Introduction

It is well-established that turbulence pervades space and astrophysical plasmas, transferring energy from the large scales at which it is injected down to the plasma microscales where it can be dissipated. The resulting plasma heating is thought to be dynamically important in a number of systems, e.g., the solar corona and solar wind1, the interstellar medium2, and galaxy clusters3, although it is not yet known which physical dissipation mechanisms are responsible. It is therefore a major open question as to how turbulent plasma heating occurs, although due to the weakly collisional nature of these plasmas, it is inevitably through a series of different microphysical plasma processes. In this paper, we apply a field-particle correlation technique to in situ spacecraft data to investigate the first step in the thermalisation process: the mechanism by which energy is transferred from the turbulent electromagnetic field to the plasma particles.

The solar wind provides an ideal opportunity to study turbulent heating, due to the high-resolution in situ measurements available, and several different mechanisms have been proposed. Early suggestions4 invoked cyclotron damping to enable perpendicular ion energisation5,6,7. The realisation that the turbulence could have a substantial k⊥ component led also to suggestions of Landau damping8,9 and later work predicted that this would be dominant over cyclotron damping10,11,12 due to the anisotropic nature of the turbulent cascade13,14,15. Many models now incorporate the effect of both ion and electron Landau damping16,17,18,19,20, although recent work has raised interesting questions about how effective this is in turbulent systems21,22,23,24. Non-resonant mechanisms have also been proposed, most notably stochastic heating25,26,27, which leads to the broadening of particle distributions in a stochastic field. It has also been suggested that dissipation is localised at structures, such as reconnecting current sheets28,29, vortices30,31, and double layers32, although the question remains which dissipation processes would occur within such structures33,34,35,36,37,38.

Various observational evidence has been presented for the above mechanisms, although to date this has been somewhat indirect. For example, evidence for cyclotron damping has been based on the wavenumber of the ion-scale break in the turbulence spectrum4,39,40,41,42,43, the shape of contours in the ion distributions44,45, or correlations between species temperatures and drifts46,47,48. Similarly, evidence for stochastic heating has been based on relationships between measured temperatures and turbulence amplitudes49,50,51. Localised enhancements in temperature28,52,53 and work done on the particles28,29,53 have also been cited as evidence for dissipation at structures.

In this paper, we present a direct measurement of the secular transfer of energy from the turbulent electromagnetic field at kinetic scales to the electrons as a function of the electron velocity. This velocity-space signature allows the different heating mechanisms to be identified, and here is found to be consistent with electron Landau damping.

Results

Data set

Data from the Magnetospheric Multiscale (MMS) mission54 were used, when the spacecraft were in the Earth’s magnetosheath on 16 October 2015 09:24:11–09:25:21. The mean plasma parameters at this time were: magnetic field strength B ≈ 39 nT, number density ni ≈ ne ≈ 14 cm−3, bulk velocity ui ≈ ue ≈ 180 km s−1, and temperatures \(T_{||{\mathrm{i}}} \approx 150\,{\mathrm{eV}}\), T⊥i ≈ 240 eV, \(T_{||{\mathrm{e}}} \approx 22\,{\mathrm{eV}}\), T⊥e ≈ 23 eV. These correspond to average plasma betas βi ≈ 0.80 and βe ≈ 0.088 (where βs = 2μ0nskBTs/B2). Magnetic field data were measured by FGM55 and SCM56, electric field data by SDP57 and ADP58, and particle data by FPI59. All data in this paper are from MMS3 and the turbulence measured during this time period was previously characterised60.

