The Fluid-like and Kinetic Behavior of Kinetic Alfvén Turbulence in Space Plasma

Standard single-fluid magnetohydrodynamics (MHD) contains four linear modes in a collisional plasma: the Alfvén wave, the fast-magnetosonic wave, the slow-magnetosonic wave, and the entropy mode. The polarization and dispersion relations identify their counterparts in the collisionless kinetic theory (Stix 1992; Gary 1993). Due to the collisionless nature of the solar wind, kinetic models have been expected to describe these fluctuations more accurately than MHD. However, Verscharen et al. (2017) compare theoretical predictions for wave polarization properties from both the MHD theory and the kinetic theory with observations of compressive fluctuations in the solar wind and find that the predictions from linear MHD agree better with the observations than the predictions from the kinetic theory for slow modes at large scales. The comparison between the fluid-like and the kinetic behavior of collisionless plasmas is of great importance for our fundamental understanding of plasma turbulence. In this context, we define a fluid-like mode as a plasma mode that follows the predictions from fluid equations with adiabatic and isotropic pressure closure.

The magnetic fluctuations in the inertial range of solar-wind turbulence exhibit Alfvénic correlations (Belcher & Davis 1971) and a scale-dependent anisotropy with ${k}_{\perp }\gg {k}_{\parallel }$ (Horbury et al. 2008; Wicks et al. 2010; Chen et al. 2011, 2012; He et al. 2013; Yan et al. 2016), where k _⊥ is the perpendicular wavenumber and k _∥ is the parallel wavenumber with respect to the background magnetic field. Kinetic Alfvén waves (KAWs) are the short-wavelength extension of the Alfvén-wave branch in the case of highly oblique propagation with respect to the background magnetic field. Therefore, it is thought that the cascade continues into the KAW-like regime for k _⊥ ρ _p  1, where ρ _p = v _th/ω _cp is the ion gyroscale, v _th is the perpendicular thermal speed, ω _cp = q _p B ₀/m _p c is the proton gyrofrequency, q _p is the proton electric charge, m _p is the proton mass, B ₀ is the magnitude of the background magnetic field, and c is the speed of light. A growing body of evidence corroborates the presence of kinetic Alfvén turbulence in the solar wind (Chandran et al. 2009; Chen et al. 2010, 2013; Sahraoui et al. 2010; He et al. 2011; Salem et al. 2012). Likewise, observations of small-scale fluctuations in the magnetosheath also suggest the presence of KAW-like turbulence (Chen & Boldyrev 2017; Breuillard et al. 2018). We extend the study of Verscharen et al. (2017) to examine KAW-like fluctuations at small scales. In Section 2, we present predictions for fluctuations in the first three velocity moments associated with KAWs from both the collisional two-fluid theory and the collisionless linear kinetic theory. In Section 3, we describe our analysis of Magnetospheric Multiscale (MMS) magnetosheath measurements. We compare observations with our predictions in Section 4. In Section 5, we discuss our results and present our conclusions. We present predictions for solar-wind measurements with Parker Solar Probe and Solar Orbiter in the Appendix.

We take the background magnetic field to be ${{\boldsymbol{B}}}_{0}=(0,0,{B}_{0})$, the electric field to be ${\boldsymbol{E}}=\delta {\boldsymbol{E}}$ ($\delta {\boldsymbol{E}}$ is the fluctuating electric field), and the wave vector to be ${\boldsymbol{k}}=({k}_{\perp },0,{k}_{\parallel })$. In our coordinate system, the z-direction is parallel to  B ₀, and ${\boldsymbol{k}}$ lies in the x–z plane. We define the dimensionless quantities ξ _s , χ _∥s , χ _⊥s , and ψ _s as the normalized amplitudes of the fluctuations in the first three velocity moments of species s through

$$\begin{eqnarray}&&\displaystyle \frac{\delta {n}_{s}}{{n}_{0s}}={\xi }_{s}\displaystyle \frac{\delta {B}_{{\rm{y}}}}{{B}_{0}},\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\delta {v}_{\parallel s}}{{v}_{{\rm{A}}}}={\chi }_{\parallel s}\displaystyle \frac{\delta {B}_{{\rm{y}}}}{{B}_{0}},\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\delta {v}_{{\rm{y}}s}}{{v}_{{\rm{A}}}}={\chi }_{\perp s}\displaystyle \frac{\delta {B}_{{\rm{y}}}}{{B}_{0}},\end{eqnarray} \tag{ 3 }$$

