and G. I. Grigor’ev3
UDC 550.388.2
We obtain dispersion relations for magnetogravity waves in the ionosphere with allowance for the combined influence of magnetic field, gravity, and finite conductivity within the framework of the hydrodynamic approximation. The required conditions are fulfilled in the ionosphere at altitudes over or about 250 km. The auroral electrojet is considered as a source of magnetogravity waves which are frequently observed as traveling ionospheric disturbances. The contribution of magnetogravity waves to the ionospheric disturbances is determined on the basis of analyzing the data from the vertical sounding of the ionospheric F2 layer and the geomagnetic disturbances along the chosen magnetic meridian and on its sides. The features of the obtained dynamic spectra of magnetogravity waves agree with the characteristic frequencies and velocities determined by the calculated dispersion curves. As a result, we confirm the fact that magnetogravity waves stipulate some traveling ionospheric disturbances and can be used for diagnostics of the ionospheric parameters.
1.
INTRODUCTION
The traveling ionospheric disturbances are the most pronounced manifestation of the ionospheric wave motions. As the main sources of such disturbances, the acoustic-gravity waves, which can be related to the terminator motion and jet streams, are excited during earthquakes, or emerge in the periods of auroralelectrojet intensification are usually considered. However, in the ionosphere, as in the conducting medium, magnetogravity waves, whose velocity is larger than that of acoustic-gravity waves, but smaller than that of magnetohydrodynamic waves, can also propagate along with the acoustic-gravity waves [1]. The existence of magnetogravity waves in the ionosphere at the F2 -layer altitudes was confirmed experimentally in [2] in which their dispersion relations for the case of a medium with infinite conductivity were analyzed. In this work, we obtain the dispersion relations for the magnetogravity waves in the ionosphere with finite conductivity. To this end, the joint influence of a magnetic field and gravity is allowed for in the hydrodynamic equations. This should be done in the case where magnetic pressure is comparable with or higher than the hydrostatic pressure and the frequency of the radiated waves is much lower than the neutral–ion collision rate. These conditions are fulfilled in the ionosphere at the altitudes exceeding or about 250 km. It is known that auroral electrojets are often considered as possible sources of ionospheric wave disturbances including the traveling ionospheric disturbances. Disturbance transfer from the auroral region to the middle and low latitudes by acoustic-gravity waves of various spatial scales is commonly accepted. In this study, we determine the contribution from magnetogravity waves, which are generated by auroral ∗
[email protected] 1
Nizhny Novgorod State Architectural and Civil-Engineering University; 2 Nizhny Novgorod State Pedagogical University; 3 Radiophysical Research Institute, Nizhny Novgorod, Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 55, No. 6, pp. 421–430, June 2012. Original article submitted June 20, 2011; accepted June 18, 2012. c 2012 Springer Science+Business Media New York 382 0033-8443/12/5506-0382
electrojets, to the ionospheric disturbances. This is performed on the basis of analyzing the data from the vertical sounding of the ionospheric F2 layer and the values of the horizontal components of the geomagnetic field at the ground-based magnetic stations. The experimentally obtained characteristic features of the dynamic spectra of magnetogravity waves are compared with the characteristic frequencies and velocities obtained from the calculated dispersion curves. As a result, we confirm the fact that magnetogravity waves determine a part of the traveling ionospheric disturbances. 2.
