Parallel Velocity Mixing Yielding Enhanced Electron Heating During Magnetic Pumping
aa r X i v : . [ phy s i c s . p l a s m - ph ] F e b Under consideration for publication in J. Plasma Phys. Parallel Velocity Mixing Yielding EnhancedElectron Heating During Magnetic Pumping
J.Egedal † , J. Schroeder , E. Lichko Department of Physics, University of Wisconsin-Madison, Madison, Wisconsin, USA Lunar and Planetary Laboratory, University of Arizona, Tucson, AZ, USA(Received xx; revised xx; accepted xx)
Magnetic wave perturbations are observed in the solar wind and in the vicinity of Earth’sbow shock. For such environments, recent work on magnetic pumping with electrons trappedin the magnetic perturbations have demonstrated the possibility of efficient energization ofsuperthermal electrons. Here we also analyze the energization of such energetic electrons forwhich the transit time through the system is short compared to time scales associated with themagnetic field evolution. In particular, considering an idealized magnetic configuration we showhow trapping/detrapping of energetic magnetized electrons can cause effective parallel velocity( v k -) diffusion. This parallel diffusion, combined with naturally occurring mechanisms knownto cause pitch angle scattering, such as Whistler waves, produces enhanced heating rates formagnetic pumping. We find that at low pitch angle scattering rates the combined mechanismenhances the heating beyond the predictions of the recent theory for magnetic pumping withtrapped electrons.
1. Introduction
The transport of matter and radiation in the solar wind and terrestrial magnetosphere is acomplicated problem involving competing processes of charged particles interacting with electricand magnetic fields. Given the rapid expansion of the solar wind within the Parker spiral, it wouldbe expected that superthermal particles originating in the corona would cool rapidly as a functionof distance to the Sun. However, observations show that this is not the case and superthermalparticles have been observed out to the termination shock Decker et al. (2008), suggesting thepresence of an additional heating/acceleration mechanism. These superthermal tails have beenobserved to follow a power-law distribution in velocity space Fisk & Gloeckler (2006).Much of the work on a possible explanation for this additional heating centers on wave-particleinteractions as the primary heating mechanism, where energy is provided by the turbulenceassociated with propagating waves Kennel & Petschek (1966); Ergun et al. (1998); Vinas et al. (2000); Tongnyeol Rhee et al. (2006); Califano & Mangeney (2008); Vocks et al. (2005). Inthese models, particles are energized at the resonant velocities, where vk cos Θ ≃ ω , and with cos Θ = v · k / ( vk ) . Particle energization is then limited to v ω/ ( k cos Θ ) . Superthermalelectrons then require energization by waves with large phase velocities v p = ω/k , such asWhistler waves Wilson et al. (2012). However, in many systems, the energy available in Whistlerwaves has been found to be insufficient to explain the observed level of electron energizationand in a recent analysis using MMS data it was found that while whistlers are effective for pitchangle scattering, the whistler bursts did not correlate well with electron energization Oka et al. (2017).Another challenge in using wave-particle interactions to explain the heating in the solar windis the near-ubiquitous observations of power-law distributions of superthermal particles. Such † Email address for correspondence: [email protected] power-law distributions are known to form in Fermi-like heating processes where the energygains of individual particles are proportional to their initial energies, but it is difficult to reproducewith a set of resonant wave-particle interactions.Previous work on magnetic pumping (such as transit-time damping Berger et al. (1971);Barnes (1966); Stix (1992); Lichko et al. (2017)), have largely concluded that pumping is notefficient for energizing superthermal electrons with v > ω/ ( k cos Θ ) . Meanwhile, the pumpingmodels of Ref. Egedal et al. (2018); Lichko & Egedal (2020) include the effects of trappingand differs significantly from earlier results, as the trapping permits especially electrons with v ≫ ω/ ( k cos Θ ) to become energized. Another interesting property that is in contrast tothe turbulent cascade where the energy is transferred from large scales to small scales beforebeing absorbed Sahraoui et al. (2009); Howes et al. (2008), in magnetic pumping the energy isprovided directly by the energy rich largest scale magnetic fluctuations.The results of the present paper can be considered an extension of the magnetic pumpingmodel by Lichko & Egedal (2020). Here we uncover an additional heating mechanism, whichis related to the particular effect of electrons becoming trapped/untrapped in magnetic mirror-structures that form in the presence of compressional wave dynamics. The process leads toparallel energy mixing (or v k -mixing) yielding a net energy gain also for electrons for whichthe magnetic moment, µ , is conserved. In turn, by adding weak pitch angle scattering to thesystem a heating model is obtained similar to that of Lichko & Egedal (2020), with the maindifference being enhanced heating rates of superthermal electrons in the limit of weak pitchangle scattering.The paper is organized as follows: In Section II we evaluate the parallel electron energizationand mixing in a system of magnetic trapping/detrapping where the magnetic moment, µ isconsider an adiabatic invariant. In the drift-kinetic limit and the limit of fast orbit bounce motion,in Section III we show how this v k -mixing leads to parallel diffusion within the trapped partof the electron distributions. In Section IV we add to the model a phenomenological pitchangle scattering and evaluate the net changes to a distribution which after complete v k -diffusionbecomes reisotropized by the pitch angle scattering. We then in Section V consider a scenario ofsimultaneous v k -mixing and pitch angle diffusion, for which we derive an evolution equation forthe slowly varying background electron distribution. In Section VI the new results are discussedand the analysis concluded with a comparison to those of Lichko & Egedal (2020), emphasizingenhanced heating rates for low values of pitch angle scattering.
