Bi-Directional Energy Cascades and the Origin of Kinetic Alfvénic and Whistler Turbulence in the Solar Wind
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Bi-Directional Energy Cascades and the Origin of KineticAlfv´enic and Whistler Turbulence in the Solar Wind
H. Che, M. L. Goldstein, and A. F. Vi˜nas
NASA/Goddard Space Flight Center, Greenbelt, MD, 20771, USA
Abstract
The observed ion-kinetic scale turbulence spectrum in the solar wind raises the question of howthat turbulence originates. Observations of keV energetic electrons during solar quiet-time suggestthem as possible source of free energy to drive kinetic turbulence. Using particle-in-cell simulations,we explore how the free energy released by an electron two-stream instability drives Weibel-likeelectromagnetic waves that excite wave-wave interactions. Consequently, both kinetic Alfv´enic andwhistler turbulence are excited that evolve through inverse and forward magnetic energy cascades.
PACS numbers: 96.60.Vg,52.35.Ra, 94.05.Lk, 52.25.Dg ∝ k − / , is replaced by a steeper [1–3] anisotropic scalinglaw B k ⊥ ∝ k − α ⊥ , where α is a number larger than 5 /
3. It is found that the observed spectralindex is α ≈ /
3, but this value is not universal and varies from interval to interval. Magneticfluctuations with about tenth of ion gyro-frequency propagating nearly perpendicularly tothe solar wind magnetic field are identified as kinetic Alfv´enic waves (KAWs) [4–9] and thebreak frequencies of the magnetic power-spectra appear to follow the ion inertial length[3, 10, 11]. The origin of the KAW turbulence is still unknown. In this letter, we address theorigin of kinetic turbulence by proposing a source of free energy that has not been exploredpreviously. For the first time we find that an inverse energy cascade appears to play a crucialrole in generating both KAW and Whistler turbulence.Observations using the STEREO spacecraft have found that even during quiet-time pe-riods, the solar wind contains a previously unknown electron population different from thecore solar wind, called “superhalo electron”, with energy ranging in ∼ −
20 keV [14, 15].One possible origin of the superhalo electrons is the escaping nonthermal electrons relatedto coronal nanoflares in the quiet solar atmosphere (Parker 1988 [16]; Lin 1997 [14]). Therelative drift of these nonthermal electrons to the background solar wind can drive an elec-tron two-stream instability in a neutral current [17], and release the free energy to the solarwind. The impact of this unstable process on the solar wind has so far not been studied. Inthis letter, using particle-in-cell (PIC) simulations, we investigate how the rapidly releasedenergy drives Weibel-like electromagnetic waves. The wave-wave interactions on ion inertialscales d i = c/ω pi and electron inertial scales d e = c/ω pe generate KAW and whistler turbu-lence through both forward and inverse energy cascades. At the end of this letter, we willcompare the testable features produced by this model with observations.We initialize the 2.5D PIC simulations in the solar wind frame of reference with a uniformmagnetic field B = B ˆ x . Both the ion and electron densities are uniform. The initial ionvelocity distribution function (VDF) is a single isotropic Maxwellian. The electron VDF isa core-beam isotropic bi-Maxwellian. The core is the solar wind electrons and the beam is2he energetic electrons. Their relative drift is along B : f e = (cid:16) m e πk (cid:17) / " − δT / c e − m e ( v e ⊥ +( v ex − v cd )) / kT c + δT / b e − m e ( v e ⊥ +( v ex − v bd )) / kT b , where v ⊥ = v ey + v ez , δ = n b /n . n is the solar wind density and the density normalizationunit, n b is the density of beam electrons. T c is the temperature of core, T b is the temperatureof beam, v cd is the drift of the core and v bd is the drift of the beam. The drift velocitiessatisfy (1 − δ ) v cd = − δv bd to maintain null current. v bd = 12 v te = 60 v A , where v A is theAlfv´en speed and v te = p kT c /m e is the thermal velocity of the core electrons. The energy ofthese beam electrons will be released and join the core electrons at energy ∼ kT c . We choose δ = 0 .