THE BASIC SYSTEM OF EQUATIONS DESCRIBING THE PROPERTIES OF MAGNETOGRAVITY WAVES AND THE DERIVATION OF DISPERSION RELATIONS
Analysis of the conditions of propagation of magnetogravity waves in a medium with finite conductivity σ in the absence of regular winds can be performed on the basis of the magnetohydrodynamics equations. In this case, the initial linearized system of equations is written in the form 1 ∂ρ ∂v = −∇p + ρg + (j × H0 ); + ρ0 div v + (v∇) ρ0 = 0; ∂t ∂t c ∂ρ 4π ∂p + (v∇) p0 = Vs2 + (v∇) ρ0 ; j; div h = 0; rot h = ∂t ∂t c 1 1 ∂h ; j = σ E + (v × H0 ) . rot E = − c ∂t c ρ0
(1)
Here, ρ is the medium density, p is the pressure, v is the medium velocity, h is the magnetic-field disturbance, Vs2 is the squared adiabatic speed of sound, γ is the adiabatic constant, g is the gravitational acceleration, and c is the speed of light. The quantities marked by the subscript “0” denote the unperturbed values of the parameters of the medium and the magnetic field H0 . Note that the third equation of system (1) assumes adiabaticity of disturbances, which occurs in the case of their weak dissipation. In the Cartesian coordinate system with the z axis directed vertically upwards (see Fig. 1), the equilibrium pressure and density are related as dp0 /dz = −gρ0 and are first of all determined by the temperature distribution T0 (z). In this case, if we consider the isothermal atmosphere (T0 (z) = const), the pressure and the density exponentially depend on the altitude, so that p0 (z) and ρ0 (z) ∝ exp(−z/H), where H = κT /(mg) is the reduced altitude of homogeneous atmosphere, κ is Boltzmann’s constant and m is the effective mass of molecules. For convenience of calculations, all the variables in the initial system of equations (1) are expressed in terms of the displacement vector ξ (v = ∂ξ/∂t) and the solutions are sought in the form of harmonic waves exp(−iωt + ikr) in the coordinate system shown in Fig. 1. In this case, ∂/∂t → −iω, ∂/∂x → ikx , ∂/∂y → iky , and ∂/∂z → ikz + (2H)−1 . Such a dependence on the vertical Fig. 1. The used coordinate system of and orientacoordinate is typical of the velocity v and the displace- tion of the gravity vector, the magnetic field, and ment vector ξ, while the unperturbed pressure p0 and the the wave vector. density ρ0 vary with altitude as exp[ikz z − z/(2H)]. The latter two conditions are adopted according to the exponential dependence of the solution of the hydrodynamic equations for acoustic-gravity waves on the vertical coordinate z in the absence of a magnetic field, 383
which was obtained in [3]. Note that for a constant-conductivity medium, the dependence of the magnetic perturbation h in a magnetogravity wave on z is the same as for the displacement vector. Finally, under the condition H0 g, for the displacement components we obtain a system of homogeneous equations, the determinant of which yields the dispersion relation 2 ) (kx2 + ky2 )] (ω 2 + Q) ω 2 [ω 2 + Q − (Vs2 + VAM
2 (kx2 + ky2 )] + (γ − 1) g2 (kx2 + ky2 ) = 0, − Vs2 [kz2 + 1/(4H 2 )] [ω 2 + Q − VAM
where VA2 = H02 /(4πρ0 ) is the squared Alfv´en velocity, 2 = VAM
VA2
1+i
c2 [k2 + ky2 + kz2 − 1/(4H 2 ) − ikz /H] 4πσω x
2 [1/(2H) + ik ]2 . is the squared Alfv´en velocity under the condition of finite conductivity, and Q = VAM z 2 The solutions of this equation are the modified Alfv´en waves with the dispersion relation ω + Q = 0 and the magnetogravity waves described by the dispersion relation
(ω 2 + Q) {ω 2 − Vs2 [kz2 + 1/(4H 2 )]} 2 2 ) (kx2 + ky2 ) + Vs2 VAM [kz2 + 1/(4H 2 )](kx2 + ky2 ) + Vs2 ωg2 (kx2 + ky2 ) = 0, (2) − ω 2 (Vs2 + VAM
where ωg2 =
(γ − 1) g γH
2 = k 2 sin2 θ, k = k cos θ, and k is the wave number. is the squared Brunt–V¨ ais¨ al¨ a frequency, kx2 + ky2 = k⊥ z In the special case where the magnetic field is absent (H0 = 0), Eq. (2) for magnetogravity waves goes over to the relation for acoustic-gravity waves in the isothermal atmosphere:
ω 4 − ω 2 Vs2 [kx2 + ky2 + kz2 + 1/(4H 2 )] + Vs2 ωg2 (kx2 + ky2 ) = 0. 3.