2. Parallel Energy Mixing
Waves including magnetic perturbations can trap electrons, and considering an idealizedstanding wave configuration Lichko & Egedal (2020) demonstrated the importance of trappingto render magnetic pumping an efficient heating mechanism for superthermal electrons. Ingeneral, however, magnetic perturbations will have a range of wave-lengths, amplitudes andphases such that at different locations along a magnetic flux-tube regions of trapped electronscan develop and interact in a range of ways not accounted for in the previous analysis. Wehere explore how the process of trapping/detapping itself leads to parallel energy mixing ( v k -diffusion), with the result of heating even for the case where the electron magnetic momentsare conserved. The electron dynamics is here well accounted for by the drift-kinetic frameworkpioneered by Kulsrud (1983). In this framework the change of the electron energy is describedby ∂ E /∂t = µ∂B/∂t − e ( v k + v D ) · E , where v k and v D are the field-aligned parallel streamingand guiding center drift, respectively.As a simplifying assumption and similar to Montag et al. (2017); Egedal et al. (2018);Lichko & Egedal (2020), in present analysis we will only consider the electron dynamics in thefast transit time limit. In this limit it is assumed that the timescale associated with the electronbounce motion, τ b , is much shorter than the timescales characterizing the evolution of themagnetic perturbations, such that both the magnetic moment µ , and the parallel action integral J = H v k dl become adiabatic invariants. In addition, we assume a 1D spatial geometry wherethe electrons are confined in a single flux-tube. Orbits of electrons are then fully characterizedby µ and J , and given the fast transit-time limit we can apply the multiple timescale methodDavidson (1972), where f ( x , v ) is approximately constant along the “instantaneous” bounceorbits. Consistent with Jean’s theorem Jeans (1915), it then follows that distributions can beexpressed on the form f = g ( µ, J ) , where g is an arbitrary function.For the present analysis, however, we find it more useful to write f ( µ, E , t ) = f ( µ, E ) , where E is the initial particle energy at a time t of an initial known distribution, f . The problem ofsolving for the distribution f ( µ, E , t ) is then reduced to obtaining a mapping between E ( t ) and E consistent with conservation of µ and J . Because we will only consider prescribed magneticperturbations there are no feedback of f ( µ, E , t ) onto the wave dynamics. Determining themapping E ( t ) → E then becomes a “single particle” problem which can be solved by basicallyconsidering one point in phase-space, ( µ, E , t ) , at a time. For general magnetic perturbations,determining the mapping E ( t ) → E is then a problem well suited for numerical orbit integrationmethods. Here, however, we will consider particularly simple magnetic geometries that allowexplicit expressions for E ( µ, E , t ) to be determined, which (as we will see below) then in turnprovides an explicit solution for the distribution function, f ( µ, E , t ) = f ( µ, E ( µ, E , t )) .To illustrate how v k -diffusion can occur for adiabatic electrons with fixed magnetic moments,in Fig. 1 we consider a magnetic flux-tube with a square-shaped magnetic perturbation charac-terized by a reduced magnetic field B . Again, throughout the analysis we will assume that theelectron transit time is fast compared to the timescale at which the magnetic field is changing.Thus, the evolution of the electron population will be adiabatic and reversible. As illustratedin Fig. 1(a), the magnetic well can trap electrons. The locations inside the magnetic well areparameterized by x ∈ [0; 1] , and at x = d a narrow region is introduced where the magnetic fieldis increased to an enhanced value B T . The width of this enhancement is assumed to be so narrowthat we can neglect any heating µ∂B/∂t that result as it builds in time. Thus, the role of B T issolely to split electron orbits with total energy E < µB T into locally trapped orbits in regions-Aand -B, indicated in Fig. 1(b).From this point the magnetic geometry can be modified in a variety of ways. For example,in appendix A, we analyse the result of changing dynamically the location d of the barrier field B T , and obtain very similar results to those to be derived now for a very different magneticevolution. In this main section we consider a mixing cycle where we slowly raise the magneticfield in region-A until it reaches the value of the barrier, B T . During this process all the electronstrapped in region-A will be energized at a rate µ∂B/∂t and become un-trapped as their totalenergies reach E = µB T . Here the electron orbits will undergo a transition from the blue orbittype to the red orbit type in Fig. 1(c). In region-B the magnetic field is constant and no heatingoccurs, and compared to the trapped orbits