Annales Geophysicae, 23, 2589–2597, 2005 SRef-ID: 1432-0576/ag/2005-23-2589 Annales © European Geosciences Union 2005 Geophysicae
Relation between magnetic fields and electric currents in plasmas
V. M. Vasyliunas¯ Max-Planck-Institut fur¨ Sonnensystemforschung, 37191 Katlenburg-Lindau, Germany Received: 16 May 2005 – Revised: 26 June 2005 – Accepted: 27 June 2005 – Published: 14 October 2005
Abstract. Maxwell’s equations allow the magnetic field B 1 Introduction to be calculated if the electric current density J is assumed to be completely known as a function of space and time. Starting with elementary courses in electromagnetism, one The charged particles that constitute the current, however, are becomes accustomed to think of the magnetic field B as de- subject to Newton’s laws as well, and J can be changed by termined by the given distribution of electric current den- forces acting on charged particles. Particularly in plasmas, sity J . Many researchers, particularly in magnetospheric where the concentration of charged particles is high, the ef- physics, continue to apply this mode of thinking (that B is fect of the electromagnetic field calculated from a given J on derived from J ) when investigating problems of large-scale J itself cannot be ignored. Whereas in ordinary laboratory plasma physics, for which the converse view – that J is de- physics one is accustomed to take J as primary and B as de- rived from B simply as (c/4π)∇×B – has been proposed rived from J , it is often asserted that in plasmas B should by Cowling (1957) and Dungey (1958) and in recent years be viewed as primary and J as derived from B simply as strongly argued by Parker (1996, 2000). From Maxwell’s (c/4π)∇×B. Here I investigate the relation between ∇×B equations (Gaussian units are used throughout this paper) and J in the same terms and by the same method as previ- ∂B/∂t = −c∇ × E , (1) ously applied to the MHD relation between the electric field and the plasma bulk flow (Vasyliunas¯ , 2001): assume that ∂E/∂t = −4πJ + c∇ × B , (2) one but not the other is present initially, and calculate what ∇ · B = 0 ∇ · E = 4πρc (3) happens. The result is that, for configurations with spatial one easily obtains an equation explicitly relating B to J only: scales much larger than the electron inertial length λe, a given ∇×B produces the corresponding J , while a given J does 1/c2 ∂2B/∂t2 − ∇2B = (4π/c) ∇ × J , (4) not produce any ∇×B but disappears instead. The reason for this can be understood by noting that ∇×B6=(4π/c)J allowing B at any point to be derived from a knowledge of implies a time-varying electric field (displacement current) J over the complete backward light cone of the point (plus which acts to change both terms (in order to bring them to- knowledge of boundary conditions at −∞). When dealing ward equality); the changes in the two terms, however, pro- with a plasma, however, there are two plasma equations in- ceed on different time scales, light travel time for B and elec- volving J that must be considered in addition to Maxwell’s tron plasma period for J , and clearly the term changing much equations: the generalized Ohm’s law (see, e.g. Vasyliunas¯ , more slowly is the one that survives. (By definition, the two 2005, and references therein) time scales are equal at λe.) On larger scales, the evolution of B (and hence also of ∇×B) is governed by ∇×E, with E X n 2 ∂J /∂t = qa na/ma (E + V a × B/c) determined by plasma dynamics via the generalized Ohm’s a law; as illustrative simple examples, I discuss the formation o − (q /m ) ∇ · κ + q n g + (δJ /δt) (5) of magnetic drift currents in the magnetosphere and of Ped- a a a a a coll ersen and Hall currents in the ionosphere. representing in its exact form (Eq. 5) the charge-weighted summation of all particle accelerations, and the plasma mo- Keywords. Ionosphere (Electric fields and currents) – mentum equation Magnetospheric physics (Magnetosphere-ionosphere inter- actions) – Space plasma physics (Kinetic and MHD theory) ∂ρV /∂t + ∇ · κ = J × B/c + ρg + f . (6)
In Eq. (5), qa, ma, na, V a, and κa are the charge, mass, con- centration, bulk velocity, and kinetic tensor, respectively, of Correspondence to: V. M. Vasyliunas¯ particle species a, (δJ /δt)coll represents the sum of all colli- ([email protected]) sion effects, and g is the gravitational acceleration (included 2590 V. M. Vasyliunas¯ : Relation between B and J for exactness but mostly unimportant in practice). Equa- 2.1 The initial-value thought experiment method tion (6) contains the parameters of the plasma as a whole: ρ is the mass density, V is the bulk velocity, and What makes this approach possible is the evolutionary char- X acter of the basic equations of classical (non-quantum) κ = κa = ρVV + P physics (Vasyliunas¯ , 2001, 2005), which allows an arbitrary a configuration, subject only to the constraint of satisfying the is the kinetic tensor; f stands for any other force, not explic- divergence equations (3) (plus the analogous gravitational itly displayed. equation), to be postulated at an initial time t=0, the equa- When the