Meaning of Ionospheric Joule Heating Vytenis M

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 110, A02301, doi:10.1029/2004JA010615, 2005

Meaning of ionospheric Joule heating Vytenis M. Vasyliunas Max-Planck-Institut fu¨r Sonnensystemforschung, Katlenburg-Lindau, Germany

Paul Song Center for Atmospheric Research and Department of Environmental, Earth and Atmospheric Sciences, University of Massachusetts, Lowell, Massachusetts, USA Received 7 June 2004; revised 18 June 2004; accepted 22 June 2004; published 3 February 2005.

[1] The possibility of relating the electric current in the ionosphere to the electric field in the frame of reference either of the neutral atmosphere (as commonly done) or of the plasma raises the question: which should be used in calculating Joule heating? The energy equations for the plasma and for the neutral medium, including collision effects, can be combined with momentum equations to separate energy transfer into work done and heating (this is not the same as separation of energy content into bulk-flow kinetic energy and thermal energy). Heating/dissipation processes can be unambiguously identified, showing that true electromagnetic (Joule) heating rate is given by J E in the frame of reference of the plasma. The major part of energy dissipation associated with J and E in the ionosphere is heating by collisions between plasma and neutrals, proportional to the square of the difference in their bulk flow velocities and distributed approximately equally between the two. The conventionally calculated ionospheric Joule heating as J E in the frame of reference of the neutral atmosphere equals the sum of total heating rate of plasma (only a small part of which is true Joule heating) plus total heating rate of neutrals, plus a term equal to work done on plasma in the neutral-atmosphere frame of reference; the latter is assumed negligible, with J B/c balanced entirely by plasma- neutral collisions and with no net work done on the plasma. This assumption implies that all the magnetic stresses exerted on the plasma at ionospheric heights are taken up by the local neutral medium; hence the dynamics of the neutral atmosphere play an important role in magnetosphere/ionosphere interactions. Citation: Vasyliunas, V. M., and P. Song (2005), Meaning of ionospheric Joule heating, J. Geophys. Res., 110, A02301, doi:10.1029/2004JA010615.

1. Introduction proportionality of current to electric field in a material medium; in this case, obviously, the velocity V must be that [2] In studies of energy input to the ionosphere [e.g., Lu of the medium, and Joule or Ohmic heating can be uniquely et al., 1995; Thayer et al., 1995; Fujii et al., 1999; Thayer, identified with J E in the frame of reference in which the 2000; Richmond and Thayer, 2000], the rate of energy medium is at rest. In the ionosphere, however, there are two conversion from electromagnetic to mechanical form, J media, the plasma with bulk velocity U and the neutral E, is commonly expressed as the sum of two terms, Joule atmosphere with u . Conventionally, the ionospheric Ohm’s heating and work done. Defining Joule heating in this n law is expressed in terms of E* with V = u , but Song et al. context, however, is not as simple as might seem at first. n [2001] have shown that on the basis of the same physical In a strict sense, Joule or Ohmic heating is that given by hJ2 assumptions, J can be written as proportional both to E + U where h is the resistivity, and it appears as part of J E B/c and to E + u B/c, with different conductivity whenever there is a linear relation between J and E. n coefficients; this is possible because U À un is proportional to Because E varies with frame of reference, while J in a J, by virtue of the plasma momentum equation. In fact, it is nonrelativistic, quasi-neutral plasma does not, J cannot easily shown that, with appropriate redefinition of the depend on E simply but only on conductivity coefficients, J can be rewritten in terms of E + V B/c with any V that satisfies E* ¼ E þ V B=c; ð1Þ V ¼ U þ zðÞu À U ; ð2Þ the electric field in a particular frame of reference. n Historically, Ohm’s law was first established as an observed where z is an arbitrary constant. If Joule heating is taken as J E*, what V should be used in equation (1)? Copyright 2005 by the American Geophysical Union. [3] In this paper we discuss the physical meaning to be 0148-0227/05/2004JA010615$09.00 attached to the various definitions of Joule heating in the

