Resonance of low-frequency electromagnetic and ion-sound modes in the solar wind
I. Y. Vasko F. S. Mozer T. Bowen J. Verniero X. An A. V. Artemyev J. W. Bonnell J. Halekas I. V. Kuzichev

Generalized Optimization Modulo Theories
    Nestan Tsiskaridze10000-0002-4729-9770
Clark Barrett10000-0002-9522-3084
Cesare Tinelli20000-0002-6726-775X
    
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Parker Solar Probe measurements have recently shown that coherent fast magnetosonic and Alfvén ion-cyclotron waves are abundant in the solar wind and can be accompanied by higher-frequency electrostatic fluctuations. In this letter we reveal the nonlinear process capable of channelling the energy of low-frequency electromagnetic to higher-frequency electrostatic fluctuations observed aboard Parker Solar Probe. We present Hall-MHD simulations demonstrating that low-frequency electromagnetic fluctuations can resonate with the ion-sound mode, which results in steepening of plasma density fluctuations, electrostatic spikes and harmonics in the electric field spectrum. The resonance can occur around the wavenumber determined by the ratio between local sound and Alfvén speeds, but only in the case of oblique propagation to the background magnetic field.  The resonance wavenumber, its width and steepening time scale are estimated, and all indicate that the revealed two-wave resonance can frequently occur in the solar wind. This process can be a potential channel of energy transfer from cyclotron resonant ions producing the electromagnetic fluctuations to Landau resonant ions and electrons absorbing the energy of the higher-frequency electrostatic fluctuations.








§ INTRODUCTION



Spacecraft measurements at 0.3–1 AU showed that circularly-polarized electromagnetic fluctuations with the frequency of the order of local proton cyclotron frequency can be continuously present in the solar wind for up to tens of minutes <cit.>. These low-frequency fluctuations are either left- or right-handed in the  spacecraft frame and propagate nearly parallel to local magnetic field lines. Refs. <cit.> interpreted these fluctuations in terms of Alfvén ion-cyclotron waves, left-handed in the plasma frame. The observations of right-handed fluctuations in the spacecraft frame were suggested to be caused by the Doppler shift of Alfvén ion-cyclotron waves propagating sunward in the plasma frame. The stability analysis of ion velocity distribution functions at 1 AU showed however that the low-frequency fluctuations can be also fast magnetosonic waves, right-handed in the plasma frame <cit.>. The occurrence of the low-frequency fluctuations is probably much higher than a few percent reported at 0.3–1 AU <cit.>, since turbulence can mask their presence <cit.>. The substantially larger occurrence would be consistent with the fact that ion velocity distribution functions in the solar wind are typically unstable to low-frequency electromagnetic fluctuations <cit.>. 

 



Parker Solar Probe (PSP) and Solar Orbiter measurements have recently shown that similar low-frequency fluctuations are abundant closer to the Sun <cit.>. The occurrence of these fluctuations is around 30% within 0.3 AU <cit.>. The analysis of particle distribution functions showed that the low-frequency fluctuations can be both Alfvén ion-cyclotron and fast magnetosonic waves produced by local ion-driven instabilities <cit.>. In contrast to earlier analyses at 0.3–1 AU, PSP has provided electric field measurements that allowed quantifying Doppler shifts and establishing that the low-frequency fluctuations can be indeed left- and right-handed in the plasma frame <cit.>. The low-frequency fluctuations may contribute to solar wind heating <cit.>, but specific dissipation mechanisms are still investigated. Using electric field measurements, Ref. <cit.> have shown that the low-frequency electromagnetic fluctuations can be accompanied by electrostatic fluctuations at higher frequencies. They hypothesized that some nonlinear process channels the energy from the electromagnetic to electrostatic fluctuations, but the physics of that nonlinear process has not been revealed. 