full set of Eqs. (1–6) is to be taken into account, tions then specifying completely the evolution of the system it is by no means obvious that a complete specification of at all other times. Thus, to investigate the question, which of J , prior to and independently of B, is possible. If we are the two quantities J and (c/4π)∇×B determines the other, dealing with a situation where the displacement current may we may assume at t=0 any initial spatial profile of J and B, be neglected and Eq. (2) reduces to the ordinary Ampere’s` subject to the sole constraint that ∇·B=0 but not satisfying law, Eq. (7), and then follow their evolution for t>0 by solving Eqs. (1–6). J = (c/4π)∇ × B , (7) 2.2 Vacuum case then the two sides of the equation must be equal, but the ques- tion remains: which side is the one that can be specified first It is instructive to consider first the case of a system in a (and hence taken as producing the other)? vacuum, where only Maxwell’s equations need to be solved. This paper attempts to determine whether and under what Assume that at t=0 the initial magnetic field has ∇×B6=0, conditions a definite physical answer to the above question even though J =0 everywhere. Obviously, the non-curl-free can be given. It is the third paper in a series dealing with magnetic field will propagate away at the speed of light, some fundamental questions, motivated in part by the contro- accompanied by appropriate electric fields as required by versy (Parker, 1996, 1997, 2000; Heikkila, 1997; Lui, 2000) Maxwell’s equations. on whether B and V or, instead, J and E are to be treated As a simple example, consider a 1-D magnetic field rever- as the primary variables. The first paper (Vasyliunas¯ , 2001) sal: considered an analogous question: given that V and E are connected by the MHD approximation, which one can be re- B = sign(z) B0 xˆ E = 0 (8) garded as producing the other? The clear answer was that, t= provided the Alfven´ speed V 2 c2 (i.e. the inertia of the at 0 in a vacuum, with no current to support the reversal. A t> plasma is dominated by the rest mass of the plasma parti- At all later times 0, the magnetic and electric fields are cles and not by the relativistic energy-equivalent mass of the given by magnetic field), V produces E but not vice versa. The sec- B = 0 E = B0yˆ |z| ˜ n 2 o An equation for B alone is obtained by taking the curl of 1 + λe ∇ × ∇× [J ∞ − (c/4π)∇ × B∞] = 0 . (28) Eq. (17) and using Eqs. (16) and (18) to eliminate E˜ and J˜ : Equations (27) and (28) are independent of initial values; 2 2 ˜ 2 2 ˜ 2 2 −λe ∇ B + 1 + s /ωp B = 1/s + s/ωp B0 hence, to order (λe/L) , E∞ vanishes and B∞ and J ∞ sat- isfy Ampere’s` law, for any initial values. For the rest, it is +λ 2∇ × ( π/c) J /s − E /c . e [ 4 0 0 ] (19) evident that if the initial values vary spatially only on scales Similarly, an equation for J˜ alone is obtained by taking the L λe, Eqs. (25) and (26) reduce to curl of the curl of Eq. (18) and using Eqs. (16), (17), and (18) = + 2 ˜ ˜ B∞ B0 O (λe/L) (29) to eliminate E and B: 2 J ∞ = (c/4π) ∇ × B0 + O (λe/L) . (30) 2 ˜ 2 2 ˜ 2 λe ∇ × ∇ × J + 1 + s /ωp J = s/ωp J 0 B maintains its initial value, while J becomes equal to 2 +λe ∇×∇×J 0/s + (c/4π) [∇×B0/s+E0/c] . (20) (c/4π)∇×B regardless of its own initial value. 2592 V. M. Vasyliunas¯ : Relation between B and J Explicit solutions of Eqs. (25), (26), and (27) can be writ- Differentiating Eq. (13) with respect to time and applying ten down for the 1-D geometry Eq. (2) gives B = B(z, t)xˆ J = J (z, t)yˆ E = E(z, t)yˆ 2 2 2 (31) ∂ J /∂t = ωp /4π (c∇ × B − 4πJ ) , (41) with initial values 2 2 from which (∂ J /∂t )0 can be evaluated in terms of initial 2 2 B(z, 0) = B0(z) J (z, 0) = J0(z) E(z, 0) = 0 (32) values; (∂ J /∂t )∞ is assumed zero, as before. With this, Eq. (40) can be rearranged to give Eq. (26). (dB /dz6=(4π/c)J ) and spatial boundary condition of all 0 0 For E , taking Eq. (38) at t→∞, inserting Eq. (13) for quantities vanishing at z→±∞. The solutions for t→∞ are: ∞ ∂J /∂t, and neglecting (∂2E/∂t2) gives Eq. (27) directly. E=0 and ∞ Although this alternative derivation may appear simpler, D E 2D E B = B0 − (4π/c) λe dJ0/dz , (33) one crucial step, neglecting the second time derivatives at t→∞, can be properly justified only by the Laplace- D E 2D 2 2E J = (c/4π) dB0/dz − λe d J0/dz transform treatment. D E D E = (c/4π) dB0/dz + J0 − J0 , (34) 2.4 Physical description where Underlying the complicated-looking mathematics of D E 1 Z ∞ dz0 |z − z0| Sect. 2.3 is a simple physical picture. Fundamentally, it ξ(z) ≡ ξ(z0) exp − (35) 2 −∞ λe λe is the motion of all the charged particles in a volume that constitutes the electric current in that volume. If the curl is a spatial average around a point, heavily weighted to the of the magnetic field does not equal the current density neighborhood within a distance λe from it; the second line of determined by the charged-particle motion (factors of 4π/c Eq. (34) follows from the first by two integrations by parts. taken as understood throughout this discussion), Maxwell’s equations say that an