arXiv:1406.2745v1 [physics.plasm-ph] 11 Jun 2014 yaiso h udt lo o oegnrlequa- general more for allow to fluid thermo- the the of generalize and dynamics MHD extended briefly scribe MHD. Hall Inertial consistent including and investigated, the is are of model term reductions MHD Various extended momentum this full ordering. the of appropriate in the retention re- neglected with that conservation often and energy is that that equation, term find a we quires Upon L¨ust [1] extended of of not. derivation two-fluid MHD do original which the and to energy returning conserve models MHD requires equation. momentum conservation will the energy We of instances, MHD. modification some usual in for that momentum that as the see same with choice the law, the being Ohm’s in equation only of differ generalization models the these to of of seem All limitations unknown. their relation- and be the models of these some between and ships 12–14], 5, [2, e.g., implemented, [9–11]). (e.g., well reduction as geometric used on been based numerically,have models and reduced analytically many im- and both ex- be models various can MHD investigated effects tended have of researchers variety many a Because portant, in- [6–8]). electron as (e.g. such ertia effects Hamiltonian other the by on work afforded resistivity significant reconnection by is induced there is but it [5]), (e.g. when well-studied this Reconnec- most that transfer. is energy tion and for regions place important is specific takes phenomenon to flux reconnection rise frozen gives (see magnetic the and 2] where breaks inclusion MHD [1, law the ideal that Refs. Ohm’s of well-known condition in of also terms work is additional it early of And, extend- the 4]). by in that [3, for as also and accounted law respects are Ohm’s many deficiencies ing in these physicists deficient of plasma is some to MHD well-known ideal is that it fusion However, nuclear and science. of geophysics, those astrophysics, including in plasmas, of relevance types to application wide of hsppri raie sflos nSc Iw de- we II Sec. In follows. as organized is paper This extended which out sort to is paper this of goal The been have models MHD extended of versions Different history long a has (MHD) magnetohydrodynamic Ideal npriua,atr ntemmnu qainta sotnn often is that v equation I when momentum i.e., the limit, energy. performed. in ideal of term is conservation the a in inertia particular, energy electron In conserve and not do terms models Hall include that ytmtcsuyo nrycnevto o xeddmagn extended for conservation energy of study systematic A e od:eeto nri,mgeoyrdnmc,MD e MHD, magnetohydrodynamics, inertia, electron Words: Key .INTRODUCTION I. neeg osraini xeddmagnetohydrodynamics extended in conservation energy On 1 iahrkw iaeco ay-u yt 0-52 Japa 606-8502, Kyoto Sakyo-ku, Oiwake-cho, Kitashirakawa eerhIsiuefrMteaia cecsKooUnive Sciences,Kyoto Mathematical for Institute Research 2 eateto hsc n nttt o uinStudies, Fusion for Institute and Physics of Department nvriyo ea tAsi,Asi,T 78712-1060 TX Austin, Austin, at Texas of University ej Kimura Keiji Dtd ue1,2014) 12, June (Dated: 1 n .J Morrison J. P. and .Fnly ecnld nSc ,weew ics some discuss we Table where in V, possibilities. Sec. summarized and in are limitations conclude results we the Finally, – I. not energy do determine conserve which inertia systematically and electron concern. we with main IV, models of MHD Sec. which are in these reason, litera- energy, this the For conserve in not models the do without IMHD-like ture model various MHD the Since extended treat full ordering. re- the to our to able but apply of independently, are sults law IMHD we Ohm’s of the Thus, conservation of energy of terms dominated. the ordering are which an HMHD in employing MHD extended by full (IMHD) in- including MHD introduce conservation we ertial Then, energy discus- thermodynamics. its its our generalized in verify the model begin and consistent we right, well-known MHD own a III Hall is consider Sec. first which We in (HMHD), Goldberger, conservation. Next, energy Chew, of of sion [15]. form and Low pressure the electron and of of inclusion pressure the anisotropic with state of tions noprt nstoi rsueit h thermodynam- can