and

$$\begin{eqnarray}&&\displaystyle \frac{\delta {P}_{s}}{{P}_{B0}}={\psi }_{s}\displaystyle \frac{\delta {B}_{{\rm{y}}}}{{B}_{0}},\end{eqnarray} \tag{ 4 }$$

where ${P}_{B0}={B}_{0}^{2}/8\pi ;$ n _0s is the average density of species s, δn _s , δv _∥s , δv _ys, δP _s , and δB _y are the amplitudes of fluctuations in the number density, the bulk velocity component parallel to ${{\boldsymbol{B}}}_{0}$, the bulk velocity component in the y-direction, the thermal pressure, and the magnetic-field component in the y-direction, respectively. We note that ξ _s , χ _∥s , χ _⊥s and ψ _s are complex and include information about the phases between the fluctuating quantities. We consider a plasma consisting of protons (s = p) and electrons (s = e) only and neglect all of the effects of temperature anisotropies.

2.1. Two-fluid Model

The two-fluid dispersion relation follows from linearizing the continuity, momentum, and energy equations and Maxwell's equations. We rewrite the two-fluid solutions of Hollweg (1999) for the dispersion relation of KAWs using our coordinate system. The KAW dispersion relation is then given by

$$\begin{eqnarray}\begin{array}{rcl}{\omega }^{2} & = & \displaystyle \frac{{k}_{\parallel }^{2}{v}_{{\rm{A}}}^{2}}{2\left(1+G+{k}_{\perp }^{2}{d}_{{\rm{p}}}^{2}\tfrac{{m}_{{\rm{e}}}}{{m}_{{\rm{p}}}}\right)}\left\{\Space{0ex}{4ex}{0ex}1+2G+{{Gk}}_{\perp }^{2}{d}_{{\rm{p}}}^{2}\right.\\ & & \left.+{\left[{(1+{{Gk}}_{\perp }^{2}{d}_{{\rm{p}}}^{2})}^{2}+4{{Gk}}_{\perp }^{2}{d}_{{\rm{p}}}^{2}\left(G-\displaystyle \frac{{m}_{{\rm{e}}}}{{m}_{{\rm{p}}}}\right)\right]}^{1/2}\right\},\end{array}\end{eqnarray} \tag{ 5 }$$

where ${v}_{{\rm{A}}}={B}_{0}/\sqrt{4\pi {n}_{0{\rm{p}}}{m}_{{\rm{p}}}}$ is the Alfvén speed, d _p = v _A/ω _cp is the ion inertial length, G = (γ _p β _p + γ _e β _e)/2,

$$\begin{eqnarray}&&{\beta }_{s}=\displaystyle \frac{8\pi {n}_{0s}{\kappa }_{{\rm{B}}}{T}_{s}}{{B}_{0}^{2}},\end{eqnarray} \tag{ 6 }$$

${\gamma }_{s}$ is the specific heat ratio of species s, m _e is the electron mass, T _s is the parallel temperature of species s, and κ _B is Boltzmann constant.