ANALYSIS OF DISPERSION PROPERTIES OF MAGNETOGRAVITY WAVES UNDER CONDITIONS OF WEAK ABSORPTION
Under conditions of finite conductivity, the squared modified Alfv´en velocity can be written as 2 = VAM
VA2
=
VA2 ≈ VA2 (1 − iχ). 1 + iχ
c2 [k2 + ky2 + kz2 − 1/(4H 2 ) − ikz /H] 4πσω x In the case of weak damping, the realationship χ 1 should be fulfilled. Using this fact, we estimate the admissible conductivity values: (a) for longitudinal propagation when kz2 1/(4H 2 ), or λz 4πH ≈ 360 km, we can choose λz ∼ 10 km = 106 cm. The requirement χ = c2 kz2 /(4πσω) 1 leads to the condition σ = c2 kz2 /(4πω) 1012 s−1 for magnetogravity waves with a period of about 30 min; 2 1/(4H 2 ), or λ 4πH ≈ 360 km, λ ∼ 100 km = 107 cm (b) for transverse propagation when k⊥ ⊥ ⊥ 2 2 2 /(4πω) 1010 s−1 for magnetogravity waves satisfies this condition. Since χ = c k⊥ /(4πσω) 1 σ = c2 k⊥ with a period of about 30 min. Under the conditions of weak absorption and restriction for the scales (the vertical length of the wave λz is much smaller than 4πH, or kz 1/(2H)), performed for the traveling ionospheric disturbances, 384
1+i
dispersion relation (2) for magnetogravity waves takes the form 2 2 2 + kz2 ) + VL2 Vs2 kz2 (kz2 + k⊥ ) + Vs2 ωg2 k⊥ = 0. ω 4 − ω 2 (Vs2 + VL2 ) (k⊥
(3)
The roots of this equation correspond to the fast (subscript “+”) and slow (subscript “−”) modes of magnetogravity waves, for which the following expressions can approximately be used: 2 2 = (Vs2 + VL2 ) (kz2 + k⊥ ), ω+ 2 ω− =
(4)
2 ) + V 2 ω2 k2 VL2 Vs2 kz2 (kz2 + k⊥ s g ⊥ . 2) (Vs2 + VL2 ) (kz2 + k⊥
(5)
Equation (4) yields the expression for the group velocity of the wave in the horizontal direction: k⊥ Vs2 (ω 2 − ωg2 ) + VL2 (ω 2 − Vs2 kz2 ) dω = 2) . dk⊥ ω 2ω 2 − (Vs2 + VL2 ) (kz2 + k⊥ In particular, the following approximate equality holds for the group velocity of the fast mode: Vs2 + VL2 dω+ ≈ k⊥ . dk⊥ k2 + k2 z
(6)
(7)
⊥
If the weak absorption due to finite conductivity is allowed for, the frequency ω will also have the imaginary part ω = ωr + iδ in addition to the real component. If δ ωr , dispersion relation (3) yields δ≈
2 ) (V 2 k 2 − ω 2 ) VL2 c2 kz2 (kz2 + k⊥ s z r 2 (V 2 + V 2 )] . 8πσωr2 [2ωr2 − VL2 kz2 − k⊥ s L
(8)
If we assume that the frequency ωr is determined by Eq. (4) and VL Vs , while kz k⊥ , we obtain the relation V 2 c2 k4 (9) δ = L 2z , 4πσωr which is equivalent to the condition c2 kz 4πσVL . This condition is a fortiori fulfilled for λz > 105 cm. 4.
