Revisiting the Role of Magnetic Moments in Heliospheric Plasmas

A&A 552, A38 (2013) Astronomy DOI: 10.1051/0004-6361/201220965 & c ESO 2013 Astrophysics

Revisiting the role of magnetic moments in heliospheric plasmas

H.-J. Fahr and M. Siewert

Argelander Institut für Astronomie der Universität Bonn, Abteilung f. Astrophysik und Extraterrestrische Forschung, Auf dem Huegel 71, 53121 Bonn, Germany e-mail: [email protected] Received 20 December 2012 / Accepted 14 February 2013

ABSTRACT

The magnetic moment is one of the most fundamental, nontrivial particle properties known in plasma physics. Also known as the ﬁrst adiabatic invariant, it is conserved under a broad range of microphysical processes and plasma boundary conditions. However, especially in modern numerical simulations, the conservation of this property, which in principle may serve as a control parameter, is no longer used explicitly. In light of this fact, as well as in a recent series of studies of the solar wind termination shock and comparable systems, where this conservation was used as a key ingredient, we now revisit this old, but standing problem. Building on our earlier arguments on when the magnetic moment of individual particles is conserved, we now carefully expand this argument further, studying nontrivial systems, such as systems where individual ions are represented by a broad distribution function that is described with the help of a transport equation. We see that the magnetic moment is conserved under an even wider variety of situations than those studied in earlier publications. When studying systems of purely kinetic equations, the magnetic moment can be converted into an additional force term in the transport equation. We also study the physics that is taken into account in shock simulations and pinpoint weaknesses in the physics used by these modern codes. Key words. plasmas – Sun: heliosphere

1. Introduction the magnetic moment conservation, which operates independent of the convection and diffusion terms (see Sect. 3). One of the most known invariants identified in plasma theory is the individual ion/electron (scalar) magnetic moment In most astrophysical plasma systems, the ion’s density is sufficiently small that binary collisional particle-particle scatter- 2 mv⊥ ing is inefficient, while plasma-wave interactions can often not μ = = const., (1) 2B be ruled out. Therefore, it is not surprizing that the (in-)validity where v⊥ is the particle velocity component perpendicular to the of μ-conservation is a somewhat controversial point in the magnetic field B,andm it’s mass. (In the following, we will plasma literature of the past. Though nowadays it is expected only talk about ions, dropping the separate notation for elec- that modern MHD shock simulations of the solar wind termina- trons.) This property essentially describes the motion of an in- tion shock (see e.g. Li & Zank 2006; Kucharek et al. 2006, 2010; dividual ion gyrating around a background magnetic field (see Giacalone & Jokipii 2006) may suggest that ions under such e.g. Chen 1977). Assuming that the motion of the ion can be conditions do not conserve their initial magnetic moment, many ff averaged over a gyroperiod, resulting in a quasi-stationary con- di erent conclusions opposite to that point have been achieved tinuous electric current density j(r), this current will generate in earlier and alternative shock literature. If, for example, one a magnetic field that can be developed in a magnetic multipole studies the paper by Sarris & van Allen (1974), one finds that development, where the first nonvanishing moment is the mag- these authors successfully represent, by theoretical calculations, netic dipole moment μ, which does not depend on the charge data on shock-processed ions near the Earth’s bowshock ob- of the gyrating ion; the absolute value of this vector then is the servationally obtained by the satellites Explorer 33 and 35. As magnetic moment (Eq. (1)). these authors can show a fairly good fit to the observational data Also known as the first adiabatic invariant, the magnetic mo- is obtained when assuming that these particles are multiply re- ment is known to be conserved when external physical param- flected at their magnetic mirror points and finally are moving eters (i.e. in the case of the magnetic moment, the background downstream of the shock with an energy gain corresponding to 2 2 magnetic field B) of the system are changing sufficiently slowly, v2/v1 B2/B1 (indices are marking upstream (1) and down- so that the motion of the isolated ion, when averaged over a gyra- stream (2) values of particle velocity and magnetic field magnitude, respectively). This result, in optimal accordance with the tion period τg, can be smeared out to a continuous circle current. An additional requirement for the validity of adiabatic invariants observed ion data, proves directly that the magnetic moments of is the absence of efficient stochastic processes, such as particle- these particles while crossing over the shock is in fact conserved. particle and particle-wave scattering. However, even though un- A similar conclusion is reached by Terasawa (1979)who der these conditions, no strict conservation of μ can be expected, compares ion spectra of MHD shock-reflected ions calculated by the basic magnetic Lorentz force still favors a gyrational motion, two different approaches, namely the “adiabatic approach” and and therefore, the final transport equation describing the system the so-called “kink treatment”. Here the first approach automat- must still be formulated using a term describing the tendency of ically should apply to the situation when the shock structure is Article published by EDP Sciences A38, page 1 of 11 A&A 552, A38 (2013) large with respect to the ion gyroradius and then evidently con- is conserved. This study here aims to close this gap in the- serves the ion magnetic moment, while the kink-field approach ory using a different approach, collecting and expanding exist- applies to the case when the shock structure is small with re- ing arguments on a more general base. We demonstrate several spect to the ion gyroradius. Interestingly enough the results for counterintuitive properties of the magnetic moment and derive the spectra of the reflected ions are nearly identical in both cases, hard criteria that allow to reevaluate existing models, and also meaning that, surprizingly enough, in both cases ions behave, as to test existing numerical implementations with respect to their if their individual magnetic moments are conserved independent physical validity. on the extent of the shock transition region. Our approach to this problem is as follows. First, we will On the other hand people have hesitations to accept the for- present general properties of the magnetic moment, as they are mulation of a tendency to conserve the ion magnetic moment in found with varying degree of accuracy in existing textbooks, transport equations, especially if – in addition to conservative pointing out those properties that are basic for understanding forces – collisions or wave-particle interactions operate which at non-textbook scenarios. Following this, we will study these non- the same time tend to not conserve this quantity. For instance, ideal systems, where some of the general assumptions com- Barakat & Schunk (1983, 1989) had developed a model of the monly found in literature are relaxed, or even dropped. In the ion distribution in the cusp region of the geomagnetic field in section following this one (Sect. 3), we carefully study the role which they used kinetic theory under the restriction of conserved of magnetic moment conservation in transport equations, with total ion energy and ion magnetic moment. This approach then an emphasis on when it makes sense to add an explicit term to later has been enlarged by Barghouthi et al. (1994)andBarakat the equations to respect the magnetic moment physics, and when & Barghouthi (1994) by including wave-particle interactions in this term is already, implicitly, present. Finally, in the conclu- the form of Fokker-Planck type diffusion terms describing a dif- sions, we briefly touch the wide field of numerical simulations, fusion in the velocity component perpendicular to B. Similarly, studying how our current results will allow to estimate the be- Le Roux & Webb (2006) have carried out a test particle calcula- havior of existing (and possibly upcoming) codes with respect tion solving the transport equation at the solar wind termination to the conservation of the magnetic moment. shock. In this equation they use, on the left side of the equation, a Liouville operator which takes care of energy and magnetic mo- 2. The magnetic moment of individual ions ment conservation at free ion motions, and at the same time on the right side use a Fokker-Planck diffusion term for pitch-angle in a plasma scattering. While the latter process definitely violates magnetic In the following sections, we study old and new aspects of the moment conservation, in the full transport equation it is never- conservation of an individual ion’s magnetic moment in a plasma theless required to have the correct operation of a Liouville op- in the test particle approach, i.e. where test ions are moving in erator that guarantees the conservation of particle invariants in a predetermined electromagnetic field generated by independent the case of absence or weakness of other stochastic processes. parts of the plasma. This is the reason why also recently Fahr & Fichtner (2011), in The historical reason why Eq. (1) is called the magnetic mo- their recent treatment of pick-up ion transport, do include terms ment of the ion (or likewise the electron) is due to the fact that the to describe both the tendency to conserve the magnetic moment gyrational motion of the ion in fact represents an electric circle and Fokker-Planck terms to describe velocity space diffusion. current j(x) that acts in an electromagnetic sense like a magnetic Finally, some more recent analytical studies do, again, make dipole μD given by explicit use of the conservation of the magnetic moment. Based eω on a collisionless (and, until now, diffusionless) transport equa- μ = −1 g 2 = − D πrg (B/B) μB(B/B), (2) tion, Fahr & Siewert (2006); Siewert & Fahr (2008)havemade c 2π extensive use of the magnetic moment conservation to derive an where ωg is the gyration frequency, rg is the gyration radius, effective force term acting on individual ions crossing an astro- and B the background magnetic field strength. Since this mag- physical shock wave, and found an analytical relation between netic moment μD is always directed opposite to the main field B, upstream and downstream distribution functions. With the same ions (and electrons as well) contribute a diamagnetic action in a assumptions, they treated the processing in velocity space of an magnetized plasma. In plasma physics, especially in test particle ion distribution function undergoing multiple passages of trav- approaches, one is often interested in the dynamic properties of elling shock fronts (Fahr et al. 2012a). Intermixed with strong an individual ion, the test particle, instead of the averaged prop- isotropisation processes, they discovered a mechanism that al- erties of the entire system. This test ion typically is sent into a lows to generate stable power law distribution functions, which predefined region of arbitrarily complicated magnetic and elec- have in fact been observationally found in most astrophysical tric fields, and the individual ion’s scalar magnetic moment μ systems. Finally, they also studied an electrostatic beam insta- (see Eq. (1)) is sometimes used as a control parameter. However, bility as a mechanism violating conservation of the magnetic studying the motion of an arbitrary ion is violating the require- moment in multicomponent systems Verscharen & Fahr (2008); ments of magnetostatics as defined in the previous section, as the Fahr et al. (2012b). individual ion will be located at different positions at different Due to the broad range of these examples (analytical mod- times, usually not generating a quasistationary electric current, eling, numerical modeling, and also inspection of space plasma not even in the line-convected frame. data), it is obvious that the conservation of the magnetic mo- Therefore, we need to apply a specific average, the gyrolimit, ment is a highly nontrivial question which, until now, has never to convert our time-dependent individual ion motion into a qua- been studied explicitly and ab initio with sufficient detail and sistationary, time-independent current density j(x), confined to generality. The only recent, related study on magnetic moments asufficiently small region of space, “small” being defined as in plasma systems was presented by Cary & Brizard (2009), small compared to any length scale of interest to an observer. who derived a Hamiltonian formulation of the classical guid- The degree of details given on this approximation varies greatly ing center theory, however without going into practical details in literature, which is why we now repeat the most relevant as- or reaching a definite conclusion when the magnetic moment pects. First, it is possible to perform a multipole expansion in