A02301 1of8 A02301 VASYLIU NAS AND SONG: IONOSPHERIC JOULE HEATING A02301 ionosphere, on the basis both of equations relating J and E defining a single effective plasma-neutral collision fre- and of the energy equations of plasma and neutrals. We note quency to write that the electromagnetic fields interact directly only with the plasma and that any effects on neutrals, including energy ðÞdrU=dt pn¼ npnrðÞun À U : ð7Þ transfer, can only occur by means of plasma-neutral collisions. We examine the common assumption that Joule [6] Equation (3) is exact as it stands. The left-hand side, heating represents the power going into thermal energy however, is negligible whenever only time scales 1/w (whether of the plasma or of the neutral gas is not always p and spatial scales le = c/wp, where wp is the (electron) clear) and that the work done represents the mechanical plasma frequency, are considered [Vasyliunas, 1996]. The energy transfer into the neutral gas, with the tacit (and gravity term rcg (rc = charge density) is practically always sometimes [Richmond and Thayer, 2000] explicit) under- negligible, and the kinetic tensor terms usually are neglected standing that it goes into kinetic energy of bulk flow of the in the ionosphere. With these approximations and with neutral gas. the further assumption of electrons and one species of ions, 2 me/mi 1, the generalized Ohm’s law (divided by nee /me 2. Equations Relating J and E in the Ionosphere for convenience) simplifies to

4 ÀÁ [ ] Because all charged particles contribute to it, J 2 appears in only those plasma equations that are derived 0 ¼ E þ U B=c À J B=nec þ me=nee ðÞdJ=dt coll: ð8Þ from the velocity moments of the Boltzmann equations summed over all species. (E, by contrast, can appear also [7] A resistive term hJ arises from any part of (dJ/dt)coll in equations referring to individual species or even to single proportional to J. From equation (5) the resistivity coeffi- particles.) Particularly important is the generalized Ohm’s cient h is law, specified precisely as the velocity moment equation that gives @J/@t [see, e.g., Rossi and Olbert, 1970; Greene, h ¼ m ðÞn þ n þ ðÞm =m n =n e2: ð9Þ 1973]: e ei en e i in e X ÈÀÁ 2 The dominant resistive effect comes from electron colli- @J=@t ¼ qana=ma ðÞE þ Va B=c a sions, with ion-neutral collisions contributing only a K correction term of order me/mi. The U À un term in À ðÞrqa=ma a þ qanaggþðÞdJ=dt coll ð3Þ equation (5) likewise contributes only a term of order me/mi to equation (8); this can be shown from the relation between where q , m , n , V , and K are the charge, mass, a a a a a U u and J that follows from equation (6) and the concentration, bulk velocity, and kinetic tensor, respec- À n momentum equation (4) (in which the time-derivative and tively, of charged particles of species a, and (dJ/dt) coll kinetic-tensor terms are assumed to be negligible). Leaving represents the sum of all collision effects. The momentum out these small corrections, we have equation for the entire plasma (subsuming into the plasma medium all the charged particles but not the neutrals) is 0 ¼ E þ U B=c À J B=nec À hJ ð10Þ K @rU=@t þr ¼ J B=c þ ðÞdrU=dt pn; ð4Þ with where r is the mass density, U is the bulk velocity, and K = P 2 aKa (more commonly written as K = rUU + P); E does h ¼ meðÞnei þ nen =nee : ð11Þ not appear because of quasi-neutrality. (In equation (4) we have omitted gravity and inertial (e.g., Coriolis) force terms The Joule heating hJ2 is equal to J (E + U B/c); it becomes which play no significant role in our discussion.) negligible if electron collision effects are negligible. [5] The collision term in equation (4) is labeled pn to [8] By contrast, ionospheric Joule heating as convention- emphasize that only collisions of plasma with neutral ally defined is taken to be equal to J (E + un B/c). Over particles contribute to the momentum equation. Given the much of the ionosphere, it depends primarily on ion-neutral electron-ion, electron-neutral, and ion-neutral collision fre- collisions and is not greatly changed if electron collisions quencies n, the collision terms are are neglected; only at altitudes below about 100 km do the electron collisions become unavoidably important. It can be derived from equation (8) if U everywhere (including the ðÞdJ=dt ¼ÀðÞnei þ nen þ ðÞme=mi nin J þ ðÞnen À nin neeðÞU À un coll U B/c term) is eliminated by using the momentum ð5Þ equation (4) (again with neglect of the time-derivative and kinetic-tensor terms): the resulting equation has the same [see, e.g., Song et al., 2004] and form as equation (10), with U replaced by un, a more complicated coefficient of J B/nec,andadifferent drU=dt n m n m n U u m =e n n J ðÞpn¼À eðÞi in þ e en ðÞÀ n À ðÞe ðÞin À en expression for h. However, while the mathematical form ð6Þ is the same, the physical meaning is not: hJ in this version does not arise from balancing the collisional effects on J [e.g., Song et al., 2001]; for later use, we will simplify against E* in the generalized Ohm’s law but instead from equation (6) by neglecting the J term (of order me/mi) and balancing the collisional effects on U À un against J B/c

