Coupling of micropulsation wave modes by the Coriolis force and centrifugal force in the Earth\u27s equatorial plane has been considered. The analytical form of electric field is derived for a particular coupled wave mode. To study the variation of electric field with geocentric distance, numerical analyses have been made which are presented graphically.Razmatramo vezanje mikropulsnih valnih modova Coriolisovom i centrifugalnom silom u ekvatorijalnoj ravnini Zemlje. Izveli smo analitički izraz za električno polje posebnih valnih modova. Za proučavanje promjena električnog polja s geocentričnom udaljenošću, načinili smo numeričke račune i ishode prikazali grafički

De, M.

De, S. S.

Dubey, S. K.

English

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ISSN1330–0008CODENFIZAE4COUPLING OF MICROPULSATION WAVE MODES BY THE CORIOLISFORCE AND CENTRIFUGAL FORCE IN THE EARTH’S EQUATORIALPLANES. S. DE, S. K. DUBEY and M. DECentre of Advanced Study in Radio Physics and Electronics, University of Calcutta,Calcutta 700 009, INDIAReceived 5 March 2001; revised manuscript received 10 May 2001Accepted 5 September 2001 Online 25 May 2002Coupling of micropulsation wave modes by the Coriolis force and centrifugal forcein the Earth’s equatorial plane has been considered. The analytical form of electricfield is derived for a particular coupled wave mode. To study the variation of elec-tric field with geocentric distance, numerical analyses have been made which arepresented graphically.PACS numbers: 94.30.Ms, 94.20.Vv UDC 537.87Keywords: Earth’s equatorial plane, ionospheric disturbances, Coriolis force and centrifu-gal force, micropulsations, magnetic pulsations1. IntroductionA remarkable variety of magnetic disturbances are observed as quasi-sinusoidalat the Earth’s surface and on board of satellites by sensitive magnetometers. Theseare geomagnetic micropulsations, various types of which appear in a wide rangefrom audio to very low frequency. Some of them are highly damped.Solar streamers, geomagnetic micropulsations and ionospheric shock waves aregenerally discussed on the basis of the theory of outer atmospheric oscillations. Asthe frequency range corresponding to geomagnetic micropulsation is far below thegyrofrequency of positive ions, the magnetohydrodynamic waves are obtained inthe ionised gas.In connection with the studies of long-period micropulsations, dimensionlessresonant frequencies of hydromagnetic modes have been calculated for a simplemodel plasmasphere including a lower ionosphere [1,2]. It has been concluded thatthe definite form of coupling between adjacent magnetic shells in the Alfvén modedepends on the ratio of the Hall conductivity to the Pedersen conductivity, knownFIZIKA A 11 (2002) 1, 1–8 1de et al.: coupling of micropulsation wave modes by the coriolis force . . .as Hall coupling. It is established that magnetospheric resonance and hydromag-netic (HM) waves are a cause of certain types of geomagnetic micropulsations. Thedamping of magnetospheric motions and HM wave propagations in the ionosphereare very important in the study of ionospheric roles during both quiet and disturbedperiods. There are various damping mechanisms which may occur for geomagneticpulsations. Hughes and Southwood [3] showed that such mechanism could be thecoupling between adjacent flux tubes in the magnetosphere due to a compressionof the magnetic field lines or due to mode coupling via the Hall conductance in theionosphere. There could also be energy loss from the hydromagnetic wave modesto the plasma populations in the magnetosphere.The investigation of magnetic pulsation is used as the indirect diagnostics ofmagnetospheric and solar wind conditions. Development of such an ability leadsto a better physical understanding of the processes involved in the solar-wind–magnetosphere interaction. However, magnetic pulsations are attenuated and mod-ified as they propagate through