1, because at ∼
10 keV, or ∼ kT c , the superhalo electrons have a density of ∼ − of the solar wind density [15], and we assume the kinetic energy flux density of beam n b v bd / c = 100 v A and the massratio is m i /m e = 100. The ion temperature T i = T c . The boundaries are periodic in bothdirections with a box size L x = L y = 32 d i . The total number of cells in each dimension is10,240 and the total number of particles is ∼ . The total simulation time ω pe t = 10 , E = v A B /c . We take kT b = 2 kT c = 0 . m i v A and β = kT c /B = 0 .
25 estimated from the solar wind β observations at 0.3 AU[11].Electron two-stream instability occurs early at ω pe t = 24 as shown in Fig. 1a, and h δE x i (solid black line) quickly reaches a peak at ω pe t ≈
50, where hi denotes the average over xy .At ω pe t = 200, the drift of the beams decreases from 60 v A to ∼ v A , and δE x decreasesby nearly a factor of 20 and then stays nearly constant. The growth rate of the electrontwo-stream instability at ω pe t = 24 is close to the cold plasma limit of γ b ∼ ( n b / n ) / ω pe ∼ . ω pe . The fastest growing mode k f,x = ω pe /v db ∼ /d i is consistent with the spectrum of | δE x ( k x , k y ) | at ω pe t = 24, as shown in Fig. 1b.The fast growth of δE x generates an inductive magnetic field B z that satisfies B z ∼ δE x ∆ yc ∆ t ∼ . B , which is close to the middle value of B z shown in Fig. 2a, where we take∆ y ∼ λ f,x = 2 π/k f,x ∼ d e , ∆ t ∼ /γ b ∼ . ω − pe . The middle value of δE x ∼
20 duringthe instability is estimated from Fig. 1. The internal energy density released per wavelengthper is ∼ m e n b (∆ v db ) ∆ y/ (2 ω pe ∆ t ) ∼ . n m i v A ω − pe λ − f,x where ∆ v db ∼ v A . Around 10% isconverted into magnetic energy B / π ∼ . n m i v A ω − pe λ − f,x at the end of the two streaminstability, while nearly 90% is converted into the thermal motion of trapped electrons [19].The electric current density j ex produced by the inductive magnetic field becomes as3 IG. 1. Panel a: Time evolution of energy of h δE x i (black line) and h δB x i (red line), h δB y i (green line), h δB z i (blue line). The embedded plot is an expanded view of the time evolution from ω pe t = 0 − ω pe t = 230. Panel b: Power spectrum of | δE x ( k x , k y ) | at ω pe t = 24 on a logarithmic scale. important as the displacement current when the two-stream instability starts to decay.Then j ex drives a Weibel-like instability that generates nearly non-propagating transverseelectromagnetic waves. The variances ( δB z ) / and ( δB x ) / in Fig. 1a reach a second peak at ω pe t ≈ δB x ) / follows ( δB z ) / closely, while the variance ( δB y ) / reachesits peak at a slightly later time. A significant change from electrostatic waves to transverseelectromagnetic waves can be seen in the evolution of j ex , shown in Fig. 3. The j ex wavevector induced by the two-stream instability is along x . Gradually, the wave vector rotatesso that it is parallel to y , which indicates the generation of electromagnetic fluctuations in B z that align along y . The wavelength of B z fluctuations increases to half d i as seen inFig. 2b at ω pe t = 480, near the peak of the Weibel-like instability.4 IG. 2. Images of B z /B at ω pe t = 24 (panel a ), 480 (panel b ), 2424 (panel c ), and 10560 (panel d ). Please refer to the movie in the supplement. The decay of the Weibel-like instability enhances the interactions between the localizedcurrents and the nearly non-propagating transverse electromagnetic waves. This processbreaks up the transverse waves and produces randomly propagating waves as shown inFig 2c. From ω pe t = 2400, the wave-wave interactions dominate the dynamics. The wave-wave interactions lead to a momentum transfer from perpendicular to parallel magnetic field.As a result, parallel propagating waves appear, which is consistent with the fact that a peakappears in ( δB y ) / at ω pe t = 2400 (Fig. 1). Finally at ω pe t = 10 , > ◦ and nearly parallel waves are excited (Fig. 2d).The wave-wave interactions drive a bi-directional energy cascade. The perpendicularmagnetic wave energy is now transferred from the electron inertial scale back to the ioninertial scale, and the parallel magnetic wave energy is transferred from the ion inertialscale down to the electron inertial scale. The 2D power spectra of δB z at ω pe t = 24, 480,5 IG. 3. The transition of j ex wave patten when Weibel-like instability occurs. Panel a : j ex at ω pe t = 96, the late stage of two-stream instability; panel b : j ex at ω pe t = 168, the transition stagefrom the two-stream instability to the Weibel-like instability; panel c at ω pe t = 240, the beginningof the Weibel-like instability. ω pe t = 24, we only see atransverse mode peaked k y d i ∼