2of8 A02301 VASYLIU NAS AND SONG: IONOSPHERIC JOULE HEATING A02301 in the plasma momentum equation (which, as noted before, energy of bulk flow plus thermal energy (or, in thermody- does not contain E). The quantity J (E + un B/c) would namic language, internal energy), the latter being kinetic thus seem to be more accurately described not as Joule energy of motion relative to bulk flow. The terms on the heating but as frictional heating due to the relative motion of right-hand side represent the rate of change of total me- plasma and neutrals. chanical energy density by electromagnetic and collisional processes (the heat-flux terms, to be neglected anyway, 3. Energy Equations should strictly speaking be on the left-hand side but have been placed on the right, for later convenience). Under [9] Of course, the mere name ‘‘Joule heating’’ is not change of frame of reference, equations (12) and (13) are important. What matters is which expression most closely covariant: their form remains the same but the values of the represents what would normally be called ‘‘heating’’ in the individual terms may change. sense of thermal energy input into the medium. To decide [11] Taking, separately for plasma and for neutrals, the this, it is necessary to examine the energy conservation dot product of the momentum equation with the bulk equations of both plasma and neutrals. Generalizing the velocity gives an equation which can be recast in conser- derivation for a fully ionized plasma presented by Rossi and vation form for the bulk flow kinetic energy density: Olbert [1970] to include plasma-neutral collisions (under the assumption that these are nonionizing and thus do @ 1 1 rU 2 þr U rU 2 þr½P U not change the densities) gives the energy conservation @t 2 2 equations: U drU ¼ÀJ B þ U ð15Þ c dt pn @ 1 1 rU 2 þ þr U rU 2 þ þ P U @t 2 2 for plasma, and dW ¼ E J þ Àrq ð12Þ dt pn @ 1 2 1 2 drnun rnun þr un run þr½Pn un ¼ un @t 2 2 dt np for plasma, and ð16Þ @ 1 2 1 2 for neutrals; by conservation of momentum in collisions rnun þ n þr un rnun þ n þ Pn un @t 2 2 dW ðÞdrU=dt pn¼ÀðÞdrnun=dt np: ð17Þ ¼ n Àrq ð13Þ dt n np The right-hand sides of equations (15) and (16) represent the work done by the electromagnetic and collisional forces for the neutral medium. Here q and q are the heat flux n acting on the bulk flows. The rate of change of bulk flow vectors (which we neglect henceforth). The quantity is the kinetic energy density is determined, however, not only thermal energy density, equal to (1/2)(Trace P) in general or by work done by these forces (right-hand side) but also by (3/2)P for isotropic pressure, similarly, in terms of P . n n work done against pressure (terms on the left). Whether Any viscous effects appear as nonisotropic terms of P or P n work done by electromagnetic or collisional stresses [e.g., Landau and Lifshitz, 1959]; see also the discussion contributes to changing the bulk flow or the thermal energy following equation (39). The collision terms on the right- depends entirely on whether the stresses are balanced by the hand sides are related, by conservation of energy in inertial or by the pressure terms in the momentum equation. collisions, as Under change of frame of reference, equations (15) and (16) as written do not transform in any well-defined way and do dW=dt dW =dt : 14 ðÞpn¼ÀðÞn np ð Þ not preserve their form. [12] One can now eliminate the kinetic energy of bulk (By assuming the validity of equation (14), we are flow from the energy equation by subtracting equations (15) neglecting transfer of energy from kinetic energy of particle and (16) from equations (12) and (13). The right-hand sides motion to any internal degrees of freedom that the particles of the resulting equations contain the electromagnetic and might have, particularly in the case of molecular species of collisional energy input minus work-done terms. The left- ions or neutrals. To first approximation, such effects can be hand sides can be transformed into a form that is invariant included by adding a term to or n to represent the under change of frame of reference, through extensive additional energy density; a proper derivation, however, manipulation (invoking also the continuity equation) given requires generalizing the concept of particle distribution in detail by Rossi and Olbert [1970, pp. 293–295]. The function beyond the usual velocity space as well as treating results are: for plasma collisions in adequate detail.) [10] The left-hand sides of equations (12) and (13) are in 3 d P U dQ standard conservation form: partial time derivative of den- P log ¼ J* E þ B þ ; ð18Þ 2 dt 5=3 c dt sity of, plus divergence of flux density of, the quantity that r pn is conserved if the right-hand side equals zero. The conserved quantity here is total mechanical energy, of the same as equation (10.147) of Rossi and Olbert [1970] plasma or of the neutral medium, respectively: kinetic except for the addition of plasma-neutral collision terms

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(and neglect of heat flux and viscous heating terms—it is at corresponding moments of the collision term (df/dt)c in the this point that they must be put on the right-hand side), and Boltzmann equation: for neutrals Z dQ 1 df ¼ dv m jjv À U 2 dt 2 i dt 3 d Pn dQn pn Z pn Pn log ¼ : ð19Þ ð22Þ 2 dt r 5=3 dt dQn 1 2 dfn n n np ¼ dv mnjjv À un : dt np 2 dt np