NUMERICAL ANALYSIS OF THE DISPERSION FOR MAGNETOGRAVITY WAVES
Calculation of dispersion curves for magnetogravity waves on the basis of the obtained dispersion relation given by Eq. (2) is performed in dimensionless quantities. As in [4], the dimensionless frequency and the wave number are adopted as W = ω/ωc and K = k/kc , respectively, where ωc = H0 g/[Vs2 (4πρ0 )1/2 ] and kc = ωc /Vs . In the dimensionless quantities, the dispersion relation of Eq. (2) is written in the form W 5 + AW 4 + BW 3 + CW 2 + DW + E = 0, where
√
2K cos θ +i A = 3/2 1/2 γ β Rem
2K 2 1 − β Rem Rem
(10)
;
√ 1/2 2 2 γ 3β 2γ γ2 2 2 2 2 2 − 3 K − + 1 K sin θ − − K cos θ + i 1/2 K cos θ; B= 4 γ β γβ 2 β √ 3 5/2 1/2 4 γ β K 2K γ 4β 2K cos θ − √ cos θ − i 2 +i ; C = − 1/2 γ β Rem 32 Rem γ β Rem 4 2 Rem 385
Fig. 2. Dispersion curves for the propagating fast (a and c) and slow (b and d ) modes of the magnetogravity wave in dimensionless variables which were obtained by solving the full dispersion relation in Eq. (3) (a and b) and abridged Eq. (4) (c and d). The solid curve corresponds to the longitudinal propagation (θ = 0◦ ), the dotted curve, to the oblique propagation (θ = 45◦ ), and the dashed curve, to the transverse propagation (θ = 90◦ ). D=
γ2 γ 2K 2 γ 5 β (γ − 1) γβK 2 2K 4 sin2 θ + K 2 − cos2 θ − + sin2 θ γβ 4 4 2 2 √ −i E=
γ 7/2 β 1/2 2γ 1/2 3 3 √ K cos θ; K cos θ − i β 1/2 4 2
(γ − 1) 4 2 (γ − 1) γ 2 β 2 2 (γ − 1) γ 1/2 β 1/2 3 2 √ K sin θ cos θ + i K sin θ − i K sin θ. γ Rem 8 Rem 2 Rem
Here, β = p0 /(H02 /8π) is the ratio of the gas-kinetic pressure to the magnetic pressure, Rem = 4πVL H ×σ/c2 , and θ is the angle between k and g H0 . The dispersion curves were calculated for the ionospheric parameters close to the actual parameters for an altitude of 250 km. Conductivity was assumed equal to σ = 1010 s−1 , the magnetic Reynolds number Rem = 3 · 102 [1], T = 103 K, the adiabatic index γ = 1.4, β = 0.02, H = 3 · 106 cm, and |H0 | = 0.5 G. The values of these parameters agree with the MSIS-E-90 ionospheric model. Figures 2a and 2b show the solutions of Eq. (10), while Figs. 2c and 2d show the dispersion curves for abridged Eq. (3) in dimensionless variables for the propagating fast and slow modes of a magnetogravity wave. Here, the solid curve corresponds to the longitudinal propagation (θ = 0◦ ), the dotted curve, to the oblique propagation (θ = 45◦ ), and the dashed curve, to the transverse propagation (θ = 90◦ ). The lines of the dispersion curves, which are not shown in Fig. 2a, fall outside the considered frequency range. In Fig. 2c, the propagation-angle dependence for the fast mode is absent. 5.