A38, page 2 of 11 H.-J. Fahr and M. Siewert: The magnetic moment in heliospheric plasmas terms of vector spherical harmonics of the magnetic vector po- moment of the plasma per volume unit. In the continuous plasma tential A(x), which can be derived from j(x). For a localized, formulation, the relation between the (global) magnetic field and divergence-free current density, the first nontrivial, nonvanish- the magnetisation is given by ing term in the mentioned multipole expansion is given by (see = + Jackson 1975) B H 4πM, (7) μ × x which correctly differentiates between the magnetic induction B A = , (3) x3 and the vacuum magnetic field H. The magnetic moment of the plasma per volume M is then defined by (see e.g. Fahr & Scherer where μ is the development coefficient called the magnetic mo- 2005) ment, which is defined by B 2 3 M = − mi fiv⊥d v. (8) 1 3 B2 μ = d x x × j(x ). (4) i 2 The summation runs over the different species i of plasma par- ffi This equation, which just represents a development coe cient, ticles with different masses and distribution functions. While in is often found as the definition of the magnetic moment. For an the inner heliosphere the contributions to M from Maxwellian arbitrary charged current system, this vector usually is not an in- d distributed electrons and protons of the solar wind may be variant. However, in the magnetostatic limit, i.e. for j = 0, it 2 2 dt fairly comparable (mi v⊥ me v⊥ ) and relatively small, is easy to demonstrate that the above integral usually reduces to i e a constant value. It should be pointed out here that this approach the Kappa-distributed pick-up ions due to their dominant pres- to define a magnetic moment is not restricted to magnetostatic sure Ppui most likely contribute the major portion given by (see situations, though; in principle, it would be possible to define Fahr & Scherer 2004) a “dynamic” magnetic moment by taking, e.g., the more gen- B M = − P . (9) eral retarded vector potential describing a nonstatic system, and 2B2 pui define a generalized magnetic moment as the development coefficient appearing in the retarded description. Even though deriv- While this diamagnetic behavior of a broad multispecies plasma ing such a generalized magnetic moment would break the limit distribution function is understood in theory, no use is made of of this paper, this proves that the basic ideas behind the concept its consequences. of the magnetic moment could be generalized. Now, aiming to approximate a stationary system, where the 2.2. Slowly varying magnetic fields charge currents are time-independent, it is assumed that the ion Next, we study the reaction of a gyrating ion to a slowly chang- is performing an unperturbed gyrational motion where the gy- ing background magnetic field strength, while keeping the orien- = −1 roperiod τg ωg is much smaller than all other timescales of tation of B fixed, i.e. B˙ B and making use of the gyroaverage interest, including the timescale for background field changes introduced above. In addition, it follows from basic electromag- = ˙ (τB B/B). In this case, it is possible to average the motion of netic concepts that the gyroradius is given by the individual ion over one gyroperiod, i.e. to derive a quasista- mv⊥c tionary circle current, j, which no longer depends on time and rg = · (10) allows to derive a magnetic moment consistent with the one used qB in plasma theory (i.e. with an absolute value identical to Eq. (1)). Using the assumptions on the magnetic field introduced above, Using such an averaged stationary electric current, Eq. (4) this parameter rg is no longer constant due to B˙ 0, and also due transforms into to induced electric forces modifying v⊥ (as we will demonstrate 1 further down). Therefore, the enclosed surface will change ac- μ = q dφ rg(φ) × u(φ), (5) cordingly – but also the electric current strength I flowing along 4π the border line of the enclosed surface. For a line-like current due where we integrate over one gyrocircle, using polar coordinates to an electric charge circling a gyrocircle at constant speed v⊥ = u = d over one gyroperiod, the current strength is given by (i.e. rg(φ) rg(cos φ, sin φ, 0), and (φ) dt rg). Evaluating this integral is trivial, and immediately leads to Eq. (1). Assuming q q I = | j| = v⊥ = = qΩg, (11) that the motion of the gyration is located in a plane (i.e. the mo- l τg tion is 2-dimensional, as for the gyrating ion discussed above), and that the current loop is closed (but not necessarily a circle), where l is the length of the enclosed surface’s border line. Then, then the scalar magnetic moment is given by (see e.g. Northrop the magnetic moment as defined by the enclosed surface relation 1963; Jackson 1975; Cary & Brizard 2009) (see Eq. (6)) is given by qv⊥A μ = I · S, (6) μ = , (12) l where I = j · dσ is the electric current and S the enclosed sur- v⊥A which is only conserved if l is constant. For an arbitrary mo- face. Therefore, the magnetic moment is conserved for an ion tion, e.g. of ions moving in strongly inhomogeneous magnetic following a fixed planar motion, in the magnetostatic gyroaver- fields, this relation will not be fulfilled. In fact, if the magnetic age limit, due to the fact that both the electric current and the field changes over a time scale that is much smaller than the gy- enclosed surface remain constant. rofrequency, the ion will not undergo a closed gyrational motion, and the concept of a constant enclosed surface breaks down. 2.1. The diamagnetic moment of the magnetized plasma In the gyrolimit, however, the magnetic moment turns out as 2 qv⊥πr qv⊥r mv2 The magnetic moment defined by Eq. (2) is a single-particle for- = g = g = ⊥ mulation of the more general magnetisation M, i.e. the magnetic μ , (13) 2πgr 2 2B A38, page 3 of 11 A&A 552, A38 (2013) which is the same as the relation introduced earlier in Eq. (1). where ei are unit vectors pointing into the direction of the sub- Differentiating this expression with respect to time, setting the script vector i (e.g. eB = B/|B|). This equation proves that the time derivative of μ to zero with some elementary algebra then induced electric force modifies v⊥, the vector product between results in the following condition: rg and Eind does not vanish, it seems like the torque is nonzero, and the angular momentum seems to be not conserved. mv⊥ ∂B q ∂B r mv˙⊥ = = rg · (14) However, one also has to consider that the gyroradius g, 2B ∂t 2 ∂t which is a function of v⊥ and B (see Eq. (10)), is not constant ff The force providing just the correct magnitude and orientation either, which, e ectively, results in an additional force acting on follows from Faraday’s law of induction, the gyrating ion. This force, which results in = ∂B x˙ r˙g 0, (21) ∇ × E = − , (15) ind ∂t contributes to the first term in the definition of the torque (Eq. (19)). As it is trivial to verify, this term compensates the which, after making use of Stoke’s theorem, trivially results in classical torque, allowing to arrive at a conserved angular mo- the induced electric force (Fahr & Siewert 2010) mentum. This demonstrates that the identification of the mag- q dB netic moment μ with the angular momentum L is nontrivial, and mv˙⊥ = −qE = r , (16) ind 2 g dt a consistent description of gyrating ions requires a full treatment of electromagnetic effects. which is just the force required to conserve the magnetic moment. One might perhaps suspect that, in these above considera- 2.4. Weakly spatially inhomogeneous magnetic fields tions, the influence of a secondary electric field component has In general, a magnetic field will never be as ideal as assumed been neglected which is due to the temporal change of the vector in our previous examples, but undergo local variations and fluc- potential A. However, as one can easily demonstrate, this con- tuations. Omitting an explicit time-dependence, the commonly tribution does not influence the gyrating ion if external electric found parameterisation of this behavior is given by currents can be neglected, since then, = +Δ B(x) B0 B(x), (22) ∂A ∂ j ds⊥ · = ds⊥ · dV = 0. (17) Δ ≡ dt ∂t r where our previous examples are characterized by B 0over the entire relevant spatial region. In a turbulent system domi- nated e.g. by plasma waves, dipolar/multipolar fields or even just 2.3. The magnetic moment and the angular momentum in case of the solar wind magnetic field dropping in intensity with solar distance, this will no longer be the case. For example, Before finishing our introduction on established concepts related when the wavelength of a plasma wave is on the order of the gy- to the magnetic moment, we point out one more useful property. roradius, the magnetic field inducing the gyrational motion may Respecting, again, that the ion performs a gyrational motion, the no longer be conserved. integrand appearing in Eq. (5) is proportional to the orbital an- We begin our study of this configuration by assuming a static gular momentum L with respect to the gyrocenter, inhomogenity, where the background magnetic field is defined q 1 q q by a magnetostatic, but not spatially constant (ΔB 0) configu- μ = dφ L(φ) = L(φ) = L. (18) 2m 2π 2m φ 2mc ration where, over one gyroperiod, the ion at its gyration is expe- riencing slightly different magnetic field strengths. Without loss Therefore, the magnetic moment is conserved exactly when the of generality, we now select a coordinate system where B0 ez, angular momentum is, providing another aspect to our study. i.e. the gyrational motion will be initially confined to the xy- Now, it is trivial to see that the angular momentum is con- plane. In addition, we assume that ∂Bz 0, i.e. that a linear τ ∂z served exactly when the torque vanishes, spatial gradient into the z-direction exists, i.e. ΔB(x) is a linear τ = L˙ function in x that is small compared to B0. In addition, it follows from the source-freeness of the magnetic field that B(x)mustbe = × u + × u m˙rg mrg ˙, (19) divergence-free, i.e. where the first term is associated with a force reducing the gyro- ∂B ∂By ∂B x + = − z · (23) radius. In most (mechanical) textbooks, this term is argued away ∂x ∂y ∂z by selecting a specific, center-of-mass reference frame that allows to keep the distance of the rotating mass points constant. Then, it follows from Eq. (23) that a magnetic field gradient However, examples exist for systems where the angular momen- in the z-direction automatically induces an additional magnetic tum is conserved even though the radius of the circular motion field component in the xy-plane, which generates a force that with respect to it’s center is not constant. Therefore, this first depends on the gyroangle φ, term is actually important for understanding the general behavior q F⊥ = u⊥(φ) × ΔB⊥(φ). (24) of the angular momentum, and therefore, the magnetic moment. c To further elaborate this claim, we now derive the classical torque acting on a gyrating ion in the presence of time-dependent This force is oriented parallel to the z direction and possesses an absolute magnitude of magnetic fields. The induced electric force derived in the previous section generates a nonzero torque acting on the gyrating q Fz = vxΔBy − vyΔBx ion, which is of the magnitude c ∂B m = q y − ∂Bx · τ = r × E ∝ e × e 0, (20) vxry vyrx (25) e g ind x v c ∂y ∂x A38, page 4 of 11 H.-J. Fahr and M. Siewert: The magnetic moment in heliospheric plasmas