The collision terms (dQ/dt) are defined by Note that (dQ/dt)pn and (dQn/dt)np do not obey an anti- symmetry relation analogous to equation (14) for (dW/dt)pn and (dW/dt)np: heat, unlike energy and momentum, is not ðÞdQ=dt pn ¼ ðÞdW=dt pn À U ðÞdrU=dt pn ð20Þ conserved. They can, however, be rewritten, by exploiting the relations in equations (14) and (17), as each the sum of two terms common to plasma and neutrals, one term for plasma, and positive for both and the other with opposite signs: dQ dQ dQ ðÞdQn=dt ¼ ðÞdWn=dt À un ðÞdr un=dt ¼ þ np np n np dt dt dt pn 0 1 ð23Þ ¼ÀðÞdW=dt pn þ un ðÞdrU=dt pn ð21Þ dQ dQ dQ n ¼ À ; dt np dt 0 dt 1 for neutrals, where the second line in equation (21) follows from conservation of energy and momentum in the where collisions. The operators d/dt are convective derivatives following the bulk flow of the respective medium: dQ 1 2 ¼ npnrjjun À U ð24Þ dt 0 2 d=dt ¼ @=@t þ U r

Z ðÞd=dt ¼ @=@t þ un r: 2 n dQ 1 1 df ¼ dv mi v À ðÞun þ U : ð25Þ dt 2 2 dt J* is the current density in the frame of reference of the 1 pn plasma, i.e., without the advection of the charge density rc, One may also rewrite the collision terms in the energy conservation equations as J* ¼ J À rcU and is equal to J for most practical purposes, by quasi- dW dW 1 ÀÁdQ n n r u2 U 2 26 neutrality. (The distinction between the two comes from the ¼À ¼ in n À þ ð Þ dt pn dt np 2 dt 1 rcE term in the plasma momentum equation, and we need it solely for the individual-fluid approach later on.) (in deriving equations (24) and (26), equation (7) has been [13] Equations (18) and (19) are the fluid counterparts of the thermodynamic equation used). [16] The term (dQ/dt)0 represents a positive net heating TdS ¼ dQ rate, supplied equally to both plasma and neutrals, proportional to the relative bulk velocity difference; it is thus and can be used to define the entropy function S in terms of readily interpreted as heating by friction from relative pressure and density. Their right-hand sides, also invariant motion. (dQ/dt)1, by contrast, represents zero net input, with under change of frame of reference, can be taken as heat supplied to one medium and removed from the other; it uniquely defining the heating (energy dissipation) rates. It is equal to the rate of energy change by collisions in the should be noted that the heating rate is related to the rate of frame of reference in which the plasma and the neutral change of entropy and is not necessarily equal to the rate of velocities are equal and opposite. If we assume that the change of thermal energy; the latter is given by heating plus collisions are elastic and that ions and neutrals are of the work against pressure forces. same species (i.e., equal masses), then, for the special case when the thermal speeds of both ions and neutrals can be [14] Joule heating can now be identified, independently of the generalized Ohm’s law, from the right-hand side of neglected in comparison to jun À Uj, evidently (dQ/dt)1 =0. equation (18) as the electromagnetic term in the heating rate With a little thought, it is apparent that this conclusion holds also with nonnegligible thermal energies as long as their of plasma: J* (E + U B/c), i.e., J E in the plasma frame of reference. distribution is the same for ions and for neutrals; (dQ/dt)1 is nonzero only when the two media differ in thermal distri- 3.1. Collision Effects bution. By dimensional arguments, we may then write [15] That the collisional heating terms (dQ/dt) have the same values in all frames of reference, as claimed above, ÀÁ 2 2 can be made manifest by expressing them in terms of the ðÞdQ=dt 1¼ ðÞ1=2 npnr wn À w x; ð27Þ