PROCESSING, ANALYSIS, AND COMPARISON OF EXPERIMENTAL DATA WITH THE PERFORMED CALCULATIONS
In this work, it is assumed that magnetogravity waves were excited by an auroral source during the substorms and propagated towards the middle latitudes. The waves were sought along the geomagnetic meridian on the basis of a comparative spectral analysis of oscillations of the AU index [5], variations in the critical frequencies of the ionospheric F2 layer at the stations of ionospheric vertical sounding [6], and variations in the horizontal component of the geomagnetic field in the diurnal intervals of March–April 2006 [7]. The data from the ionospheric stations (with removed diurnal variation) and the magnetic stations with a resolution of 15 min (see Table 1) were used for the analysis. Geographical positions of the stations 386
TABLE 1 Ionospheric and magnetic stations L Aquila Furstenfeldbruck HEL Lvov Juliusruh/Rugen San Vito
Abbreviated notation of stations AQU FUR HLP LVV JR055 VT139
Geogr. latitude, deg 42.38 48.17 54.61 49.90 54.60 40.60
Geogr. longitude, deg 13.32 11.28 18.82 23.75 13.40 17.80
Geomagn. latitude, deg 58.7 64.2 69.5 66.3 69.3 57.0
Geomagn. longitude, deg 1.9 1.6 4.0 4.7 2.2 2.7
L (McIlvain parameter) ) 1.50 1.90 2.50 2.05 2.50 1.47
Fig. 3. Geographic location of the considered ionospheric and magnetic stations. Crosses and black circles show magnetic stations and the ionospheric stations, respectively. The shaded area corresponds to the conditional location of the eastward electrojet (AU). Thin curves are the lines of the geographic latitudes (N) and longitudes and thick curves are the lines of the magnetic latitudes (NM) and longitudes.
and the eastward auroral electrojet are shown in Fig. 3. Figure 4 shows examples of matched dynamic spectra of the AU index, and the ionospheric and magnetic disturbances for April 4 and 5, 2006. Coincidence of the spectral features of dynamic spectra in the interval 16:15–19:00 UT for April 4 and 11:15–15:15 UT for April 5 indicate propagation of the density and magnetic-field disturbances from the auroral source to the middle latitudes. Analysis of the dynamics of these spectra (Fig. 4) shows the actual absence of the time shift among both the density disturbances of the ionospheric F2 layer and the geomagnetic-field disturbances at all considered stations. This is indicative of higher velocities of magnetogravity wave compared with the classical internal gravity waves. Using the known distances from auroral electrojets to the ionospheric and magnetic stations at which the density and magnetic-field disturbances are recorded, we can estimate the propagation velocities of the considered waves. For distances about 2000 km, the time-shift absence which is observed for the dynamic spectra obtained from the data with a 15-min resolution, indicates that the magnetogravity-wave velocities exceed 2000 m/s. Analysis of the dynamic spectra, which are similar to those shown in Fig. 3, allows one to estimate typical recorded frequencies of magnetogravity waves. They fall within the range 10−4 –4 · 10−4 Hz and correspond to the dimensionless frequencies W = 5 · 10−4 –210−3 . This means that it makes sense to pay attention to the dimensionless frequencies below 2 · 10−2 on the plots of the dispersion curves (Fig. 2). 387
Fig. 4. Examples of the dynamic spectra of the AU index and the data from the ionospheric stations JR055 and VT139 and the magnetic stations HLP and AQU (from top to bottom) for (a) April 4, 2006 and (b) April 5, 2006. The vertical curves denote the regions of coincidence of the spectral features for April 4, 16:15–19:00 UT and April 5, 11:15–15:15 UT. According to Fig. 2, in this range, one can observe the fast (second) mode of magnetogravity waves during longitudinal propagation and slow (fourth) mode during propagation at an angle to the magnetic field. In this case, the values of the dimensionless wave vector are in the interval 0–5·10−3 , which corresponds to the wavelengths above 4000 km. Table 2 shows the calculated phase and group veTABLE 2 locities of the fast (“+”) (Vp+ and Vg+ , respectively), and Vp Vg Mode θ, rad slow (“−”) (Vp− and Vg− , respectively) modes of magne8.0 — “+” 0 togravity waves in the considered wavelength range. The 5.5 5.5 “+” π/4 dimensionless phase velocity Vp+ is about 8, while Vp− is — 6.0 “−” 0 about 5.5. The dimensionless group velocity Vg+ is about 5.5 4.5 “−” π/4 5.5, while Vg− is about 6 for longitudinal propagation and about 4.5 for propagation at the angle θ = 45◦ . The actual values