in region-A, the red orbit types are subject to reducedheating rates related the reduced fraction of time a given electron is present in region-A. Then,as illustrated in Fig. 1(d) we again lower the magnetic field in region-A and also eliminate theprevious magnetic barrier at x = d . During this process, all the electrons initially confined toregion-B will observe orbit changes corresponding to the transition from the magenta to the redorbit types in Fig. 1(d). All electrons traversing regions-A and -B will be cooled proportionallythe relative faction of the time they spend in region-A.As indicated above, the heating of the various orbit types can in principle be computed byintegrating the orbit averaged value of µ∂B/∂t . Meanwhile, the task of evaluating the energy a)B x b) xB B T R e g i on A R e g i on B c) xB B T B d) xB B F IGURE
1. Sequence of magnetic perturbations considered for parallel velocity mixing, with the coloredarrows indicating distinct trapped orbit types. The deep magnetic square well of a) is in b) modified by aspatially narrow magnetic barrier at x = d with B = B T , separating Regions A and B. In c) the floorof Region A is enhanced until B = B T is reach. In d) the floor of Region A (and the barrier) is reducebringing the configuration back the the initial state in a). changes is significantly simplified by considering the parallel action integral J = r m I v k dl . (2.1)Here the integral is taken over the trapped orbits’ bounce motion, and the unimportant factor p m/ is included to ease the notation below. In addition, the use of J makes the analysis moregeneral as this framework also applies to less idealized configurations where the − e ( v k + v D ) · E -term (mentioned above) becomes important to the energization process Montag et al. (2017).For orbits bouncing through both region-A and region-B we may differentiate between thecontributions from the two regions J = J A + J B , where J A = r m Z d v k dl , J B = r m Z d v k dl . Introducing the present energy E and initial energy E of an electron in the configuration, using v k = p /m √E − µB , the present and initial values of these action contributions are triviallyevaluated as J A = d p E − µB , J A = d p E − µB , and J B = (1 − d ) p E − µB , J B = (1 − d ) p E − µB . In general, the action integrals will be conserved while the magnetic configuration is evolvingslowly in time. An exception to this occurs during the orbit transition in Fig. 1(c), where anew contribution, J BT , from region-B is acquired. Because the energy of a newly transitionedelectron is E = µB T , in region-B the parallel energy will be µ ( B T − B ) and we find J BT = (1 − d ) p µ ( B T − B ) . It then follows that electrons initially trapped in region-A, will after their transition be character-ized by J A + J B = J A + J BT , from which we can relate the initial energy to the present energy E = 1 d h (1 − d ) (cid:16)p E − µB − p µ ( B T − B ) (cid:17) + d p E − µB i + µB . (2.2)While the electrons are confined in region-A they experience the full heating provided by µ∂B/∂t . This heating is stronger than the average cooling they observe when they transit bothregion-A and region-B and B is decreasing in region-A. Consistently, from Eq. (2.2) it is readilyshown that E < E and it follows that all electrons originally in region-A will gain energy duringthe mixing sequence.We may consider the orbit transition in Fig. 1(d) where orbits confined to region-B transitioninto orbits passing through both regions. This transition is different from that described above(where electrons cleared a barrier µB T and fell into a region of lower magnetic field B < B T ,yielding a parallel energy boost). In the present transition there is no localized barrier, and thenewly transitioned electrons will have v k ≃ during their initial traversal of region-A, and thereare no new contribution to the action integral. Thus, for the electrons transitioning out of region-Bwe have J A + J B = J B , such that E = (cid:2) d √E − µB + (1 − d ) √E − µB (cid:3) (1 − d ) + µB . (2.3)From Eq. (2.3) it can be shown that E > E and all electrons which have undergone a transitionout of region-B will observe a net cooling. This is consistent with our expectation because theseelectrons only reach region-A during the period where ∂B/∂t is negative.With Eqs. 2.2 and 2.3 we have obtained expressions for the initial energy as a function ofthe present energy for the orbits which have undergone orbit transitions. Similar expressions arealso readily obtained for the orbits which have not undergone transitions simply by imposing J A = J A , J B = J B , and J A + J B = J A + J B , for the “blue”, “magenta”, and “red” orbittypes, respectively. Thus, the application of the action integral permits a very effective evaluationof E as a function of E for all orbit classes continuous during the mixing sequence.In Fig. 2 we illustrate the evolution of E ( E ) at selected