4of8 A02301 VASYLIU NAS AND SONG: IONOSPHERIC JOULE HEATING A02301 where w and wn are the mean thermal speeds of plasma and is to treat the plasma and the neutral particles as two neutrals, respectively, and x is a dimensionless number separate fluids, each with its own density, bulk velocity, which may be a function of the dimensionless ratio and pressure. The equations derived in the previous section ÂÃÀÁ are directly applicable. Equations (29) and (30) are the 2 2 2 jjun À U = c1 wn þ w þ c2wnw ; ð28Þ energy balance equations; they can be added together to get the energy balance of the whole medium (plasma together with neutrals). The only net source of energy is J E, where c1, c2 are constants; x is expected to be of order unity, and it in any case remains finite when the ratio in electromagnetic energy supplied to the plasma; in addition, equation (28) approaches either 0 or 1. there is energy exchange between plasma and neutrals, an equal amount being taken from one and added to the other [17] The energy equations (12) and (13) can now be written as by collisions. Equations (31) and (32) are the dissipation equations; in general they cannot be added to each other or to the energy balance equations. Energy is dissipated in the @ 1 1 rU 2 þ þr U rU 2 þ þ P U plasma by J* (E + U B/c), which is Joule heating as @t 2 2 ÂÃÀÁ properly defined (J E in the plasma frame of reference) 1 2 2 2 2 and, according to the discussion following equation (10), ¼ E J þ npnr un À U þ x wn À w ð29Þ 2 generally is small except at the lowest altitudes; in addition, energy is dissipated by collisions, the relative flow between (plasma), and plasma and neutrals contributing the same positive amount to both, and the difference between plasma and neutral @ 1 2 1 2 temperatures contributing equal amounts of opposite sign. rnun þ n þr un run þ n þ Pn un @t 2 2 (Note that the expression J (E + u B/c), Joule heating ÂÃÀÁ n 1 2 2 2 2 as conventionally defined by J E in the neutral frame of ¼À n r u À U þ x w À w ð30Þ 2 pn n n reference, does not appear anywhere.) (neutrals). The dissipation equations (18) and (19) become 4.2. Plasma and Neutral Atmosphere as One Fluid [20] As an alternative, it is possible to treat the plasma 3 d P U and the neutral medium as a single fluid, with total mass P log ¼ J* E þ B 2 dt r5=3 c density rT given by the sum of plasma and neutral compo- hi 1 ÀÁ nents and total bulk velocity UT defined by the linear þ n r jju À U 2 þ x w2 À w2 ð31Þ 2 pn n n momentum of the entire medium:

(plasma), and rT ¼ r þ rn rT UT ¼ rU þ rnun; ð33Þ hi 3 d P 1 ÀÁhence P log n ¼þ n r jju À U 2 À x w2 À w2 2 n dt r 5=3 2 pn n n n n UT ¼ ðÞaU þ un =ðÞ1 þ a ; ð34Þ ð32Þ where a r/rn. With this approach there are no collision (neutrals), independent of frame of reference as they should terms in the momentum and energy equations for the be. medium as a whole (as long as collisions conserve energy and momentum and are nonionizing). The energy conserva- 4. Application to the Ionosphere//Thermosphere tion equation is System @ 1 1 r U 2 þ þr U r U 2 þ þ P U [18] At ionospheric heights, several different gases coex- @t 2 T T T T 2 T T T T T ist in the same volume of space: the plasma and the neutral ¼ E J Àrq ð35Þ atmosphere, each in turn containing several different spe- T cies, interacting through collisions between charged and neutral particles. We have mostly ignored the multineutral Equation (35) is the sum of equations (12) and (13), which and the multi-ion aspects in this work, but for plasma one together with the definitions (33) and (34) yields the must always include at least the electrons as a species expressions for the thermal quantities of the medium as a different from the ions. That the bulk velocities of electrons whole: and ions differ is implied by the presence of electric current. r 2 ¼ þ þ jju À U ð36Þ The plasma as a whole and the neutral atmosphere also T n 21ðÞþ a n have, in general, different bulk velocities; this fact underlies r the ambiguities of defining an Ohm’s law for the iono- PT ¼ P þ Pn þ ðÞun À U ðÞun À U ð37Þ sphere, as discussed earlier. 1 þ a u À U rðÞ1 À a 4.1. Plasma and Neutral Atmosphere as q ¼ n a À À jju À U 2 Two Separate Fluids T 1 þ a n 21ðÞþ a n u À U [19] The most common approach in dealing with the þ n ðÞaP À P : ð38Þ electrodynamics and energetics of the ionospheric regions 1 þ a n