electric field will develop, having a curl 2.3.1 Alternative direct derivation that implies a change in the magnetic field in turn, tending Equations (25), (26), and (27) can also be derived without to make the curl of the magnetic field equal to the current explicit use of the Laplace transform. For B∞, take the time density; the whole process takes place at the speed of light. derivative of Eq. (4), replace the resulting ∂J /∂t on the right- These changing electric and magnetic fields, however, also hand side from Eq. (13) and the resulting ∇×E from Eq. (1) change the motion of charged particles and hence affect the to obtain current density, tending to make it equal to the curl of the magnetic field; the time scale of this process is the period of h 2 2 2 2 i 2 2 ∂/∂t 1/c ∂ B/∂t −∇ B =− ωp /c ∂B/∂t . (36) electron plasma oscillations. Which way an initially imposed difference between J Integrating Eq. (36) with respect to t from 0 to ∞ gives and (c/4π)∇×B resolves itself depends thus on the ratio h i 1/c2 ∂2B/∂t2 − ∂2B/∂t2 of two times scales: light travel time across a typical spa- ∞ 0 tial scale L vs. the inverse of the electron plasma frequency 2 2 2 ≡ −∇ (B∞−B0) =− ωp /c (B∞−B0) , (37) ωp. When L/c 1/ωp (or L c/ωp λe), J hardly changes during the time interval it takes for B to reach adjustment 2 2 (∂ B/∂t )0 can be evaluated in terms of initial values from with Ampere’s` law, Eq. (7). It is then convenient to treat J 2 2 Eq. (4), and (∂ B/∂t )∞ is assumed zero. Inserting these as given and to calculate B (neglecting the usually ignorable values into Eq. (37) and rearranging terms gives Eq. (25). retardation effects) from Eq. (7); this is the environment of For J ∞, the electric-field counterpart of Eq. (4) is needed: the elementary EM laboratory with its circuits and devices. In space and astrophysical plasmas, on the other hand, typi- 2 2 2 2 1/c ∂ E/∂t + ∇ × ∇ × E = − 4π/c ∂J /∂t . (38) cally L λe; then J can change to satisfy Ampere’s` law in a time interval during which B hardly changes. Equation (7) Apply the differential operator on the left-hand side of can now be used only to calculate J from (c/4π)∇×B; the Eq. (38) to Eq. (13) and use Eq. (38) to replace the right- magnetic field itself must be determined from other consid- hand side, to obtain erations (some examples are discussed in Sect. 3). h i 1/c2 ∂2/∂t2 + ∇ × ∇× ∂J /∂t = 2.5 Plasma response and magnetic field evolution − ω 2/c2 ∂J /∂t . (39) p So far the evolution of B and J from given initial values has Integrating Eq. (39) with respect to t from 0 to ∞ gives been calculated assuming that direct acceleration of charged particles by the electric field is the dominant mechanism for 2 h 2 2 2 2 i 1/c ∂ J /∂t − ∂ J /∂t changing the current on the short (∼1/ωp) time scales in- ∞ 0 volved – the assumption underlying the reduction of the gen- 2 2 +∇×∇× (J ∞−J 0) =− ωp /c (J ∞−J 0) . (40) eralized Ohm’s law to the approximate form Eq. (13). The V. M. Vasyliunas¯ : Relation between B and J 2593 consequences of all the other field and plasma effects that The physical meaning of the procedure is now apparent. change the current, contained in the E∗ term of Eq. (12), Provided the spatial variations of E∗ and of the initial values can be studied by including that term (treated formally as a B0, J 0, and E0 are only on scales L λe, the solutions of 2 given function of space and time) when taking the Laplace Eqs. (47), (50), and (51) are, to order (λe/L) , transforms. The transformed Eqs. (16) and (17) remain un- = ∗ changed while Eq. (18) is replaced by E∞ E∞ , (52) ∗ B∞ = B∞ , (53) ˜ 2 ˜ ˜ ∗ s J − J 0 = ωp /4π E − E , (42) ∗ ∗ J ∞ = (c/4π) ∇ × B∞ − (1/4π) ∂E /∂t ∞ , (54) from which it follows that the previously derived Eqs. (19), i.e. the asymptotic mean values, reached for times τ 1/ωp, (20), and (21) remain valid provided one substitutes ∗ ∗ are determined solely by E∞ and B∞ and do not depend at ∗ all on the initial values. But it is a property of plasmas on J → J − ω 2/4π E˜ (43) 0 0 p large spatial and temporal scales whenever J appears in the equations. Applying the limit 0 L λ and τ 1/ω (55) defined by expression (24), needed to obtain the long-time e p solutions, to the substitution (43) requires some care because that the generalized Ohm’s law can be reduced to ∗ of the singularities of E˜ at s→0. Using integration by parts 2 ∗ and the definition of the Laplace transform as needed, one 0 ≈ ωp /4π E − E (56) can show that (Vasyliunas¯ , 2005), determining the electric field which then h 2 ˜ ∗i 2 ∗ lim s J 0 − ωp /4π E = ωp /4π E∞ , (44) governs, via Maxwell’s equations, the evolution of the mag- s→0 ∗ nh ∗i o netic field and hence the value of the current. Thus, E 2 ˜ ∗ lim s J 0 − ωp /4π E /s equals the electric field and B is the magnetic field, both cal- s→0 Z ∞ culated from approximate equations valid in the large-scale 2 ∗ = J 0 − ωp /4π dt E (r, t) , (45) plasma limit (55) discussed by Vasyliunas¯ (2005). What 0 Eqs. (52), (53), and (54) show is that the electromagnetic nh 2 ˜ ∗i o lim s J 0 − ωp /4π E s fields and currents calculated from the exact evolutionary s→0 equations with given initial values for all the quantities be- 2 ∗ = − ωp /4π ∂E /∂t ∞ . (46) come, on time scales longer than the electron plasma os- cillation period (and provided the initial values satisfy the With the use of expression (44) in place of J 0 in Eq. (21), large-scale condition), indistinguishable from those calcu- the equation for E∞, replacing Eq. (27), becomes lated from the approximate equations. 