Now, inserting the usual expressions for a circular motion (rx = This identification now allows us to study the conservation of rg cos φ, ry = rg sin φ, vx = −v⊥ sin φ and vy = v⊥ cos φ), we the magnetic moment for a system where the magnetic field vec- obtain tor is changing it’s orientation, while keeping both μ and Φ constant. Assuming that the plane of gyration changes it’s orienta- q 2 ∂By 2 ∂Bx Fz = − v⊥rg sin φ + cos φ · (26) tion so that the local magnetic field vector B is always normal c ∂y ∂x to this plane, this means that the angular momentum L can be written as (see Eq. (18)) Averaging this over φ, i.e. over one full gyroperiod, we obtain 2mc d ⊥ = × = q ∂By ∂Bx L(B,v ) rg p μeB, (31) F = − v⊥r + e dt z 2 g ∂y ∂x where, again, eB = B/|B| is a unit vector. The derivative with 2 = 1 mv⊥ ∂Bz respect to time then is given by 2 B ∂z d 2mc dμ deB ∂Bz L(B,v⊥) = e + μ · (32) = μ · (27) dt e dt B dt ∂z If we now repeat the results from earlier sections, namely that In other words, the (average) force is proportional to the mag- the magnetic moment is conserved for B˙ 0and˙e = 0, the netic moment and the spatial field gradient into the z-direction, B first term in this expression vanishes, and we obtain and it is oriented parallel to the magnetic field. This force, which does not change the full kinetic energy of d 2mc deB 2 = − 2 L(B,v⊥) = μ = Lv(eB · ∇)eB. (33) the particle due to dv dv⊥, allows us to solve the equation dt e dt of motion parallel to B by This means that the angular momentum of the gyrating / dv dv ion electron changes with the magnetic line element s as the field m = mv B L dt dz itself is doing, so that both vectors and and remain parallel to each other, while the ion is propagating along the field line. d m 2 = v This relation now allows to study the conservation of the dz 2 scalar magnetic moment for an arbitrary magnetic field behav- d m 2 ior. If we assume that the first-order correction to the angular = − v⊥ dz 2 momentum is given by dB = F = −μ z · (28) dL z dz L(t +Δt) = L(t) + Δt, (34) dt t Obviously, this equation is solved by any solution fulfill- 2 then obviously the scalar angular momentum is conserved when ing d(v⊥/B) = 0, which in turn is equivalent to the conservation of the magnetic moment μ. This means that the motion 2 dL dL of the gyrating ion along a field line takes place such that μ is |L(t +Δt)| = |L|2(t) + Δt + 2L · Δt, (35) conserved. This result also leads to the well established phe- dt t dt t nomenon of mirror points and particle reflections at critical fields = − 2 which would imply that the scalar magnetic moment is con- Bc B/(1 ξ ), for particles with pitchangle cosines ξ at a cer- served as well. Evaluating this expression is highly nontrivial tain point where the field has a strength of B. and probably requires numerical evaluation of the quantities appearing in this expression. Nevertheless, given a specific (dy- 2.5. Magnetic induction in the guiding center frame namic) magnetic field configuration, Eq. (33) allows to study the conservation of the magnetic moment of ions gyrating under Next, we study the conservation of the magnetic moment in these magnetic conditions. a case where, in addition to a time-dependent magnetic field strength, also a reorientation of the magnetic field is introduced. Due to mathematical complications related to transforming local 2.6. Magnetic moment in more general cases gyrational velocities, we will perform this part of the study in the The cases studied until here are by no means complete; un- framework of guiding center theory. fortunately, an arbitrarily complicated, inhomogeneous and We begin with a variation of previously introduced con- dymnamic magnetic field can not be studied easily using fun- Φ= 2 cepts by introducing the magnetic flux πrg B. The temporal damental electrodynamics. However, before we continue to the derivative in the guiding center rest frame is given by complicated field of transport theory, we would like to summa- dr rize a few general arguments that can be used easily to study d Φ= g + 2 dB· ff 2πrg B πrg (29) more general systems using di erent methods than those pre- dt dt dt sented here. Obviously, this flux is not constant for an arbitrary behavior of rg First, the possibility to identify a magnetic moment with and B. For this expression to be constant, we need that an angular momentum provides a promizing tool that can be adapted to test the conservation of the magnetic moment with 1 dv⊥ 1 dB e.g. numerical codes. However, a full treatment using this test − = 0, (30) v⊥ dt 2B dt parameter requires a full and correct treatment of all microphysical electromagnetic forces, including those generated by the in- which, again, is a formulation of the requirement that the mag- 2 duction terms in Maxwell’s equations, which is a challenging = mv⊥ netic moment μ 2B is conserved. problem even to modern supercomputers. A38, page 5 of 11 A&A 552, A38 (2013)

