Electrostatic steepening of whistler waves
Ivan Y. Vasko, Oleksiy V. Agapitov, Forrest S. Mozer, John W. Bonnell, Anto V. Artemyev, Vladimir V. Krasnoselskikh, Yuguang Tong
EElectrostatic steepening of whistler waves
I.Y. Vasko , ∗ O.V. Agapitov , , F.S. Mozer , J.W. Bonnell , A.V. Artemyev , , V.V. Krasnoselskikh , and Y. Tong Space Sciences Laboratory, University of California, Berkeley, USA Space Research Institute RAS, Moscow, Russia National Taras Shevchenko University of Kyiv, Ukraine University of California in Los Angeles, Los Angeles, USA and LPC2E, University of Orleans, France (Dated: January 31, 2018)We present surprising observations by the NASA Van Allen Probes spacecraft of whistler waveswith substantial electric field power at harmonics of the whistler wave fundamental frequency. Thewave power at harmonics is due to nonlinearly steepened whistler electrostatic field that becomespossible in the two-temperature electron plasma due to whistler wave coupling to the electron-acoustic mode. The simulation and analytical estimates show that the steepening takes a few tensof milliseconds. The hydrodynamic energy cascade to higher frequencies facilitates efficient energytransfer from cyclotron resonant electrons, driving the whistler waves, to lower energy electrons.
Whistler waves play fundamental role in electron ac-celeration in space [1, 2], solar wind [3] and astrophysical[4] plasmas and continuously stimulate laboratory plasmaexperiments [5, 6]. In particular, whistler waves controlthe dynamics of the Van Allen radiation belts [7], wherethey are generated via the cyclotron resonant instabilitywith regularly injected ∼
10 keV anisotropic electrons[8]. Whistler waves mediate the energy of the injectedelectrons to higher and lower energy electrons via theresonant interaction resulting in electron acceleration upto relativistic energies [1, 2] and electron losses to theatmosphere [9].Whistler waves are typically observed in the form ofquasi-monochromatic wave packets propagating quasi-parallel to the background magnetic field [10]. Thewhistler wave field is decomposed into the electrostaticfield along the wave vector k and elliptically polarizedelectromagnetic field perpendicular to k [11]. Whistlerwaves are fundamentally different from compressiblesound waves in fluids and plasmas [12, 13] in that thewhistler electrostatic field has been argued not to steepen[14]. In accordance, the reported whistler waves in theVan Allen radiation belts typically have quasi-sinusoidalwaveforms even at the highest observed amplitudes [15].Slightly non-sinusoidal waveforms have been attributedto electrons trapped within the whistler electrostatic field[16].In this Letter we present whistler waves with sur-prisingly significant electric field power at harmonics ofthe fundamental frequency that is due to highly non-sinusoidal waveform of the whistler electrostatic field.The wave energy cascade to higher frequencies is dueto the classical hydrodynamic steepening that becomespossible in the two-temperature electron plasma.The twin NASA Van Allen Probe spacecraft launchedon August 30, 2012 into the Van Allen radiation beltsprovide wave and particle measurements with unprece-dent time resolution. We present Van Allen Probe Ameasurements on May 1, 2013 near the Earth magnetic dipole equator at the geocentric radial distance of 5.5Earth radii. During the considered time interval thebackground magnetic field and electron density are about87 nT and 3.2 cm − . The electron cyclotron and plasmafrequencies are f c ∼ . f p ∼
16 kHz.Figure 1 shows one second of the waveform of whistlerwaves continuously present for more than ten secondsaround 11:27:25 UT. The left panels show the waveformin the coordinate system related to the background mag-netic field. Surprisingly, the waveform of the parallelelectric field is highly non-sinusoidal, in contrast to theother electric and magnetic field components. The mid-dle panels present 30 ms of the waveform in the coordi-nate system with the Z axis along the whistler wave vec-tor that is the direction delivering minimum to the rootmean square of div B [17]. In this coordinate system thewhistler wave field is decomposed into the electrostaticfield E z and the electromagnetic field in the XY plane(Fig. 2).The selected 30 ms whistler wave packet propagatesat the wave normal angle θ ∼ ◦ . The electromagneticfields have quasi-sinusoidal waveforms and their spectraare peaked at f ∼