We calculate the quantities ξ _p, χ _∥p, χ _⊥p, and ψ _p as

$$\begin{eqnarray}\begin{array}{rcl}{\xi }_{{\rm{p}}} & = & \displaystyle \frac{2i({\omega }^{2}-{k}_{\parallel }^{2}{v}_{{\rm{A}}}^{2})}{{k}_{\perp }{d}_{{\rm{p}}}\Space{0ex}{2ex}{0ex}[{\omega }^{2}(2{m}_{{\rm{e}}}/{m}_{{\rm{p}}}-{\gamma }_{{\rm{p}}}{\beta }_{{\rm{p}}})-{\gamma }_{{\rm{e}}}{\beta }_{{\rm{e}}}{k}_{\parallel }^{2}{v}_{{\rm{A}}}^{2}\Space{0ex}{2ex}{0ex}]}\\ & & \times \displaystyle \frac{\omega }{{v}_{{\rm{A}}}\left({k}_{\parallel }-{k}_{\perp }\tfrac{\delta {E}_{{\rm{z}}}}{\delta {E}_{{\rm{x}}}}\right)},\end{array}\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}\begin{array}{rcl}{\chi }_{\parallel {\rm{p}}} & = & \left[\displaystyle \frac{i{\omega }_{\mathrm{cp}}}{\omega }+\displaystyle \frac{i{\gamma }_{{\rm{p}}}{\beta }_{{\rm{p}}}{\omega }_{\mathrm{cp}}{k}_{\parallel }^{2}{v}_{{\rm{A}}}^{2}}{\omega ({\gamma }_{{\rm{e}}}{\beta }_{{\rm{e}}}{k}_{\parallel }^{2}{v}_{{\rm{A}}}^{2}-2{\omega }^{2}{m}_{{\rm{e}}}/{m}_{{\rm{p}}})}\right]\displaystyle \frac{\delta {E}_{{\rm{z}}}}{\delta {E}_{{\rm{x}}}}\\ & & \times \displaystyle \frac{\omega }{{v}_{{\rm{A}}}\left({k}_{\parallel }-{k}_{\perp }\tfrac{\delta {E}_{{\rm{z}}}}{\delta {E}_{{\rm{x}}}}\right)},\end{array}\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}\begin{array}{rcl}{\chi }_{\perp {\rm{p}}} & = & \left[-1-\displaystyle \frac{i\omega \tfrac{\delta {E}_{{\rm{y}}}}{\delta {E}_{{\rm{x}}}}}{{\omega }_{\mathrm{cp}}}-\displaystyle \frac{{\gamma }_{{\rm{p}}}{\beta }_{{\rm{p}}}{k}_{\perp }{k}_{\parallel }{v}_{{\rm{A}}}^{2}\tfrac{\delta {E}_{{\rm{z}}}}{\delta {E}_{{\rm{x}}}}}{{\gamma }_{{\rm{e}}}{\beta }_{{\rm{e}}}{k}_{\parallel }^{2}{v}_{{\rm{A}}}^{2}-2{\omega }^{2}{m}_{{\rm{e}}}/{m}_{{\rm{p}}}}\right]\\ & & \times \displaystyle \frac{\omega }{{v}_{{\rm{A}}}\left({k}_{\parallel }-{k}_{\perp }\tfrac{\delta {E}_{{\rm{z}}}}{\delta {E}_{{\rm{x}}}}\right)},\end{array}\end{eqnarray} \tag{ 9 }$$

and

$$\begin{eqnarray}&&{\psi }_{{\rm{p}}}={\gamma }_{{\rm{p}}}{\beta }_{{\rm{p}}}{\xi }_{{\rm{p}}},\end{eqnarray} \tag{ 10 }$$

where

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{\delta {E}_{{\rm{y}}}}{\delta {E}_{{\rm{x}}}} & = & \displaystyle \frac{-i\omega {\omega }_{\mathrm{cp}}({\gamma }_{{\rm{p}}}{\beta }_{{\rm{p}}}+{\gamma }_{{\rm{e}}}{\beta }_{{\rm{e}}})}{\left[\Space{0ex}{2ex}{0ex}{\omega }^{2}({\gamma }_{{\rm{p}}}{\beta }_{{\rm{p}}}-2{m}_{{\rm{e}}}/{m}_{{\rm{p}}})+{\gamma }_{{\rm{e}}}{\beta }_{{\rm{e}}}{k}_{\parallel }^{2}{v}_{{\rm{A}}}^{2}\Space{0ex}{2ex}{0ex}\right]}\\ & & \times \displaystyle \frac{{\omega }^{2}-{k}_{\parallel }^{2}{v}_{{\rm{A}}}^{2}}{{\omega }^{2}-{k}_{\parallel }^{2}{v}_{{\rm{A}}}^{2}-{k}_{\perp }^{2}{v}_{{\rm{A}}}^{2}}\end{array}\end{eqnarray} \tag{ 11 }$$

and

$$\begin{eqnarray}&&\displaystyle \frac{\delta {E}_{{\rm{z}}}}{\delta {E}_{{\rm{x}}}}=\displaystyle \frac{({\omega }^{2}-{k}_{\parallel }^{2}{v}_{{\rm{A}}}^{2})({\gamma }_{{\rm{e}}}{\beta }_{{\rm{e}}}{k}_{\parallel }^{2}{v}_{{\rm{A}}}^{2}-2{\omega }^{2}{m}_{{\rm{e}}}/{m}_{{\rm{p}}})}{{k}_{\parallel }{k}_{\perp }{v}_{{\rm{A}}}^{2}\left[\Space{0ex}{2ex}{0ex}{\omega }^{2}(2{m}_{{\rm{e}}}/{m}_{{\rm{p}}}-{\gamma }_{{\rm{p}}}{\beta }_{{\rm{p}}})-{\gamma }_{{\rm{e}}}{\beta }_{{\rm{e}}}{k}_{\parallel }^{2}{v}_{{\rm{A}}}^{2}\Space{0ex}{2ex}{0ex}\right]}.\end{eqnarray} \tag{ 12 }$$