In this letter we demonstrate that electrostatic fluctuations reported by Ref. <cit.> are produced due to resonance of slightly oblique electromagnetic (fast magnetosonic) fluctuations with the ion-sound plasma mode. The resonance results in steepening of low-frequency plasma density fluctuations, electrostatic spikes phase-correlated with the electromagnetic fluctuations and harmonics in the electric field spectrum. The resonance condition, its width and steepening time scale are estimated. The potential effect of this nonlinear process in the dissipation of low-frequency electromagnetic fluctuations is discussed.







§ DATA AND OBSERVATIONS



We use magnetic and electric field measurements provided at respectively about 146 and 293 S/s (Samples per second) by the FIELDS instrument suite <cit.>, ion and electron moments provided at respectively 0.2 and 28s cadence by the SWEAP instrument suite <cit.>, and plasma density estimates at 7s cadence delivered by the quasi-thermal noise spectroscopy <cit.>. The electric field aboard PSP is measured by four 2-meter long cylindrical whip sensors mounted at the edges of the forward heat shield and providing two electric field components in the XY plane perpendicular to the Z axis that is the sunward radial direction. The electric field is calibrated through a least-squares fit to -V_i×B_0, where ion flow velocity V_i and background magnetic field B_0 are averaged over 12s <cit.>. The effective antenna length at the frequencies of interest, 1–10 Hz, is between about 0.5 and 2 of its DC value revealed by the least squares fit <cit.>. 

Figure <ref> overviews a one hour interval of PSP observations on April 1, 2019. Panels (a) and (b) show that the magnetic field of about 55 nT was directed almost radially toward the Sun, while the plasma was flowing radially outward at about 340 km/s. The plasma density was around 230 cm^-3 during the entire interval according to the quasi-thermal noise spectroscopy (not shown). The electron and ion temperatures were both around 30 eV (not shown), though ion velocity distributions were not Maxwellian <cit.>, the corresponding electron and ion betas were β_e≈β_i≈ 1. Panels (c) and (d) present power spectral densities (PDS) of magnetic field components in the XY plane and along the radial direction. The PSDs demonstrate that narrow-band magnetic field fluctuations around 4 Hz were continuously present for about 20 minutes. These fluctuations must have propagated within a few degrees of the background magnetic field, since the PSD of the radial component was a hundred times lower than that of the components in the XY plane. Panel (e) presents the PSD of two electric field components and demonstrates the expected narrow-band electric field fluctuations around 4 Hz. These fluctuations are accompanied by the wave power at harmonics of the fundamental frequency of 4 Hz. Panel (f) shows distinct harmonics in the electric field PSD averaged over the 20 minute interval, but no similar harmonics in the averaged magnetic field PSD. There are only three distinct harmonics visible, but panel (g) demonstrates up to ten harmonics in a PSD computed over a 3s interval. 













Figure <ref> presents electric and magnetic field waveforms observed over that 3s interval. Panel (a) demonstrates electric field waveforms, while panels (b) and (c) show its low- and high-frequency constituents computed by band-pass filtering over 2–6 Hz and 6–146 Hz, respectively. Panel (d) presents similarly computed low-frequency magnetic field. The fluctuations around 4 Hz are circularly-polarized and right-handed, both electric and magnetic fields rotate in the direction of electron gyration. In contrast, the high-frequency fluctuations consist of linearly-polarized electric field spikes phase-correlated with the low-frequency electromagnetic fluctuations. It is these electrostatic spikes that create harmonics in the electric field spectra. Panels (e) and (f) demonstrate that the low-frequency electric and magnetic fields in the XY plane are correlated and the best-fit slopes are E_X/B_Y≈-715 km/s and E_Y/B_X≈ 1070 km/s. Since the low-frequency fluctuations propagate almost perpendicular to the XY plane, these correlations imply outward propagation from the Sun in the spacecraft frame. The actual phase speed estimate is however dependent on the effective antenna length to be used.





