Khodachenko, M. L.

Langmayr, D.

Rucker, H. O.

Elektronisches Publikationsportal der Ãsterreichischen Akademie der Wissenschaften

On the Structure of Electromagnetic Field Generated by a Moving External Current Source in a Magnetized Plasma

Elektronisches Publikationsportal der Österreichischen Akademie der Wissenschaften

ON THE STRUCTURE OF ELECTROMAGNETICFIELD GENERATED BY MOVING EXTERNALCURRENT SOURCE IN A MAGNETIZEDPLASMAM. L. Khodachenko∗†, D. Langmayr∗‡, and H. O. Rucker∗‡AbstractWe present the first step of our theoretical analysis of the electromagnetic struc-ture excited by a conductor, moving with the velocity V0 in a hot magnetized plasma.The case of slow motion (Vi << V0 << Ve) is considered within the frame of plasmakinetics. Of the primary interest for us are the fields, which are steady–state in thereference frame co–moving with the conductor. These fields are mainly due to thelow–frequency quasi–stationary inductive electromagnetic mode. A characteristicfrequency of the studied electromagnetic fields is ω = k ·V0. This means that theplasma dielectric properties appear to be in a strong dependence of the orientationof the wave vector k with respect to the velocity vector of the conductor, V0. Asa first step of our investigation of the spatial structure of the electromagnetic envi-ronment created by a slowly moving conducting body in a magnetized plasma wereplace the conductor by an elementary external non–self–consistent current source(thin line current). In this work we describe the mathematical methods, physicalassumptions, and obtained solutions for this particular case. The presented problemhas many applications for antenna and probe operations in plasmas as well as inspace science, e. g., tethered satellites systems and the Io–Jupiter electromagneticinteraction. For our particular calculations in this paper we use as input parame-ters the specific circumstances in the Io torus. We demonstrate that the generatedelectromagnetic fields show a stretched configuration along the line of motion of theconductor, and have an oscillatory behavior with decay along the external magneticfield. Collision–less energy dissipation rates for the external line current moving ina magnetized plasma are calculated. They demonstrate a resonant behaviour withthe maximum value reached for V0 ∼ 0.7 Ve.∗Space Research Institute, Austrian Academy of Sciences, Schmiedlstrasse 6, A–8042 Graz, Austria†on leave from Institute of Applied Physics, Russian Academy of Sciences, Ulyanova str. 46,Nizhny Novgorod, Russia‡Institute for Geophysics, Astrophysics, and Meteorology, University of Graz, Universita¨tsplatz 5,A–8010 Graz, Austria12 M. L. Khodachenko et al.1 IntroductionThis paper is devoted to the theoretical analysis of the electromagnetic interaction betweenan external current source moving with the velocity V0 and a hot magnetized plasma.This problem is usually considered within the frame of MHD–theory for applications inlaboratory experiments, antennas, as well as in space physics [McKenzie, 1991]. Ourinterest in this rather general problem is inspired by some questions of Io–Jupiter physics.Indeed, the problem of the electromagnetic interaction between the satellite Io and thesurrounding Jovian magnetospheric plasma can be formulated as a self–consistent problemof the interaction between a conducting body and a magnetized plasma in relative motion.The key point is that the specific plasma properties in the Io torus cause the situationthat the collisions of plasma particles are not important up to several tens of Io radii, sayup to (60− 80) RIo. Therefore, MHD approximation is inapplicable and the problem hasto be treated within the frame of kinetic theory. Considering the Io–Jovian application ofthe stated problem implies that we appear to be in the situation of relatively slow motionof the conductor with respect to the Alfve´n velocity, VA, and thermal electron velocity, Ve,whereas the thermal ion velocity, Vi, remains to be less than the velocity of the conductor(Ve À V0 À Vi).The slow motion of an external current source means that in this situation the effectsof plasma spatial dispersion, connected with the particle thermal motion, together