2.2. Kinetic Theory

The linear kinetic hot-plasma dispersion relation follows from linearizing the Vlasov equation and Maxwell's equations (see, e.g., Stix 1992, Chapter 10). We assume that both protons and electrons have isotropic Maxwellian background distribution functions. The dispersion relation is then given by the non-trivial solutions to the wave equation,

$$\begin{eqnarray}&&\displaystyle \frac{{\boldsymbol{k}}c}{\omega }\times \left(\displaystyle \frac{{\boldsymbol{k}}c}{\omega }\times \delta {\boldsymbol{E}}\right)+\epsilon \delta {\boldsymbol{E}}\equiv {\boldsymbol{D}}\delta {\boldsymbol{E}}=0,\end{eqnarray} \tag{ 13 }$$

where is the dielectric tensor. We note that ω is complex. The electric current density fulfills

$$\begin{eqnarray}&&{\boldsymbol{j}}=\displaystyle \sum _{s}{{\boldsymbol{j}}}_{s}=-\displaystyle \frac{i\omega }{4\pi }\displaystyle \sum _{s}{{\boldsymbol{\sigma }}}_{s}\cdot \delta {\boldsymbol{E}},\end{eqnarray} \tag{ 14 }$$

where ${{\boldsymbol{j}}}_{s}$ is the current density contribution from species s, ${{\boldsymbol{\sigma }}}_{s}$ is the susceptibility tensor of species s, and $\epsilon =1+{\sum }_{s}{{\boldsymbol{\sigma }}}_{s}$.

The linearized continuity equation connects the fluctuations of the density with the fluctuations of the current density,

$$\begin{eqnarray}&&-\omega {q}_{s}{n}_{0s}\delta {n}_{s}+{\boldsymbol{k}}\cdot {{\boldsymbol{j}}}_{s}=0.\end{eqnarray} \tag{ 15 }$$

Therefore, we find

$$\begin{eqnarray}&&{\xi }_{s}=\displaystyle \frac{-{{iB}}_{0}}{4\pi {n}_{0s}^{2}{q}_{s}}{\boldsymbol{k}}\cdot \left({{\boldsymbol{\sigma }}}_{s}\cdot \displaystyle \frac{\delta {\boldsymbol{E}}}{\delta {B}_{{\rm{y}}}}\right).\end{eqnarray} \tag{ 16 }$$

The electric current density also fulfills

$$\begin{eqnarray}&&{\boldsymbol{j}}=\displaystyle \sum _{s}{{\boldsymbol{j}}}_{s}=\displaystyle \sum _{s}{q}_{s}\int {\boldsymbol{v}}\,\delta {f}_{s}\,{d}^{3}{\boldsymbol{v}},\end{eqnarray} \tag{ 17 }$$

leading to

$$\begin{eqnarray}&&{\chi }_{\parallel s}=\displaystyle \frac{-i\omega {B}_{0}}{4\pi {n}_{0s}{q}_{s}{v}_{{\rm{A}}}}{\left({{\boldsymbol{\sigma }}}_{s}\cdot \displaystyle \frac{\delta {\boldsymbol{E}}}{\delta {B}_{{\rm{y}}}}\right)}_{{\rm{z}}}\end{eqnarray} \tag{ 18 }$$

and

$$\begin{eqnarray}&&{\chi }_{\perp s}=\displaystyle \frac{-i\omega {B}_{0}}{4\pi {n}_{0s}{q}_{s}{v}_{{\rm{A}}}}{\left({{\boldsymbol{\sigma }}}_{s}\cdot \displaystyle \frac{\delta {\boldsymbol{E}}}{\delta {B}_{{\rm{y}}}}\right)}_{{\rm{y}}}.\end{eqnarray} \tag{ 19 }$$