2 ∗ E∞ + λe ∇ × ∇ × E∞ = E∞ . (47) The equation for B∞ retains the form of Eq. (25) if J 0 in it 3 Examples of magnetic field evolution is replaced by the expression (45). Noting that only the curl of J appears, one may carry out what looks at first like a For plasmas on scales large in the sense of inequalities (55), 0 ∇× purely mathematical simplification: given E∗(r, t), define a only B can be independently specified and not J : if (un- field B∗(r, t) as the solution of the equation constrained) initial values for both are assumed, J evolves within a time of order of 1/ωp to the value required by ∗ ∗ ∂B /∂t = −c∇ × E (48) Ampere’s` law while ∇×B remains essentially unchanged. Since this notion – that B determines J rather than the other subject to the boundary condition B∗(r, 0)=B (r). Then, 0 way around – runs counter to so much conventional thinking, with the use of expression (45), the curl of the substitution it is useful to illustrate it by some specific examples. In the (43) (multiplied by λ 2 for convenience) can be rewritten as e following, I consider a few cases where a priori specification 2 2 ∗ λe ∇ × J 0 → λe ∇ × J 0 + (c/4π) B∞ − B0 (49) of the current might seem “intuitively obvious” and, by trac- ing the evolution from initial conditions, show that in fact the and inserted into Eq. (25) gives current develops from deformation of the magnetic field by 2 2 ∗ 2 B∞ − λe ∇ B∞ = B∞ + λe ∇ × (4π/c) J 0 . (50) plasma dynamics. Finally, the equation for J ∞ differs from Eq. (26) both in 3.1 Magnetic drift currents replacing J 0 by expression (45), handled as above, and in having an added term, arising out of the first term on the Charged-particle drifts in inhomogeneous magnetic fields are right-hand side of Eq. (20), equal to expression (46) divided very well known; they are, for instance, generally considered 2 by ωp . The resulting equation is the primary contributors to the ring current in the magneto- sphere during magnetic storms. If a population of moderately + 2∇ × ∇ × = ∇ × ∗ J ∞ λe J ∞ (c/4π) B∞ energetic ions were suddenly placed on a shell of dipole field ∗ 2 − (1/4π) ∂E /∂t ∞ + λe ∇ × ∇ × J 0 . (51) lines, filling it, shouldn’t a current carried by gradient and 2594 V. M. Vasyliunas¯ : Relation between B and J curvature drifts appear (on a time scale something like an ion At t=0, the only term in all these equations that is out of gyroperiod), with J given by ion concentration times drift balance is the radial pressure gradient. Equation (58) then speed and ∇×B adjusting itself accordingly? implies that plasma begins to flow radially, the flow velocity To obtain the current density, however, one must consider initially increasing linearly with time: all the motions of all the charged particles. A fundamental re- sult derived by Parker (1957) is that J obtained by summing Vr ∼ (1/ρ)(∂P /∂r) t . (61) all the single-particle drifts satisfies the plasma momentum Given the radial V , Eq. (59) implies an azimuthal E, with a equation, rewritten from Eq. (6) as non-zero curl that implies a time-varying B. The change in ∂ρV /∂t + ∇ · (ρVV + P) = J × B/c . (57) B can be estimated directly from Eq. (60): Gradient, curvature, and magnetization drifts add up to Z δB ∼ dt V B/L ∼ (1/ρ)(∂P /∂r)(B/L) t2 (62) the pressure tensor term; the time-derivative and inertial r terms come from the so-called polarization drifts (see, e.g. ∇× Northrop, 1963); and E×B drifts, of course, carry no net with represented simply as 1/L. It is only the curl of δB current. (The gravitational term ρg has been neglected as from Eq. (62) that finally gives rise to the (azimuthal) current 2 unimportant in most plasma contexts; it can be derived by density, which initially increases as t : summing the currents carried by the g×B drifts.) 2 2 2 Consider a simple model of the symmetric ring current: Jφ ∼ (c/4π) δB/L ∼ [(c/B) ∂P /∂r] VA t /L . (63) ions with isotropic pressure, electrons with negligible pres- × sure, over a region of finite radial extent. The asymptotic As J B/c increases, it approaches and ultimately reaches ∇ steady state of this configuration is well known: eastward balance with P , putting an end to the radial flow of the (for the Earth’s dipole field orientation) current at the inner plasma and thus stopping the further increase of δB and Jφ. edge (where ∂P /∂r>0), larger westward current at the outer The equilibrium value of Jφ is equal to the quantity in [ ] edge (∂P /∂r<0), current carried by ions (in the frame of on the right-hand side of Eq. (63); the order of magnitude of reference where E=0), hence corresponding eastward and the time scale to reach balance is therefore L/VA, the