The other major tool that should be useful for future stud- inhomogeneous magnetic guide fields. This operator is of the ies of a possible magnetic moment conservation is the gyrolimit. form Considering that the magnetic moment usually is associated with ∂ f ∂ f v ∂ f a quasistationary (gyro-)limit, it is possible to study the aver- L( f ) = + ξv + 1 − ξ2 , (37) aged conservation of the magnetic moment by considering the ∂t ∂z 2L ∂ξ gyrolimit. If the forces and related angular torques acting on a where the distribution function f = f (z,v,ξ,t) depends on gyrating ion vanish when averaged over one gyroperiod, then it the line coordinate z, velocity magnitude v, pitchangle cosine is reasonable to claim that the magnetic moment is conserved, ξ = cos ϑ, and the time t. L is the inhomogeneity scale of the on average, as well. This requires a full, consistent treatment of background magnetic field. all microphysical forces. Now it must be noted that, at the absence of collisions or stochastic interactions, the Liouville theorem always requires that the phasespace density of the function f , whatever it’s form 3. The magnetic moment in transport theory may be, is constant along each particle trajectory s = s(t), mean- 3.1. The Vlasov equation ing that d f /ds = 0 must be valid. It is now interesting to recognize that this will automatically be fulfilled, if the distribution Finally, having demonstrated that the magnetic moment is con- function can be written as a function of particle invariants of the served under a large variety of assumptions, and also deriving motion. For example, assuming that the distribution function can an improved condition on when it is conserved exactly, we study be written as a function of the kinetic energy (∝u2)andthemag- how the magnetic moment is treated in existing models for the netic moment μ = mv2(1 − ξ2)/B, f i = f (v2,μ), the function is transport of cosmic rays through the heliosphere and the local constant when both parameters are particle invariants, because of interstellar medium. Since μ represents a derived parameter that can be interpreted quite differently, we now need to study how ∂ f i dv2 ∂ f i dμ d f i/ds = + = 0. (38) a term associated with the conservation of μ can be identified in ∂v2 ds ∂μ ds existing transport equation. First, as an illustrative example, we look into a Vlasow equa- For motions in stationary magnetic fields, it is evident that due to tion that describes the distribution function f = f (z,v,v⊥)as d du u2 ∝ u · = (e/mc)(u · (uxB)) = 0, (39) function of coordinates z, v = vξ,andv⊥ = v 1 − ξ2,where dt dt ξ = cos ∠(u, B). Then, taking into account the force k due to z the kinetic energy is conserved. Furthermore, if the inhomogene- the inhomogeneity of the B-field and all other forces k parallel = B ity scale L LB of the background field is large with respect to to , one finds for the stationary case the following form of a Ω differential equation (see Fahr et al. 1977): the gyroradius rg or with respect to the gyrolength vμ/ g,then it can be assumed that also the magnetic moment of the particle 2 ∂ f 1 1 mv⊥ dBz ∂ f v⊥v dB ∂ f is invariant at the motion along the guide field B, and the distri- v + k − + = 0. (36) bution function should then satisfy the Liouville theorem. ∂z m 2 B dz ∂v 2B dz ∂v⊥ Now, we can immediately show below that the above men- Here, k represents all other than magnetic (e.g. electric or grav- tioned Liouville operator for stationary fields (Eq. (37)), when i 2 itational) forces acting parallel to the field B. It is interesting applied to functions parameterized as f = f (v ,μ), automati- to see that in the case of k = 0, this equation immediately re- cally fulfills the Liouville theorem, i.e. we will demonstrate that, duces to the equation which is used by Siewert & Fahr (2008) in this case, to describe the changing ion distribution function at the plasma ∂ f i v ∂ f i passage over the solar wind termination shock. This latter equa- L f i = ξv + 1 − ξ2 = 0 (40) tion was derived under the assumption of the conservation of the ∂z 2L ∂ξ magnetic moment, demonstrating that a term that can be identi- is valid. This is verified easily by expanding the partial deriva- fied with this effect can easily appear in a kinetic equation. Even tives appearing in the operator, because then, the following is though this term does not necessarily have to be the only force evident: acting on the ion, it nevertheless suggests that a trend towards ∂ f i ∂ f i dB dB ∂ f i ∂μ dB ∂ f i μ the conservation of μ is implicitly present in many applications = = = − (41) of transport theory. ∂z ∂B dz dz ∂μ ∂B dz ∂μ B and 3.2. The Liouville transport operator in a focussed transport ∂ f i ∂ f i dμ ∂ f i 2ξμ = = − · (42) equation ∂ξ ∂μ dξ ∂μ (1 − ξ2) We now study the implicit conservation of the magnetic mo- Now, we remind the reader that the field inhomogenity length 1 1 dB ment in kinetic transport equations using a more general ar- scale LB is defined through = − (see Earl 1976). This LB B dz gument. In papers dealing with ion phasespace transport like allows us to write those by e.g. Roelof (1969); Earl (1976); Kunstmann (1979); ∂ f i v ∂ f i Spangler & Basart (1981); Le Roux et al. (2001, 2002); Le Roux ξv + (1 − ξ2) &Webb(2007); Schlickeiser & Shalchi (2008); Schlickeiser ∂z 2L ∂ξ