450 Hz that is about 0 . f c . Thereis a good correlation between E y and B x in accordancewith the Faraday’s law, E y /B x provides the phase veloc-ity estimate of about 15000 km/s that is in reasonableagreement with c (cid:2) f ( f c cos θ − f ) /f p (cid:3) / ∼ E z has highly non-sinusoidal waveformwith pronounced negative electric field spikes showing upin the wave spectrum as electric field power at harmonicsof ∼
450 Hz. Van Allen Probe measurements show thatthe whistler waves are associated with the presence of thetwo-temperature electron population.Figure 3 presents cuts of the electron spectrum corre-sponding to fluxes of electrons with pitch angles around0 ◦ and 90 ◦ . The energy spectrum is anisotropic abovea few keV as required to excite whistler waves via thecyclotron resonant instability [8]. The high-energy part a r X i v : . [ phy s i c s . s p ace - ph ] J a n of the electron spectrum is fitted to the κ − distributionresulting in density ∼ − and temperature ∼ . − indicates the density of thelow-energy electron population (below ∼ − . The temperature of the low-energy populationis below a few hundred eV.We address the whistler wave dynamics in the two-temperature electron plasma using the hydrodynamicand Maxwell equations. Because of the high whistlerwave frequency ions can be considered as immobilecharge neutralizing background [11]. In the coordinatesystem shown in Fig. 2 the hydrodynamic equations forthe two electron populations can be written as ddt (cid:20) u j − eA x mc (cid:21) = − πf c v j cos θ,ddt (cid:20) v j − eA y mc (cid:21) = 2 πf c ( u j cos θ + w j sin θ ) ,dw j dt = em ∂ Φ ∂z − m n j ∂ ( T j n j ) ∂z − πf c v j sin θ −− emc (cid:20) u j ∂A x ∂z + v j ∂A y ∂z (cid:21) ,dn j dt = − n j ∂w j ∂z , ddt ≡ ∂∂t + w j ∂∂z , where j = l, h corresponds to the low- and high-energy populations, n j , ( u j , v j , w j ) and T j are elec-tron densities, bulk velocities and temperatures, − e and m are electron charge and mass. The Maxwellequations for the electrostatic and vector potentialsare ∂ Φ /∂z = 4 πe (cid:16)(cid:80) j n j − n (cid:17) and ∂ A /∂z =(4 πe/c ) (cid:80) j ( n j u j ˆ x + n j v j ˆ y ), where n is the unper-turbed electron density and the displacement currentis neglected, because ∂ /∂t (cid:28) c ∂ /∂z . We ne-glect the thermal spread of the low-energy population asnot qualitatively affecting the whistler wave dynamics.The high-energy population is assumed to be isothermal T h = const as in the theory of ion-acoustic waves [13],because its thermal velocity is higher than the whistlerwave phase velocity. By linearizing the equations we ob-tain the dispersion relations of linear waves propagatingin the observed plasma.Figure 4 shows that the whistler wave dispersionrelation is not affected by the thermal spread andcoincides with the cold dispersion relation, f ≈ f c cos θ k c / ( k c + 4 π f p ) [11]. The two-temperatureelectron plasma supports electrostatic electron-acousticwaves, which dispersion relation at long wavelengths is2 πf ≈ k v EA cos θ , where v EA = ( T h /m ) / ( n c /n ) / is the electron-acoustic velocity [18, 19]. Whistler andelectron-acoustic waves turn out to be coupled in the crossover points at f ∼ . f c and ∼ . f c . The mostpronounced effect around the crossover frequencies is seenin the electron compressibility that is the ratio of ampli-tudes of compressional w j and non-compressional bulkvelocities. The hot and cold electron populations be-come highly compressible around the crossover frequen-cies, although the full electron compressibility remainsnegligible.We address the nonlinear evolution of whistler wavesby solving the hydrodynamic and Maxwell equations us-ing the energy conserving numerical scheme based on theFast Fourier Transform previously used for analysis ofsteepening of electron-acoustic waves [20]. The initialcondition is a monochromatic whistler wave of a realis-tic amplitude. We have found that whistler waves withthe frequency far from the crossover frequencies remainundisturbed in accordance with the previous simulations[14]. On the contrary, whistler waves with the frequencyaround the crossover frequencies exhibit a fundamentallydifferent behavior.Figure 5 presents evolution of a whistler wave with thefrequency f around the first crossover