Alfven´ westward plasma bulk flows. Of interest is to trace the pro- wave travel time across a typical spatial gradient. (The actual cess by which this steady state is reached if the following approach to stress balance is likely to be oscillatory.) initial state is assumed at t=0: J =0, ∇×B=0 (the magnetic Injecting a population of (potentially) drifting ions thus field is purely dipolar), E=0 (in the chosen frame of refer- does not by itself create a current: the initial effect is a radial ence), and the plasma ion pressure is enhanced (within a ra- displacement of the plasma by the imposed pressure imbal- dially limited shell of field lines), with a velocity distribution ance, and it is only as the magnetic field becomes deformed function that is purely isotropic in the E=0 frame (consistent by the plasma bulk flow that its curl gives rise to the cur- with J =0 and implying also V =0). (Slow loss processes, rent. The current density reaches its full drift value only after e.g. precipitation or charge exchange, are ignored.) an elapsed time comparable to the Alfven´ wave travel time, The evolution of the system is governed by Faraday’s law as the magnetic stresses approach balance with the pressure (1), Ampere’s` law (7) (neglect of the displacement current, gradients. implied by the use of Eq. (7) instead of Eq. (2), is easily But this is not the end of the story yet, because drifting shown to be equivalent to neglect of B2/4πc2 in comparison ions imply a sustained azimuthal bulk flow of the plasma, 2 2 whereas all the plasma bulk flows discussed here so far have to ρ in the momentum equation, valid as long as VA /c 1), the momentum equation (57) with isotropic pressure been radial and transient. Given the azimuthal J as above and no azimuthal V , Eq. (59) implies a radial E, the curl of ρ (∂V /∂t + V · ∇V ) + ∇P = J × B/c (58) which could in principle be zero. In that case, in the cho- sen frame of reference (in which E=0 initially) the current (supplemented by continuity and energy equations as is carried entirely by the electrons – the gradient, curvature, needed), and the generalized Ohm’s law, which can be ap- and magnetization drifts of the ions are offset by E×B drifts proximated by from (inhomogeneous) electric fields that have appeared as 0 = E + V × B/c − J × B/neec (59) part of the dynamical evolution. To change this situation requires ∇×(J ×B/nee)6=0, which from Eq. (59) allows a with neglect of any initial transients on times scales non-zero Bφ to evolve: a Bφ that in general varies along the O(1/ωp), electron pressure, and O(me/mi) terms. Equa- field line (e.g. it may have opposite signs on the two sides tions (1) and (59) can be combined immediately to give of the magnetic equatorial plane) and thus gives (through the magnetic curvature force) an azimuthal component to the ∂B/∂t = ∇ × [V × B − (J × B/nee)] (60) magnetic stress in the momentum equation, with consequent which does not refer to E explicitly. Solving this set of equa- azimuthal acceleration of plasma bulk flow. (The equivalent tions is in general very difficult, but the various time scales conventional description in terms of currents is that Bφ vary- can be estimated by order-of-magnitude arguments (e.g. Va- ing along the field line implies a radial component of J and syliunas¯ , 1996). hence an azimuthal component of J ×B/c.) V. M. Vasyliunas¯ : Relation between B and J 2595 One obvious source of the non-zero curl is the increased is by interaction with the ionosphere, as described above, electron concentration ne at and near the ionosphere, unre- that eventually the azimuthal flow of the ions is established lated to the magnetospheric plasma pressure so that and the inhomogenous electric field eliminated. This also resolves an apparent paradox: If the initial state has, as as- × × ∇ ∼ ∇ × ∇ 6= (J B) ne c P ne 0 . sumed, zero bulk flow and hence zero angular momentum Representing ∇× as 1/Lz (to emphasize that the relevant about the magnetic dipole axis, where does the angular mo- length scale now is predominantly along the magnetic field mentum of the drifting ions in the final state come from? Ob- line), we can estimate Bφ from Eq. (60) viously, it has been transferred from the ionosphere (and ulti- mately, of course, from the atmosphere and the planet) during Z the transient phase of non-zero B . Bφ ∼ dt JφB/neeLz (64) φ 3.2 Pedersen and Hall currents and Vφ from Eq. (58) Z V ∼ dt BB /4πρL . (65) A recurrent question in studies of magnetosphere-ionosphere φ φ z interaction is how the horizontal electric currents in the iono- With the use of Eqs. (63) and (64), the estimated azimuthal sphere evolve in response to temporal changes within the bulk flow becomes magnetosphere and the solar wind. Traditionally, the change in the electric field (or else of the plasma bulk flow) just 4 4 2 2 neeVφ ∼ [(c/B) ∂P /∂r] VA t /L Lz , (66) above the ionosphere is taken as given by magnetospheric processes, and the corresponding electric current is calcu- written in this form for direct comparison with Jφ in Eq. (63), lated by treating the ionosphere as a resistive medium, with the difference Vφ−(Jφ/nee) being what determines the ra- the resistivity due primarily to collisions between plasma and dial component of E from Eq. (59). Equation (66) shows neutral particles and hence with J proportional to the elec- 4 that neeVφ increases from zero initially as t and is at the be- tric field in the frame of reference of the neutral atmosphere. 