PROTON HEATING IN SOLAR WIND COMPRESSIBLE TURBULENCE WITH COLLISIONS BETWEEN COUNTER-PROPAGATING WAVES

Turbulence is a common phenomenon in various environments, e.g., laboratory (Howes et al. 2012b), space (Goldstein et al. 1995; Tu & Marsch 1995; Bruno & Carbone 2013), and astrophysical plasmas (Zhuravleva et al. 2014). In the heliosphere the solar wind is deemed to be an ideal natural laboratory to investigate plasma turbulence, since particles and fields can be measured in situ and the solar wind is approximately boundless for the turbulent eddies/fluctuations therein. Magnetic fluctuations are often correlated with velocity fluctuations. Together with the weak compressibility of the magnetic field strength, this indicates that Alfvénic fluctuations dominate in the turbulence (Belcher & Davis 1971). The so-called Elsässer variables, which are defined as a sum or difference of fluctuations for the velocity and magnetic field in Alfvén units (${{\boldsymbol{Z}}}_{\pm }={\boldsymbol{v}}\pm {\boldsymbol{b}}$), have often been employed in theoretical calculations to study the counter-propagating waves in Alfvénic turbulence. From the space observation point of view, the nature of the minor (subdominant) Elsässer variable remains unknown, although it is assumed to represent sunward/inward Alfvén waves (AWs) in the scenario of Alfvénic turbulence. However, this notion has not yet been confirmed. Nevertheless, sunward propagating AWs in the absence of anti-sunward AWs are identified in upstream regions of the bow shock (Wang et al. 2015) and in some streams of the pristine solar wind (He et al. 2015a).

The minor Elsässer field may also be associated with convected magnetic structures (Tu & Marsch 1993; Bruno et al. 2001), which would also contribute to the observed excess of magnetic energy over kinetic energy (Chen et al. 2013). Therefore, it is still unclear whether ${{\boldsymbol{Z}}}_{{\rm{minor}}}$ represents inward waves and/or involves mostly nonlinear interaction with the outward ${{\boldsymbol{Z}}}_{{\rm{major}}}.$ The compressibility of solar wind turbulence usually manifests itself in pressure-balanced structures of various types (Yao et al. 2011), which may come in the form of spaghetti-like flux tube stemming from the Sun (Thieme et al. 1990; Borovsky 2008), quasi-perpendicular kinetic slow magnetosonic waves (SMWs; Howes et al. 2012a; Zhang et al. 2015), or mirror-mode structures (Zhang et al. 2009). The nature of ${{\boldsymbol{Z}}}_{{\rm{minor}}}$ and its relation to compressibility will be investigated here.

Turbulence is thought to dissipate through the damping of small-scale kinetic waves or via currents in coherent structures, and can thereby heat the plasma through energy conversion (e.g., Voitenko & Goossens 2004; Marsch 2006; Markovskii & Vasquez 2011). For example, both linear and nonlinear resonances between waves and particles can break the adiabatic behavior of particles (Marsch et al. 1982; Hollweg & Isenberg 2002). Resonance diffusion plateaus of various types, e.g., Landau resonance and left/right cyclotron resonance, seem to be consistent with the measured contours of proton velocity distributions in fast solar wind streams (Marsch & Tu 2001; He et al. 2015b), where turbulence is dominated by AWs from the Sun. Non-resonant (stochastic) heating might be important in regions with extremely low plasma beta, e.g., close to the solar corona (Chandran et al. 2010). Dissipation of strong current density within coherent structures may also be an important source of local heating (Parashar et al. 2009; Osman et al. 2011; Karimabadi et al. 2013; TenBarge & Howes 2013; Wang et al. 2013). But how particles are heated in compressible MHD turbulence involving counter-propagating waves still remains to be understood. This question will also be studied here in detail by use of Wind in situ particle and field measurements.