the ionosphere.It is known that certain geomagnetic micropulsations could be hydromagneticresonances [4,5]. Near the Earth, in the equatorial plane, Coriolis force and cen-trifugal force have important effects in the process of coupling of micropulsationwave modes.Micropulsation activity obtained in the equatorial plane of the Earth is verymuch dependent on the time in the solar cycle and seasonal effect. These are alsocontrolled by overall level of magnetic activity and controlled by particles and elec-tromagnetic radiations emitted by the Sun. Micropulsation modes are known to berelated to geomagnetic storm activity and have been connected with storm suddencommencement (SSC), storm sudden impulses (SSI) and low-frequency oscillations(Bays).Various results of micropulsation observations in different period ranges havebeen summarised through statistically analysed data of different latitudes. The am-plitude and periods of micropulsation activities from magnetograms are analysed,power spectra of these occurrences have been computed by means of rubidium-vapour magnetometer measuring the total field [6]. Statistically stable averagespectra are observed from data collected from magnetograms and also throughdifferent geostationary satellites.Hydromagnetic waves in the period range 150 to 600 s (PC5) may be producedby the interaction of the solar wind with the magnetospheric volume. Particularly,near the dawn and dusk meridians, the turbulence-modified solar wind at the mag-netopause produces tangential stress which can give rise to waves generated by theKelvin-Helmholtz instability. An increase in the solar wind pressure causes com-pression of the magnetosphere, observed as a sudden commencement (SC) on amagnetogram, and as a result, certain parts of the magnetosphere oscillate due tothe impulse.Some workers [7–10] have noted that amplitudes of waves in the period range40 to 150 s (Pi2) are maximal in the auroral zone, although some of the laterdata indicate a sub-auroral maximum amplitude [11]. Apart from times of very low2 FIZIKA A 11 (2002) 1, 1–8de et al.: coupling of micropulsation wave modes by the coriolis force . . .magnetic activity (Kp = 0), there are some records which show that the waveformsand power spectra of Pi2, observed in the auroral zone, are different from thoseobserved simultaneously at mid and low latitudes [12]. Supporting evidence for thislatitude dependence is known [11]. Usually a predominant period can be found inthe wave-trains observed at mid and low latitudes, while the waveforms of auroralzone Pi2 are irregular. It has been concluded that power spectra are of a random-noise type [12].In the auroral regions, charged particles are accelerated to very high energiesby electric fields parallel to the magnetic field. The effect of parallel electric fieldhas a stabilising effect on the electrostatic ion cyclotron instability and reduces theenergy of resonant particles [13].It is found that the gradual growth of wave amplitude with time initiates aconvective plasma instability within the magnetosphere rather than an externallyapplied impulse. This event was detected as an almost pure transverse wave atcollege, Alaska, which is quite close to the geomagnetic field line that links ATS 1satellite to the Earth’s surface [14].Experimentally, many micropulsation studies have suggested that the magne-tospheric system has certain resonances through the frequency of occurrence [15].Under resonant condition, during wave-particle interaction, particles can lose en-ergy to the wave so that it is amplified. Particles can also gain energy from thewave giving thereby Landau damping. Also amplification occurs within the plas-masphere [16]. The electric field plays an important role in the dynamics of theplasma in the plasmasphere as well as in the magnetosphere and in the solar wind.In this paper, some theoretical investigations are carried out in the