2 ginning much smaller than Jφ which increases as t . Thus, While there is hardly any doubt concerning the numerical re- relative to the chosen frame of reference, the ions indeed un- sults, there are some subtle points concerning the physical dergo almost no net drift at first, the current being carried interpretation. mostly by the electrons. Only after an elapsed time compara- Consider, for simplicity, the region of the ionosphere ble to Lz/VA has the azimuthal plasma flow been accelerated above an altitude of ∼120 km, where only ion-neutral col- to Vφ≈(Jφ/nee), the electric field has decreased to zero, and lisions are significant and electron collisions can be ne- the ions now do carry the current as envisaged in the simple glected. The plasma momentum equation (horizontal com- drift picture. ponents only) then simplifies to Another way of looking at this process is to note that isotropic pressure is constant along a field line, hence the ρ∂V /∂t = J × B/c + νinρ (V n − V ) , (68) curl of the magnetic field deformed by plasma pressure gra- dients varies in a smooth, well-defined manner along the field where νin is the ion-neutral collision frequency, V n is the line. If the current is carried predominantly by electrons, the bulk velocity of the neutral atmosphere, and the kinetic ten- rapid increase in electron concentration near and in the iono- sor terms (usually assumed unimportant in the present con- sphere implies a corresponding decrease in the electron bulk text) have been dropped. The generalized Ohm’s law (see Va- velocity required to carry the current. But Eq. (59) implies, syliunas¯ and Song, 2005, and references therein for a more as is well known, that the magnetic field is frozen to the bulk detailed discussion) is still well approximated by Eq. (59); flow of electrons. If the electrons carrying the azimuthal cur- only at altitudes below ∼100 km do electron collision terms rent are flowing much more slowly in the ionosphere than become important. in the magnetosphere, the field lines become bent, with the The first point to note is that the electric field by itself does curvature being in the sense to accelerate the plasma in the almost nothing: in the ionosphere, νin ωp and therefore the direction of the current (opposite to the electron flow). result derived by Vasyliunas¯ (2001), that V produces E but The question may be raised: How is the magnetic drift E does not produce V , applies – although his calculation current carried entirely by the ions established if there is no did not include the effect of collisions, the adjustments he ionosphere? The simple answer is: It isn’t! As long as the describes occur on a time scale of electron plasma frequency, condition ∇P ×∇ne=0 can be maintained, the asymptotic so the presence or absence of ion-neutral collisions makes steady-state solution of Eqs. (58) and (59) is given by no difference. To study the development of currents in the ionosphere one must thus posit an initial condition of plasma ∇P = J × B/c V = 0 E = ∇P /n e ; (67) e flow. the current implied by the deformation of the magnetic field Assume then that at t=0 there is bulk flow of plasma (both continues to be carried entirely by the electrons, and the ion in the ionosphere and above it in the magnetosphere) relative bulk flow remains zero – a configuration sustainable because to the frame of reference of the neutrals, but no electric cur- the associated inhomogeneous electric field has zero curl. It rent: V 6=V n, J =0. If one assumes also E=0 at t=0, the 2596 V. M. Vasyliunas¯ : Relation between B and J electric field goes from zero to the value −V ×B/c (consis- that depend on the parameter regime. In the ordinary labora- tent with Eq. (59) when J =0) on the very short time scale tory environment of confined currents surrounded by a vac- of 1/ωp (Vasyliunas¯ , 2001), so one may equally well assume uum (or the electromagnetic equivalent thereof), J clearly this as the initial value of E. In either case, the further evo- produces B. In a plasma that is sufficiently dense so that the lution proceeds as described below. electron inertial length λe L (the length scale of the spatial With J =0 initially, the plasma-neutral collisional friction gradients), on the other hand, ∇×B produces J , in the pre- term in Eq. (68) is not balanced, and the plasma flow is de- cise sense that J changes, if necessary, from its given initial celerated (relative to the neutrals) on a time scale 1/νin: value to become equal to (c/4π)∇×B (with B not changing significantly from its initial value). The agent of the change δV ∼ −νin (V − Vn) t . (69) in both cases is the electric field of the displacement cur- rent term: its effect on B