The solar wind observations presented here are from the Wind spacecraft, and are based on the plasma and magnetic field data provided, respectively, by the 3DP and MFI instruments (Lepping et al. 1995; Lin et al. 1995). The time interval for our study is [15:00, 18:00] on 2012 December 31. Time profiles of parameters, i.e., proton number density (${N}_{{\rm{P}}}$), bulk velocity components (${V}_{\parallel }$, ${V}_{\perp 1},$ and ${V}_{\perp 2}$), and magnetic field components (${B}_{\parallel }$, ${B}_{\perp 1},$ and ${B}_{\perp 2}$), are illustrated in the left panels of Figure 1. The parallel and the first and second perpendicular directions are defined as the directions of ${{\boldsymbol{b}}}_{{\rm{0}}}$, ${{\boldsymbol{b}}}_{{\rm{0}}}\times {\boldsymbol{x}}$, and ${{\boldsymbol{b}}}_{{\rm{0}}}\times ({{\boldsymbol{b}}}_{{\rm{0}}}\times {\boldsymbol{x}}),$ where ${{\boldsymbol{b}}}_{{\rm{0}}}$ represents the mean magnetic field direction pointing anti-sunward in this case, and ${\boldsymbol{x}}$ is the GSE x-direction. It is surprising to find the following points: ${V}_{\parallel }$ is positively correlated with ${B}_{\parallel },$ indicating sunward SMWs propagating obliquely to ${{\boldsymbol{b}}}_{{\rm{0}}}$; ${V}_{\perp 1}$ and ${V}_{\perp 2}$ are anti-correlated, respectively, with ${B}_{\perp 1}$ and ${B}_{\perp 2},$ indicating anti-sunward propagating AWs. The detection of sunward propagating SMWs is also confirmed by the anti-correlation between ${N}_{{\rm{P}}}$ and ${V}_{\parallel }.$

Zoom In Zoom Out Reset image size

The right panels of Figure 1 illustrate the cross-correlation spectra calculated by means of the formula ${{\rm{CC}}}_{{ab}}={\rm{Re}}({\tilde{W}}_{a}{\tilde{W}}_{b}^{*})/(\left|{\tilde{W}}_{a}\right|\cdot \left|{\tilde{W}}_{b}\right|),$ where ${\tilde{W}}_{a}$ and ${\tilde{W}}_{b}$ are the wavelet coefficient spectra of a and b. Spectra of negative ${\rm{CC}}({N}_{{\rm{P}}},|{\boldsymbol{B}}|)$, ${\rm{CC}}({N}_{{\rm{P}}},{V}_{\parallel }),$ and positive ${\rm{CC}}({V}_{\parallel },{B}_{\parallel })$ are clear evidence of sunward oblique SMWs. For oblique SMWs, the density fluctuation is driven by longitudinal compression/expansion, and the magnetic field pressure is almost in balance with the thermal pressure. The anti-sunward AWs are characterized by spectra having negative ${\rm{CC}}({V}_{\perp },{B}_{\perp }).$ The evidence of counter-propagating waves is solid and not affected by the definition of ${{\boldsymbol{b}}}_{{\rm{0}}},$ whether it is a global mean field averaged over the whole time interval (Figures 1(a)–(e)) or a local mean magnetic field direction as introduced by Horbury et al. (2008; Figures 1(f)–(j)).

The Elsässer variables ${{\boldsymbol{Z}}}_{\pm }$ are calculated by ${{\boldsymbol{Z}}}_{\pm }={\boldsymbol{V}}\pm {{\boldsymbol{V}}}_{{\rm{a}}}={\boldsymbol{V}}\pm {\boldsymbol{B}}/\sqrt{{\mu }_{{\rm{0}}}{\rho }_{{\rm{0}}}},$ with ${\rho }_{{\rm{0}}}$ being the proton density averaged over the interval of [15:00, 18:00] UT. The trace power spectral densities (PSD) of ${{\boldsymbol{Z}}}_{+}$, ${{\boldsymbol{Z}}}_{-}$, ${\boldsymbol{V}},$ and ${{\boldsymbol{V}}}_{{\rm{a}}}$ are illustrated in Figures 2(a) and (b). The related spectral indices are found by power-law fitting to be −1.55 ± 0.04, −1.54 ± 0.03, −1.51 ± 0.03, and −1.52 ± 0.03.