of the velocities can be obtained by multiplying the dimensionless values by the sound speed (500–600 m/s). Therefore, the mode velocities are in the interval 3000–4000 m/s. Experimental checking of the determined phase-velocity values can be performed by comparing the dynamic spectra of the horizontal components of the geomagnetic field for the considered stations, which are obtained with a 1-min resolution. With allowance for the distance between the magnetic stations and the velocity estimates, the time shift of the dynamic spectra in the region responsible for the passage of magnetogravity waves should be 5–15 min. Figure 5 shows the dynamic spectra of the AU index and the horizontal components at the HLP and AQU stations. Ovals in each spectrum denote the regions which are assumed to correspond to the southward passage of magnetogravity waves excited by the polar electrojet. 388
Fig. 5. The time shift among the intensity jumps in the dynamic spectra of the AU index, which are related to the magnetogravity-wave passage in the southward direction, and the horizontal components at the AQU, FUR, and LVV stations (April 4, 2006). When comparing such regions in the spectra of the AU index and the above-mentioned stations, one can see the time shift among them. In the considered case, the shift value is 10–15 min, which agrees with the performed analytical estimates. The propagation direction of magnetogravity waves is also of interest. To this end, analysis of the data for the above-mentioned magnetic stations has been augmented by the data for two additional stations, namely, FUR (48◦ N, 11◦ E) and LVV (50◦ N, 24◦ E), which are located to the left and to the right of the assumed wave- propagation direction, respectively (Fig. 3). Comparing the dynamic spectra for all considered stations, we see that the spectrum intensity weakens at the HLP and LVV stations in the region of magnetogravity-wave propagation, whereas the spectrum intensity is significantly higher at the AQU and FUR stations (see Fig. 5). This seems to indicate the directional propagation of magnetogravity waves from the auroral source to the middle latitudes. Indeed, the AQU and FUR magnetic stations are at the same magnetic longitude, while the HLP and LVV stations are at another longitude which is 2◦ to the east (see Table 2). Therefore, in this experiment, the magnetogravity wave predominately propagates along the magnetic longitudes 1.6◦ –1.9◦ .
389
6.
CONCLUSIONS
This study has been undertaken to clarify the contribution from magnetogravity waves generated by auroral electrojets to the mid-latitude ionospheric disturbances. To this end, the dispersion relations have been obtained and dispersion curves for the fast and slow magnetogravity modes have been plotted from the hydrodynamic equations with allowance for the joint influence of the magnetic field and gravity in the case where the ionospheric conductivity is finite. The intervals of the ionospheric-conductivity values for which magnetogravity waves can propagate have been obtained. Comparison of the results of the dynamic spectral analysis of the traveling ionospheric disturbances, obtained using the data of vertical sounding of the ionospheric F2 layer, and the geomagnetic disturbances with typical frequencies and velocities estimated from the calculated dispersion curves shows agreement of the results. The range of the characteristic experimental frequencies is 10−4 –4 · 10−4 Hz for the propagation velocity exceeding 2000 m/s. The values of the analytically obtained phase velocities amount to about 4000 m/s and 3000 m/s for the fast and slow modes, respectively. Analysis of the dynamic spectra of the horizontal components of the geomagnetic field at the stations located at the close geomagnetic meridians demonstrated propagation of magnetogravity waves from the auroral region to the low latitudes within the magnetic-longitude range 1.6◦ –1.9◦ . Therefore, we confirmed that magnetogravity waves stipulate a part of the traveling ionospheric disturbances and can be used for diagnostics of the ionospheric parameters. This work was supported by the Russian Foundation for Basic Research (project Nos. 12–05–00425 and 12–02–31043) and the Ministry of Education and Science within the framework of the program “Development of modern methods for predicting the magnetosphere–ionosphere state to ensure successful communications on the basis of search for fundamental laws of the solar-activity influence.” REFERENCES
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