(2009)orLitvinenko & Schlickeiser (2011), a transport oper- dB ∂ f i μ v ∂ f i 2ξμ = ξv − − (1 − ξ2) = 0, (43) ator for focussed particle motions has been applied. This form dz ∂μ B 2L ∂μ (1 − ξ2) of a transport description is always associated with the action of the same typical Liouville operator L on the particle dis- and proves our above mentioned claim that the scatter-free tribution function f which describes free particle motions along Liouville operator used in a transport equation automatically

A38, page 6 of 11 H.-J. Fahr and M. Siewert: The magnetic moment in heliospheric plasmas and necessarily implies the conservation of the magnetic mo- present that decelerates the protons, while at the same time leads ment, even when this conservation can not be concluded from a to an (initial) acceleration of the electrons. Taken together, these suffiently long inhomogenity scale. two effects generate a local space charge separation which in This result was already envisaged at kinetic transport equa- turn generates the electric field. All these effects must be present tions written down by Northrop (1963), Kulsrud (1982)or in a consistent numerical (or even analytical) description. Toptygin (1983). Thus Le Roux & Webb (2007) clearly and cor- Now, we are not interested in the initial excitation of the rectly stated that the scatter-free version of this form of a trans- shock wave. Instead, we aim to understand how existing nu- port equation necessarily implies the conservation of the particle merical simulations handle the problem of quasineutrality, it’s magnetic moment μ in the plasma flow frame along the charac- possible breaking and the reaction of gyrating ions to quasi- teristics and fully contains the physics of particle mirroring. instantaneous magnetic transitions. In a recent paper on the MHD shock structure (Fahr et al. 2012b), we have shown that the electron overshooting in the downstream plasma frame is a 3.3. Some critical remarks on scattering terms highly important physical process that leads to strong electron Using the arguments presented above, we are also able to gener- heating, to entropy generation and to high compression ratios. alize these results to a few aspects of scattering terms appearing As we have shown in that paper, the electron overshoot velocity in the transport operator. Assuming that the scattering processes in the downstream region is given by: are elastic, no kinetic energy is transfered during the collisions, 2 and u is an invariant. Then, for large inhomogenity scales, the mp 1 δU = U − U + − e 2 1 1 1 2 (44) magnetic moment is conserved as well, and the transport opera- me s tor, even after including elastic scattering terms, should conserve the magnetic moment. where U1,2 are the plasma bulk velocity at the upstream and This result is slightly complicated for arbitrary inhomogen- downstream side of the shock, respectively. ity scales. Since the magnetic field inducing the gyrational mo- From this formula, one evidently sees the importance of the tion is always the local field at the current position of the ion, mass ratio me/mp = μe,i for the magnitude of the resulting over- elastic scattering might conserve the kinetic energy, but not the shoot velocity. For example, adopting a physically correct ratio magnetic moment, as a modification of the pitch angle ξ clearly of μe,i = 1/1840 will lead to influences the magnetic moment μ ∝ (1 − ξ2)v2.

On the other hand, one has to be really careful when studying m 1 Fokker-Planck-like scattering and diffusion terms, which only = − + p − δUe U2 U1 1 1 2 represent the first terms in a Taylor expansion of the magnetic me s field, which, may not be consistent with the source-free require- m ment for magnetic fields or similar electromagnetic boundary − p 2 − ≥ (U1/s) 1 s 1 43U1. (45) conditions. In a more accurate description, particle-wave inter- me actions (which are the microphysical source of the scattering On the other hand, adopting a smaller ratio, which is done by terms), are not a discrete, but a continuous process, which in many numerical shock simulation studies, likewise results in a principle must be treated fully consistent with Maxwell’s equa- strongly different behavior. In the most extreme case, taking a tions. As demonstrated earlier in this study, taking the induced highly artificial ratio of μ = 1 (i.e. like in an electron-positron electric field into account allows to fully demonstrate the conser- e,i pair plasma), this would yield a much lower value of vation of the magnetic moment. However, when these induced fields are not present, e.g. because one of the two electromag- √ 1 1 2 δUe = U2 −U2 1 + s − 1 =U1 − 1 =U1 1 − ≤ U1. (46) netic field components is “weak” and, therefore, ignored, one s s must expect that the underlying equations do not describe a conservation of the magnetic moment, even when this conservation This difference is sufficient to demonstrate that modern shock should be present. This might be able to explain why the con- MHD simulations, which often use artificial values of this ra- servation of the magnetic moment usually does not appear in tio μe,i, thereby may react sensitively to the choice of this mass modern numerical simulations. ratio, especially in handling the electron overshoot situation. At the first look, it seems like that the ratio of the thermal energy E of the electron overshoot and the kinetic energy E 4. The magnetic moment in numerical simulations t,e kin,i would be independent of the mass ratio μe,i, which is given by