propagates at the speed of light, The strong altitude dependence of 1/νin leads to a cor- but in the presence of a free-electron concentration ne it also responding altitude dependence of the deceleration (effec- changes J on the time scale 1/ωp, and the ratio (time to tively, the plasma slows down in the ionosphere but not in change J )/(time to change B) is c/ωpL≡λe/L. the magnetosphere), implying from Eq. (60) a changing mag- The criterion on spatial scales L λ , together with netic field: e the criterion on time scales τ 1/ωp, has previously (Va- Z syliunas¯ , 2005) been shown to define the large-scale plasma ∼ L ∼ − − L 2 δB dt δV B/ z νin (V Vn)(B/ z) t , (70) regime in which the time evolution of J cannot be calcu- lated directly, but only, through Ampere’s` law, from the the curl of which gives rise to the current density: time evolution of B, the latter being governed by Fara- day’s law with the electric field determined (in this large- J ∼ (c/4π) δB/L z scale plasma regime) by plasma dynamics through the gen- 2 2 2 ∼ [(V − Vn) cρνin/B] VA t /Lz . (71) eralized Ohm’s law. The present paper extends this con- clusion from ∂J /∂t to J itself: in a fundamental physi- Initially J increases as t2 and ultimately, after a time com- cal sense, J in the large-scale plasma regime is determined parable to Lz/VA, reaches the value in [ ] on the right-hand by (c/4π)∇×B−(1/4π)(∂E/∂t). This has, of course, side of Eq. (71), at which the J ×B/c force and the plasma- long been a familiar concept within magnetohydrodynamics neutral collisional friction are in balance, with no further de- (Cowling, 1957; Dungey, 1958), recently emphasized partic- celeration of bulk flow. In the steady-state limit, Eq. (59) and ularly by Parker (1996, 2000); what the present paper does is Eq. (68) with ∂V /∂t=0 may be combined to eliminate V and to exhibit in detail how it follows from the basic equations. express J as a linear function of E+V n×B/c, with Peder- The phrase “in a fundamental physical sense” requires sen and Hall conductivities – the conventional ionospheric some explanation. There are many instances in magneto- Ohm’s law (see Song et al. (2001) and Vasyliunas¯ and Song spheric physics of J being calculated not from (c/4π)∇×B, (2005) for critical discussions of its derivation and meaning). but in essentially all cases these are currents required in or- The actual approach to steady state is somewhat more der for J ×B/c to balance a mechanical stress (e.g. plasma complicated than the preceding sketch suggests. In addition pressure gradient in the magnetosphere, or drag force from to the current in the ionosphere given in Eq. (71), δB from the difference between plasma and neutral bulk flows in the deceleration of the plasma also launches a transient Alfven´ ionosphere), under the a priori assumption that stress balance wave into the magnetosphere, which acts to decelerate the holds. If, to determine how and on what time scale these cur- plasma flow in the magnetosphere as well. Only if there are rents came about, one posits an initial state of the mechanical dynamical processes in the magnetosphere that act to main- stress only, without current, and follows the evolution of the tain the initial flow, launching appropriate Alfven´ waves to- system, the result is that at first the stress imbalance produces ward the ionosphere in turn, can the velocity difference be- a change in plasma flow, which deforms the magnetic field, tween the plasma and the neutral atmosphere be sustained in the curl of the deformed field being what finally makes the a steady state. current; the time scale for the process is typically related to the Alfven´ wave travel time. The physical sequence of what determines what in large- 4 Summary and conclusions scale plasma systems may be summarized as follows: Although Ampere’s` law, with the displacement current term 1. The electric field is determined directly by the general- neglected, equates J and (c/4π)∇×B, treating both quanti- ized Ohm’s law (neglecting the ∂J /∂t term): a combi- ties on an equal footing, they can be distinguished, as shown nation of flows and kinetic tensors of the various particle in this paper, by positing an initial state with only one of the species. two present and then using the basic equations (including the complete Maxwell’s equations) to follow the subsequent de- 2. The time derivative of the magnetic field is determined velopment of both. By this method, one of the two can be by the curl of the electric field, hence equivalently by the unambiguosly identified as producing the other, with results curl of the generalized Ohm’s law, which can be written V. M. Vasyliunas¯ : Relation between B and J 2597 as References n ∂B/∂t=∇× V ×B− (J ×B/nee) − (∇·κe) /nee Cowling, T. G.: Magnetohydrodynamics, Section 1.3, Interscience Publishers, Inc., New York, 1957. o + [chδneδEi+hδ (neV −J /e) ×δBi] /ne , (72) Dungey, J. W.: Cosmic Electrodynamics, Section 1.4, Cambridge University Press, London, 1958. a more general approximation than Eq. (60), con- Heikkila, W. J.: Comment on “The alternative paradigm for magne- tospheric physics” by E. N. Parker, J. Geophys. Res., 102, 9651– taining as additional terms (for a plasma containing 9656, 1997. only electrons and one species of singly charged ions Lui, A. T. Y.: Electric current approach to magnetospheric physics with mi me) the electron kinetic tensor and the aver- and the distinction between current disruption and magnetic re- aged small-scale fluctuation terms (see Eq. (19) of Va- connection, in: Magnetospheric Current Systems, edited by: syliunas¯ , 2005, and associated discussion). Ohtani, S.