Here, we focus on the energy transfer to the electrons, which were measured at 30 ms resolution, resulting in a total of 2333 three-dimensional velocity distributions. The average of these, f0e = 〈fe〉, is shown in Fig. 1a, in the frame in which the mean electron bulk flow is zero and in a coordinate system in which v|| is parallel to the global mean field B0 = 〈B〉, \(v_ \bot = \sqrt {v_{ \bot 1}^2 + v_{ \bot 2}^2}\), and \(v_{{\mathrm{th,e}}} = \sqrt {2k_{\mathrm{B}}T_{\mathrm{e}}/m_{\mathrm{e}}}\) is the isotropic electron thermal speed. In the conversion from measured energy bin to particle velocity, the mean spacecraft potential (relative to the plasma) of +4.2 V was subtracted to compensate for the energy gain of the electrons arriving at the positively charged spacecraft. Note that data are unavailable for the central part of the distribution with \(v \lesssim 0.5v_{{\mathrm{th,e}}}\).

Fig. 1
figure 1

Measured average electron distribution and field-particle energy transfer rates. a Average electron distribution f0e. b Alternative energy transfer rate \(C_{E_{||},{\mathrm{e}}}^\prime ({\mathbf{v}})\) using fe and unfiltered E||. c Alternative energy transfer rate using δfe and high-pass-filtered (at 1 Hz) E||

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Measuring secular energy transfer

The energy transfer was measured by calculating \(C_{E_{||},{\mathrm{e}}}^\prime ({\mathbf{v}}) = \langle {q_{\mathrm{e}}v_{||}E_{||}f_{\mathrm{e}}} \rangle\) (see Equation (5) of the Methods section) at each point of the measured electron distributions, with the average taken over the whole interval. For the parallel electric field E||, the time series of electric field vectors (measured at ≈0.12 ms resolution) was first Lorentz transformed to the zero mean bulk velocity frame61, averaged down to 30 ms resolution, then the component parallel to B0 taken. The E|| measurement here remains above the instrumental noise level for frequencies \(\lesssim 100\,{\mathrm{Hz}}\), which covers the range used for the correlation measurement. Since FPI was operating in interleave mode, in which alternate distributions were sampled at different points in velocity space59, \(C_{E_{||},{\mathrm{e}}}^\prime ({\mathbf{v}})\) was calculated separately for each of the two sets of distributions. This results in an effective lower time resolution of 60 ms (corresponding to a maximum wavenumber kρi ≈ 34, where ρi is the ion gyroradius, under the Taylor hypothesis) but greater coverage in velocity space when recombined. The resulting energy transfer measure, combined, binned, and averaged in \((v_{||},v_ \bot )\) space, is shown in Fig. 1b. To ensure reliability, distribution measurements with fewer than three particle counts and greater than 20% data gaps in time were excluded, leading to the reduced coverage.

Figure 1b shows a clear signature roughly antisymmetric about v|| = 0. However, this is likely due to the large-scale wave-like oscillation that dominates the energy transfer62,63,64. As discussed in the Methods section, part of the technique is to average out this oscillation to leave the secular transfer; however, in a turbulent spectrum, averaging over longer times leads to larger-scale oscillations dominating the transfer measurement. Instead, the E|| time series was high-pass filtered at 1 Hz to allow sufficient averaging for fluctuations above this frequency, but eliminate contamination from lower-frequency oscillations. This filtering means that any form of energy transfer in modes below 1 Hz is not measured by the technique. Together with the finite time resolution of the data discussed earlier, this means that the method is sensitive only to energy transfer in a specific range of spacecraft-frame frequencies, corresponding to \(2 \lesssim k\rho _{\mathrm{i}} \lesssim 34\) under the Taylor hypothesis, which covers the majority of the kinetic range between the ion and electron gyroscales. In addition, the fluctuating distribution δfe = fe − f0e was used, which removes the constant velocity-space structure that does not contribute to the small-scale energy transfer. The result is shown in Fig. 1c. It can be seen that the peak is more than an order of magnitude smaller, as expected for the secular transfer, and a qualitatively different pattern emerges: a symmetric pair of bipolar signatures at the thermal speed, evocative of Landau damping. As discussed in the Methods section, other mechanisms would produce a qualitatively different signature.