withthe anisotropy of the plasma dielectric properties due to the external magnetic field leadto a modification of the Alfve´nic and magnetoacoustic wing–like structure of the gener-ated fields near the moving current source [McKenzie, 1991], and produce some specificsteady–state electromagnetic environment, formed by the non–propagating inductive elec-tromagnetic fields [Gubchenko, 1989; Khodachenko and Gubchenko, 1990, 1997; Barnettand Olbert, 1986] co–moving with the source. These fields decay in space due to a colli-sionless energy dissipation and can be considered as a kind of a local magnetosphere of themoving current source [Barnett and Olbert, 1986; Khodachenko and Gubchenko, 1997].Along with the influence the energy losses of the moving external current [Gubchenko,1989; Khodachenko and Gubchenko, 1990; Khodachenko et al., 1998], inductive fields canbe responsible for the appearance of electromagnetic structures near the source, whereresonant charged particles will be effectively accelerated, and in their turn would manifestin the radiation features of the source.The task of obtaining a self–consistent solution describing the structure of the electromag-netic fields and plasma in the vicinity of the conductor can be formulated in the form ofan integral equation [Barnett and Olbert, 1986; Khodachenko et al., 1998]. The numericalanalysis of this equation requires information about the plasma dielectric properties interms of the dielectric tensor. The consideration of the non self–consistent problem, i.e.,the investigation of the excited electromagnetic fields due to a given external elementarycurrent source, provides solutions serving as tests for the general problem. The main aimof the present paper is to calculate the spatial structure of the excited electromagneticenvironment assuming the external current source to be an infinite thin line current.We are interested in the fields which are comoving with the conducting body. The charac-teristic frequency of such fields being steady–state in the reference frame of the conductorOn the structure of electromagnetic fields 3is ω = k · V0. Thus, the frequency range of the generated fields is determined by thedirection of the wave vector k with respect to V0, as well as by their total strengths[Khodachenko et al., 1998].2 Basic equationsIn the linear case, the electric field generated by any external current in a plasma isdetermined by the following wave equationΛˆE˜(k, ω) =4piωic2j˜s(k, ω). (1)Here E˜(k, ω) and j˜s(k, ω)E˜(k, ω) =1(2pi)3∫dt∫d3reiωt−ikrE(r, t), (2)j˜s(k, ω) =1(2pi)3∫dt∫d3reiωt−ikrjs(r, t), (3)correspond to the Fourier images of the electric field E(r, t) and the current source js(r, t),respectively. The tensor Λˆ(k, ω) is connected with the dielectric tensor εij(k, ω), andcharacterizes the plasma dispersion propertiesΛˆ = k2δij − kikj −ω2c2εij(k, ω). (4)The corresponding magnetic field can be calculated via the electric fieldB˜(k, ω) =cω[k× E˜(k, ω)]. (5)The specific circumstances of the Io–Jupiter interaction allow us to introduce some basicassumptions, i.e., the direction of the conductor (external current source) motion is per-pendicular to the external magnetic field. Thus, we assume V0 to be directed along thex–axis and the external magnetic field B0 along z. As a next step, we take the externalthin line current source only having an y–component non–zero. The Fourier image of sucha current source is of the following form [Khodachenko et al., 1998]j˜sy(kx, ky, ω) = −I0(2pi)2δ(ω − kxV0)δ(ky). (6)Due to the two δ–functions, the two integrations over ω and ky can be trivially doneand we obtain the expressions, which describe the respective components of the comovingelectromagnetic fieldsEx(x, z) =16piV0I0c2(2pi)2∞∫0∞∫0sin(kxx) sin(kzz)k2xkzf1(kx, kz) dkxdkz4 M. L. Khodachenko et al.