the one ics. into how the pressure show completing anisotropic we by incorporate addition, conserva- L¨ust’s model in energy extend and, for we thermodynamics necessary that Next, is equation and momentum tion. derivation neglected one-fluid His often the mass. in is electron term the a of yields smallness ion expand- the and and in quasineutraility, electron ing enforcing individual equations, first subtracting model fluid the one-fluid begin and a be we for adding to law but by Ohm’s appears generalized 16]), who the L¨ust 13, derive [1], to of [2, derived results e.g. be the (see can with theory model one-fluid kinetic a from such well-known, is As nti eto efis tt h xeddMDmodel. MHD extended the state first we section this In soiyadrssiiyaeneglected. are resistivity and iscosity sosre htcmol used commonly that observed is t xtended I XEDDMDAND MHD EXTENDED II. thdoyai MD models (MHD) etohydrodynamic getdi ent enee for needed be to seen is eglected 2 THERMODYNAMICS rsity, n. 2
A. Extended MHD thermodynamic representation is the one in terms exten- sive variables, with the intensive quantities determined The assumptions of quasineutrality and smallness of by the electron mass compared to the ion mass leads to a extended MHD ∂Uα 2 ∂Uα model that we will refer to as . It is given Tα = and pα = ρα . (4) by the following: ∂sα ∂ρα
For isothermal processes Uα = κ(sα) ln(ρα), while for a the continuity equation, γ−1 polytropic equation of state, Uα = κ(sα)ρα /(γ − 1), γ ∂ρ whence pα = κ(sα)ρα. For these choices, one can substi- = −∇ · (ρV ) , (1) ∂t tute the variable pα in lieu of sα, as is more common in plasma physics. the momentum equation, For the ideal, energy conserving, fluid the Fourier and other heat flux terms are dropped and the two entropies ∂V obey the advection equations, ρ + (V · ∇)V = −∇p + J × B (2) ∂t ∂sα m J + vα · ∇sα =0 . (5) − e (J · ∇) , ∂t e en Since entropy is extensive it is natural to introduce the and the generalized Ohm’s law total entropy for one-fluid extended MHD as follows: J 1 E + V × B − = (J × B − ∇p ) s = (misi + mese)/m . (6) σ en e m ∂J In addition, if Hall MHD is to have a complete set of + e + ∇ · (V J + JV ) e2n ∂t thermodynamic variables, one must retain the electron entropy, se. Using V = (mivi + meve)/m and J = me J − (J · ∇) , (3) en(vi − ve), a simple calculation gives e2n en ∂s where ρ is the mass density of plasma, V the bulk veloc- = −V · ∇s , (7) ∂t ity, p the pressure, B the magnetic field, J = ∇× B/µ0 the current density, me the electron mass, e the elemen- and tary charge, n the number density of each species of ∂s 1 charged particles, E the electric field, σ the conductivity, e = −V · ∇s + J · ∇s , (8) ∂t e en e and pe the electron pressure.
Although this generalized Ohm’s law arises upon sub- where we have dropped terms of order me/m. tracting the individual electron and ion fluid momentum Again, appealing to the extensive property of energy, v v equations, with velocity fields e and i, respectively, the total internal energy per unit volume of the two and contains electron momentum dynamics, we will refer species is given by to this system of Eqs. (1), (2), and (3), supplemented by the thermodynamics of Sec. II B, as a single fluid model. ρU = ρiUi(ρi,si)+ ρeUe(ρe,se) . (9)
Because of quasineutrality, n is the only density variable; B. Extended thermodynamics for our purposes it is sufficient to rewrite this, again cor- rect to order me/m, as
In the original article of L¨ust [1], and to our knowledge nU(n,s,se)= nUi(n,s)+ nUe(n,se) , (10) elsewhere, the thermodynamics of the two-fluid to one- fluid extended MHD reduction has not been treated in where mU =: U and mαUα =: nUα. Evidently, the pres- 2 2 full generality. L¨ust assumes polytropic laws for pi and sures satisfy p = ρ ∂U/∂ρ = n ∂U/∂n = pi + pe. Con- pe, and constructs the extended MHD pressure as p = sistent with the derivation of extended MHD, we can ex- pi + pe, in accordance with Dalton’s law on the addition pand (10) in the smallness of me/mi. Because the only of partial pressures. Because of quasineutrality, ρ = ρi + thermodynamic deviation from single fluid MHD occurs ρe = min + men = mn, where m = mi + me. in the Hall term ∇pe, it is