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Basic Physics of Magnetoplasmas-I: Single Particle Drift Montions

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Basic Physics of Magnetoplasmas-I: Single Particle Drift Montions 310/1749-5 ICTP-COST-CAWSES-INAF-INFN International Advanced School on Space Weather 2-19 May 2006 _____________________________________________________________________ Basic Physics of Magnetoplasmas-I: Single Particle Drift Montions Vladimir CADEZ Astronomical Observatory Volgina 7 11160 Belgrade SERBIA AND MONTENEGRO ___________________________________________________________________________ These lecture notes are intended only for distribution to participants LECTURE 1 SINGLE PARTICLE DRIFT MOTIONS Vladimir M. Cade·z· Astronomical Observatory Belgrade Volgina 7, 11160 Belgrade, Serbia&Montenegro Email: [email protected] Gaseous plasma is a mixture of moving particles of di®erent species ® having mass m® and charge q®. UsuallyP but not necessarily always, such a plasma is globally electro-neutral, i.e. ® q® = 0. In astrophysical plasmas, we often have mixtures of two species ® = e; p or electron-proton plasmas as protons are the ionized atoms of Hydrogen, the most abundant element in the universe. Another plasma constituent of astrophysical signi¯cance are dust particles of various sizes and charges. For example, the electron-dust and electron-proton-dust plasmas (® = e; d and ® = e; p; d resp.) are now frequently studied in scienti¯c literature. Some astrophysical con¯gura- tions allow for more exotic mixtures like electron-positron plasmas (® = e¡; e+) which are steadily gaining interest among theoretical astrophysicists. In what follows, we shall primarily deal with the electron-proton, electro- neutral plasmas in magnetic ¯eld con¯gurations typical of many solar-terrestrial phenomena. To understand the physics of plasma processes in detail it is necessary to apply complex mathematical treatments of kinetic theory of ionized gases. For practical reasons, numerous approximations are introduced to the full kinetic approach which results in simpli¯ed and more applicable plasma theories. This lecture will show what can be learned about plasma dynamics by looking at motions of individual particles in external or prede¯ned magnetic and electric ¯elds assuming that the induced electromagnetic ¯elds, produced by such mov- ing charged plasma particles, are negligible in comparison with ¯elds externally prescribed. Such an approximation of una®ected external ¯elds signi¯cantly simpli¯es the analysis of particle dynamics and can be applied in many low plasma density con¯gurations including those existing in the solar corona and in planetary magnetospheres and ionospheres. Let us now consider a series of examples of di®erent external magnetic and electric ¯eld con¯gurations. 1 1.1 Case with E=const~ and B~ = 0 The equation of motion for a particle with mass m and charge q is d~v m = qE~ (1) dt whose solution for the particle velocity q ~v = (t ¡ t )E~ + ~v m 0 0 represents a uniformly accelerated particle motion along a constant E~ -¯eld. In this example, particles with charges of di®erent sign move in opposite directions which results in electric currents: ~j = qe~ve + qp~vp . 1.2 Case with B=const~ and E~ = 0 Motion of a charged particle in a constant magnetic ¯eld is described by d~v m = q~v £ B~ (2) dt which immediately yields d~v ~v ¢ = 0 ) j~vj = v = const (3) dt i.e. constancy of the velocity vector intensity. This further tells us that a charged particle does not gain any kinetic energy form the considered magnetic ¯eld. Decomposing the velocity vector into two components ~v = ~vk + ~v? in directions parallel and normal to B~ , the equation of motion (2) reduces to two equations for each of the velocity components: d~v d~v m ? = q~v £ B~ and k = 0: (4) dt ? dt ~ We see that the velocity component ~vk along B remains constant in this case, i.e. una®ected by the presence of magnetic ¯eld and its magnitude vk is simply prescribed as the initial condition. Now, if vk=const the same must also be true 2 2 2 for the magnitude of the normal component v? since v ´ vk + v? = const as already obtained in Eq (3). Thus: q 2 2 v?; vk; v ´ v?; vk = const (5) 2 To ¯nd the shape of the trajectory the particle moves along, we take the ~ Cartesian geometry with B = (0; 0;B), ~v? = (vx; vy; 0) and ~vk = (0; 0; vz) and write Eq (4) as: dv dv dv m x = qBv ; m y = ¡qBv ; m z = 0: dt y dt x dt After some elementary rearrangements, we obtain the following set of equations: d2v x = ¡!2 v ; v2 + v2 = v2 ; v = v : (6) dt2 L x x y ? z k where vk and v? are constants