frequency corre-sponding to the wave number k ∼ πf p /c . The evolutionof the electrostatic potential exhibits signatures of theclassical hydrodynamic steepening (overtaking) inherentto sound waves in fluids and plasmas [12, 13]. The steep-ening produces the negative electric field spikes in theelectrostatic component. The spikes become quite pro-nounced after about 40 ms and resemble those in obser-vations. In accordance with observations, the magneticfield B x and the other electromagnetic components re-main practically undisturbed.The steepening of the whistler electrostatic field isidentical to the steepening of electron-acoustic waves[20]. In fact, the physics of the effect is equivalent.An initially monochromatic sufficiently long electron-acoustic wave ( f, k ) produces harmonics ( (cid:96)f, (cid:96)k ) that arein phase with the fundamental wave due to the lineardispersion relation at long wavelengths. Similarly, thewhistler wave ( f , k ) produces electron-acoustic waves( (cid:96)f , (cid:96)k ) that are in phase until ( (cid:96)k ) − becomes com-parable to the dispersive scale of the electron-acousticmode that is the Debye length λ D = ( T h / πn e ) / .Therefore, the number of harmonics may not exceed (cid:96) ∼ ( k λ D ) − ∼ c ( T h /m ) − / ∼
10 that is consis-tent with the observed spectrum. Because the steep-ening of long electron-acoustic waves is known to bedescribed by the Korteweg-de Vries equation [21], wecan estimate the steepening time of the whistler elec-trostatic field as τ s ∼ A ( mT h ) / ( eE ) − , where E isthe initial amplitude of the whistler electrostatic field and A − = ( n h /n c ) / (3+ n c /n h ) [20, 21]. Assuming the typ-ical amplitude E ∼
10 mV/m we find τ s ∼
50 ms thatis consistent with the simulation results.After about 60 ms the energy leaks out of the steep-ening region due to the electron-acoustic wave dispersionconverting the electric field spikes into oscillations that isinconsistent with observations. However, we have verifiedthat inclusion of the collisional Burgers dissipation [12]does provide the persistency of the spikes. In collision-less plasma the dissipation is provided by the resonantwave-particle interaction. The wave-particle interactioneffects were originally included into the hydrodynamicdescription of ion-acoustic waves [22]. The recent sim-ulations of electron-acoustic [20] and Alfven [23] waveshave confirmed that the wave-particle interaction doesprovide persistency of the electric field spikes and mag-netic field pulses, respectively.A whistler wave can efficiently exchange energy withelectrons with the parallel velocity v (cid:107) satisfying the res-onance condition, ω − k (cid:107) v (cid:107) = nω c , where ω = 2 πf , ω c = 2 πf c , k (cid:107) = k cos θ and n = 0 , ± , ... [24]. Thesteepening of the whistler electrostatic field provides thewave energy cascade to higher frequencies and wavenumbers opening the door for many more resonances: ω − k (cid:107) v (cid:107) = nω c /(cid:96) . This facilitates the efficient energymediation from the cyclotron resonant electrons, drivingthe whistler wave, to other electron populations, in par-ticular, to lower energy electrons.In summary, whistler waves propagating in the two-temperature electron plasma can exhibit the classicalsteepening producing pronounced spikes in the whistlerelectrostatic field. The steepening occurs due to whistlerwave coupling to the electron-acoustic mode and becomesnoticeable for whistler wave packets with non-negligiblewave power at frequencies f satisfying 4 πn c T h /B ∼ f /f c cos θ − ( f /f c cos θ ) . The steepening explains sur-prising observations of whistler waves with significantelectric field power at harmonics of the whistler wavefundamental frequency in the Van Allen radiation belts.The work of I.V., O.A., F.M. and J.B. was performedunder JHU/APL Contract No. 922613 (RBSP-EFW). ∗ [email protected] [1] R. M. Thorne et al., Nature (London) , 411 (2013).[2] F. S. Mozer et al., Phys. Rev. Lett. , 035001 (2014).[3] S. P. Gary et al., Astrophys. J. , 142 (2012).[4] V. Petrosian and S. Liu, Astrophys. J. , 550 (2004).[5] B. Van Compernolle et al., Phys. Rev. Lett. , 145006(2014).[6] B. Van Compernolle et al., Phys. Rev. Lett. , 245002(2015).[7] C. F. Kennel and H. E. Petschek, J. Geophys. Res. ,1 (1966).[8] Y. Omura et al., in Dynamics of the Earth’s RadiationBelts and Inner Magnetosphere , American GeophysicalUnion, edited by D. Summers, I. U. Mann, D. N. Baker,and M. Schulz (2013) pp. 243–254.[9] R. M. Thorne et al., Nature (London) , 943 (2010).[10] O. Santol´ık et al., J. Geophys. Res. , 1278 (2003).[11] R. A. Helliwell,