−∞∫0∞∫0cos(kxx) sin(kzz)k2xkzf2(kx, kz) dkxdkz , (7)Ey(x, z) =16piV0I0c2(2pi)2∞∫0∞∫0sin(kxx) cos(kzz) · kxf3(kx, kz) dkxdkz+∞∫0∞∫0cos(kxx) cos(kzz)kxf4(kx, kz) dkxdkz , (8)Ez(x, z) = −16piV0I0c2(2pi)2∞∫0∞∫0sin(kxx) cos(kzz)kxλxxf2(kx, kz) dkxdkz+∞∫0∞∫0cos(kxx) cos(kzz)kxλxxf1(kx, kz) dkxdkz , (9)Bx(x, z) = −16piI0c(2pi)2∞∫0∞∫0cos(kxx) sin(kzz)kzf3(kx, kz)dkxdkz−∞∫0∞∫0sin(kxx) sin(kzz)kzf4(kx, kz) dkxdkz , (10)By(x, z) =16piI0c(2pi)2∞∫0∞∫0sin(kxx) cos(kzz)[kxλxx − kxk2z ]f2(kx, kz) dkxdkz+∞∫0∞∫0cos(kxx) cos(kzz)[kxλxx − kxk2z ]f1(kx, kz) dkxdkz , (11)Bz(x, z) =16piI0c(2pi)2∞∫0∞∫0sin(kxx) cos(kzz)kxf3(kx, kz) dkxdkz (12)+∞∫0∞∫0cos(kxx) cos(kzz)kxf4(kx, kz) dkxdkz , (13)wheref1(kx, kz) =Im[λzy]Re[detλˆ]− Re[λzy]Im[detλˆ]Re2[detλˆ] + Im2[detλˆ], (14)f2(kx, kz) =Re[λzy]Re[detλˆ] + Im[λzy]Im[detλˆ]Re2[detλˆ] + Im2[detλˆ], (15)f3(kx, kz) =(λxxRe[λzz]− k2xk2z)Re[detλˆ] + λxxIm[λzz]Im[detλˆ)Re2[detλˆ] + Im2[detλˆ], (16)f4(kx, kz) =λxxIm[λzz]Re[detλˆ]− (λxxRe[λzz]− k2xk2z)Im[detλˆ]Re2[detλˆ] + Im2[detλˆ]. (17)We note that while integrating in the (kx, kz)–space one should take into account differentasymptotic expressions for λij(ω = kxV0, ky = 0, kz) in dependence on the position in the(kx, kz)–plane. We use a semi–analytical method to solve these integrals, thereby applyingOn the structure of electromagnetic fields 5the concept of asymptotic expansion of the relevant functions. This method allows us toobtain analytic expressions for the spatial behaviour of the electric and magnetic fieldscomponents far from the source, yielding a power–law decay along the line of motionof the conductor and an oscillatory behavior along the z–direction. Figure 1 shows theobtained solution for the x–component of the magnetic field Bx(x, z) and the total valueof the electric field E2(x, z) = E2x + E2y + Ez2. We note that these plots only correspondto the numerical value of the integrals (7)–(12) as a function of the spatial coordinates incm. Taking into account that we take the source current I0 equal to the current in theIo flux tube, the constant factors are A1 = 120.9 statvolt/cm for the electric field andA2 = 5.1 · 105 gauss for the magnetic field.-20000-40000-60000-80000-100000x-500000-2500000250000500000z-0.00002-0.0000100.000010.00002BxA2-100000-80000-60000-40000-20000x-100000-50000050000100000z00.000050.00010.000150.0002EA1Figure 1: Plot of the x–component of the magnetic field (left) and the total amount of theelectric field (right). Along the z–direction, i.e., the direction of the external magnetic field, weobtain a decaying oscillatory behaviour with a gradual damping.The figures show a power law decay of the fields along the direction of the current sourcemotion, i.e., along the x-direction. The electromagnetic environment just behind thesource shows a stretched structure co-moving with the body.3 Energy dissipationIn the previous section we obtained solutions for the electric and magnetic fields structureshowing a stretched configuration. These fields decay in space due to a collisionless energydissipation. This process is analogous to the anomalous skin–effect in a collisionlessisotropic plasma. The important point is that this is a pure kinetic effect and it isimpossible to detect it within the frame of MHD. The external current source’s energydissipated via this collisionless mechanism is equal to the work produced by the electricfield E applied to the current source , js, taken with another sign. Thus, in our case wehave the following expression for the energy dissipation rate calculated per unit length6 M. L. Khodachenko et al.along the y–axisPunit =16piI20V0c2(2pi)2∞∫0∞∫0kx (f4(kx, kz))ky=0; ω=V0kx dkxdkz. (18)The behaviour of this energy dissipation rate as a function of the current source velocityis shown in Figure 2.0 10 20 30 40V0VIo01234PunitPunit,IoFigure 2: Energy dissipation rate as a function of the current source velocity. This rate isnormalized to the energy dissipation rate when the conductor moves with the Io’s velocity(Punit,Io = 3.8 · 106 erg/s), whereas the velocity is normalized to the velocity of Io (VIo =5.2 · 106 cm/s).In our 2D problem we deal with the infinite in y–direction current sources, then the totalenergy dissipated is infinite, too. Since, production of such an infinite current sourcerequires an infinite energy source. All these difficulties are just a result of a theoreticallyidealized infinite current source structure considered here, which in fact can’t be createdin reality. At the same time, formula (18) well demonstrates the role and importanceof the collisionless energy dissipation effects in