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Whenever there is a bulk velocity difference between more than one (at the very least, electrons and one species plasma and neutrals, un À U 6¼ 0, the combined plasma- of positive ions). neutral medium has a ‘‘thermal’’ energy density that includes a contribution from kinetic energy of relative bulk motion, and it also has a nonisotropic pressure and a 5. Assumptions Underlying the nonzero heat flux even if these are absent both in the plasma Conventional View and in the neutral medium considered by themselves. (The [22] In none of the general statements of the energy unavoidable presence of these terms must be counted as a conservation and dissipation equations, as formulated here disadvantage of the single-fluid approach, offsetting the with several alternative approaches, does the conventionally advantage of having no collision terms.) The dissipation used expression for ionospheric Joule heating, J E in the equation is frame of reference of the neutral atmosphere, appear as a significant quantity. Any basis for the applicability of this "# ! 3 d P U expression must lie therefore not in the general formulation P log T ¼ J* E þ T B but in some specific assumption. 2 T dt 5=3 c T rT [23] Let us add together the heating rates of plasma and of X @ðÞUT neutrals, designating by Q the sum of the right-hand sides Àrq À S l ; ð39Þ T lk @ of the plasma and neutral dissipation equations (31) and lk rk (32). (It should be noted that the physical meaning of such a sum is not entirely clear; see discussion following where now, of course, equation (45)). Neglecting the difference between J* and J, ðÞd=dt T ¼ @=@t þ UT r U 2 Q¼J E þ B þ npnrjjun À U : ð40Þ and c

J* ¼ J À rcUT : With the use of equations (7) and (4) we then obtain hi is the traceless part of the pressure tensor un @rU S J E þ B ¼QþðÞU À u þrK : ð41Þ c n @t PT ¼ PT I þS The expression on the right-hand side is a hybrid: a heating with PT = (1/3)(Trace PT), and the entire last term represents term plus a term that equals the work done on the plasma in viscous heating, which has been included together with the the frame of reference of the neutrals. This term can be heat flux term, in accordance with the discussion above. dropped under the assumption (already mentioned in our The electromagnetic dissipation in equation (39) is J* (E + discussion of J À E relationsasthebasisforthe UT B/c), i.e., J E in the frame of reference of the conventional formulation of ionospheric Ohm’s law) that total medium, which is close to the conventional value J all terms on the left-hand side of the plasma momentum (E + un B/c) numerically when a 1 but different equation (4) are negligible, reducing the equation to conceptually; furthermore, the meaning of ‘‘heating’’ is now ambiguous, kinetic energy of relative bulk motion counting 0 ¼ J B=c þ ðÞdrU=dt pn; ð42Þ as part of thermal energy. that is, the linear momentum supplied by the magnetic stress 4.3. Each Plasma Component as Separate Fluid on the plasma is assumed to be entirely transferred by [21] At the other extreme, it is possible to treat, in collisions to the neutral atmosphere, with negligible transfer addition to the neutral medium, each charged particle to the plasma itself. Then the conventional expression for species, including the electrons, as a separate fluid with the ionospheric Joule heating becomes its own density, bulk velocity, and pressure. The energy hi balance and the energy dissipation equations can be derived u J E þ n B ¼Q; ð43Þ for each fluid separately, by manipulating the moment c equations as before but without summing over species. The resulting equations are identical in form to those which does indeed represent the complete combined heating previously derived, with each quantity referring to species rate of plasma plus neutrals. That heating rate, however, is a and with the replacements the sum of the true Joule heating (in the plasma frame of reference) J (E + U B/c) and of twice the common J ! qanaVa rc ! qana; collisional heating term (dQ/dt)0 (equation (24)), once for plasma and once for neutrals. As noted before, the true Joule which immediately implies that J* ! 0 for all a.Inthis heating depends on electron collisions (equation (11)) and is approach, therefore, for any individual species, all electro- small over much of the ionosphere. The term (dQ/dt)0 comes magnetic energy input represents work done—there is no from collisional effects in the dissipation equations, electromagnetic heating. Electromagnetic dissipation thus associated with relative bulk flow and entirely independent appears only as a consequence of interactions among of E or J (other than that these happen to be associated with charged-particle species, of which there must therefore be the existence of the bulk flow difference between plasma