sufficient to retain only the Here we generalize this by using entropy as the sec- leading order in this expansion in order to ensure energy ond thermodynamic variable for each species. With this conservation. This will be shown in Sec. III A where we choice, the thermodynamics of species α ∈{e,i} is deter- examine the total energy conservation for Hall MHD. mined from an internal energy function Uα(ρα,sα), the Since it is not widely known, we review here the ther- internal energy per unit mass mα, where sα is the entropy modynamics for anisotropic pressure in a magnetofluid per unit mass mα. Since 1/ρα is the specific volume, this model, which to our knowledge was first given in [17] 3
(see also [18]). The generalization follows upon adding Upon calculating dHH /dt it is readily seen that the B = |B| as an additional thermodynamic variable, i.e., MHD terms cancel as usual and that the Hall term J ×B now U(ρ,s,B), and the parallel and perpendicular pres- produces the energy flux B × (B × J)/(µ0en). Conse- sures are given by quently, only the thermodynamic terms are of concern for the energy of (13) to satisfy a conservation law. Assum- 2 2 ∂U ∂U ing U(ρ,s,se) and p = ρ ∂U/∂ρ, one obtains the usual p|| = ρ and ∆p = −ρB . (11) ∂ρ ∂B internal energy flux of MHD, (p + ρU)V . Thus, we are left with the following upon neglect of surface terms: where ∆p = p|| − p⊥. Expressions (11) can be seen to be consistent with the dHH 3 1 J ρ ∂U J natural extensive thermodynamic variables (ρ−1,s,B), = d x · ∇pe + · ∇se , (14) dt ZD en en ∂se all being specific quantities. The intensive thermody- namic dual variables associate with this set are (pk,T,M) where use has been made of (7) and (8). with the magnetization M being given by For barotropic electron pressure, U has no dependence on the electron entropy se and, consequently, only the ∆p ∂U ′ M := = − . (12) first term of (14) is present and ∇pe/(en) = ∇(ρU) /e, ρB ∂B where prime denotes d/dn. Then, upon integration by parts we obtain In (12), ∆p = p|| − p⊥ is related to the work done when magnetic flux is held fixed. Note the magnetic moment dHH 3 ′ 3 ′ 2 = d x (J · ∇(ρU) /e)= d x ∇ · (J(ρU) /e) , µ = mv⊥/(2B); thus the magnetization per unit volume dt 2 ZD ZD is M = nmv⊥/(2B), or one can argue macroscopically (15) M ∼ p/B. Therefore, the specific magnetization would using ∇ · J = 0. Thus, for this case the energy flux is be M ∼ p/(ρB); thus (12) makes sense as a relative mag- −J(ρU)′/e. netization. Alternatively, one can use H as the variable The barotropic model is incomplete since electron pres- thermodynamically conjugate to B by introducing the sure can change at fixed electron density. To account 2 ‘total’ energy, Utot = ρU + B /2µ0. Then, the conjugate for this we have included the electron entropy in the 2 to B is given by ∂Utot/∂B = H = −M + B/µ0, as ex- dynamics via U. Using (10) with pe = n ∂Ue/∂n and pected. In [17] it was shown that this way of introducing ρ∂U/∂se = n∂Ue/∂se, (14) becomes anisotropy is energy conserving. For simplicity, in the U U following we will restrict to isotropic pressure, but the dHH 1 3 1 2 ∂ e ∂ e = d x J · ∇ n + J · ∇se , generalization is straightforward. dt e ZD n ∂n ∂se 1 ∂U ∂U = d3x ∇ · J n e + e J · ∇n III. ENERGY CONSERVATION e ZD ∂n ∂n ∂Ue + J · ∇se Because the examination of general energy conserva- ∂se tion of extended MHD is complicated, we divide the cal- U 1 3 J ∂ e U culation into two parts. We first consider HMHD and = d x∇ · n + e , (16) e ZD ∂n then its complement IMHD. Results for total energy fol- low upon superposing the calculations. In addition, we yielding −J (n∂Ue/∂n + Ue) /e as another contribution give an ordering for IMHD, an energy conserving model to the energy flux. in its own right. Thus, we conclude that HMHD has the integrand of (13) as an energy density, say EH , and this quantity sat- isfies a conservation law of the form ∂EH /∂t+∇·J H =0 A. Hall MHD for an energy flux J H given by
2 B 1 ∂Ue Setting me = 0 in (2) and (3) gives resistive HMHD. J H = J MHD + J ⊥ − n + Ue J , (17) Since electron inertia is absent, the energy is expected to µ0en e ∂n be composed of the sum of kinetic, internal, and mag- J netic, i.e. where MHD is the usual MHD energy flux.