times during the mixing sequence.In this figure the y -axes represent the present total kinetic energy E of electrons with magneticmoments µ , while the color contours describe their spectrum of initial energies E . Fig. 2(a)displays the initial range of relevant energies, where we simply have E = E . The black linesare the energy barriers due to the imposed magnetic field structure. Those electrons with presentenergy E > µB T will be able to overcome the barrier separating region A and region B . Thechanges in color from one panel to the next then describes the evolution of the relationshipbetween E and E , as expressed in the derived equations. Because µ is conserved during thewhole mixing process, and both E and E are proportional to µ , the results of the figure becomeapplicable to any value of µ . At the time of panel (d) the magnetic field in region-A has reachedthat of the barrier B = B T , and all the electrons originally in region-A now have energies largerthan those of region-B. In panels (e,f) the magnetic field of region-A is reduced again and themagnetic barrier at x = d is eliminated.In Fig. 2(g) the magnetic field has returned to its initial state. Consider an electron originallymarginally trapped in region-B with E = µB T , using Eq. 2.3 with B = B we can solve for E to obtain the transition energy between the two electron populations after the mixing cycle iscomplete: E T = (1 − d ) µ ( B T − B ) + µB . Further, using Eqs. 2.2 and 2.3 with B = B we obtain the mapping between E and E after one F IGURE
2. a-g) Color contour plots of the initial energy E as functions of position x and present energy E as observed during the evolution of the magnetic well outlined in Fig. 1 with d = 0 . and B T /B = 8 .Panel h) illustrates the results of two complete mixing cycles. complete mixing cycle E = E − µB (1 − d ) + µB for µB < E < E T d h √E − µB − (1 − d ) p µ ( B T − B ) i + µB for E T < E < µB T E for µB T < E . (2.4)As is evident from Eq. (2.4) and Fig. 2 the energy gain ∆ E = E − E depends on the initial E as well as the initial location of the electrons. The electrons which gain the most energyare initially located in region-A with vanishing parallel energy, such that E = µB . After onemixing cycle these electrons will then have a total energy of E = E T , marked in Fig. 2(g). On theother hand, the electrons that will be cooled the most are originally barely trapped in region B (i.e. E ∼ µB T ). After the mixing cycle, these will also have a present energy E = E T . Meanwhile,region-A electrons with E = µB T as well as region-B electrons with E = µB observe nochance in their energies.To characterize the effect of multiple mixing cycles we can evaluate Eq. 2.4 recursively. Weintroduce E N as the electron energy after N cycles and Eq. 2.4 then implies that E N − = E N − µB (1 − d ) + µB for µB < E N < E T d h √E N − µB − (1 − d ) p µ ( B T − B ) i + µB for E T < E N < µB T E N for µB T < E N . (2.5)With Eq. 2.5 we have now established a direct mapping between the energy E N after N mixingcycles and the initial energy E . An example of E ( E N ) with N = 2 is given in Fig. 2(h).
3. Parallel diffusion of f e As discussed in Section II, the electrons are governed by the drift-kinetic equation, which, forthe considered limit of slow magnetic field evolution and well magnetized electrons (such that F IGURE
3. For the initial distribution in a), the distributions resulting from 1, 2 and 5 mixing cycles areshown in panels b-d), respectively. The distributions are calculated using Eqs. 2.5 and 3.1 with d = 0 . and B T /B = 8 . dµ/dt = 0 ), simply takes the form df e ( E , µ ) /dt = 0 Montag et al. (2017). Assuming an initialelectron distribution f e ( E , µ ) , with Eq. 2.5 we then obtain the distribution that results after N cycles as f e ( E N , µ ) = f e ( E ( E N ) , µ ) , (3.1)where E ( E N ) can be obtained from the recursion relation given in Eq. 2.5. As an example,starting with an initial Maxwellian f e ( E , µ ) shown in Fig. 3(a), the results of 1, 2 and 5 mixingcycles are illustrated in Figs. 3(b-d), respectively.The changes in f e induced by the parallel mixing are fully reversible. However, we note howthe number of stripes in the f e grows like N such that at sufficiently large N the smallestamount of scattering will be sufficient to smooth out the exponentially narrowing stripes. Thiswill render f e independent of E k , such that for the trapped ranges affected by the pumping wehave f e = f e ( µ ) . At this point the mixing process has run its course and no further changes willoccur in f e by parallel mixing alone.While the effect of parallel mixing above was calculated for a highly idealized magneticgeometry it is clear that the cause of the mixing of f e is the orbit transitions of the type introducedwith Fig. 1(c). Therefore, any wave activity that leads to similar orbit transitions will causeequivalent mixing in naturally occurring systems.