Figure <ref> presents properties of the low-frequency electromagnetic fluctuations. For each 10s subinterval of about 20 minute interval, we estimated E_X/B_Y and E_Y/B_X, and also applied the Minimum Variance Analysis <cit.> to compute unit vector k̂ along the minimum variance direction that is equivalent to the propagation direction. Panels (a) and (b) show that -E_X/B_Y and E_Y/B_X are consistent and typically within 400–900 km/s, while the propagation direction is within 175^∘–180^∘ of the background magnetic field. The electromagnetic fluctuations are either fast-magnetosonic waves propagating radially outward or Alfvén ion-cyclotron waves propagating sunward in the plasma frame. The phase speed of slightly oblique fast magnetosonic and Alfvén ion-cyclotron waves in the plasma frame is respectively larger and smaller than the Alfvén speed v_A (Section <ref>). Therefore, the phase speed in the spacecraft frame would exceed V_i·k̂+v_A≈ 420 km/s in the case of fast magnetosonic waves, while would be smaller than V_i·k̂-v_A≈ 260 km/s in the case of Alfvén ion-cyclotron waves, where v_A≈ 80 km/s and V_i·k̂≈ 340 km/s. The observed values of E_Y/B_X≈ -E_X/B_Y≈ 400–900 km/s indicate that the interpretation in terms of fast magnetosonic waves is consistent with the effective antenna length varying within a factor of two of its DC value. This interpretation is also consistent with linear instability of fast magnetosonic waves for several ion velocity distribution functions in the considered interval <cit.>. Panels (c) and (d) present wavenumber kλ_i, spacecraft and plasma frame frequencies, f_sc/f_ci and f/f_ci, where f_ci and λ_i are local ion cyclotron frequency and inertial length. The parameter f_sc/f_ci≈ 4 was determined by the peak value of the PSD, while f/f_ci and kλ_i were related through the cold plasma dispersion relation (Section <ref>) and computed using the Doppler-shift relation, f_sc/f_ci=f/f_ci+kλ_iV_i·k̂/v_A. Resolving this relation we found plasma frame frequency f/f_ci≈ 1.2 and wavenumber kλ_i≈ 0.75. In the next section we reveal the mechanism resulting in the formation of electrostatic spikes associated with fast magnetosonic waves. The schematics of the wave propagation direction in the coordinate system XYZ is shown in Figure <ref>.   









§ THEORETICAL INTERPRETATION AND NUMERICAL SIMULATIONS


The electromagnetic fluctuations and electrostatic spikes observed aboard PSP have frequencies below ion plasma frequency and spatial scales above electron inertial length. In this case the plasma dynamics can be described within Hall-MHD approximation, which neglects electron inertia and assumes identical electron and ion densities (e.g., Ref. <cit.>). The Hall-MHD equations can be written as follows

    ∂ N/∂ t   +   ∇ (NV)=0,
    
    ∂V/∂ t+(V∇)V   =   -∇ P/m_iN+ curl B× (B_0+B)/μ_0m_iN,
    
    ∂B/∂ t   =   - curl E,
    
    E+V× (B_0+B)   =   -∇ P_e/eN+ curl B× (B_0+B)/μ_0 eN,

where N and V are the plasma density and flow velocity, P=P_e+P_i is the total of electron and ion thermal pressures assumed to be isotropic, E and B_0+B are electric and magnetic fields, B_0 is a uniform background magnetic field. These equations are complemented by equations of state, P_e/N^γ_e= const and P_i/N^γ_i= const, where γ_e and γ_i are polytrope indexes. We will demonstrate that it is the electron pressure gradient term in the Ohm’s law that results in the electrostatic spikes (Section <ref>). 


Ref. <cit.> presented the linear theory of waves in warm uniform plasma using the Hall-MHD approximation. In the plasma rest frame the corresponding linear dispersion relation between angular frequency ω and wavenumber k of monochromatic waves propagating at wave normal angle θ to the background magnetic field reads