V 2 B 2 3 | | | | HH := d x ρ + ρ U + , (13) B. Inertial MHD ZD 2 2µ0 where it remains to determine the function U that will Now let us consider the remaining terms of extended −1 ensure conservation of HH when the resistivity σ is set MHD. The last term on the right-hand-side of the mo- to zero. Determination of U will be tantamount to the mentum equation (2) exists due to electron inertia, and determination of the entropy dynamics. we will see that its retention is crucial when electron 4 inertia terms are included in Ohm’s law. We will call parameters have value unity. The above equations are to the second term on the left-hand-side of the generalized be solved with the pre-Maxwell’s equations, Ohm’s law (3) the “nonlinear term,” the third term on the left-hand-side the “collision term,” the first line on ∂B ∇× E = − and ∇× B = µ J , (22) the right-hand-side the “Hall term,” and the remaining ∂t 0 terms on the right-hand-side the “electron inertia terms.” To compare the size of these terms, we use the following with the initial condition ∇·B = 0. Note, consistent with dimensionless numbers: quasineutrality and the neglect of the Maxwell displace- ment current, the current density is solenoidal, ∇·J = 0. Nonlinear term RM := = σµ0UL, We stress that the energy conservation results we ob- Collision term tain do not depend on the IMHD ordering, but exist with Hall term σB the inclusion of the Hall terms, provided one extends the CH := = , Collision term en thermodynamics as in Sec. III A. For IMHD one only Electron inertia term σme need consider U(ρ,s), but it could also be generalized to C := = , I Collision term e2nτ include B. Upon considering a candidate energy of this IMHD where U, L, B, and τ are the characteristic velocity model by taking the scalar product of V and the mo- scale, length scale, magnitude of magnetic field, and time mentum equation, the scalar product of J and the gen- scale of current change, respectively. Here R is usual M eralized Ohm’s law, and using the pre-Maxwell equations, magnetic Reynolds number or Lundquist number as it is we obtain the following energy relation: sometimes called. The two Hall terms are comparable in size if B2 ∼ p , i.e., β ∼ 1. e e ∂ ρ m |J|2 |B|2 For IMHD we focus on the situation where the electron V 2 e | | + ρ U + ǫ 2 + inertia term is larger than the collision and Hall terms; ∂t 2 e n 2 2µ0 2 however, the nonlinear term is still considered to be com- ρ me |J| + ∇ · |V |2 + p + ρ U + ǫ V (23) parable with the electron inertia term, that is, 2 e2n 2
me me E × B CI +ǫ (V · J)J − δ |J|2J + =0. R ≫ 1, C ≫ 1, ≫ 1. 2 3 2 M I e n 2e n µ0 CH Since the last inequality is equivalent to Observe the new term of the energy density of (23), 2 2 me|J| /(2e n), which arises from electron inertia and C m 1 1 I = e = ≫ 1, represents the electron kinetic energy density. Note that, CH eB τ Ωeτ from the generalized Ohm’s law, because of the depen- dence of E×B in the energy flux of (23), the flux includes where Ωe is the electron gyro-frequency, this relation can 2 the time derivative term ǫme∂J/∂t/(e n), so the above be interpreted as saying that the characteristic time scale formulation is not in the usual conservation form. How- of the current change is much shorter than the gyro- ever, upon integrating the above energy relation over the period of the electron. whole domain D with appropriate boundary conditions, With the above ordering, extended MHD reduces to it is revealed that the total energy H, which is defined as the IMHD model given by the following set of equations: ρ m |J|2 |B|2 ∂ρ H := d3x |V |2 + ρU + ǫ e + , = −∇ · (ρV ), (18) 2 ∂t ZD 2 e n 2 2µ0 ∂V ρ + (V · ∇)V = −∇p + J × B is conserved. ∂t Note, if we were to consider the governing equations m J − ǫ e (J · ∇) , (19) (18)–(21) with the full Maxwell’s equations, then the fol- e en lowing energy relation applies: m ∂J E + V × B = ǫ e + ∇ · (V J + JV ) e2n ∂t ∂ ρ m |J|2 |B|2 ǫ 0= |V |2 + ρU + ǫ e + + 0 |E|2 m J ∂t 2 e2n 2 2µ 2 − δ e (J · ∇) , (20) 0 e2n en J 2 ρ V 2 me | | V ∂s + ∇ · | | + p + ρU + ǫ 2 = −V · ∇s, (21) 2 e n 2 ∂t E B me V J J me J 2J × +ǫ 2 ( · ) − δ 3 2 | | + , where s is the entropy per unit mass of the plasma and e n 2e n µ0 the last equation means the plasma is adiabatic. Note, we have artificially inserted book keeping parameters ǫ and this relation is of the usual conservation form since and δ in order to identify terms – in reality, both of these E is now a dynamical variable. 5