4.1. Electron-proton shocks and the role of secondary shock 2 1 mp components me U1 n2,e E 2 me t,e = = s2. (47) Shock simulations are a rather complicated, and in fact often 1 2 Ekin,i mpU n2,i physically untransparent, theoretical business. They, however, 2 2 often are generally considered to give a self-consistent view of However, the resulting compression ratio s again strongly de- the plasma-particle interactions at shock structures. Here, we re- pends on the mass ratio. Making use of the formalism devel- strict ourselves to a careful overview of complications related to oped by Fahr et al. (2012b), we present (in Fig. 1) resulting such a fully consistent modeling. compression ratios following the overshooting shock formal- We begin with a (non-numerical) example demonstrating the ism, as a function of the mass ratio μe,i and the magnetic field subtle complications introduced with a more consistent mod- strength. These results easily demonstrate that the compres- eling. We start with the classical, but nontrivial example of a sion ratio strongly depends on the electron-proton mass ratio. quasineutral electron-proton system crossing an MHD shock Therefore, any numerical simulation using unphysical mass ra- wave. Due to the deceleration of the bulk plasma flow at the tios (which are typically found in the intervall 1/200 <μe,i < 1) shock, there must be a local, spaceccharge-induced electric field might be flawed due to just this approximation. In addition,

A38, page 7 of 11 A&A 552, A38 (2013)

4 In addition, if one were to consider, individually, the spe- 3.95 cific length scales of both electrons and protons, the electrons would require a significantly higher resolution than the pro- 3.9 tons, introducing an additional complication. In most numeri- 3.85 cal simulations, the time scale problem is addressed by selecting an electron-to-proton mass ratio that reduces the differences 3.8 in characteristic length scales. Other numerical situations treat s 3.75 electrons as a background fluid, which may eliminate the electron overshooting effect discussed earlier, and likewise result in 3.7 realistic typical numerical a contradiction. value choice 3.65 parallel shock perpendicular shock 4.2.2. Reference frames 3.6 inclined shock inclined + stronger B 3.55 Another complication that appears in a numerical simulation is 0.0001 0.001 0.01 0.1 1 the choice of the reference frame in which specific plasma pro- μ cesses are considered. Typically, dispersion relations describing plasma waves and nonlinear particle-wave interactions are for- Fig. 1. Compression ratios obtained for a variety of shock inclinations mulated in the reference frame of the bulk flow of the back- and magnetic field strengths. Default magnetic field strength (first three ground plasma. The latter, however, in these simulations is an curves) corresponds to a Parker field at 90 AU, the fourth line enhances the field by a factor of 2. undefined frame due to the resulting time- and space-variable bulk velocities occuring during the run of the simulation. The usual Alfvénic wave mode and cyclotron resonances (see e.g. whether this contribution is actually pronounced depends on the Scholer 1993, 1995; Scholer & Kucharek 1999) are thus asso- magnetic field strength, which, in the case of the solar wind ter- ciated with a highly variable magnetic field and Alfvén velocity mination shock, decreases with distance. Therefore, one must propagating in a highly variable plasma bulk frame. What in fact expect that the relative strength of the overshooting effect de- then counts for the study of ion-ion instabilities and creation of pends on the position of the shock system being studied. For ex- diffuse upstream ions is the relative ion motion with respect to ample, the solar wind termination shock has a solar distance be- resonant Alfvén modes. It is hard to imagine how hereby a sat- tween approximately 90 and 200 AU, while interplanetary shock isfactory degree of consistency can be achieved. waves experience much stronger magnetic fields. In the following section, we study selected properties of nu- 4.2.3. The electron-proton mass ratio merical simulations and estimate their respective potential for conflicts with a correct treatment of the magnetic moment. Furthermore, most simulations are run as hybrid simulations, meaning that electrons are treated as massless fluids while ions 4.2. Hybrid shock simulations are treated kinetically as particles. This means that a consistent value of electrical currents in such a code can only be achieved 4.2.1. Characteristic length- and timescales if the hydrodynamic value of the electron current je = −eneUe is determined in a comparable accuracy as the kinetically de- The main complication appearing in a numerical treatment termined ion current given by ji =+e dniui, while the to- of underlying electrodynamic processes appears with counting tal electric current is determined by solving Maxwell’s equa- Ω−1 times in units of ion gyroperiods τi,i.e.of p , and measuring ∇ × = = Ω tion B 4π j. In order to reach high enough statistics to distances in units of inertial lengths given by li vA/ p (e.g. see safely define electrical currents in a box of size l3, one either has areviewbyLembege et al. 2004). This property is closely re- i to go to huge particle numbers in relevant time units τi,orto lated to the magnetic moment conservation studied earlier here, huge box sizes, i.e. resulting in a correspondingly coarse current as the magnetic moment is, essentially, equivalent to gyrational resolution. motion. Faced with these problems (and limited computational re- Looking to the fact that both ion density and magnetic field sources), one commonly finds the usual restriction to a 4-d phas- magnitude change from upwind to downwind, and also at time- espace, i.e. only one space dimension x is admitted, which means and space-variable transient structures during the run of the that the total current j is restricted to one direction perpendicular simulation, thus means that these units are variable as well. to x given by Concerning upstream/downstream conditions, they change by c ∂2 = jy = − Ay (51) τi,up/τi,down Bdown/Bup s (48) 4π ∂x2 and where A = Ay is the magnetic vector potential (see Sgro & √ Nielson 1976; Leroy et al. 1982 or Liewer et al. 1993). = Ω Ω li,up/li,down (vA/ p)up/(vA/ p)down s, (49) Furthermore, these codes usually assume strict electric charge neutrality, meaning that the first moments of the elec- where s denotes the usual MHD compression ratio defined as tron and ion distribution functions are taken to be identical, i.e. n = n . However, seen in the frozen-in field frame the = ndown = Bt,down · e i s (50) shock is induced by an electrostatic ramp connected with elec- nup Bt,up tric space charges η,i.e.ne < ni, that as a first step deceler- This means that both time- and space-resolution usually are ates the ions as it accelerates the electrons and thereby consis- much coarser on the upwind side compared to the downwind tently creates the required space charges (see Verscharen & Fahr side in these simulations. 2008). During the action of this space-charge-induced electric