    (cos^2θ-ω^2/k^2v_A^2)·(cos^2θ-ω^2/k^2c_s^2-ω^2/k^2v_A^2(1-ω^2/k^2c_s^2))=ω^2/ω_ci^2cos^2θ(1-ω^2/k^2c_s^2),

where ω_ci=2π f_ci=eB_0/m_i is the angular ion cyclotron frequency. The sound speed c_s, Alfvén speed v_A and their ratio are given as follows

    c_s=(γ_eT_e+γ_iT_i/m_i)^1/2,      v_A=(B_0/μ_0 N_0 m_i)^1/2,      c_s/v_A=(γ_eβ_e+γ_iβ_i/2)^1/2,

where N_0, T_e and T_i are background plasma density, electron and ion temperatures. In cold plasma, ω/kc_s→∞, the dispersion relation simplifies to (1-ω^2/k^2v_A^2)(cos^2θ-ω^2/k^2v_A^2)=ω^2cos^2θ/ω_ci^2 and describes fast magnetosonic and Alfvén ion-cyclotron modes. In warm plasma the dispersion relation already delivers three wave modes. In the case of strictly parallel propagation these are a purely electrostatic ion-sound mode, whose dispersion relation is ω=kc_s, and purely electromagnetic fast magnetosonic and Alfvén ion-cyclotron modes. In the case of oblique propagation the three modes are named fast, intermediate and slow, whose phase speeds v_F, v_I and v_S at ω/ω_ci→ 0 are ordered as v_S<v_I<v_F and delivered by the classical expressions, v_I=v_Acosθ and v_F,S^2=(v_A^2+c_s^2)/2±((v_A^2+c_s^2)^2/4-v_A^2c_s^2cos^2θ)^1/2. The asymptotic analysis of Eq. (<ref>) at small wave normal angles, sin^2θ≪ 1, and low frequencies, sin^2θ≪ω/ω_ci≪ 1, shows that in the case of c_s>v_A the fast mode is practically dispersionless, while the intermediate and slow modes are dispersive

    (ω^2/k^2)_F≈ v_F^2,     (ω^2/k^2)_I≈ v_I^2(1+ω/ω_cicosθ),     (ω^2/k^2)_S≈ v_S^2(1-ωcosθ/ω_ci).

Note that at small wave normal angles we have v_F≈ c_s(1+v_A^2sin^2θ/2(c_s^2-v_A^2))≈ c_s, v_I=v_Acosθ≈ v_A, and v_S≈ v_A(1-c_s^2sin^2θ/2(c_s^2-v_A^2))≈ v_A. 












Figure <ref> presents the dispersion curves of fast, intermediate and slow modes propagating at wave normal angle of θ=5^∘ in a warm plasma with c_s/v_A≈ 1.4, which corresponds to β_e=β_i=1 and polytrope indexes of γ_e=1 and γ_i=3. Panel (a) shows ω/ω_ci versus kλ_i, while panel (b) demonstrates ω/kv_A versus kλ_i, where λ_i is the ion inertial length. Both panels also present the dispersion curves of fast magnetosonic and Alfvén ion-cyclotron waves in cold plasma. The slow mode can be clearly associated with the Alfvén ion-cyclotron wave, while neither fast nor intermediate modes alone can be associated with the fast magnetosonic wave. The dispersion curve of the intermediate mode is almost identical with that of the fast magnetosonic wave at k≲ k_*, while becomes close to ω=kc_s at k≳ k_*. In turn, the dispersion curve of the fast mode is close to ω=kc_s at k≲ k_* and almost identical to that of the fast magnetosonic wave at k≳ k_*. The intermediate and fast modes are closely approaching each other at k≈ k_* that can be named a resonance point between the ion-sound and fast magnetosonic waves. Assuming (ω^2/k^2)_F≈ (ω^2/k^2)_I and using Eq. (<ref>) we estimate the wavenumber and frequency of the resonance point 

    ω_*/ω_ci≈cosθ(v_F^2/v_I^2-1)≈c_s^2/v_A^2-1,       k_*λ_i≈v_F/v_I-v_I/v_F≈c_s/v_A-v_A/c_s,

where we have taken into account that sin^2θ≪ 1. In the considered case of c_s/v_A≈ 1.4 we obtain ω_*≈ω_ci and k_*λ_i≈ 0.7 that is consistent with panels (a) and (b). 