The fluctuations in pressure follow from the second moment of the fluctuations in the distribution δf _s ,

$$\begin{eqnarray}&&{\psi }_{\parallel s}=\displaystyle \frac{{n}_{0s}{m}_{{\rm{p}}}{B}_{0}}{{P}_{B0}\delta {B}_{{\rm{y}}}}\int {v}_{\parallel }^{2}\delta {f}_{s}\,{d}^{3}v\end{eqnarray} \tag{ 20 }$$

and

$$\begin{eqnarray}&&{\psi }_{\perp s}=\displaystyle \frac{{n}_{0s}{m}_{{\rm{p}}}{B}_{0}}{2{P}_{B0}\,\delta {B}_{{\rm{y}}}}\int {v}_{\perp }^{2}\delta {f}_{s}\,{d}^{3}v.\end{eqnarray} \tag{ 21 }$$

We define fluctuations in the total pressure as

$$\begin{eqnarray}&&{\psi }_{s}=\displaystyle \frac{1}{3}{\psi }_{\parallel s}+\displaystyle \frac{2}{3}{\psi }_{\perp s}.\end{eqnarray} \tag{ 22 }$$

The New Hampshire Dispersion relation Solver (NHDS) code (Verscharen & Chandran 2018) solves the hot-plasma dispersion relation of the linearized Vlasov–Maxwell system and determines the polarization properties numerically. Figure 1 shows NHDS solutions for ω _r (the real part of the frequency) of KAWs with k _⊥ ρ _p = 2 as functions of β _p for β _e/β _p = 1 and β _e/β _p = 0.2. In addition, we show KAW solutions from our two-fluid theory with the same parameters and γ _p = 5/3 and γ _e = 1. The dispersion relations agree quantitatively well, especially for β _p  1.

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We analyze data from the four MMS spacecraft (Burch et al. 2016). In burst mode, the Fast Plasma Investigation (FPI) instrument provides proton moments every 150 ms and electron moments every 30 ms (Pollock et al. 2016). The fluxgate magnetometer (FGM) measures the magnetic field with a resolution of <1 ms. We select intervals in the magnetosheath from 2015 October 1 to 2018 February 28 based on the quicklook archive of MMS. We split every interval into 10 s intervals with an overlap of 5 s. We remove data intervals in which $\max [| {n}_{{\rm{p}}}-{n}_{{\rm{e}}}| /{n}_{{\rm{p}}}]\gt 0.1$.

Magnetosheath plasma exhibits fluctuations not related to pure turbulence, e.g., instability-generated waves (such as mirror modes, ion-cyclotron waves, and whistler waves) and various other non-turbulent structures (see, e.g., Lucek et al. 2005). In order to investigate the turbulence itself, we eliminate intervals with $\max ({\beta }_{{\rm{p}}})-\min ({\beta }_{{\rm{p}}})\gt {\beta }_{0{\rm{p}}}$ and $\max ({n}_{{\rm{p}}})-\min ({n}_{{\rm{p}}})\,\gt 0.2\,{n}_{0{\rm{p}}}$, where the subscript 0 represents the average over the 10 s interval. We also remove data intervals in which v _0p < v _A, where v _0p is the average ion bulk velocity so that Taylor's hypothesis applies (Taylor 1938; Klein et al. 2014). For each interval, we calculate ${\beta }_{0{\rm{p}}}$ and β _0e. We find that the average ratio β _0e/β _0p ≈ 0.21 with a standard deviation of 0.09 and, therefore, set β _e/β _p = 0.2 in our study. The averaged normalized amplitude of the fluctuations in δB _⊥ is δB _⊥/B ₀ = 0.05 with a standard deviation of 0.04, and the averaged normalized amplitude of the fluctuations in magnetic-field amplitude is $\delta | {\boldsymbol{B}}| /{B}_{0}=0.07$ with a standard deviation of 0.05. The Columb collision frequency ν _ei is 7 × 10⁻⁵ Hz under the conditions in our data set: n _e ≈ 25 cm⁻³, T _e ≈ 50 eV, corresponding to a mean free path of λ ≈ 2 × 10⁷ km for v _0p ≈ 150 km s⁻¹. Since λ/d _p ≈ 1500, we consider the plasma to be collisionless.