To check whether this signature is coherent over time (which it should be for secular transfer and not for oscillatory transfer), the period was divided into ten subintervals and the same analysis applied to each. Since the structure is mainly in v||, a reduced energy transfer measure was calculated, \(C_{E_{||},{\mathrm{e}}}^\prime (v_{||}) = {\int} C_{E_{||},{\mathrm{e}}}^\prime ({\mathbf{v}})\,{\mathrm{d}}^2{\mathbf{v}}_ \bot\), which is shown in Fig. 2a as a function of time. Due to the significant amount of averaging resulting in a less noisy signal, this could now be converted to the energy transfer rate specified in Equation (4) using the relation

$$C_{E_{||{\mathrm{,e}}}}(v_{||}) = - \frac{{v_{||}}}{2}\frac{{\partial C_{E_{||},{\mathrm{e}}}^\prime (v_{||})}}{{\partial v_{||}}} + \frac{{C_{E_{||},{\mathrm{e}}}^\prime (v_{||})}}{2},$$
(1)

which is shown in Fig. 2b. It can be seen that the symmetric bipolar pattern is indeed coherent over time, consistent with secular energy transfer to the electrons. The time average is shown in Fig. 2c, where the signatures consistent with electron Landau damping are present at velocities \(\sim \pm v_{{\mathrm{th,e}}}\).

Fig. 2
figure 2

Reduced energy transfer rate measurements. a Alternative 1D energy transfer rate \(C_{E_{||},{\mathrm{e}}}^\prime (v_{||})\) as a function of time t. b 1D energy transfer rate \(C_{E_{||{\mathrm{,e}}}}(v_{||})\) obtained from Equation (1). c \(C_{E_{||{\mathrm{,e}}}}(v_{||})\) averaged over time; a signature consistent with Landau damping can be seen

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Finally, the curve in Fig. 2c was integrated over \(v_{||}\) to obtain the net rate of secular transfer of energy density to the electrons \(C_{E_{||},{\mathrm{e}}} \approx 3.4 \times 10^{ - 12}\,{\mathrm{kg}}\,{\mathrm{m}}^{ - 1}\,{\mathrm{s}}^{ - 3}\). Comparing this to the electron thermal energy density, \(\frac{3}{2}n_{\mathrm{e}}k_{\mathrm{B}}T_{\mathrm{e}} \approx 7.7 \times 10^{ - 11}\,{\mathrm{kg}}\,{\mathrm{m}}^{ - 1}\,{\mathrm{s}}^{ - 2}\), gives a transfer timescale of 23 s, and comparing to the total thermal energy density, which is ten times larger, gives 230 s. This value of \(C_{E_{||},{\mathrm{e}}}\) is six times larger than the equivalent perpendicular quantity, \(C_{E_ \bot ,{\mathrm{e}}}\), indicating a dominant parallel energy transfer to electrons in this interval. It can also be compared to previously computed magnetosheath turbulent cascade rates65, where a wide variation of cascade rates were reported there in the range \(\sim [10^{ - 16},10^{ - 12}]{\kern 1pt} {\mathrm{kg}}\,{\mathrm{m}}^{ - 1}\,{\kern 1pt} {\mathrm{s}}^{ - 3}\). The value of \(C_{E_{||},{\mathrm{e}}}\) obtained here is at the upper end of this range, consistent with the turbulence amplitude here being comparable to the upper end of the range of amplitudes65. This raises the possibility that a significant fraction of turbulent energy is being transferred to electrons at kinetic scales.