IV. CLASSIFICATION BY ENERGY the following equations: CONSERVATION OF IMHD ∂V ρ + (V · ∇)V = −∇p + J × B 0 ∂t In this section we sort IMHD models into energy con- m serving and non-energy conserving classes. e J J − ǫmom 2 ( · ∇) , We first consider the compressible IMHD model com- e n0 m ∂J m posed of the pre-Maxwell equations and the following: E + V × B = ǫ e + ǫ e (V · ∇)J ti e2n ∂t ad e2n ∂ρ 0 0 + ∇ · (ρV )=0, me J V + ǫohm 2 ( · ∇) ∂t e n0 ∂V m ρ + (V · ∇)V = −∇p + J × B e J J − δ 3 2 ( · ∇) , ∂t e n0 me J − ǫmom (J · ∇) , together with ρ = ρ0 = constant or equivalently n = e en n = constant. Note that ǫ does not occur in the above J 0 cp me ∂ me equations because of incompressibility. For this system, E + V × B = ǫti + ǫad (V · ∇)J e2n ∂t e2n energy conservation law is as follows: me me + ǫcp J(∇ · V )+ ǫohm (J · ∇)V J 2 B 2 e2n e2n ∂ ρ0 V 2 me | | | | | | + ǫti 2 + m J ∂t 2 e n0 2 2µ0 − δ e (J · ∇) , e2n en ρ m |J|2 + ∇ · 0 |V |2 + p + ǫ e V ∂s ad 2 + (V · ∇)s =0 . 2 e n0 2 ∂t m m |J|2 E × B +ǫ e (V · J)J − δ e J + Recall, the parameters δ and ǫ were artificially inserted ohm 2 3 2 e n0 e n0 2 µ0 into (18)–(21) as book keeping parameters, but now ǫ has me V J J been replaced by several parameters that track the vari- = (ǫohm − ǫmom) 2 · · ∇) . e n0 ous effects: ǫti (current time derivative), ǫad (current ad- Thus, it is revealed that total energy is conserved if vection), ǫcp (compressibility), ǫmom (term in momentum ǫ = ǫ = 0 or 1, and there are no other condi- equation that was ǫ), ǫohm (term in Ohm’s law partnered ohm mom with that in the momentum equation). These parame- tions on ǫti and ǫad. ters are useful for determining how the various terms in All of our results on energy conservation for IMHD the calculation of the energy may cancel. models are summarized in Table I. Proceeding, we see the various terms involved in energy conservation combine as follows: V. CONCLUSION ∂ ρ m |J|2 |B|2 |V |2 + ρ U + ǫ e + ti 2 2 ∂t 2 e n 2 2µ0 One might argue that the term −me(J · ∇)J/(e n) ρ m |J|2 of (2) may be neglected without consequence, since it is + ∇ · |V |2 + p + ρ U + ǫ e V 2 ad e2n 2 small. However, doing so would amount to the introduc- tion of nonphysical dissipation. Since the whole point J 2 E B me V J J me | | J × of reconnection studies is that a small physical dissipa- +ǫohm 2 ( · ) − δ 3 2 + e n e n 2 µ0 tion can have important consequences, one should view m |J|2 ∇ · (nV ) any reconnection calculation with this nonphysical dissi- = (ǫ − ǫ ) e ti ad e2n 2 n pation with caution. Note, however, in some geometries m this term may vanish. + (ǫ − ǫ ) e |J|2(∇ · V ) (24) ad cp e2n Equations (3.7.1)–(3.7.4) of Ref. [19] do not conserve m J energy, whether or not Maxwell’s displacement current is + (ǫ − ǫ ) e V · J · ∇) . ohm mom e en retained. In this reference and elsewhere it is described how the neglected term in the momentum equation can Since our main interest here is with electron inertial mod- be recognized by reverting to a two-species kinetic theory, els, we do not consider the case where ǫti vanishes. Thus, where the pressure tensors for each species are given by from Eq. (24) we find that the total energy is conserved only when all the epsilon terms are non-vanishing or ǫ , α 3 ti P = mα d vfα (v − vα) ⊗ (v − vα) (25) ǫad and ǫcp are non-vanishing. Note, we have conserva- Z tion for any value of δ. Therefore, we conclude that the with α ∈ {e,i}, fα being the phase space density of epsilon term in the momentum equation is essential for species α, and the fluid velocities given as usual by energy conservation in IMHD models. 3 v Second, we consider the incompressible IMHD model v d vfα α = 3 . (26) which is governed by the pre-Maxwell equations and by R d vfα R 6
ǫti ǫad ǫcp ǫohm Ohm’s law E + V × B = ǫmom Conserved?