In order to study the scale-dependent behavior of the fluctuations, we apply a continuous wavelet transform based on the Morlet wavelet (Torrence & Compo 1998) to n _p, all components of ${{\boldsymbol{v}}}_{{\rm{p}}}$ and ${\boldsymbol{B}}$, and to every element of the proton pressure tensor ${{\boldsymbol{P}}}_{{\rm{p}}}$. We obtain the fluctuation amplitudes δn _p, $\delta {{\boldsymbol{v}}}_{{\rm{p}}}$, $\delta {\boldsymbol{B}}$, and $\delta {{\boldsymbol{P}}}_{{\rm{p}}}$ from the absolute wavelet coefficients as functions of both scale ℓ (16 logarithmically spaced scales) and time t. We calculate the local magnetic field and the local velocity at wavelet scale ℓ and time t _n by weighting the time series with a Gaussian curve centered at time t _n (Horbury et al. 2008; Podesta 2009) as

$$\begin{eqnarray}&&{{\boldsymbol{C}}}_{n}=\displaystyle \frac{{\displaystyle \sum }_{m=0}^{N-1}{{\boldsymbol{C}}}_{m}\exp \left[-\tfrac{{({t}_{n}-{t}_{m})}^{2}}{2{{\ell }}^{2}}\right]}{{\displaystyle \sum }_{m=0}^{N-1}\exp \left[-\tfrac{{({t}_{n}-{t}_{m})}^{2}}{2{{\ell }}^{2}}\right]},\end{eqnarray} \tag{ 23 }$$

where N is the number of data points and ${\boldsymbol{C}}$ is either ${\boldsymbol{B}}$ or ${\boldsymbol{v}}$. In this way, we obtain the local mean magnetic field ${{\boldsymbol{B}}}_{0}$, the local mean velocity ${{\boldsymbol{v}}}_{0{\rm{p}}}$, the angle between local mean velocity and local mean magnetic field θ _vB , as well as β _p and ρ _p as functions of ℓ and t _n .

We find that $\delta {B}_{{\rm{y}}}\gg \delta {B}_{{\rm{x}}}$ for KAWs in both the two-fluid theory and the kinetic theory. Therefore, we use δB _⊥ ≈ δB_y and exploit the azimuthal symmetry of a gyrotropic distribution of the fluctuating energy. We calculate the parallel and perpendicular fluctuations by

$$\begin{eqnarray}&&\delta {{\boldsymbol{C}}}_{\parallel }=\displaystyle \frac{| \delta {\boldsymbol{C}}\cdot {{\boldsymbol{B}}}_{0}| }{| {{\boldsymbol{B}}}_{0}| }\end{eqnarray} \tag{ 24 }$$

and

$$\begin{eqnarray}&&\delta {{\boldsymbol{C}}}_{\perp }=\displaystyle \frac{| {{\boldsymbol{B}}}_{0}\times (\delta {\boldsymbol{C}}\times {{\boldsymbol{B}}}_{0})| }{| {{\boldsymbol{B}}}_{0}{| }^{2}},\end{eqnarray} \tag{ 25 }$$

where $\delta {\boldsymbol{C}}$ is either $\delta {{\boldsymbol{v}}}_{{\rm{p}}}$ or $\delta {\boldsymbol{B}}$.

We obtain the transformation matrix α _ij from the Geocentric Solar Ecliptic coordinate system to the field-aligned coordinate system with respect to ${{\boldsymbol{B}}}_{0}$ and then transform the tensor of pressure fluctuations $\delta {P}_{{\rm{p}}{ij}}$ to the field-aligned coordinate system by

$$\begin{eqnarray}&&\delta {P}_{{\rm{p}}{ij}}^{{\prime} }={\alpha }_{{ki}}\,\delta {P}_{{\rm{p}}{km}}\,{\alpha }_{{mj}}.\end{eqnarray} \tag{ 26 }$$

The total proton thermal pressure is given by

$$\begin{eqnarray}&&\delta {P}_{{\rm{p}}}=\displaystyle \frac{1}{3}\Space{0ex}{2ex}{0ex}(\delta {P}_{{\rm{p}}11}^{{\prime} }+\delta {P}_{{\rm{p}}22}^{{\prime} }+\delta {P}_{{\rm{p}}33}^{{\prime} }\Space{0ex}{2ex}{0ex}).\end{eqnarray} \tag{ 27 }$$

For each scale ℓ and time t _n , we determine k _⊥ as

$$\begin{eqnarray}&&{k}_{\perp }{\rho }_{{\rm{p}}}=\displaystyle \frac{2\pi {\rho }_{{\rm{p}}}}{{\ell }\,{v}_{0{\rm{p}}}\sin ({\theta }_{{vB}})},\end{eqnarray} \tag{ 28 }$$

and retain only coefficients with 1.8 ≤ k _⊥ ρ _p ≤ 2.2 and $60^\circ \,\lt {\theta }_{{vB}}\lt 120^\circ$.