10, i.e., k y d e ∼
1, which is consistent with the wavelengthof the inductive magnetic field B z that was produced by the two-stream instability. At ω pe t = 48, the Weibel-like instability generates a transverse electromagnetic magnetic fieldwith longer wavelengths. At ω pe t > k x d i <
1, and the parallel branch with k y d i <
2. We study the time evolution of the magnetic components of waves, the resultsshow that both wave types are right-hand polarized. During the evolution, the magneticwave-wave interactions forms localized thin current sheets with widths from several d e to d i .Some of which might be caused by magnetic reconnections (supplementary Fig. 2).The frequency of the nearly perpendicular wave is around 0 . − . i where Ω i is the ion6 IG. 4. Power spectra | B z ( k x , k y ) | on logarithmic scale at ω pe t = 24 (panel a), 480 (panel b),2424 (panel c), and 10560 (panel d). cyclotron frequency. From the dispersion relation of KAW given by two-fluid equation [20] ω k x v A = 1 + k y ρ s k y d e (1)where ρ s = d e v te /v A , we estimate k y d i < k x d i ∼ .
01 and the electron thermal velocityis larger than the initial velocity v te > T c /m e = 25 v A . The resulting KAW k y d i is consistentwith the spectrum shown in Fig. 4d. The frequency of the parallel branch is ω ∼
10 Ω i andthe wavenumber is kd i ∼ ω pe t = 10 ,
560 , which satisfies the whistler wave dispersionrelation ω/ Ω i = v A ( kd i ) cos θ [21] for θ ∼
0. But at the transition time ω pe t = 2424, kd i ∼
4, then θ ∼ and k x d i ∼ .
5. Thus the oblique whistler wave evolves to parallel.The ratio of δB x /δB y ∼ k ± k = k is the dominant process in wave-wave interactionsand leads to the simultaneous generation of KAWs and whistler waves. For perpendicular7 IG. 5. The 1D spectra of δB ( k ) vs. k x d i and k y d i . The blue short-dashed line is the k y spacerange for magnetic energy injection. d i = 2 ρ i and d e = 2 ρ e in the simulation where ρ i,e are the ion(electron) gyro-radius, thus k x,y ρ i = 2 and k x,y ρ e = 20 interactions, the major contribution is from k kaw ⊥ , ± k whistler ⊥ , = k kaw ⊥ , , where | k whistler ⊥ , | ≪ | k kaw ⊥ , | ,thus k kaw ⊥ moves to smaller wavenumbers and the magnetic energy transfers from small scaleto large scale. For parallel interactions, the major contribution is from k kaw k , ± k whistler k , = k whistler k , , where | k kaw k , | ≪ | k k , | whistler , thus k whistler k moves to larger wavenumbers and themagnetic energy cascades down to the small scales.In Fig. 5, we show 1D power spectra of the magnetic energy δB ( k ) vs. k x (parallelspectrum) and δB ( k ) vs. k y (perpendicular spectrum) at ω pe t = 10560. Whistler waveenergy cascades from ion to electron scales and it is clear that the contribution to the parallelspectrum in k x on ion scale is from whistler waves. The perpendicular spectrum has a bumpat k y d i ∼ − ω pe t = 24 (Fig. 4 a). Then the wave-wave interactions inversely transferthe KAW energy to ion scale smaller than d i and generate the whistler waves (Fig. 4d).8hus, both the whistler waves and KAWs contribute to the perpendicular spectrum on ionscale while only KAWs contribute to the perpendicular spectrum on electron scale L e with d e > L e > ρ e . The spectrum is much steeper on L e since the wave-particle interactions aremuch more stronger. The parallel spectrum on scale smaller than d e and the perpendicularspectrum on scale smaller than ρ e suggest exponential decays that imply the dissipationprocesses are less space and time correlated. The plateaus between the power law and theexponential decays indicate that the energy is