Below we use coordinate system xyz shown in panel (c) with unit vector z along the propagation direction k̂ and unit vector x in the plane of k̂ and B_0. Note that unit vector k̂ is oriented at angle θ to -B_0. The electric and magnetic field fluctuations E and B in Eqs. (<ref>) can be decomposed into electromagnetic components E_⊥ and B_⊥ in the xy plane, and the electrostatic component E_|| along the wave vector. Note that B_||=0, because divB=0. The  plasma flow velocity is similarly decomposed into compressible and incompressible components, V_|| and V_⊥. The analysis of linear monochromatic fluctuations, whose quantities are proportional 
to e^i(kz-ω t), reveals the following relations

    N/N_0=1+kv_A/ωV_||/v_A,    V_||/v_A=-kv_Asinθ/ω/1-k^2c_s^2/ω^2B_x/B_0,
    
          V_⊥/v_A=-k v_Acosθ/ωB_⊥/B_0,   E_⊥/v_AB_0=-ω/kv_Az×B_⊥/B_0,
    
          B_y/B_0=iωcosθ/ω_ci/ω^2/k^2v_A^2-cos^2θB_x/B_0,    
    
                    E_||/v_AB_0=tanθ(
        ω/kv_A+γ_eβ_e/2kv_A/ωω^2-k^2v_A^2cos^2θ/ω^2-k^2c_s^2)B_y/B_0.

We consider the intermediate mode at k≲ k_* and the fast mode at k≳ k_*, whose combination will be referred as the fast magnetosonic wave. Eq. (<ref>) shows that the fast magnetosonic wave is right-handed B_y∝ iB_x, because ω^2> k^2v_A^2cos^2θ, and should be almost circularly polarized at ω/ω_ci≫sin^2θ as in cold plasma, because B_y/B_x is independent of the sound speed. The sound speed enters into electrostatic component E_|| through the electron pressure gradient term in the Ohm's law and into the compressible velocity component, making the ratio V_||/V_⊥ and plasma density fluctuations by a factor of (1-k^2c_s^2/ω^2)^-1 larger compared to cold plasma. The electrostatic component and compressibility are therefore expected to enhance at k≈ k_*.





The nonlinear evolution of 1D planar fast magnetosonic waves propagating oblique to the background magnetic field is addressed by numerically solving Eqs. (<ref>) with (V∇)→ V_|| ∂ / ∂ z, (B_0∇)→ B_0cosθ ∂/∂ z and (B∇)→ 0. The initial condition is a monochromatic fast magnetosonic wave with B_x|_t=0=δ B cos(k z) and other quantities delivered by Eqs. (<ref>)–(<ref>) of the linear theory. All the computations are carried out in the reference frame of the fast magnetosonic wave that is accomplished by replacing ∂ /∂ t→∂ /∂ t-ω/k. The Hall-MHD equations describing evolution of plasma density N, flow velocity V and magnetic field B_⊥  are solved in the spatial domain of 0≤ z≤λ=2π/k with periodic boundary conditions. The numerical scheme based on the pseudo-spectral method that was previously applied for similar equations <cit.> conserves the total energy up to 0.01%. The numerical simulations were carried out for fast magnetosonic waves with initial amplitude δ B=0.04B_0, propagating at wave normal angle of θ=5^∘ in a warm plasma with β_e=β_i=1 and polytrope indexes of γ_e=1 and γ_i=3. The initial wavenumber kλ_i was chosen close to the coupling point of k_*λ_i≈ 0.7 (Figure <ref>).