Zoom In Zoom Out Reset image size

The characteristic directions of ${{\boldsymbol{Z}}}_{\pm }$ fluctuations are estimated by applying the Singular-value-decomposition method (SantolíK et al. 2003) to the spectral matrix of ${{\boldsymbol{Z}}}_{\pm }.$ The directions with maximum singular values are regarded as the oscillation directions. The direction with minimum singular value is deemed a good proxy for the propagation direction, if the Elsässer fluctuations are ideally incompressible. The spectra of the angles between the derived "oscillation" or "propagation" direction and the local mean magnetic field directions ($\theta ({{\boldsymbol{Z}}}_{+},{{\boldsymbol{b}}}_{{\rm{0}}})$, $\theta ({{\boldsymbol{Z}}}_{-},{{\boldsymbol{b}}}_{{\rm{0}}})$, $\theta ({{\boldsymbol{k}}}_{+}^{*},{{\boldsymbol{b}}}_{{\rm{0}}})$, and $\theta ({{\boldsymbol{k}}}_{-}^{*},{{\boldsymbol{b}}}_{{\rm{0}}})$) as a function of time and period are illustrated in Figures 2(c)–(f). The angle $\theta ({{\boldsymbol{Z}}}_{+},{{\boldsymbol{b}}}_{{\rm{0}}})$ in Figure 2(c) is basically smaller than 30°, indicating longitudinal oscillation of ${{\boldsymbol{Z}}}_{+}.$ Large angles $\theta ({{\boldsymbol{Z}}}_{-},{{\boldsymbol{b}}}_{{\rm{0}}})$ in Figure 2(d) suggest a transverse oscillation of ${{\boldsymbol{Z}}}_{-}.$

The oscillation difference between ${{\boldsymbol{Z}}}_{+}$ and ${{\boldsymbol{Z}}}_{-}$ demonstrates the intrinsically different natures of ${{\boldsymbol{Z}}}_{+}$ and ${{\boldsymbol{Z}}}_{-},$ with the former and latter pertaining to the sunward SMWs and anti-sunward AWs, respectively. Figure 2(e) shows that the majority of $\theta ({{\boldsymbol{k}}}_{+}^{*},{{\boldsymbol{b}}}_{{\rm{0}}})$ is larger than 70°, while $\theta ({{\boldsymbol{k}}}_{-}^{*},{{\boldsymbol{b}}}_{{\rm{0}}})$ in Figure 2(f) displays a scattered probability distribution. To be cautious, $\theta ({{\boldsymbol{k}}}_{+}^{*},{{\boldsymbol{b}}}_{{\rm{0}}})$ may not be the real propagation angle of SMWs due to the compressibility of ${{\boldsymbol{Z}}}_{+}.$ For AWs with transverse magnetic field oscillations, the fluctuating magnetic field component vector is orthogonal to both the wavevector and the background magnetic field vector, along both of which directions there is ideally no oscillation of magnetic field. Therefore, the derived ${{\boldsymbol{k}}}_{-}^{*}$ for AWs may be either the real propagation direction ${{\boldsymbol{k}}}_{-}$ or ${{\boldsymbol{b}}}_{{\rm{0}}}.$ The patches of $\theta ({{\boldsymbol{k}}}_{+}^{*},{{\boldsymbol{Z}}}_{-})\ne 90^\circ$ in Figure 2(g) may allow one to infer the nonlinear action of AWs (${{\boldsymbol{Z}}}_{-}$) on SMWs (${{\boldsymbol{Z}}}_{+}$) (${{\boldsymbol{Z}}}_{-}\cdot {{\rm{\nabla }}}_{+}{{\boldsymbol{Z}}}_{+}\ne 0$). The wave fronts of AWs may be distorted by the SMWs when $\theta ({{\boldsymbol{k}}}_{-}^{*},{{\boldsymbol{Z}}}_{+})\ne 90^\circ$ (Figure 2(h)), which may lead to a transverse cascade of AWs.

The proton 2D velocity distribution functions (VDFs) in the top panels of Figure 3 illustrate the bi-directional asymmetric beams along the ${{\boldsymbol{b}}}_{{\rm{0}}}$ direction. Such a twofold beam pattern has, to our knowledge, not been reported before. As compared to the anti-sunward beam, which appears prolonged in ${V}_{\parallel }$ and broad in ${V}_{\perp },$ the sunward beam looks relatively shorter in ${V}_{\parallel }$ and narrower in ${V}_{\perp }.$ We interpret this new observational pattern as probable evidence of wave–particle interaction between protons and counter-propagating waves.