space nearthe Earth in the equatorial plane to determine the electric field and the influence ofCoriolis force and centrifugal force in the process of coupling of micropulsation wavemodes. The expression of electric field in the equatorial plane has been deduced for aspecific wave mode, considering the influence of Coriolis force and centrifugal force.Numerical analyses are made for the coupled wave mode to study the variation ofelectric field with geocentric distance.2. Mathematical formulationThe equations giving the dynamics are chosen as∇× ~E = −1c∂ ~H∂t, (1)∇× ~H = −1c∂ ~E∂t+4πec(ni~v(1) − ni~v(2)) , (2)∇ · ~E = 4πe (ni − ne) , (3)∇ · ~H = 0 . (4)FIZIKA A 11 (2002) 1, 1–8 3de et al.: coupling of micropulsation wave modes by the coriolis force . . .The equations of motion can be written asm(1)[∂~v(1)∂t+ 2~ω0 × ~v(1) + ~ω0 × (~ω0 × ~r)]= e[ ~E + ~v(1) × ~B0] , (5)m(2)[∂~v(2)∂t+ 2~ω0 × ~v(2) + ~ω0 × (~ω0 × ~r) + (~v(2) · ∇)~v(2)]= −e[ε ~E + ~v(2) × ~B0] ,(6)where the angular velocity of the Earth ~ω0 has components (0, 0, ω0), ~B0 is themagnetic field with components (0, 0, B0), ~r is the radius vector from the centreof the Earth, ~E the induced electric field, ~H the induced magnetizing field, e theelectronic charge, c the velocity of light in free space, ε the electrical permittivityof the medium, m(1) the mass of the ion, m(2) the mass of the electron, ni the ionnumber density, ne the electron number density, ~v( 1) the velocity of the ion, and~v( 2) is velocity of the electron. Here, superscript 1, and superscript 2 are for theions and electrons, respectively. One can express~v(s)⊥ = ~v(s)x + j~v(s)y ~v(s)‖ = ~v(s)z ê3, (s = 1, 2)where, ê3 = ~B0/| ~B0|. The suffixes ⊥ and ‖ are used to denote vectors perpendicularand parallel to ~B0, respectively. For any vector ~ρ which is perpendicular to ~B0,ê3 × ~ρ = j~ρ . (7)The electric current can be written as~J = e(ni~v(1) − ni~v(2)) . (8)Assuming time dependence of the form eiωt, one obtains∇× (∇× ~E) = ω2c2~E − 4πiωc2~J , (9)~v(1)‖ = −iω(1)m(e ~E‖ + m(1)ω20~r‖) , (10)~v(2)‖ =iω(2)m(eε ~E‖ − m(2)ω20~r‖) , (11)~v(1)⊥ =1m(1)e ~E⊥ + m(1)ω20~r‖iω + j(2ω0 + ω1), (12)~v(2)⊥ =1m(2)−eε ~E⊥ + m(2)ω20~r‖iω + j(2ω0 + ω1), (13)4 FIZIKA A 11 (2002) 1, 1–8de et al.: coupling of micropulsation wave modes by the coriolis force . . .~J‖ = − ieωB0(niω1 + εneω2) ~E‖ − ieω20ω(ni − ne)~r‖ , (14)~J⊥ =[iωω1+ j(2ω0ω1− ω2ω21)] [eB0(ni + εne) ~E⊥ +ω20B0(nim(1) − nem(2))~r‖],(15)where ω1 = eB0/m(1) and ω2 = eB0/m(2). Writing( ~E)⊥ = Ex + jEy and ~E‖ = Ez ê , (16)one obtains[∇× (∇× ~E)]⊥ = [αEx + iβEy + γz] + j[αEy − iβEx + izδ] , (17)where[∇× (∇× ~E)]‖ =[εω2c2− 4πec2B0(niω1 + εneω2)]Ez − 4πeiω20c2(ni − ne)z , (18)withα =ω2c2[1 +4πec2B0(ni + εne)],β =ω2c2[4πeB0ωω1(2ω0 − ω2ω1)(ni + εne)],γ =ω2c2[4πω20B0ω1(nim(1) − nem(2))],δ =ω2c2[4πiω20B0ω1(2ω0 − ω2ω21)(nim(1) − nem(2))].For a plane wave, field components depend only on the coordinate z. Hencefrom (17), one can derive the following two equationsd2Exdz2+ αEx + iβEy = γz , (19)d2Eydz2+ αEy − iβEx = izδ . (20)From (18), the solution for the z-component of the electric field can be deduced asEz =4πeω20(ni − ne)B0zεω2B0 − 4πe(niω1 + εneω2) . (21)FIZIKA A 11 (2002) 1, 1–8 5de et al.: coupling of micropulsation wave modes by the coriolis force . . .Assuming furtherF+ = Ex + iEy, F− = Ex − iEy , (22)one can getdF+dz2+ (α + β)F+ = −(γ − δ)z , (23)dF−dz2+ (α − β)F+ = −(γ + δ)z . (24)The dispersion relation has been derived asK2 =ω2c2+4πeω2c2B0ω1(ni + εne) ± 4πeωc2B0ω1(2ω0 − ω2ω1)(ni + εne) = α ± β . (25)The nature of the dispersion can be known from the roots of (25). The possiblecombinations of K would be (K1,K2), (K1,K4), (K2,K3) and (K3,K4), whereK1 = +√α + β, K2 = +√α − β, K3 = −√α + β and K4 = −√α − β .When β = 0, one obtains K1 = K2, K3 = K4 and ω2 = ω0ω1, and there willbe no coupling of different micropulsation wave modes.For the (K1,K2) mode of the dispersion relation, the field components