Figure <ref> demonstrates the nonlinear evolution of plasma density N/N_0, compressible velocity component V_||/V_A, electrostatic field E_||/v_AB_0 and magnetic field B_⊥/B_0. The simulation results are presented in the domain encompassing three wavelengths to stress periodicity. Panels (a) show the evolution of the intermediate mode with kλ_i=0.7, while panels (b) demonstrate similar evolution of the fast mode with kλ_i=0.75. In both cases the waves are right-handed and circularly-polarized, the initial amplitude of the electrostatic component E_||/v_AB_0≈ 5· 10^-3 is one order of magnitude smaller than the amplitude of the electromagnetic component, E_⊥/v_AB_0=(ω/kv_A) B_⊥/B_0≈ 0.05. The compressible component of the plasma flow velocity is initially comparable with the incompressible component, V_||/v_A≈ 0.04 and V_⊥/v_A≈ (kv_Acosθ/ω) B_⊥/B_0 ≈ 0.03, and the amplitude of density fluctuations is around 0.04N_0. Both these quantities are about ten times larger than would be in cold plasma. Panels (a) and (b) show that the magnetic field B_⊥ is essentially not evolving with time and the same is true for electromagnetic components E_⊥ and incompressible velocity component V_⊥ (not shown). In quite a contrast, a substantial evolution has occurred by the time of about 40 ω_ci^-1 in the plasma density, compressible velocity component and electrostatic field. The observed evolution of N/N_0 and V_||/V_A resembles steepening of sound waves in fluids and plasmas (e.g., Ref. <cit.>). The steepened plasma density fluctuations result in spikes in the electrostatic field E_|| due to the electron pressure gradient term ∇ P_e/eN in the Ohm's law. Numerical computations well beyond 40 ω_ci^-1 led to development of gradient scales smaller than the grid size of λ/300. The use of a smaller grid size allowed extending simulations only slightly beyond 40 ω_ci^-1 and resulted in development of even smaller gradient scales and larger-amplitude spikes (not shown). This indicates of wave breaking within finite time (e.g. Ref. <cit.>). 





The critical question is whether steepening would occur for initial wavenumbers sufficiently far from the resonance point. Panels (c) and (d) present the evolution of the intermediate and fast modes with respectively kλ_i=0.5 and 0.9. In both cases the amplitude of plasma density fluctuations is about 0.01N_0. The amplitude of the compressible velocity component V_||/v_A≈ 0.01 is smaller than the amplitude of the incompressible component, V_⊥/v_A≈ 0.03, but still larger than in cold plasma, where according to Eqs. (<ref>) and (<ref>) we would have V_||/V_⊥≈tanθ≈ 0.1. Both panels demonstrate that even by the time of 500 ω_ci^-1 none of the quantities exhibit nonlinear evolution.  This strongly indicates that steepening and development of electrostatic spikes is a resonant phenomenon with a finite resonance width, Δ k=|k-k_*|. Since additional simulations carried out for the intermediate and fast modes with respectively kλ_i=0.6 and 0.8 revealed steepening, while similar simulations for kλ_i=0.55 and 0.85 revealed no steepening (not shown), we can state that in the considered case we have 0.1≲Δ kλ_i≲ 0.15. 






The physics of the process revealed in Figure <ref> can be clarified by noticing that the nonlinear evolution occurs only for the plasma density and compressible velocity component. The evolution of these quantities can be described by two of the Hall-MHD equations

    ∂ N/∂ t+∂(N V_||)/∂ z=0,      m_iN[∂ V_||/∂ t+V_||∂ V_||/∂ z]=-∂ P/∂ z+F(χ),

where P=N_0T_e (N/N_0)^γ_e+N_0T_i (N/N_0)^γ_i, while F(χ)= curl B_⊥× (B_0+B_⊥)/μ_0 is determined by the initial magnetic field of the fast magnetosonic wave and depends only on χ=kz- ω t. Eqs. (<ref>) without F(χ) would predict that any initially monochromatic wave propagates at the sound speed c_s and breaks within a finite time of τ_S=[ max |∂ V_||/∂ z|_t=0]^-1=(kδ V)^-1, where δ V is the initial amplitude of V_|| (e.g., <cit.>). The simulations demonstrated that similar steepening occurs in the presence of F(χ) provided that it is resonant, ω/k≈ c_s. The time scale of 40 ω_ci^-1 revealed in Figure <ref> is quite close to τ_S=(kδ V)^-1≈ 35 ω_ci^-1, where we used kλ_i=0.7 and δ V/V_A=0.04. The steepening is expected to vanish, when F(χ) is not resonant, |ω/k-c_s|τ_S≳λ/2 or |ω/k-c_s|≳πδ V, so that perturbations propagating at ω/k and c_s cannot remain in phase over the time scale of τ_S. Since according to Eq. (<ref>) we have δ V∝δ B (1-k^2c_s^2/ω^2)^-1, the resonance width Δ k resulting from |ω/k-c_s|≈πδ V is more appropriate to be expressed through the magnetic field amplitude 