Comparison to expected resonant velocity

The question now arises as to whether this signal occurs at the velocity expected for Landau damping. To answer this, Fig. 3 shows numerical solutions of the linear Vlasov−Maxwell system for the kinetic Alfvén wave (KAW) obtained from the PLUME dispersion solver66. The measured mean plasma parameters were used, along with \(k_{||}\rho _{\mathrm{i}} = 10^{ - 3}\) (the results are not very sensitive to this number as long as it is ≪1). Previous analysis of the data interval has suggested that the kinetic range fluctuations are low-frequency \(( {\omega \ll k_ \bot v_{{\mathrm{th,i}}}} )\) and anisotropic \(({k_ \bot \gg k_{||}})\), consistent with kinetic Alfvén turbulence60. Figure 3a shows the total KAW damping rate normalised to the wave frequency, −γ/ω, along with its separate contributions from the ions and electrons. It can be seen that the electron damping becomes strong \(({ - \gamma /\omega \sim 0.1})\) at a wavenumber \(k_ \bot \rho _{\mathrm{i}}\sim 20\). Figure 3b shows the resonant velocity \(( {v_{{\mathrm{res}}} = \omega /k_{||}})\), which can be seen to be \(v_{{\mathrm{res}}}\sim v_{{\mathrm{th}}||{\mathrm{,e}}}\) at \(k_ \bot \rho _{\mathrm{i}}\sim 20\). Therefore, the locations of the energy transfer in Fig. 2 are consistent with expectations for Landau damping.

Fig. 3
figure 3

Numerical linear kinetic Alfvén wave (KAW) solutions. a Damping rates, b resonant velocity, and c phase angles for the KAW for the measured parameters. The dotted lines mark where the damping becomes strong \(\left( { - \gamma /\omega \sim 0.1} \right)\)

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Quality checks

Several checks were performed to ensure that the field-particle correlation technique produced a meaningful result. Firstly, the analysis was repeated, but with a phase-randomised version of the electric field measurement. To produce this, the electric field time series was Fourier transformed, a different random value chosen uniformly in the range [0, 2π] was added to the phase at each frequency, and then the inverse Fourier transform was applied. This results in a surrogate electric field time series with the same power spectrum and autocorrelation properties as the original67. The results of the method using one realisation of the phase randomisation are shown in Fig. 4a, b. It can be seen that the pattern is quite different to Fig. 2a, b: the bipolar signatures are not present and the signal is not coherent over time. This suggests that the signals presented in Figs. 1 and 2 are indeed physical. An ensemble of 20 realisations of the phase randomisation were performed and the mean and standard deviation σ of the resulting \(C_{E_{||},{\mathrm{e}}}(v_{||})\) are shown in Fig. 4c. This number of realisations was chosen to allow sufficient convergence of the derived statistical quantities. The mean is close to zero as expected and the amplitude of the real signal is large compared to the standard deviation, ~2σ for v|| < 0 and ~4σ for v|| > 0. Figure 3c shows that the phase angle between j|| (≈j||e at these scales) and E|| is close to zero where the electron damping becomes strong, so indeed we would expect the phase randomisation to produce, on average, a smaller signal. Figure 4d shows \(C_{E_{||},{\mathrm{e}}} = {\int} C_{E_{||},{\mathrm{e}}}(v_{||})\,{\mathrm{d}}v_{||} = \langle {j_{||{\mathrm{e}}}E_{||}} \rangle\) as a function of time in comparison to the phase randomisations. It can be seen that the real signal is consistently positive (indicating net energy transfer to the particles), whereas the phase randomisations are distributed about zero.

Fig. 4
figure 4

Reduced energy transfer rate measurements using a phase-randomised electric field. a, b Same as in Fig. 2 except with randomised E||. c Energy transfer rate \(C_{E_{||{\mathrm{,e}}}}(v_{||})\) (blue) with the mean of the ensemble of phase randomisations (black) and shaded areas representing ±1σ (dark grey) and ±2σ (light grey). d Same for \(C_{E_{||},{\mathrm{e}}}\) (integrated over \(v_{||}\)) shown as a function of time t