4. Changes in f e due to combined E k and pitch angle mixing In the following section we will derive a model for the heating that occurs when pitch anglescattering is included during the continuous mixing described above. We will formulate thismodel in terms of a slowly evolving 1D velocity distribution g ( v, t ) . Any distribution as afunction of speed can be written as an isotropic distribution in ( v ⊥ , v k ) , and we denote an initial2D distribution as f ( v ⊥ , v k ) ≡ h g i ξ , and example of which is shown in Fig. 4(a) for the casewhere g ( v ) is a simple Maxwellian. Furthermore, in our manipulations we will also use h ... i ξ as an operator, which for any 2D distribution yields a distribution fully scattered in the cosine-pitch-angle variable ξ = v k / p v k + v ⊥ .The distribution in Fig. 4(b) represents the result of the v k -diffusion described above forelectrons trapped by B T . We denote this distribution as h g i k corresponding to a distributioncompletely mixed in the v k -direction for the electrons within the trapped region. Here the trappedregion is outlined by the green lines characterized by v k < hv ⊥ , where h = ( B T /B − and B T is the value of the barrier introduced in Fig. 1(b). Mathematically, h g i k is obtained from h g i ξ by particle conservation. In particular, we require that for any v ⊥ the rectangular type areas in the F IGURE
4. Illustration of how the 2D distributions h g i k and D h g i k E ξ are determined from h g i ξ . In a) and b)the green lines are the trapped passing boundaries characterized by E k = h E ⊥ , where h = ( B T /B − .The E k -mixed distribution h g i k is determined from h g i ξ by requiring particle conservation for the velocityphase-space elements of the type encircled in cyan. In turn, D h g i k E ξ in c) is determined from h g i k byrequiring particle conservation for the velocity phase-space elements of the type encircled in red in panelsb) and c). trapped regions of differential width dv ⊥ , as outlined by the areas encircled in cyan in Figs. 4(a)and 4(b), h g i k and h g i ξ contain identical number of particles.We next consider the scenario where h g i k is completely isotropized in pitch angle yieldingthe distribution here denoted D h g i k E ξ . Mathematically, as outlined in Figs. 4(b) and 4(c) thisdistribution is also determined by imposing particle conservation, this time requiring that for any v the differential speed elements dv , as outlined by the areas encircled in magenta in Figs. 4(b)and 4(c), contain identical number of particles in h g i k and D h g i k E ξ . In Appendix B we show that D h g i k E ξ ≃ h g + δg i ξ , (4.1)where δg = h (cid:18) h h (cid:19) / v ∂∂v v ∂∂v g , (4.2)and, repeated for convenience, h = B T /B − .
5. Evolution of the Background Distribution
Above we introduced the 1D distribution g = g ( v, t ) for characterizing the isotropic compo-nent of the background plasma. The main goal of the present section is to derive an evolutionequation that describes the slow evolution of g . To accomplish this we need to consider the full2D distribution, which we approximately describe as a linear combinations of the fully v k -mixeddistribution h g i k and the fully pitch angle scattered distribution h g i ξ , such that f = (1 − α ) h g i ξ + α h g i k . (5.1)The parameter α will be determined below and is dependent on the drive frequency, ν k , of theparallel mixing compared to the characteristic frequency, ν ξ , of the pitch angle diffusion.We further approximate the parallel mixing and pitch angle diffusion in terms of Krook-typeoperators, allowing us to write the kinetic equation as ∂f∂t = ν k (cid:16) h f i k − f (cid:17) + ν ξ (cid:16) h f i ξ − f (cid:17) , (5.2)which we through numerical analysis (not included) find is a reasonable approximation for ν k & ν ξ . Here ν k describes the characteristic frequency of the v k -diffusion process, whichwill be on the order of the frequencies describing the magnetic perturbations. Similarly, ν ξ is thecharacteristic frequency of the pitch angle scattering process. -2 0 2-7 -5 -3 -1 B T /B = 1.1, 1.2, 1.5, 2.3, 5v || -mixing Magnetic Pumpinglog ( / || ) l og ( G ) F IGURE
5. Blue lines: The energization rate G by v k -mixing as a function of ν ξ /ν k , calculated usingEq. (5.9) for B T /B ∈ { . , . , . , . , } . Indicated by full lines, the theory is expected to be valid for ν ξ /ν k < / . Red lines: For comparison the efficiency of magnetic pumping the red lines represent thesimilar G in Eq. 6 of Ref. Lichko & Egedal (2020), evaluated with ν/f pump = ν ξ /ν k and C K = 1 , andconsidering the same magnetic perturbations as applied for the v k -mixing. Inserting Eq. (5.1) into Eq. (5.2) yields ∂f∂t = − K (cid:16) h g i k − h g i ξ (cid:17) + ν ξ α h δg i ξ , (5.3)with K = ν ξ α − ν k (1 − α ) , (5.4)where we have used Eq. (4.1) together with the rules D h g i ξ E ξ = h g i ξ , D h g i ξ E k = h g i k , D h g i k E k = h g i k . Note that the two first of these rules follow because g ( v ) is isotropic such that h g i ξ = g .Next we use that direct differentiation of Eq. (5.1) with respect to time yields ∂f∂t = ˙ α (cid:16) h g i k − h g i ξ (cid:17) + (1 − α ) h ˙ g i ξ + α h ˙ g i k , (5.5)where we used the notation ˙ g = ∂g/∂t and ˙ α = ∂α/∂t . Matching the terms in Eqs. (5.3) and(5.5) proportional to (cid:16) h g i k − h g i ξ (cid:17) we find ˙ α = − K = ν k (1 − α ) − ν ξ α . This provides an evolution equation of the level of anisotropy parameterized by α . At a timescaleon the order of /ν k , the value of α will approach the steady state solution described by K = 0 for which α = ν k ν k + ν ξ . (5.6)Inserting Eq. (5.6) and ˙ α = K = 0 into Eqs. (5.3) and (5.5) and equating the resulting twoexpressions for ∂f /∂t we find0 ν ξ ν k ν k + ν ξ h δg i ξ = ν ξ ν k + ν ξ h ˙ g i ξ + ν k ν k + ν ξ h ˙ g i k . Then, by taking the h ... i ξ -average and using the approximation that D h ˙ g i k E ξ ≃ h ˙ g i ξ the righthand side simplifies and we find h ˙ g i ξ = ν ξ ν k ν k + ν ξ h δg i ξ . (5.7)By inspection of Eq. (5.2) it becomes clear that the approximation D h ˙ g i k E ξ ≃ h ˙ g i ξ abovecorresponds to the neglect of a second order time-derivative term of the approximate size (1 /ν k ) ∂ ˙ g/∂t .Given Eq. (5.7), we may now apply the form in Eq. (4.2) to obtain the desired evolutionequation for g ( v, t ) ∂g∂t = 1 v ∂∂v v D ∂∂v g , D = ν k v G , (5.8)where G = ν ξ ν k + ν ξ h (cid:18) h h (cid:19) / , h = B T B − . (5.9)We have boxed Eq. (5.8) as it represents the main result of our analysis.