Whistlers and Related Ionospheric Phe-nomena, by R.A. Helliwell. Stanford, CA: Stanford Uni-versity Press, 1965 (1965).[12] L. D. Landau and E. M. Lifshitz,
Course of theoreticalphysics, Oxford: Pergamon Press, 1959 (1959).[13] R. Z. Sagdeev, Reviews of Plasma Physics , 23 (1966).[14] Yoon et al., J. Geophys. Res. , 1851 (2014).[15] C. Cattell et al., Geophys. Res. Lett. , L01105 (2008).[16] P. J. Kellogg et al., Geophys. Res. Lett. , L20106(2010).[17] B. U. ¨O. Sonnerup and M. Scheible, ISSI Scientific Re-ports Series , 185 (1998).[18] K. Watanabe and T. Taniuti, J. Phys. Soc. Jap. , 1819(1977).[19] S. P. Gary and R. L. Tokar, Phys. Fluids , 2439 (1985).[20] C. S. Dillard et al., Phys. Plasmas accepted (2018).[21] R. L. Mace et al., J. Plasma Phys. , 323 (1991).[22] E. Ott and R. N. Sudan, Phys. Fluids , 2388 (1969).[23] M. V. Medvedev et al., Phys. Rev. Lett. , 4934 (1997).[24] D. Shklyar and H. Matsumoto, Surveys in Geophysics , 55 (2009).[25] J. R. Wygant et al., Space Sci. Rev. , 183 (2013).[26] C. A. Kletzing et al., Space Sci. Rev. , 127 (2013).[27] H. O. Funsten et al., Space Sci. Rev. , 423 (2013). FIG. 1. The left panels present electric and magnetic field waveforms measured with cadence 16384 samples/s by the ElectricField Instrument [25] and Electric and Magnetic Field Instrument Suite and Integrated Science [26] aboard the Van Allen ProbeA on May 1, 2013. The waveforms are in the coordinate system related to the background (DC) magnetic field: E ⊥ , B ⊥ areone of the electric and magnetic field components perpendicular to the background magnetic field, while E (cid:107) , B (cid:107) are parallel toit. The middle panels present the waveform over 30 ms in the natural coordinate system shown in Figure 2. The right panelpresents the spectrum of the electrostatic E z and electromagnetic E x , B x fields for the selected 30 ms. B θ Z X B x B y E z FIG. 2. The coordinate system with the Z axis along thewave vector directed at wave normal angle θ with respect tothe background magnetic field B . The whistler wave fieldis decomposed into the electrostatic field E z and electromag-netic field in the XY plane. FIG. 3. The spectrum (phase space density) of elec-trons with pitch angles around 0 ◦ (streaming parallel tothe background magnetic field) and around 90 ◦ (stream-ing perpendicular to it) computed by converting the fluxesmeasured by the Helium Oxygen Proton Electron (HOPE)detector [27] over 10 seconds around 11:27:25 UT. Thehigh-energy part of the spectrum is fitted to F ( E ) = n C κ ( m/ πκ E ) / [1 + E /κ E ] − ( κ +1) , where C κ = Γ( κ +1) / Γ( κ − / n is the density and T = 2 κ E / (2 κ −
3) isthe temperature. The fitting parameters κ , n and T are pre-sented in the panel.FIG. 4. The dispersion relation of linear waves propagating inthe observed two-temperature electron plasma below the elec-tron cyclotron frequency and above the low-hybrid frequency(ions are considered as immobile charge neutralizing back-ground). The dispersion relations are computed for the wavenormal angle θ = 15 ◦ . The dashed green curve shows the dis-persion relation of whistler waves in a cold plasma computedby setting zero thermal spread of the high-energy popula-tion. The small panel presents the compressibility of the high-energy population and the full electron compressibility de-fined as the ratio of amplitudes of the compressional velocities w h and w and non-compressional bulk velocity ( u + v ) / ,where ( u , v , w ) = (cid:80) j n j ( u j , v j , w j ) /n is the full electronbulk velocity. FIG. 5. The evolution of initially monochromatic whistler wave with a realistic amplitude and frequency f around the firstcrossover frequency ∼ . f c . The simulation results are presented in the whistler wave reference frame. The electrostaticpotential and electrostatic field are normalized to m ( cf c cos θ/f p ) /e and m ( cf c cos θ/f p ) /ed e , where d e = c/ πf p is theelectron inertial length. The magnetic field B x is normalized to the background magnetic field B . The electromagnetic fieldcomponents E x , E y and B y have profiles identical to B xx