accumulated by the strong thermalization.After ω pe t = 10560, the free energy is almost fully released and the induced turbulentscattering produces a nearly isotropic electron halo superposed over the core electrons. Theenergy exchange between particles and waves reaches balance. The turbulence reaches itsnew steady state with P + B / π = constant , P is the total pressure of ions and elec-trons.The ratio of amplitude of the magnetic fluctuations and background magnetic field isabout 0.2 and matches the current observations of solar wind kinetic turbulence. The decayrate of fluctuations is ≪ − Ω i in the last 3000 ω − pe ∼ − i , estimated from the simulation.This suggests that the kinetic turbulence will be preserved for a long time. If superhaloelectrons are produced in the sun, then kinetic turbulence is produced within a few solarradii. The resulting turbulence should be stable enough to travel to 1AU based on the decayrate estimated from our simulation if we take Ω i ∼ d e . The spectral index of -2.2 in our spectrum agrees with those found inobservations. In our simulation the KAW as well as the parallel propagating high frequencywhistler waves contribute to the power spectrum. In ref. [22], the authors suggest that ∼
10% of the solar wind data they analysed consists of parallel propagating whistler waves asdetermined by their right-handed polarization, but more advanced observations are needed.2) Since the growth rate of two-stream instability is related to ω pe , the spectral breakpointsof KAWs follow the ion inertial length. This agrees with observations of KAW turbulence[3,10, 11]. 3) At the final stage, enhanced by the relic parallel electric field from the two-streaminstability, h| E k | / | E ⊥ |i ∼ −
3, consistent with the observations that the parallel electricfield is larger than the perpendicular electric field expected for KAW [23]. 4) During theevolution of the turbulence, microscopic current sheets with widths varying from several d e to d i are produced (supplementary Fig. 2.), consistent with the observations of kinetic scale9urrent sheets discovered in solar wind turbulence[24, 25]. 5) Our simulations show that anearly isotropic halo is produced at the finale stage. Such halos are observed from 0.3-1 AUin slow wind[18] (supplementary Fig. 1). The formation of superhalo requires the electronbeam energy extend by more than an order of magnitude higher, rendering computationsrather expensive due to the higher c/v A ratio and higher temporal and spatial resolutionsrequired.The observations of kinetic turbulence on electron scale would be more challenging sincethe power-spectra on electron scale is much steeper and quickly become exponential. Theongoing Magnetospheric Multiscale Mission might be able to detect the kinetic process onelectron scale at 1AU.It is important to know how other possible turbulent processes in the solar wind affectthe KAWs and whistler waves when the they travel to 1 AU. The index of the powerspectra might be affected more while other features produced in our model may be slightlyor unaffected: the frequency breakpoints determined by ion inertial length, the enhancedparallel electric field and the electron halo. The current observations of solar wind can reach0.3 AU. The near future space missions Solar Probe Plus and Solar Orbiter can reach 10solar radii and hence provide more rigorous constraints on this model.This model proposed is motivated by the observations of superhalo. It is encouraging thatthe model could potentially link the existing observations of solar wind kinetic turbulence,the halo formation, and the electron acceleration and heating processes in solar corona intoa coherent picture. More advanced studies will be carried out in the near future. ACKNOWLEDGMENTS
This research was supported by the NASA Postdoctoral Program at NASA/GSFC ad-ministered by Oak Ridge Associated Universities through a contract with NASA. The simu-lations and analysis were carried out at the NASA Advanced Supercomputing (NAS) facilityat the NASA Ames Research Center, and on Kraken at the National Institute for Compu-tation Sciences. 10
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