6. Discussion and Conclusion
The expression for ∂g/∂t in Eq. (5.8) has the form of velocity diffusion, where the diffusioncoefficient D ∝ ν k v describes a process with a diffusive step-size proportional to velocity δv ∝ v . Equivalently, the diffusive energy step-size is proportional to energy, as is characteristicof a Fermi-heating process. It is readily seen that a power-law distribution of the form g ∝ v − γ with γ = 3 , represents a steady state solution to Eq. (5.8). For a more realistic representation ofa physical system, particle sources and sinks can be added to Eq. (5.8). In general, this will leadto power-law solutions with γ > Montag et al. (2017). The diffusion equation in Eq. (5.8) (aswell as the similar form obtained in Ref. Lichko & Egedal (2020)) is therefore consistent withthe power law distributions recorded in situ by spacecraft through out the solar wind and theEarth’s magnetosphere.For wave dynamics with a typical magnitude ˜ dB = dB/B we may approximate B T ≃ B + dB and it follows that h ≃ ˜ dB . Given the dependency of Eq. (5.8) on h , the efficiency ofthe energy diffusion for small ˜ dB scales as ˜ dB / , but falls off to a linear scaling for larger orderunity wave amplitudes. The present model is obtained using the Krook-model in Eq. 5.2, whichfrom preliminary numerical results (not included here) is found to be a good approximationwhen the system is characterized by weak pitch angle scattering, ν ξ /ν k . / . In Fig. 5 the bluelines illustrate the predictions of Eq. (5.9) for G , evaluated for the amplitudes of B T listed in thefigure, and the full lines for ν ξ /ν k < / correspond the range where the model is expected to beaccurate. For ν ξ /ν k & / the Krook model in Eq. (5.2) becomes inaccurate because the pitchangle scattering will cause the region-A and region-B electrons (see Fig. 1(b)) to mix withoutthe separate µ∂B/∂t heating/cooling of the two regions. In fact, for ν ξ /ν k & we expect thatheating by magnetic pumping will be more efficient than heating by v k -mixing.For comparison, the red lines in Fig. 5 is obtained from the model of magnetic pumpingwith trapped electrons developed in Lichko & Egedal (2020), where in Eq. 6 an expression isgiven for the form of G pump due to magnetic pumping, and we evaluate G pump for the same1magnetic perturbations as yielded the blue lines in Fig. 5. Furthermore, in this comparison G pump is obtained assuming ν/f pump = ν ξ /ν k . The model for G pump also includes a factor, C K , thatcalibrates the efficiency of a Krook scattering model to the efficiency of the Lorentz scatteringoperator. The Krook scattering model implemented here in Eq. (5.2) is equivalent to C K = 1 ,and is thus the value used in calculating the red lines.We observe that the predicted heating from v k -mixing is up to two orders of magnitude largerthan that expected from magnetic pumping. Physically, this result is reasonable because netenergization in the magnetic pumping model requires pitch angle scattering during each pumpingcycle. In contrast, the v k -mixing yields finite E k -energization even if ν ξ = 0 (corresponding tothe changes in the distributions between Figs. 4(a) and 4(b)). As in Landau damping, for thelimit of ν ξ = 0 the process is fully reversible for the hypothetical case where the mixing cycle isexactly reversed. But given the fine scales structures that develops in velocity space after just afew mixing cycles, such are revesal is unlikely to occur in any physical system. As emphasizedin Lichko & Egedal (2020), the electron energization is caused by mechanical work through theterm p ⊥ ∇ · v ⊥ and is linked to the development of pressure anisotropy, which in the pumpingmodel is continuously being isotropized by pitch angle scattering. Meanwhile, for the v k -mixingthis anisotropy can build during multiple mixing cycles and becomes more pronounced than theanisotropy that develop during a single magnetic pump cycle.The regime with ν ξ /ν k ≪ / is likely to be relevant to the solar wind for which a recentanalysis of the Strahl-electrons show that pitch angle scattering is mostly limited to the low levelprovided by Coulomb collisions between electrons and ions Horaites et al. (2019). Meanwhile,for MMS bow-shock encounter analyzed in Lichko & Egedal (2020) we estimate that ν ξ /ν k ≃ / , whereas the analysis of a similar MMS bow-shock event Amano et al. (2020) infer muchlarger values of ν ξ /ν k . In future studies of insitu spacecraft data, to help determine the relevantvalue of ν ξ /ν k of a given dataset, we note that Eqs. (5.1) and (5.6) can be fitted to electron dataas provided by for example NASA’s Magnetospheric Multiscale (MMS) mission Burch et al. (2016) and may prove useful for inferring ν ξ /ν k directly from the observations.The magnetic configurations considered here are highly idealized. This adds to the needfor developing new analytical and numerical techniques for simultaneously evaluating heatingby both v k -mixing and magnetic pumping for more general magnetic perturbation geometries.Nevertheless, while the configurations considered are useful for providing physical insight tothe heating mechanism, we expect that the energization theoretical rates obtained will proverepresentative also for naturally occurring systems. This is emphasized by the result that the verydifferent scenarios considered in the main text and Appendix A, respectively, provide similarlevels of v k -mixing.