energetics of a moving external currentsource and can be used in a qualitative analysis.4 ConclusionWe consider the Io–Jupiter interaction as one of possible applications of the general plasmakinetics problem of a moving conducting body in a collisionless plasma. This model al-lows to develop methods for the interpretation of some aspects of the Jovian observationaldata, in particular, for some features of the Jovian decametric radio emission and mea-surements of the fields. The specific circumstances in the Jovian system oblige us toconsider particular case of the model with a slowly moving conducting body, taking intoaccount the effects of plasma particles thermal motion (Ve À V0 À Vi). In this veryOn the structure of electromagnetic fields 7case the low–frequency inductive electromagnetic fields will be effectively generated inthe vicinity of the conductor. To emphasize the importance of these low-frequency induc-tive electromagnetic fields, which form some kind of a local magnetosphere around themoving conductor, and influence its energetic and radiative features, is the main goal ofthe present paper. We have shown that taking into account the particles thermal motioncauses additional mechanism of energy dissipation with regard to the radiative energyloss mechanism. In spite of the non–self–consistent formulation of the problem of a givenexternal current moving in a magnetized plasma, the results obtained here, can serve fortesting the self–consistent solutions in the limiting cases. Besides, these elementary casesare also of interest for a spacecraft or antenna motion in the magnetized plasma of theionosphere and magnetosphere.Acknowledgements: This work has been supported by the Committee for International Co-operation of the Austrian Academy of Sciences, by the INTAS–ESA project 99–01277, andby the Austrian “Fonds zur Fo¨rderung der wissenschaftlichen Forschung” under projectP12761–TPH. M. L. Khodachenko is grateful also to the Max–Planck–Gesellschaft for theresearch fellowship supporting his work during the years 1999–2001.ReferencesBarnett, A., and S. Olbert, Radiation of plasma waves by a conducting body movingthrough a magnetized plasma, J. Geophys. Res., 91, 10117–10135, 1986.Gubchenko, V. M., Generation of quasi–steady electromagnetic fields by a distributedmagnetic dipole moving in a collisionless plasma, Sov. J. Phys., 14, 10, 1989.Khodachenko, M. L., and V. M. Gubchenko, Generation of inductive electromagneticfields by an oblique magnetic rotator in a hot collisionless plasma, Proceedingsof the Workshop on Plasma Astrophysics, Telavi, Georgia, USSR, (ESA SP–311.-Noordwijk: ESA Publication Division, 163–168, 1990.Khodachenko, M. L., and V. M. Gubchenko, Slow magnetic rotator in a collisionlessplasma. Towards the theory of magnetospheres, in Planetary Radio Emissions IV,edited by H. O. Rucker, S. J. Bauer, and A. Lecacheux, Austrian Academy ofSciences Press, Vienna, 183–193, 1997.Khodachenko, M. L., V. M. Gubchenko, and H. O. Rucker, On the electromagnetic fieldsgenerated by a slowly moving conducting body in a magnetized plasma. Possibleapplications for the Io–Jovian system, spacecraft, and plasma probes, Radiophysicsand Quantum Electronics, 41, 10, 1998.McKenzie, J. F., Stationary MHD waves generated by a source in a moving plasma, J.Geophys. Res., 96, 9491, 1991.

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On the Structure of Electromagnetic Field Generated by a Moving External Current Source in a Magnetized Plasma

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Elektronisches Publikationsportal der Ãsterreichischen Akademie der Wissenschaften

Elektronisches Publikationsportal der Österreichischen Akademie der Wissenschaften

On the Structure of Electromagnetic Field Generated by a Moving External Current Source in a Magnetized Plasma

Abstract

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Full text

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Elektronisches Publikationsportal der Ãsterreichischen Akademie der Wissenschaften

Elektronisches Publikationsportal der Österreichischen Akademie der Wissenschaften

Elektronisches Publikationsportal der Ãsterreichischen Akademie der Wissenschaften