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The existence of Landau damping as the cascade proceeds towards electron scales can also be checked against the magnetic field spectrum; if energy is being removed from the turbulence, a steeper spectrum should result. The spectrum of magnetic fluctuations PB, and its local power-law index α (calculated over a sliding window of one decade) are shown in Fig. 5. If the turbulence is sufficiently low-frequency, which would be consistent with previous analysis60, the Taylor hypothesis can be used to interpret this frequency spectrum as a wavenumber spectrum. In the first decade of the kinetic range, α is comparable to predictions for kinetic Alfvén turbulence (−7/3 for a regular cascade12 and −8/3 for an intermittent one68), but by kρi ≈ 15 (kde ≈ 0.4; kρe ≈ 0.1) it has steepened to a value of −3.3. This is significantly steeper than any current prediction for a dissipation-free cascade at these scales, consistent with a damping mechanism being in operation. Finally, as the cascade passes through the electron inertial scale kde = 1, the spectrum steepens again, consistent with expectations for an inertial kinetic Alfvén turbulence cascade60,69. Note also that for most of the frequency range, α is gradually decreasing rather than constant; while this is partly due to the finite width of the sliding window and the smallness of the frequency ranges, it is also consistent with the presence of damping progressively steepening the spectrum.

Fig. 5
figure 5

Magnetic field properties for the data interval. a Trace magnetic field power spectrum PB and b local spectral index α; horizontal dotted lines mark asymptotic cascade predictions −7/312 and −8/368 for kinetic Alfvén turbulence and −11/3 for inertial kinetic Alfvén turbulence60,69, the vertical dotted line marks where the damping becomes strong

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Discussion

The results presented in this paper constitute direct evidence for the presence of Landau damping in a turbulent space plasma, and suggest that it plays a significant role in the dissipation process. The secular energy transfer from the electric field to the electrons has been isolated from the oscillatory component (which has been measured previously in a KAW70) and the electron distribution is seen to be gaining energy density above the resonant velocity and losing it below, with an overall net gain (Fig. 2). The resonance appears rather broad, with the signal covering a width comparable to or greater than the thermal speed (rather than a small fraction of it), as might be expected in strong turbulence10,71. The observed velocity-space signature is consistent with simulations of Landau damping in both a single KAW and strongly nonlinear kinetic Alfvén turbulence64, and in the absence of other processes would correspond to a flattening of the distribution at the resonant velocity. However, additional processes, e.g. a velocity-space cascade12,72,73 and/or collisions12,18, would act to thermalise the distribution, so such plateaus may not be observable in practice.

An important question for understanding the kinetic turbulence itself is the degree to which Landau damping steepens the energy spectrum11,17,68,72,73,74,75. While fully answering this is beyond the scope of the current work, we note that the magnetic spectrum is significantly steeper than the cascade predictions at the scale where damping becomes strong (Fig. 5). It is also of interest to note that the energy transfer is not uniform, fluctuating significantly in magnitude (Fig. 4d) but maintaining a Landau-like signature (Fig. 2b). This is consistent with suggestions that Landau damping is stronger at turbulent structures34,38, and that in general dissipation in turbulence is intermittent in nature76,77,78,79.

Finally, we note that the results of the application of this technique are promising for its use in identifying the processes involved in turbulent dissipation. Although here we have determined the parallel energy transfer to electrons, future work could explore the perpendicular transfer and also the transfer to ions. With sufficiently advanced instrumentation on future spacecraft, this would allow the relative importance of the different mechanisms to be understood, as well as the energy partition between species and the route by which heating is achieved in space and astrophysical plasmas.