A38, page 8 of 11 H.-J. Fahr and M. Siewert: The magnetic moment in heliospheric plasmas x E = − Φ = e n − n x s ≤ field η grad η 0 ( e i)d ´, which is not taken into Hence in view of compression ratios 4 there is expected account in shock simulations, the inertial forces at ions and elec- a slight decrease of the plasma β at the plasma passage from u d e,i trons are on the same order of magnitude me,i eEη , while upstream to downstream by a factor of roughly 0.6. For the case dt of the parallel shock (α = 0) one would obtain instead: in all simulations where space charges are not admitted the in- γ tertial forces acting on electrons are simply neglected because of n2 γ P1 P2 n1 n2 γ their small masses. β2 = = = β1 · = β1 · (s) . (53) 2 2 n1 Because of this reason, simulations using the commonly B2/8π B1/8π found quasineutrality condition, i.e. without actively studying the behavior of electrons, are inherently unable to control the In comparison to this expectation from which practically all conservation of the magnetic moment. shock simulations start, an alternative behavior is obtained for 2 ions conserving their magnetic moments μ = mv⊥/2B at the 4.2.4. Initial and boundary conditions shock passage. Starting from Liouville’s theorem of continuous differential phasespace flows applied to ions with isotropic dis- Another point of a puzzling astonishment is connected with the tribution functions passing from upstream to downstream of the initial conditions from which these simulations are generally shock one obtains (see Fahr & Siewert 2011, Eq. (27)): started. Nearly all of these simulations (see e.g. Leroy et al. 2 = 2 1982; Leroy 1984; Liewer et al. 1993; Omidi et al. 1990; Scholer 2πU1 f1(v1)v1dv1dcosβ1 2πU2 f2(v2,β2)v2dv2dcosβ2, (54) 1993, 1995) are carried out in a so-called “downstream box” where β are magnetic pitch angles of the upstream and down- which by itself defines the frame of the downstream plasma bulk 1,2 stream ion velocity space. After some elementary algebra, which flow. To initialize the simulation upstream ions are then injected we will not repeat here in full, this can be converted into (Fahr from the left side of the box and are elastically reflected at the et al. 2012b) right border of the simulation box. This primarily induces a sym- metrical ion counterflow situation and clearly would disturb the P = 2π(m/2) f (v ,β )v4dv dcosβ initial injection conditions at the left border, if the equation of ion 2 2 2 2 2 2 2 motion in these simulations would not contain a friction term Π = 2 2 describing an isotropisation of the counterflowing ions. 2π(m/2)(U1/U2) f1(v1)v1v2dv1dcosβ1, (55) This term was originally understood as anomalous resistiv- ity due momentum transfered from electrons to ions by means with P2 denoting the downstream thermal ion pressure, and fur- of Coulomb collisions. This type of friction of course only thermore then leads to (for a definition of the functions A(α) works under conditions of dense plasmas (see Sgro & Nielson and B(α), see Fahr & Siewert 2010) 1976; Leroy et al. 1982). For space plasmas characterized by P2 = 2π(m/2)(U1/U2) small densities this term has meanwhile been understood as due to ion-resonant conditions with respect to plasma waves 4 2 2 × f1(v1)v sin β1A(α) + cos β1 B(α) dv1dcosβ1. (56) (Krauss-Varban & Omidi 1991, 1995; Scholer 1995). However, 1 no satisfactory consistency with respect to the description of this Assuming that the ion distribution function on the upstream side term has been achieved up to now, and therefore, existing nu- is pitchangle isotropic, the integrals with respect to v1 and β1 merical simulations of MHD shock waves are not self-consistent become independent of eachother, yielding with respect to microphysical forces acting on the individual v ions. This means that simulations of shock waves carried out us- = 4 ing the “downstream box” initial boundary conditions can not be P2 2π(m/2)(U1/U2) f1(v1)v1dv1 used to study, reliably and consistently, the (non-)conservation β1 2 2 of the magnetic moment. × sin β1A(α) + cos β1B(α) dcosβ1 (57)

4.2.5. Reaching an asymptotic state which can be evaluated to β One classical problem with numerical simulations is the time re- 1 1 P = P · s · sin2 β A(α) + cos2 β B(α) dcosβ , (58) quired to reach a stable asymptotic state. Typically, this time di- 2 1 2 1 1 1 rectly scales with the quality of the initial conditions, i.e. the less the starting conditions deviate from the final state, the faster and where P1 logically denotes the upstream thermal ion pressure. more reliably does the code converge. Therefore, most simula- Ultimately, this expression then transforms into tions typically start with initial conditions derived from upstream 2 1 values for the moments of the ion distribution (e.g. density and P2 = P1 · s · A(α) + B(α) , (59) pressure) that have been derived using the usual MHD Rankine- 3 3 Hugoniot fluid relations. This approach implicitly assumes adi- which allows one to relate the upstream and downstream pres- = abatic pressure reaction with polytropic index γ 5/3anda sures. This solution is different from what one would obtain from frozen-in magnetic field (see Leroy et al. 1982). This a priori the classical, adiabatic Rankine-Hugoniot equations. prescription of a polytropic shock behavior therefore assumes For the commonly studied configuration of a perpendicular the following relation between upstream (index 1) and down- shock (i.e. α = π/2), this simply transforms to [(2/3)A(α) + stream (index 2) ratios of the thermal to the magnetic pressure, (1/3)B(α)]π/2 = (2/3)s + (1/3) (see Fahr & Siewert 2011), and or of upstream and downstream beta-values for the perpendicu- results in the following equation connecting the plasma β-values lar shock: on both sides of the shock: γ n2 γ−2 2 P1 B /8π + P2 n1 n2 −1/3 P2 1 2 1 2s 1 β2⊥ = = =β1 · =β1 ·(s) . (52) β2⊥ = = β1 · s · s + = β1, (60) 2 2 2 n1 2 2 3 3 3s B2/8π B1/8π (n2/n1) B2/8π B2/8π A38, page 9 of 11 A&A 552, A38 (2013)

100 moment conserved, perp shock electrons at the passage of the plasma over the shock structure moment conserved, parallel shock (see Fahr & Siewert 2011). These relations deliver connections adiabatic, perp shock adiabatic, parallel shock between downstream and upstream values of the ﬁelds, densities and temperatures clearly deviating from those relations elabo- 10 rated on the basis of an adiabatic behavior of ion and electrons pressures. It might then be interesting to study whether these

1 new initial starting conditions are closer or more distant from the β / 2 asymptotic state found by the simulations compared to the con- β ventional approach. Also it might be of interest whether the tur- 1 bulent fluctuation amplitudes are different from the conventionally found ones and perhaps the asymptotic state is even different from the conventionally obtained one.