areobtained asEx =z2β[{(cos K1z + cos K2z)(γ cos K1z + δ cos K2z)−(sinK1z − sinK2z)(γ sinK1z − δ sinK2z) − pβ}2 (26)+{(cos K1z − cos K2z)(γ cos K1z − δ cos K2z)+(sinK1z + sin K2z)(γ sin K1z + δ sin K2z)}]1/2,Ey =z2β[{(cos K1z + cos K2z)(−γ sinK1z + δ sin K2z)+(sinK1z − sinK2z)(γ cos K1z + δ cos K2z)}(27)+{(cos K1z − cos K2z)(γ sin K1z + δ sin K2z)−(sinK1z + sin K2z)(γ cos K1z − δ cos K2z) − qβ}2]1/2,Ez =z2β8πeβω20(ni − ne)B0εω2B0 − 4πe(niω1 − εneω2) , (28)wherep =2(αγ + βδ)α2 − β2 , q =2(βγ + αδ)α2 − β2 .6 FIZIKA A 11 (2002) 1, 1–8de et al.: coupling of micropulsation wave modes by the coriolis force . . .The resultant field can been derived from the above field components as|E| = [E2x + E2y + E2z ]1/2 . (29)Similarly, for (K1,K4), (K2,K3) and (K3,K4), the resultant fields can be cal-culated.3. Results and discussionPower spectra derived from the micropulsation activity are complex. They re-veal certain peaks which are localised in space. This suggests that the energy haspropagated in a mode which is guided. Other spectral peaks have a maximum am-plitude at a particular latitude which can be observed over a wide range of latitudeand longitude, which suggests that the propagation has been in a mode which ismore nearly isotropic.Fig. 1. Variation of the horizontal electric field with geocentric distance for differentvalues of E‖.In the equatorial plane, the influence of the Coriolis force and centrifugal forcein the coupling of micropulsation wave modes can be obtained from the study ofthe dispersion relation (25). The electric field for the (K1,K2) mode at differentgeocentric distances has been evaluated from (29). The results are shown in Fig. 1.Numerical analyses reveal that E‖ is very small in comparison to E⊥. It is alsofound that for micropulsations in the range of the period 21 to 42 seconds, theseFIZIKA A 11 (2002) 1, 1–8 7de et al.: coupling of micropulsation wave modes by the coriolis force . . .forces are important within a distance 2.05 a (approximately) from the Earth’scentre. Here a is the radius of the Earth. The results of analysis reveal that forcertain periods and magnetic field strengths, coupling of different micropulsationwave modes will be absent.References[1] W. Allan and F. B. Knox, Planet. Space Sci. 27 (1979) 79.[2] A. H. Craven and J. A. Lawrie, Planet. Space Sci. 27 (1979) 211.[3] W. J. Hughes and D. J. Southwood, J. Geophys. Res. 81 (1976) 3234.[4] J. W. Dungey, Ionos. Res. Lab. Pennsylvania State Univ. Sci. Rep. No. 69 (1954).[5] T. Tamao, Rep. Ionos. Space Res. Japan 18 (1964) 16.[6] T. J. Herron, J. Geophys. Res. 72 (1967) 759.[7] J. A. Jacobs and K. Sinno, J. Geophys. Res. 65 (1960a) 107.[8] J. A. Jacobs and K. Sinno, Geophys. J. 3 (1960b) 333.[9] O. M. Raspopov, Geomag. and Aeron. 8 (1968) 257.[10] L. T. Afanasyeva, O. M. Raspopov, R. V. Shchepetnov, V. K. Koshelevskiy and M. D.Nazarov, Geomag. and Aeron. 10 (1970) 600.[11] J. C. Gupta and R. J. Stenning, Can. J. Phys. 49 (1971) 2338.[12] H. Fukunishi and T. Hirasawa, Rep. Ionos. Space Res. Japan (1970) 24.[13] V. K. Bajaj and M. S. Tiwari, Indian J. of Radio and Space Phys. 21 (1992) 53.[14] L. J. Lanzerotti and N. A. Tartaglia, J. Geophys. Res. 77 (1972) 1934.[15] T. Saito, Sci. Rep. Tohoku Univ., fifth series, Geophysics. 14 (1962) 81.[16] R. Gendrin, Space Sci. Rev. 11 (1970) 54.VEZANJE MIKROPULSNIH VALNIH MODOVA CORIOLISOVOM ICENTRIFUGALNOM SILOM U EKVATORIJALNOJ RAVNINI ZEMLJERazmatramo vezanje mikropulsnih valnih modova Coriolisovom i centrifugalnomsilom u ekvatorijalnoj ravnini Zemlje. Izveli smo analitički izraz za električno poljeposebnih valnih modova. Za proučavanje promjena električnog polja s geocen-tričnom udaljenošću, načinili smo numeričke račune i ishode prikazali grafički.8 FIZIKA A 11 (2002) 1, 1–8

Vezanje mikropulsnih valnih modova coriolisovom i centrifugalnom silom u ekvatorijalnoj ravnini zemlje

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