    (∂ω/∂ k-ω/k)_k=k_*^2Δ k^2/k_*^2≈π v_A^2 δ Bsinθ/B_0ω/k/ω/k+c_s,

where we took into account that ω/k-c_s= ω/k-ω_*/k_*≈(∂ω/∂ k-ω/k)_k=k_*Δ k/k_*. Restricting the analysis to sin^2θ≪ 1 we use the dispersion relation of strictly parallel fast magnetosonic waves, ω^2=k^2v_A^2 (1+ω/ω_ci), to estimate the left-hand side of Eq. (<ref>). Substituting ω/k=c_s on the right-hand side of Eq. (<ref>) and using Eqs. (<ref>) for k_*λ_i and ω_*/ω_ci, we obtain the resonance width

    Δ kλ_i≈(π/2)^1/2(1+v_A^2/c_s^2)(δ Bsinθ/B_0)^1/2.

For the case presented in Figure <ref>, where δ B/B_0=0.04, θ=5^∘ and c_s≈ 1.4 v_A, Eq. (<ref>) delivers the resonance width of Δ kλ_i≈ 0.11 that is consistent with the simulation results. Similar calculations for the steepening time scale τ_S=(kδ V)^-1 lead to

    τ_S=π/ω_ci (1+v_A^2/c_s^2)/k_*λ_i Δ kλ_i·|k-k_*|/Δ k.

Note that Eq. (<ref>) is valid only for |k-k_*|≲Δ k, while no steepening occurs for |k-k_*|≳Δ k. For the case presented in Figure <ref>, Eq. (<ref>) predicts τ_S≈ 60 ω_ci^-1· |k-k_*|/Δ k that is consistent with the simulation results.





§ DISCUSSION AND CONCLUSION



The circularly-polarized electromagnetic fluctuations with the frequency of the order of local proton cyclotron frequency are abundant in the solar wind <cit.>. These low-frequency electromagnetic fluctuations may contribute to solar wind heating <cit.>, while corresponding dissipation mechanisms are still investigated. PSP measurements have recently shown that the low-frequency electromagnetic fluctuations can be accompanied by higher-frequency electrostatic fluctuations, which was assigned to some unknown nonlinear process potentially channeling the energy from lower to higher frequencies <cit.>. Parametric instabilities <cit.> can be excluded, since no daughter waves were observed, while locally induced electrostatic instabilities <cit.> would not be consistent with highly coherent electric field waveforms. 



We presented Hall-MHD simulations demonstrating that electrostatic fluctuations reported by <cit.> can be produced due to the resonance of low-frequency (fast magnetosonic) fluctuations with the ion-sound mode. The resonance occurs only in the case of oblique propagation at wavenumbers |k-k_*|≲Δ k, where k_* and Δ k are delivered by Eqs. (<ref>) and (<ref>). It manifests in the steepening of low-frequency plasma density fluctuations and results in electrostatic spikes due to the electron pressure gradient term in the Ohm's law (Eq. (<ref>)). In electric field spectra these electrostatic spikes, phase-correlated with the low-frequency electromagnetic fluctuations, correspond to wave power at harmonics of the fundamental frequency. No steepening or harmonics are observed in the magnetic field, because curl (∇ P_e/N)=0 and, hence, the electron pressure gradient term does not affect the magnetic field through ∂B/∂ t=- curl E. The steepening time scale delivered by Eq. (<ref>) corresponds to the classical expression τ_S≈ (kδ V)^-1 (e.g., Ref. <cit.>), where δ V is the amplitude of low-frequency compressible velocity fluctuations. Note that for fast magnetosonic fluctuations the resonance can occur only when the sound speed is larger than the Alfvén speed, c_s>v_A, but similar resonance can occur for Afvén ion-cyclotron fluctuations, when c_s<v_A (not shown). In contrast to three-wave resonance <cit.>, the revealed process can be named two-wave resonance. Similar two-wave resonance was previously reported between quasi-parallel whistler and electron-acoustic waves in the Earth's magnetosphere <cit.>. 