7. Appendix A
In this Appendix we consider a modified v k -mixing scenario, which turns out to yield resultsvery similar to those of Sections 2 and 3. As outlined in Fig. 6(a) we again consider a magneticbarrier at x = d with height µB T , separating the spatial dimension into region A and region B.We then examine the v k -mixing that occurs as region A is expanded at the expense of region B,corresponding to the location of the barrier being moved from d ′ = d towards d ′ = 1 . Requiringagain that the parallel action integrals be conserved ( J A = J A ) , it follows that the region Aelectrons are being cooled with E = (cid:18) d ′ d (cid:19) ( E − µB ) + µB . (7.1)2 F IGURE
6. Illustration of v k -mixing by changing the location of a magnetic barrier initially located at x = d . In a-d) the color-contours represent E as a function of x and E , with the initial profile in a), while b ) and c are computed with the barrier moved to d ′ = 0 . and d ′ = 1 , respectively. d) correspond to theresult of two complete mixing cycles. e-g) Electron distributions computed using Eq. (7.6) for 1, 2, and 5complete mixing cycles, respectively. Meanwhile, the region B electrons are being heated and the initial energy E and present energy E are similarly described by E = (cid:18) − d ′ − d (cid:19) ( E − µB ) + µB . (7.2)As region B is contracting the electrons confined to this region are all subject to v k heating andwill eventually reach the energy E = µB T where they can overcome the barrier. After clearingthe barrier they will immediately experience the cooling of region A and will therefore becometrapped in region A. For an initial value of E we obtain from Eq. (7.2) the value of d ′ = d T when this transition occurs (1 − d T ) = (1 − d ) (cid:18) E − µB µ ( B T − B ) (cid:19) . (7.3)To further established the relationship between E and E after the transition, we characterisethe subsequent cooling in region A for d T d ′ . Considering Eq. (7.1), it follows that thiscooling must be governed by µB T = (cid:18) d ′ d T (cid:19) ( E − µB ) + µB . (7.4)Combining Eqs. (7.3) and (7.4) by eliminating d T while solving for E we obtain E = µ ( B T − B )(1 − d ) − d ′ s E − µB µ ( B T − B ) ! + µB . (7.5)We further introduce the transition energy E T = d d ′ µ ( B T − B ) + µB , obtained by solving Eq. (7.1) for E with E = µB T . The electrons in region A are thencharacterized by Eq. (7.1) for µB E E T , and by Eq. (7.5) for E T E µB T . Theelectrons in region B are characterized by Eq. (7.2) for the full interval µB E µB T .With the initial barrier at d = 0 . , the derived relationship between E and E is illustrated in3Figs. 6(b,c) evaluated for d ′ = 0 . and d ′ = 1 , respectively. Note that electrons with E > µB T are not affected by the changes in the location of the magnetic barrier.In the present scenario, a mixing cycle is complete when d ′ = 1 and all electrons are thencharacterized by the region-A expressions. Similar to the derivation in Section 2, we readilyobtain recurrence relations for the impact of N complete mixing cycles: E N − = E N − µB d + µB for µB < E N < E T µ ( B T − B )(1 − d ) (cid:18) − r E N − µB µ ( B T − B ) (cid:19) + µB for E T < E < µB T E N for µB T < E N , (7.6)and because d ′ = 1 the transition energy is here characterized by E T = d µ ( B T − B ) + µB . In Fig. 6(d) we display the predictions of Eq. (7.6) computed for 2 mixing cycles, and similarto the distributions in Section 3, Fig. 6(e-g) display the distributions that result after 1, 2, and5 cycles. Although the mixing process here is different from that of Sections 2 and 3, the finalresult is again a rapid v k mixing and diffusion for the magnetically trapped electrons.