Methods

Field-particle correlation technique

The method for measuring the energy transfer is based on a field-particle correlation technique38,62,63,64,80 and briefly summarised here. The Vlasov equation,

$$\frac{{\partial f_{\mathrm s}}}{{\partial t}} + {\mathbf{v}} \cdot \nabla f_{\mathrm s} + \frac{{q_{\mathrm s}}}{{m_{\mathrm s}}}\left( {{\mathbf{E}} + {\mathbf{v}} \times {\mathbf{B}}} \right) \cdot \frac{{\partial f_{\mathrm s}}}{{\partial {\mathbf{v}}}} = 0,$$
(2)

describes the evolution of the particle distribution function fs in a collisionless plasma, where qs and ms are the charge and mass of species ‘s’, v is the velocity, and E and B are the electric and magnetic fields. Multiplying by the particle kinetic energy, an equation for the rate of change of phase-space particle energy density \(w_{\mathrm s} = \frac{1}{2}m_{\mathrm s}v^2f_{\mathrm s}\) is obtained,

$$\frac{{\partial w_{\mathrm s}}}{{\partial t}} = - {\mathbf{v}} \cdot \nabla w_{\mathrm s} - \frac{{q_{\mathrm s}v^2}}{2}{\mathbf{E}} \cdot \frac{{\partial f_{\mathrm s}}}{{\partial {\mathbf{v}}}} - \frac{{q_{\mathrm s}v^2}}{2}\left( {{\mathbf{v}} \times {\mathbf{B}}} \right) \cdot \frac{{\partial f_{\mathrm s}}}{{\partial {\mathbf{v}}}}.$$
(3)

When integrated over both position and velocity, only the second term on the right-hand side of Equation (3) is non-zero, showing that any net change in the particle energy is due to the electric field. This term has contributions from all electric field components; however, here we focus on the energy transfer parallel to the magnetic field associated with Landau damping.

The average rate of change of phase-space energy density for species ‘s’ due to the parallel electric field E|| is given by

$$C_{E_{||},{\mathrm s}}({\mathbf{v}}) = \left\langle { - \frac{{q_{\mathrm s}v_{||}^2}}{2}E_{||}\frac{{\partial f_{\mathrm s}}}{{\partial v_{||}}}} \right\rangle ,$$
(4)

where the angular brackets denote an average over space and/or time. It can be seen that this is effectively an un-normalised correlation between E|| and the parallel gradient of the distribution function. Since such gradients are challenging to measure, we also define an alternative correlation,

$$C_{E_{||},{\mathrm s}}^\prime ({\mathbf{v}}) = \left\langle {q_{\mathrm s}v_{||}E_{||}f_{\mathrm s}} \right\rangle .$$
(5)

When integrated over velocity space, Equations (4) and (5) are equivalent and correspond to the average net electromagnetic work done on the particles by E||,

$${\int} C_{E||,{\mathrm s}}({\mathbf{v}})\,{\mathrm{d}}^3{\mathbf{v}} = {\int} C_{E_{||},{\mathrm s}}^\prime ({\mathbf{v}}){\kern 1pt} {\mathrm{d}}^3{\mathbf{v}} = \left\langle {j_{||{\mathrm s}}E_{||}} \right\rangle ,$$
(6)

where \(j_{||{\mathrm s}}\) is the parallel current density of species ‘s’.

An important part of the technique is the separation of the oscillatory transfer of energy back and forth between particles and fields due to undamped wave-like motions and the secular transfer due to damping (or instability). This is achieved by ensuring that the averaging period is much larger than the relevant wavelength and/or wave period.

In their unintegrated form, these correlation measures provide the crucial information about where in velocity space the secular energy transfer is occurring. Their application to simulations has shown that: (a) the oscillatory transfer can be successfully averaged out to leave the secular transfer, (b) a bipolar signature at the resonant velocity is produced for Landau damping of a single wave, (c) a qualitatively similar signature persists in strong low-frequency turbulence, and (d) the alternative measure (Equation (5)) indicates where in velocity space the transfer happens, although with a different characteristic signature62,63,64. Energy transfer mechanisms other than Landau damping are expected to produce significantly different correlation signatures, e.g., cyclotron damping and stochastic heating would appear as perpendicular structure in the perpendicular correlations. Therefore, this technique allows the different mechanisms to be distinguished observationally.