-I., Fujii, R., Hesse, M., and Lysak, R. L., AGU Geo- physical Monograph 118, Washington D.C., 31–40, 2000. 3. The current density is determined by Ampere’s` law, Northrop, T. G.: The Adiabatic Motion of Charged Particles, Inter- Eq. (7) (the displacement current may usually be ne- science Publishers, New York, 1963. glected). Parker, E. N.: Newtonian development of the dynamic properties of ionized gases of low density, Phys. Rev., 107, 924–933, 1957. 4. In many cases, the dominant term on the right-hand side Parker, E. N.: The alternative paradigm for magnetospheric physics, × of Eq. (72) is the V B term, the time derivative of J. Geophys. Res., 101, 10587–10625, 1996. which is determined by the plasma momentum equa- Parker, E. N.: Reply, J. Geophys. Res., 102, 9657–9658, 1997. tion; the magnetic field can then be visualized as de- Parker, E. N.: Newton, Maxwell, and magnetospheric physics, in: formed by being bodily carried by the plasma bulk flow Magnetospheric Current Systems, edited by: Ohtani, S.-I., Fujii, (MHD approximation). In the next approximation, the R., Hesse, M., and Lysak, R. L., AGU Geophysical Monograph, J ×B term is added; the magnetic field is then deformed 118, Washington D.C., 1–10, 2000. by the electron bulk flow (Hall MHD approximation). Song, P., Gombosi, T. I., and Ridley, A. J.: Three-fluid Ohm’s law, The needed value of J is available from step 3) (and J. Geophys. Res., 106, 8149–8156, 2001. often may be advantageously replaced by (c/4π)∇×B Vasyliunas,¯ V. M.: Mathematical models of magnetospheric con- vection and its coupling to the ionosphere, in: Particles and in the equations). The sequence embodied in steps 1) Fields in the Magnetosphere, edited by: McCormack, B. M., D. through 4) remains valid, however, even when addi- Reidel Publishing Co., Dordrecht–Holland, 60–71, 1970. tional terms in the generalized Ohm’s law, such as elec- Vasyliunas,¯ V. M.: The interrelationship of magnetospheric pro- tron kinetic tensor or fluctuation correlations, need to cesses, in: The Earth’s Magnetospheric Processes, edited by: be included. Even when the MHD approximation does McCormack, B. M., D. Reidel Publishing Co., Dordrecht– not apply, any change in V still leads to a change in B Holland, 29–38, 1972. – merely one more complicated than a simple deforma- Vasyliunas,¯ V. M.: Time scale for magnetic field changes after sub- tion by the flow. (That the extra terms in Eq. (72) should storm onset: constraints from dimensional analysis, in: Physics change precisely so as to offset the change in V ×B is of Space Plasmas (1995), Number 14, edited by: Chang, T., most improbable!) and Jasperse, J. R., MIT Center for Geo/Cosmo Plasma Physics, Cambridge, Massachusetts, 553–560, 1996. The above is the correct description (deduced from anal- Vasyliunas,¯ V. M.: Electric field and plasma flow: What drives ysis of the exact dynamical equations) of the sequence in what?, Geophys. Res. Lett., 28, 2177–2180, 2001. which the various quantities are physically determined, in Vasyliunas,¯ V. M.: Time evolution of electric fields and currents the approximation appropriate to large-scale plasmas. The and the generalized Ohm’s law, Ann. Geophys., 23, 1347–1354, equations of each step also provide mathematical equalities 2005, SRef-ID: 1432-0576/ag/2005-23-1347. between the quantities. These equalities can, of course, be Vasyliunas,¯ V. M., and Song, P.: Meaning of iono- read in either direction and should not be taken as giving spheric Joule heating, J. Geophys. Res., 110, A02301, the direction of physical determination, one way or the other. doi:10.1029/2004JA010615, 2005. Particularly when dealing with quasi-equilibrium or slowly Wolf, R. A.: The quasi-static (slow-flow) region of the magne- varying formulations, of which the conventional magneto- tosphere, in: Solar-Terrestrial Physics, edited by: Carovillano, sphere/ionosphere coupling theory (e.g. Vasyliunas¯ , 1970, R. L., and Forbes, J. M., D. Reidel Publishing Co., Dordrecht– 1972; Wolf, 1983) is a notable example, one must take care Holland, 303–368, 1983. not to overlook this distinction between physical determining relation and mere quantitative equivalence. Acknowledgements. In writing this paper I profited from ideas and suggestions by E. N. Parker concerning a previous publication (Va- syliunas¯ , 2005). I am grateful to the referees for useful comments that led to clarification of some points. Topical Editor T. Pulkkinen thanks V. Semenov, N. Tsyganenko and another referee for their help in evaluating this paper.