8. Appendix B
We will here derive the expression for δg given in Eq. (4.2). For this we apply the procedureoutlined in Fig. 4 imposing particle conservation between h g i ξ , h g i k and D h g i k E ξ for the differ-ential velocity region encircled by the cyan and magenta lines, respectively. In our analysis wewill consider the distributions on the form f ( E k , E ⊥ ) normalized such that n = R f ( E k , E ⊥ ) d v .Because dv k = d E k mv k = d E k q m E k , πv ⊥ dv ⊥ = 2 πm d E ⊥ we have n = π √ m / Z Z f ( E k , E ⊥ ) 1 E k / d E k d E ⊥ . (8.1)First, the background distribution h g i ξ is isotropic, but during the mixing process rapid dif-fusion occur in E k for all the electrons trapped by B T . Again, the trapped electrons are thosewith E k < h E ⊥ , where h = B T /B − , and within this fully diffused velocity region of h g i k is independent of E k ; we will characterize this part of the distribution as f ⊥ ( E ⊥ ) , i.e. f ⊥ ( E ⊥ ) = h g i k for E k < h E ⊥ . From Eq. 8.1, particle conservation for the differential velocityregions outlined in cyan in Figs. 4(a,b) then imposes that ∆ E ⊥ Z h E ⊥ f ⊥ ( E ⊥ ) 1 E k / d E k = ∆ E ⊥ Z h E ⊥ g ( E ) 1 E k / d E k or f ⊥ ( E ⊥ ) = 12( h E ⊥ ) / Z h E ⊥ g ( E ) 1 E k / d E k . For approximate evaluation of this integral we use that E = E k + E ⊥ and Taylor expand g about4 E ⊥ , such that g ( E ) ≃ g ( E ⊥ ) + g ′ ( E ⊥ ) E k + 12 g ′′ ( E ⊥ ) E k , and it then follows that f ⊥ ( E ⊥ ) ≃ h E ⊥ ) / (cid:20) g E k / + 23 g ′ E k / + 15 g ′′ E k / (cid:21) h E ⊥ , ≃ g ( E ⊥ ) + 13 h E ⊥ g ′ ( E ⊥ ) + 110 ( h E ⊥ ) g ′′ ( E ⊥ ) . (8.2)For what comes next, Taylor expansion of f ⊥ ( E ⊥ ) to second order becomes useful f ⊥ ( E ⊥ ) ≃ f ⊥ ( E ) − f ⊥′ ( E ) E k + 12 f ⊥′′ ( E ) E k . We then use Eq. 8.2 to obtain expressions for the derivatives of f ⊥ such that f ⊥′ ( E ) ≃ (cid:18) h (cid:19) g ′ ( E ) + h E g ′′ ( E )) + 15 h E g ′′ ( E ) , and f ⊥′′ ( E ) ≃ (cid:18) h h (cid:19) g ′′ ( E ) . Using the cosine-pitch-angle variable introduced above we have E k = ξ E , and it follows that f ⊥ ( E ⊥ ) ≃ g ( E ) + 13 h E g ′ ( E ) + 110 ( h E ) g ′′ ( E ) − ξ E (cid:18)(cid:18) h (cid:19) g ′ ( E ) + (cid:18) h h (cid:19) E g ′′ ( E ) (cid:19) + ξ E (cid:18) h h (cid:19) g ′′ ( E ) . (8.3)With Eq. 8.3 we now have an expression for the diffused region of h g i k in terms of g , readilyevaluated as a function of E and ξ . To continue and obtain an expression for δg = D h g i k − h g i ξ E ξ we apply that the number of particles in the differential speed intervals encircled in magentain Figs. 4(b,c) must be identical. In general, the number of particles in an interval dv can becomputed as πv dv R f ( v, ξ ) dξ . Meanwhile, h g i k and h g i ξ only differ in the trapped regioncharacterized by ξ k , where k = h/ (1 + h ) . It then follows that δg Z dξ = Z k ( f ⊥ ( E ⊥ ) − g ( E )) dξ . Here, of course, R dξ = 1 , and we proceed to evaluate directly the right-hand-side using Eq. 8.3well suited for the required integration over ξ at constant E : δg ≃ kh E g ′ ( E ) + 110 k ( h E ) g ′′ ( E ) − k E (cid:18)(cid:18) h (cid:19) g ′ ( E ) + (cid:18) h h (cid:19) E g ′′ ( E ) (cid:19) + 110 k E (cid:18)(cid:18) h h (cid:19) g ′′ ( E ) (cid:19) , δg ≃ A E g ′ ( E ) + B E g ′′ ( E ) , (8.4)with Ak = 13 h − k (cid:18) h (cid:19) , and Bk = 110 h − k h − k h + 110 k (cid:18) h h (cid:19) . Using k = h/ (1 + h ) , the expressions for A and B reduce to A = 29 h (cid:18) h h (cid:19) / , B = 25 A + 475 h (cid:18) h h (cid:19) / . We further notice that Eq. (8.4) can also be written as δg ≃ A v ∂g∂v + B (cid:18) − v ∂g∂v + v ∂ g∂v (cid:19) . To within the order and accuracy of the applied Taylor expansions we have B = 2 A/ and theresult stated in Eq. (4.2) then follows from simple manipulations: δg ≃ A
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