6of8 A02301 VASYLIU NAS AND SONG: IONOSPHERIC JOULE HEATING A02301 and neutrals). What is conventionally called ionospheric momentum equations, in standard conservation form, for Joule heating is thus primarily frictional heating—mechan- both plasma and the neutral medium: ical and not electromagnetic dissipation. K [24] Applied to the kinetic energy equations (15) and @rU=@t þr ¼ J B=c þ ðÞdrU=dt pn (16), the assumption expressed by equation (42) gives zero on the right-hand side of the plasma equation. This is simply a restatement of the assumption and has no further physical K @rnun=@t þr n ¼ ðÞdrnun=dt np¼ÀðÞdrU=dt pn: meaning (in particular, it does not imply that temporal changes of plasma flow are due to pressure effects alone!). In each equation, the right-hand side represents the forces The equation for neutrals may be rewritten as acting on the respective medium; the left-hand side represents hithe momentum transfer and contains only time and space @ 1 2 1 2 un derivatives. The assumption that leads to equation (42) is rnun þr un run þr½Pn un ¼ J B ; @t 2 2 c of course just the neglect of the left-hand side of the ð44Þ plasma equation, possible because the right-hand side contains two terms, which can balance each other and each a mathematical restatement of work done by collisions as of which by itself can be large in comparison to the left- electromagnetic. As discussed above, whether this work hand side. By contrast, the right-hand side of the neutral changes the flow or the thermal energy of the neutrals atmosphere equation contains only one term, and hence the cannot be determined without detailed calculation. Only if left-hand side can never be neglected. Furthermore, as long the spatial gradients are assumed negligible does all the as the assumption (42) holds, the one term on the right- work go into increasing the neutral bulk-flow kinetic hand side of the neutral atmosphere equation equals the energy; this assumption is frequently made, with the J B/c force. Thus somewhat paradoxically perhaps, the argument that only large-scale horizontal flows are electromagnetic stresses exerted from the magnetosphere involved, but there may exist viscosity effects (nonisotropic on the ionosphere must be balanced by nonzero time Pn) which can couple horizontal flows with the (steep) and/or space derivatives of the neutral medium, while the vertical gradients (an example is discussed by Song et al. corresponding derivatives of the plasma are assumed [2004]). negligible. 5.1. Neutral-Wind Versus Center-of-Mass Frame 6. Summary and Conclusions [25] As long as most of the mass density is in the neutrals, r/rn a 1, the bulk velocity un of the neutral medium is [27] The electric current density in the ionosphere is nearly the same as UT, the bulk velocity of the entire usually regarded as driven by the electric field in the frame medium plasma-plus-neutrals (sometimes referred to as of reference of the neutral atmosphere, with resistivity the center-of-mass velocity). The conventional Joule heat- arising primarily from ion-neutral collisions. As shown by ing should then also be nearly the same if calculated in the Song et al. [2001], however, it can equally well be regarded frame of reference UT instead of un. From equations (43), as driven by the electric field in the frame of reference of the (42), and (7) it is easily shown that plasma, with a much smaller resistivity due almost entirely to electron collisions only. There is a corresponding ambi- U 1 guity about how ionospheric Joule heating should be J E þ T B ¼Qþ J B ðÞu À U c c n T calculated. Using the rigorous generalized Ohm’s law, we a ¼QÀ n rjju À U 2; ð45Þ have shown that a current driven by the electric field in the 1 þ a pn n strict sense (i.e., an applied E produces a J that increases until it is limited by collisions) is proportional to E in the which, as expected, differs negligibly from equation (43) if frame of reference of the plasma bulk flow and depends on a 1. Note that the small difference is always negative: electron collisions. The ionospheric current as convention- the heating rate in equation (45) (using the center-of-mass, ally described, proportional to E in the frame of reference of one-fluid bulk velocity) is smaller than the heating rate (43) the neutral atmosphere and depending on ion-neutral colli- (shown to equal the sum of the separate plasma and neutral sions, comes not from the generalized Ohm’s law but from heating rates); by contrast, the thermal energy density in the the plasma momentum equation: it is the current of which one-fluid formulation (36) is larger than the sum of plasma the J B/c force balances the frictional force from the and neutral thermal energy densities, because it now also relative bulk motion between plasma and neutrals. includes kinetic energy of relative bulk motion. As indicated [28] We have analyzed the energy equations both for in the first line of equation (45), the difference is simply the plasma and for neutrals, taking collisions into account, in work done on the neutrals in the center-of-mass frame. two ways: (1) conservation equations for total energy, valid Although negligible in practice as noted already, it suggests in any frame of reference, show a net input J E of that there may be some ambiguity about distinguishing electromagnetic energy into the plasma, together with heating and work done in the case of ad hoc constructions energy transfer (no net input) by collisions between plasma such as equation (40). and neutrals, all the amounts depending on frame of reference; (2) heating/dissipation equations, derived by 5.2. Implications for the Neutral Atmosphere subtracting the work done and independent of frame of [26] The consequences of the assumption expressed by reference, show Joule heating J (E + U B/c) of plasma, equation (42) can be seen more generally if we restate the plus an equal amount of heating by collisions of plasma and