Compressible plasma
me ∂J 111 1 + ∇· (VJ + JV ) 1 Yes e2n ∂t me ∂J 1 1 1 + ∇· (VJ) Yes e2n ∂t 2 me ∂J me |J| ∇· (nV ) 1 e2n ∂t e2n 2 n
me ∂J me 2 1 1 +(V · ∇)J |J| (∇· V ) e2n ∂t e2n me ∂J me J VJ JV V J 111 1 2 + ∇· ( + ) · ( · ∇) e n ∂t e en
Incompressible plasma − me ∂J 1 2 Yes e n0 ∂t me ∂J − V J 1 1 2 +( · ∇) Yes e n0 ∂t me ∂J − VJ JV 1 1 1 2 + ∇· ( + ) 1 Yes e n0 ∂t me ∂J me − VJ JV V J J 1 1 1 2 + ∇· ( + ) 2 · ( · ∇) e n0 ∂t e n0
TABLE I. Classification of energy conserving IMHD models. The values of the epsilons in the generalized Ohm’s law are listed in the first four columns, and the generalized Ohm’s law is described by the fifth column. The epsilon values in the momentum equation are listed in the sixth column. When the total energy is conserved “Yes” is written in the last column, otherwise the deficit terms are written in the last column. Note that for incompressible plasma, there is no ǫcp-term in the generalized Ohm’s law; consequently, we write “ − ” in the third column for this case.
If we follow L¨ust’s example for scalar pressure, then the manifest in the resulting violation of energy conservation total pressure is the sum of the partial pressures, i.e., when this procedure is employed. In this paper we have used physical reasoning and P Pi Pe = + . (27) direct calculation to obtain conserved energy densities. However, energy should emerge from time translation in accordance with Dalton’s law. However, it is some- symmetry by means of Noether’s theorem. That this times suggested that one use pressures defined in terms is indeed the case will be reported in future work [20] by of the center of mass velocity according to deriving the action for extended MHD and then using the Galilean group to construct the usual conservation laws. α 3 P = mα d vfα (v − V) ⊗ (v − V) (28) cm Z With this formalism one also obtains the noncanonical Poisson brackets for this model akin to that of [21] (see P Pi Pe and define the total pressure by cm = cm + cm. Upon [22] for review). This leads to the Casimir invariants and inserting opens up the possibility of applying Hamiltonian tech- niques for stability such as in [23–26]. J J me mi Finally, we point out that our starting point was two- vi = V + and ve = V − (29) m en m en fluid theory and gyroviscous effects due to strong mag- in (27), an easy calculation gives netic fields have not been incorporated [23, 27, 28]. This will also be the subject of a future publication. m m m P = P − e i J ⊗ J ≈ P − e J ⊗ J (30) cm me2n cm e2n Thus, one could replace the first and last terms on the ACKNOWLEDGEMENTS right hand side of (2) by −∇ · Pcm and obtain a tidy equation. However, if one further makes conventional thermodynamic closure assumptions on Pcm, e.g., that it We would like to acknowledge support and hospitality is isotropic and either barotropic or adiabatic, then one of the 2011 Geophysical Fluid Dynamics Program held would be in essence saying that the current is dependent at the Woods Hole Oceanographic Institution, where the on density, which is unphysical. This unphysical nature is bulk of this research was undertaken. K.K. was sup- 7 ported by a Grant in-Aid from the Global COE Program like to acknowledge useful conversations with Francesco “Foundation of International Center for Planetary Sci- Pegoraro; he was supported by U.S. Dept. of Energy Con- ence” from the Ministry of Education, Culture, Sports, tract # DE-FG05-80ET-53088. Science and Technology (MEXT) of Japan. P.J.M. would
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