Zoom In Zoom Out Reset image size

The anti-sunward beam may be the result of Landau and right-cyclotron resonance with the outward oblique AWs at kinetic scales (generalized kinetic AWs), the wave signatures of which had been reported in previous studies (Bale et al. 2005; Sahraoui et al. 2010; He et al. 2011, 2012; Podesta & Gary 2011; Roberts et al. 2013; Klein et al. 2014). The sunward beam may be formed via Landau resonance with kinetic SMWs, the wave features of which have been studied in linear theory (Zhao et al. 2014; Narita & Marsch 2015). In other time intervals, without clear signatures of counter-propagating MHD waves, the bi-directional asymmetric beams in the VDFs are absent. The proton 1D reduced VDFs in the bottom panels of Figure 3 show a plateau and an extended tail on the sunward and anti-sunward sides respectively. It is exciting to find that the plateau positions coincide well with the field-projected phase speed of quasi-perpendicular SMWs (vertical dotted line in panels), which indicates a Landau resonance between protons and sunward SMWs. The dispersion relation for quasi-perpendicular SMW yields approximately ${{C}_{{\rm{S}}}^{*}}^{2}={(\omega /{k}_{\parallel })}^{2}\sim {V}_{{\rm{A}}}^{2}/(1+{V}_{{\rm{A}}}^{2}/{C}_{{\rm{S}}}^{2}),$ with ${V}_{{\rm{A}}}$ and ${C}_{{\rm{S}}}$ being the Alfvén speed and ion sound speed, respectively. The propagation of quasi-perpendicular SMWs can also be described by the equations derived for the generalized Elsässer fields (Marsch & Mangeney 1987). But they have never before been analyzed observationally. SMWs in the extreme limit of ${k}_{\parallel }\ll {k}_{\perp }$ are also named as pseudo-Alfvénic waves, which are passive scattered/mixed by AWs with ${k}_{\parallel }\ll {k}_{\perp }$ (Schekochihin et al. 2009). AWs with considerable ${k}_{\parallel }$ may also be distorted by the quasi-perpendicular SMWs leading to larger ${k}_{\perp }$ due to the nonlinear term ${Z}_{\parallel ,{\rm{SMW}}}\cdot {\rm{\nabla }}{Z}_{\perp ,{\rm{AW}}}$ in the convective derivative.

On the basis of the observational findings presented here, a new scenario of compressible plasma turbulence is proposed and illustrated in Figure 4. Dominant AWs are propagating anti-sunward and quasi-perpendicular SMWs are propagating upstream against the AWs. The nonlinear interaction during collisions between these counter-propagating waves may lead for both waves to a cascade of wave energy toward plasma kinetic scales. Meanwhile, in response to the counter-propagating waves, bi-directional asymmetric beams emerge in the proton VDFs through resonant wave–particle interactions. In this way, the turbulence energy cascading and dissipation as well as the solar wind ion heating are included in this common scenario. This new type of turbulence, which is unusual in the solar wind fast streams, is different from the traditional paradigm, where the counter-propagating waves are both Alfvénic but have not yet been confirmed observationally.

Zoom In Zoom Out Reset image size

Some new challenges are raised along with these findings. (1) How are the sunward SMWs generated in the solar wind turbulence? Is it by compression between streams, the parametric decay from AWs (Verscharen et al. 2012), or by other processes? However, in the traditional decay paradigm, a pump AW nonlinearly decays into a SMW propagating along the same direction as the pump wave. Recently, a more generalized theory involving more types of decay processes has been developed under the assumption of larger ${k}_{\perp }$ (${k}_{\perp }\gg {k}_{\parallel }$) and small plasma beta (${m}_{{\rm{e}}}/{m}_{{\rm{i}}}\ll \beta \ll 1$) (see Zhao et al. 2015, and references therein). In the theory, as one of the three types of so-called vector decay mechanisms, the pump AW is able to decay into forward Alfvén and backward slow-mode waves, when the perpendicular wavelength is relatively shorter than the travel distance at ion thermal speed over the pump wave period. (2) The observed compressible MHD turbulence and the concurrent collisionless damping of kinetic waves, together call for a more comprehensive study, involving numerical modeling and observational statistical analysis. (3) What is the radial evolution in the solar wind of the compressible turbulence, involving AWs and SMWs, which have also been identified in the solar wind source region? (4) Besides the local wave–particle interactions, the effects of large-scale evolution (e.g., anisotropic cooling during expansion, gravity, charge-separation electric field, mirror force, and differential flows between species) need to be taken into account for the proton kinetics (e.g., Cranmer 2014; Hellinger & Trávníček 2015; Isenberg & Vasquez 2015). These are issues that may well be addressed by the upcoming missions Solar Orbiter and Solar Probe Plus, which will be "sailing against the solar wind" into the inner heliosphere and outer corona.

This work at Peking University is supported by NSFC under contracts 41222032, 41174148, 41574168, 41231069, 41274172, 41474148, and 41421003. The work at UC Berkeley is supported by NASA grants NNX13AL11G and NNX14AC07G. J.S.H., C.Y.T., E.M., C.H.K.C., and L.H.W. are members of ISSI/ISSI-BJ Team 304. We thank the 3DP team and MFI team on WIND for sharing data with us.