7of8 A02301 VASYLIU NAS AND SONG: IONOSPHERIC JOULE HEATING A02301 of neutrals, plus some heat transfer between plasma and same degree of approximation, all the electromagnetic neutrals if their temperatures differ. All these heating rates stresses exerted from the magnetosphere on the ionosphere add up to the quantity J (E + un B/c), the conventional must be balanced by the neutral atmosphere. Magneto- expression for ionospheric Joule heating, which thus would sphere-ionosphere interactions thus always involve a seem to be more accurately describable as heating by dynamical response of the neutral atmosphere, a response friction between plasma and neutrals. If the plasma and that should not be neglected unless there is an adequate the neutral atmosphere are treated as one fluid with one bulk justification for doing so. velocity (instead of as two fluids, each with its own bulk velocity), with the relative velocity between plasma and [33] Acknowledgments. This work was supported by the National neutrals then counting as ‘‘thermal’’ velocity, a slightly (but Science Foundation under award NSF-ATM0318643. not significantly) different heating rate is obtained, which [34] This paper was reviewed by editor Arthur Richmond. describes the heat input into the entire medium without References distinguishing between plasma and neutrals. Fujii, R., S. Nozawa, S. C. Buchert, and A. Brekke (1999), Statistical [29] The main conclusions from our work are the characteristics of electromagnetic energy transfer between the magneto- following: sphere, the ionosphere, and the thermosphere, J. Geophys. Res., 104, [30] 1. Joule heating is properly defined in the plasma 2357–2365. Greene, J. M. (1973), Moment equations and Ohm’s law, Plasma Phys., 15, frame of reference. The ‘‘ionospheric Joule heating’’ as 29–36. conventionally defined is not primarily Ohmic or Joule Landau, L. D., and E. M. Lifshitz (1959), Fluid Mechanics, chap. 2, sect. 15, heating in the physical sense but is for the most part simply Pergamon, New York. Lu, G., A. D. Richmond, B. A. Emery, and R. G. Roble (1995), Magneto- frictional heating from the relative motion of plasma and sphere-ionosphere-thermosphere coupling: Effect of neutral winds on the neutrals. That it has the mathematical form of Joule heating energy transfer and field aligned current, J. Geophys. Res., 100, 19,643– is largely a coincidence (resulting from the assumption 19,659. Richmond, A. D., and J. P. Thayer (2000), Ionospheric electrodynamics: A discussed in conclusion 3 below): the current density J is tutorial, in Magnetospheric Current Systems, Geophys. Monogr. Ser., related physically to U À un by the plasma momentum vol. 118, edited by S.-I. Ohtani et al., pp. 131–146, AGU, Washington, equation, and U À un is related to E by the generalized D.C. Ohm’s law, but there is no direct physical relation between J Rossi, B., and S. Olbert (1970), Introduction to the Physics of Space, chap. 10 and 12, McGraw-Hill, New York. and E (unlike Ohm’s law in ordinary conducting materials). Song, P., T. I. Gombosi, and A. J. Ridley (2001), Three-fluid Ohm’s law, [31] 2. The energy equation can be formulated in several J. Geophys. Res., 106, 8149–8156. ways, in terms either of total energy input, of work done, or Song, P., V. M. Vasyliunas, and L. Ma (2004), A three-fluid model of solar wind-magnetosphere-ionosphere-thermosphere coupling, in Multiscale of heating, but choosing among the different forms of the Coupling of Sun-Earth Processes, edited by A. T. Y. Lui, Y. Kamide, equation is not the same as separating mechanical energy and G. Consolini, Elsevier Sci., New York, in press. into bulk-flow kinetic or thermal energy. The common Thayer, J. P. (2000), High-latitude currents and their energy exchanges with assumption that (1) u J B/c goes into flow and the ionosphere-thermosphere system, J. Geophys. Res., 105,23,015– n 23,024. (2) heating goes into thermal energy is in general not Thayer, J. P., J. F. Vickrey, R. A. Heelis, and J. B. Gary (1995), Interpreta- justified; partitioning into flow or thermal energy depends tion and modeling of the high-latitude electromagnetic energy flux, for case 1 on whether the inertial or the pressure terms J. Geophys. Res., 100, 19,715–19,728. Vasyliunas, V. M. (1996), Time scale for magnetic field changes after sub- balance the force and for case 2 on whether the heating or storm onset: constraints from dimensional analysis, in Physics of Space the work done against pressure is more important. Plasmas (1995), vol. 14, edited by T. Chang and J. R. Jasperse, pp. 553– [32] 3. A crucial assumption both for the ionospheric 560, MIT Cent. for Geo/Cosmo Plasma Phys., Cambridge, Mass.

Ohm’s law and for the conventional formulation of iono- ÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀ spheric Joule heating is that, to a good approximation, the P. Song, Center for Atmospheric Research and Department of Environ- J B/c force equals the drag force between plasma and mental, Earth and Atmospheric Sciences, University of Massachusetts, 600 Suffolk Street, Lowell, MA 01854-3629, USA. ([email protected]) neutral flow. There is no particular reason to question the V. M. Vasyliunas, Max-Planck-Institut fu¨r Sonnensystemforschung, validity of the assumption, but it does imply that, to the 37191 Katlenburg-Lindau, Germany. ([email protected])

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