given as initial conditions. Three velocity components follow from Eq (6) as: dx v ´ = v cos(! t) x dt ? L dy v ´ = ¡v sin(! t) (7) y dt ? L dz v ´ = v z dt k where: qB ! ´ (8) L m is known as Larmor frequency (also the gyro or cyclotron frequency). Eqs (9) ¯nally yield the particle trajectory equation after one time-integration: x ¡ x0 = rL sin(!Lt) y ¡ y0 = rL cos(!Lt) (9) z ¡ z0 = vkt; with v? rL ´ (10) !L known as the Larmor radius(also gyro-radius). The trajectory (9) is a helix along the z-axis (i.e. in the direction of B~ ) with the pitch angle ® given through the relation v? = vktan®. The vector of the gyration angular velocity ~!L follow also directly from the ¯rst equation Eqd4 integrated over time: Z Z d~v m ? dt = q ~v dt £ B~ ) ~v = ~! £ ~r (11) dt ? ? L ? where qB~ ~! ´ ¡ : L m 3 The considered charged particle motion is therefore a superposition of gyra- tion (with the gyration radius and gyration frequency rL and !L respectively) in a plane normal to magnetic ¯led lines, and a uniform motion of the center of gyration, the so called guiding center, along the ¯eld lines. In the guiding center description, such charged particle motion is identi¯ed as the motion of its guiding center. According to Eqd5c, the positively charged particles gyrate about the mag- netic ¯eld line in the clockwise direction while particles with a negative charge move in the opposite direction. 1.3 Case with constant B~ and E~ If a constant electric ¯eld is added to the previous con¯guration, the dynamics of a charged particle is governed by the equation d~v m = qE~ + q~v £ B~ (12) dt which can be analyzed and solved in a similar way as done in Case 1.2. Thus, we decompose ~v and E~ ~ ~ ~ ~v = ~v? + ~vk; E = E? + Ek; substitute these into Eq (12) and obtain two equations for propagations parallel and normal to the magnetic ¯eld: d~v d~v m k = qE~ ; and m ? = qE~ + q~v £ B~ (13) dt k dt ? ? The ¯rst equation in Eq (13) says that the particle is accelerated along the magnetic ¯eld by the parallel component of the electric ¯eld in the same way as it happens in Case 1.1: q ~v = (t ¡ t )E~ + ~v : (14) k m 0 k 0 To solve the second equation, we switch to a new frame of reference moving with some constant speed V~E in a direction normal to B~ so that ~v? = V~E + ~u? (15) where ~u? is the normal velocity component relative to the moving frame. The second Eq (13) now becomes: Ã ! d~u dV~ m ? + E = qE~ + qV~ £ B~ + q~u £ B~ (16) dt dt ? E ? where dV~E=dt = 0 as assumed in this case. 4 Since V~E has not been speci¯ed so far, we shall so choose it now that the ¯rst two terms on the right hand side of Eq (16) mutually cancel out: E~ £ B~ E~ £ B~ qE~ + qV~ £ B~ = 0 ) V~ = ? = : (17) ? E E B2 B2 The remaining part of Eq (16) is then: d~u m ? = q~u £ B~ (18) dt ? which is the same type of equation as Eq (4) in Case 1.2. The velocity compo- nent ~u? therefore describes an orbiting motion with Larmor frequency !L and Larmor radius rL around magnetic ¯eld lines as viewed in the frame of reference moving with a constant velocity V~E. Finally, the particle velocity components in the rest frame are given by Eqs (14)-(15) indicating an accelerated guiding cen- ter motion along magnetic ¯eld lines with ~vk and a superimposed perpendicular drift motion with E~ £ B~ V~ = : (19) E B2 This drift, called the 'E£B' drift, is charge independent and therefore induces no electric currents as both the positive and negative charges move in the same direction as seen in Eq (19) for V~E. 1.4 Case with constant B~ and F~ If some additional constant external force F~ acts on a charged particle moving in a constant magnetic ¯eld we start from the equation of motion: d~v m = F~ + q~v £ B~ (20) dt Comparing this equation with the equation of motion (12) in Case 1.3 we see that the only di®erence between them is the replacement of qE~ by F~ . This means that all derivations performed in Case 1.3 can now be repeated here by taking F~ =q instead of E~ . Thus, we conclude that a charged particle spirals around magnetic ¯eld lines and its guiding center velocity has two components ~ describing an accelerated motion along the magnetic ¯eld lines due to Fk, and a drift motion across the ¯led lines with velocity V~F : F~ £ B~ F~ £ B~ V~ = ? = (21) F qB2 qB2 whose orientation is charge dependent. As a result, this drift, also called the force drift, produces electric currents as charges with opposite signs drift in opposite directions. 5 One of interesting examples of a force drift is the gravitational drift occurring in presence of a uniform gravitational ¯eld ~g when F~ = m~g.
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