0.1 5. Conclusions 1 1.5 2 2.5 3 3.5 4 s In this study, we have investigated the question of the conser- Fig. 2. Downstream plasma beta enhancement factor as a function of the vation of the magnetic moment of ions gyrating in an arbitrary MHD compression ratio for the magnetic moment-conserving approach external magnetic field. We have also collected and expanded and the adiabatic approach. general properties of the magnetic moment, it’s connection to the angular momentum and forces acting on the gyrocenter of the ion. This ultimately allowed us to understand the conserva- or, for the case of a parallel shock with [(2/3)A(α) + tion of the magnetic moment in arbitrary situations. In general, 2 (1/3)B(α)]0 = (2/3) + (1/3)s , a full electromagnetic treatment of the gyrating ion is required, which can not be carried out for the most general case; instead, (B2/8π) + 3 numerical models are required to investigate this. = P2 = 1 · · 2 + 1 2 = 2s s β2 β1 s s β1. (61) An innovative induction in our present approach is the in- B2/8π B2/8π 3 3 3 2 2 troduction of overshoot kinetic energy at dealing with shock- heated electrons (see Eqs. (44) through (44)). The induction of To give an idea on the order of magnitude by which the plasma-β this physical process as was demonstrated by Fahr et al. (2012b) can increase or decrease, we present a comparison between the leads to strong nonadiabatic heating of electrons at their pas- magnetic moment-conserving relations (60 and 61) and the adi- sage to the downstream side of the shock. This anomalous heat- abatic relations (52 and 53)inFig.2. These results demonstrate ing may be studied a little more critically in the light of more that the difference between both approximations is on an order conventional approaches to describe shock-heated electrons. In of 2, which can be sufficient to decide whether the downstream a paper by Tokar et al. (1986) it was found from simulations plasma is stable or unstable with respect to e.g. the mirror and that electrons are heated to significantly higher temperatures firehose instabilities. than that expected for adiabatic compression when the shock At the adiabatic jump, the plasma β-value in case of α = − transition happens over only a few ion inertial lengths. As the π/2 is always decreased by values between 1 ≥ s 1/3 ≥ 0.6 authors state this heating cannot be due to cross field instabili- at the passage to the downstream side, while for α = 0itis ties, but is due to direct electron acceleration in the cross shock increased by a factor sγ. In the case of the moment-conserving electric field supplying overshoot kinetic energies. This form jump, and for the perpendicular shock (α = π/2), the plasma of heating seems to occur at HNM’s (i.e. High Mach Number beta is also decreased, however, by slightly less strong values 2s+1 shocks) when upstream electrons can be considered as cold. In between 1.0 ≥ ≥ 0.75, i.e. the magnetic energy always − 3s these simulations a mass ratio μ = m /m = 10 2 is adopted, is preferentially increased with respect to the thermal energy at e,i e i and we emphasize here that applying the correct mass ratio this shock passage. In the case of a parallel moment-conserving − μ = 5.4×10 4 would lead to even stronger electron heating (see shock with α = 0, the downstream β-value is increased by strong e,i 3 Eq. (45)). However, it is interesting to see that these authors find ≤ 2s+s ≤ factors between 1.0 3 24.0. in the HNM-limit of shocks that electrons are energized such To summarize this part of our study, we can point out that that the downstream thermal energy represents a large portion of ff these di erences should definitely impact the generation of dif- the upstream kinetic ion ram energy, i.e. ferent downstream instabilities that automatically appear at these types of shocks. These instabilities then must be treated in a fully 3 1 3s 3 P m U2 n = P ≤ P (62) consistent way, or one will almost certainly produce unstable e,2 16 2 i 1 e,2 16 kini,1 4 kini,1 downstream conditions, where no asymptotic boundary state is obtained. Interestingly enough, just this unstable downstream re- a result, fairly similar to that presented in Eq. (47)ofthis gion is produced by the vast majority of simulation codes, which work here. In a simulation study carried out more recently by might be a hint that the electromagnetic conditions are not imple- Lembège et al. (2003), who were using a mass ratio μe,i = 0.024 mented correctly, and that it is consequently not possible to study it was meanwhile shown that the result showing over-adiabatic the (non-)conservation of the magnetic moment with existing heating for electrons not only applies for HNM shocks, but is numerical methods. also found in shocks with moderate Mach numbers. In order to check the sensitivity of the shock simulation code The results found by Schwartz et al. (1988) point into a sim- with respect to MHD starting conditions, we could imagine it to ilar direction. Investigating fast-mode collisionless shocks start- be useful to start the run of the shock simulations from starting ing from Rankine-Hugoniot relations on the basis of an adia- conditions that are provided by MHD shock relations derived un- batic heating with a polytrope γ = 5/3, they recognize in their der the requirement of conserved magnetic moments of ions and studies that downstream electrons appear to adapt to a much

A38, page 10 of 11 H.-J. Fahr and M. Siewert: The magnetic moment in heliospheric plasmas more effectively heated status with an effective polytropic index Fahr, H.-J., & Siewert, M. 2011, A&A, 527, A125 of γeff = 2.7 through 3.0. They conclude that electron heating is Fahr, H. J., Bird, M. K., & Ripken, H. W. 1977, A&A, 58, 339 nonadiabatic, but largely consistent with the conservation of the Fahr, H.-J., Chashei, I. V., & Siewert, M. 2012a, A&A, 537, A95 Fahr, H.-J., Siewert, M., & Chashei, I. 2012b, Ap&SS, 341, 265 quantity Te,⊥/B, clearly pointing to something like the conserva- Giacalone, J., & Jokipii, J. R. 2006, in Physics of the Inner Heliosheath, tion of the elctron magnetic moment. This seems to indicate that eds. J. Heerikhuisen, V. Florinski, G. P. Zank, & N. V. Pogorelov, AIP Conf. our considerations with respect to the electron overshoot kinetic Ser., 858, 202 energy are not out of any rational context. Jackson, J. D. 1975, Classical electrodynamics Krauss-Varban, D., & Omidi, N. 1991, J. Geophys. Res., 96, 17715 Following this theoretical study, we have investigated com- Krauss-Varban, D., & Omidi, N. 1995, Geophys. Res. Lett., 22, 3271 monly performed numerical simulations and the approximations Kucharek, H., Möbius, E., & Scholer, M. 2006, in Physics of the Inner used by them. As we have demonstrated, there are (at least) five Heliosheath, eds. J. Heerikhuisen, V. Florinski, G. P. Zank, & N. V. different aspects that prevent existing numerical situations from Pogorelov, AIP Conf. Ser., 858, 196 allowing one to study the conservation of the magnetic moment: Kucharek, H., Pogorelov, N., Moebius, E., & Lee, M. A. 2010, in AIP Conf. Ser. 125, 1302, eds. J. Le Roux, G. P. Zank, A. J. Coates, & V. Florinski 1) The small electron mass does not allow to study electrons Kulsrud, R. M. 1982, Phys. Scripta T, 2, 177 and protons on a self-consistent unique length scale. 2) Most nu- Kunstmann, J. E. 1979, ApJ, 229, 812 merical simulations do not allow to define the correct reference Le Roux, J. A., & Webb, G. M. 2006, in Physics of the Inner Heliosheath, frame used to study (electromagnetic) plasma-wave interactions. eds. J. Heerikhuisen, V. Florinski, G. P. Zank, & N. V. Pogorelov, AIP Conf. Ser., 858, 190 3) The electon-to-proton mass ratio used by most simulations Le Roux, J. A., & Webb, G. M. 2007, ApJ, 667, 930 is artificially high and does not allow to reproduce a consistent Le Roux, J. A., Matthaeus, W. H., & Zank, G. P. 2001, AGU Fall Meeting electromagnetic behavior of the individual ions. 4) Many simu- Abstracts, A731 lations of shock waves use boundary conditions that do not con- Le Roux, J. A., Zank, G. P., Matthaeus, W. H., & Milano, L. J. 2002, AGU Fall sistently allow to calculate microphysical forces acting on the Meeting Abstracts, A11 Lembege, B., Giacalone, J., Scholer, M., et al. 2004, Space Sci. Rev., 110, 161 individual ions. 5) Many existing numerical simulations do not Lembège, B., Savoini, P., Balikhin, M., Walker, S., & Krasnoselskikh, V. 2003, seem to arrive at an asymptotically stable state. J. Geophys. Res., 108, 1256 Taking these arguments together, even if a specific code al- Leroy, M. M. 1984, Adv. Space Res., 4, 231 lows to overcome one or more of these underlying complica- Leroy, M. M., Winske, D., Goodrich, C. C., Wu, C. S., & Papadopoulos, K. 1982, J. Geophys. Res., 87, 5081 tions, we are not aware of any numerics that allows to eliminate Li, G., & Zank, G. P. 2006, in Physics of the Inner Heliosheath, all five of these aspects, and therefore, we have to come to the eds. J. Heerikhuisen, V. Florinski, G. P. Zank, & N. V. Pogorelov, AIP Conf. conclusion that no existing numerical plasma code is sufficiently Ser., 858, 183 consistent to study the (non-)conservation of the magnetic mo- Liewer, P. C., Goldstein, B. E., & Omidi, N. 1993, J. Geophys. 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