The electrostatic spikes observed in the simulations are basically identical to those observed aboard PSP. It is noteworthy that according to the simulations the spikes should occur only in the electrostatic component E_|| that is along the propagation direction. Since the observed low-frequency electromagnetic fluctuations propagate almost perpendicular to the XY plane (Figure <ref>c), the spikes of about 2 mV/m observed aboard PSP (Figure <ref>) corresponds to a small projection of E_|| onto the XY plane. The actual amplitude of the electrostatic spikes, whose major electric field is along the Z-axis and not measured by PSP, is expected to be one order of magnitude larger than observed in the XY plane.

The parameters assumed in the Hall-MHD simulations (θ=5^∘, δ B/B_0=0.04, and f/f_ci≈ 1) are quite typical of circularly-polarized low-frequency fluctuations observed in the solar wind <cit.>, implying thereby that the revealed resonance can occur frequently. Several comments are in order about our simulations. First, the assumed polytrope indexes of γ_e=1 and γ_i=3 allowed reproducing the steepening of fast magnetosonic waves with kλ_i≈ 0.75 observed aboard PSP. Fully kinetic simulations would be needed to avoid the use of these ad hoc values. Note however that as long as c_s>v_A the specific values of the polytrope indexes would not affect the steepening of fast magnetosonic waves with k≈ k_*. Second, we assumed a fixed amplitude of fast magnetosonic waves, while the steepening time scale τ_S≈ 40 ω_ci^-1 is actually comparable with typical linear growth time τ_γ≈ 10^2–10^4 ω_ci^-1 of such low-frequency fluctuations in the solar wind <cit.>. Future simulations should incorporate the effect of linear growth. Third, wave breaking occurs in our simulations (∇ N/N→∞ within finite time), because in the frame of Hall-MHD the ion-sound mode is basically dispersionless <cit.>. The dispersion scale of the ion-sound mode is around electron Debye length λ_D <cit.>, whose typical value in the solar wind is λ_D≈ 10^-3λ_i. Before the dispersion set in, the steepening could, in principle, produce a thousand of harmonics in electric field spectra. In reality, dissipation processes set in much earlier, since the efficiency of Landau damping of ion-sound waves increases with frequency <cit.>. 


We believe it is the dissipation processes (e.g., Landau damping by ions and electrons) that limit the number of harmonics in electric field spectra observed aboard PSP (Figure <ref>). The spatial scale of the electrostatic spikes remains much larger than local electron Debye length, because the dissipation processes can efficiently absorb the energy of high-frequency ion-sound fluctuations. This is a potential channel of energy transfer from cyclotron resonant ions driving circularly-polarized electromagnetic fluctuations to Landau resonant ions and electrons absorbing the energy of electrostatic fluctuations produced by the two-wave resonance. The presented results have wider applications, since electrostatic fluctuations associated with electromagnetic fluctuations, including whistler waves <cit.>, highly oblique fast magnetosonic waves <cit.>, electromagnetic ion-cyclotron waves <cit.>, and kinetic
Alfvén waves <cit.>, have been reported in other space plasma environments.




Acknowledgments:
The of I.Y.V. was supported by the National Science Foundation grant No. 2026680. The work of X.A. and A.V. A. was supported by NASA grant No. 80NSSC22K1634. The work of F.S.M and I.V.K. was supported by NASA grant No. 80NSSC21K0730. J.V. acknowledges support from NASA LWS grant 80NSSC22K1014 and NASA PSP-GI grant 80NSSC23K0208. The data used in this study are publicly available at https://sprg.ssl.berkeley.edu/data/psp/. 




unsrt



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