In order to remove noise in our measurements, we only retain coefficients when $\delta {v}_{\parallel {\rm{p}}}\gt 1.0$ km s⁻¹ and δv _⊥p > 1.0 km s⁻¹. We then construct a set of 600 bins in logarithmic space of β _p and the value of the coefficients, count the number of occurrences in each bin, normalize each bin by its width, and then column-normalize the histogram.

Figure 2 shows the zeroth proton velocity moment ξ _p depending on β _p in the two-fluid and kinetic theory and our MMS observations for k _⊥ ρ _p = 2. The two-fluid theory and kinetic theory predict similar values for ξ _p under these parameters. Therefore, the observations of the zeroth moment do not favor the applicability of either theory unambiguously. However, the data agree well with the theoretical predictions as well as previous observations (Chen et al. 2013; Chen & Boldyrev 2017) and thus support the applicability of our data analysis technique and the model comparisons.

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Figure 3 shows the first parallel proton velocity moment χ _∥p depending on β _p in both theories and our MMS observations. For all shown values of β _p, χ _∥p is significantly greater in our results from the two-fluid theory than in our results from the kinetic theory. The two-fluid theory predicts a value that is approximately the observed value of χ _∥p. The kinetic theory, on the other hand, underestimates the observed value by a factor between about 3 and 30 in the shown range of β _p. This observation suggests that the plasma exhibits fluid-like behavior at k _⊥ ρ _p ≈ 2.

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Figure 4 shows the first perpendicular proton velocity moment χ _⊥p depending on β _p in both theories and our MMS observations. The predictions from our two-fluid model are greater than the predictions from the kinetic theory by a factor of about three. The observations lie between both predictions with a small bias toward the two-fluid theory.

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Figure 5 shows the second proton velocity moment ψ _p depending on β _p in both theories and our MMS observations. Like in Figure 2 for ξ _p, the two-fluid theory and kinetic theory predict similar values for ψ _p, and both predict the observations well.

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We note that an adjustment of γ _p and γ _e within reasonable values does not lead to a significant change in our results. Moreover, an adjustment of θ _kB within the range of 85°  θ _kB < 90° does not lead to a significant change in our results, confirming our assumption that the observed fluctuations are indeed consistent with highly oblique KAW turbulence.

The behavior of KAWs departs from the behavior of Alfvén waves mainly due to two effects. The ion motion is affected by compression and introduces a polarization-drift term in the equation of motion. Furthermore, the parallel component of the electric field is non-zero and electrons move in the field-parallel direction to neutralize the ion density perturbations. We derive predictions for the proton polarization of KAWs using a collisional two-fluid theory and a collisionless kinetic theory. In the linear kinetic theory, the fluctuations are represented by fluctuations in the distribution function, δf _s , so that all moments are included and generally non-zero. Our two-fluid theory, on the other hand, assumes an adiabatic and isotropic closure for the moment hierarchy, i.e., there are no fluctuations in heat flux and the pressure tensor is isotropic. Apart from these differences, the fluid theory and the kinetic theory are equivalent. Measurements with even higher velocity-space resolution may be capable of showing the heat-flux suppression in the future.

Both theories predict similar behaviors for density and pressure fluctuations, but the parallel and perpendicular velocity fluctuations show clear differences: these fluctuations are greater in the two-fluid theory than in the kinetic theory. Due to the noise in the velocity observation, we cannot rule out the possibility that fluctuations with very small amplitude exhibit a behavior consistent with the kinetic theory. However, our comparison of fluctuations in the magnetosheath above the noise level with our theoretical predictions shows that KAW turbulence behaves fluid-like at ion scales, suggesting that some of the fluid-like behavior found by Verscharen et al. (2017) extends to the ion-scale fluctuations. We note that relaxing our assumption of temperature isotropy may improve the agreement between our theory and observations. In addition, a study based on a superposition of KAW turbulence with other modes at small scales may modify our results since our present method does not distinguish the contributions from different wave types than KAWs. A comparison with alternative approximations to the dispersion relation (Hunana et al. 2013; Sulem & Passot 2015; Told et al. 2016) may give further insight into the physics of the observed modes. However, these extensions are beyond the scope of this work.

Our finding of fluid-like behavior in KAW turbulence suggests that some yet unknown mechanism creates conditions similar to the adiabatic and isotropic closure applied in our two-fluid theory, even at small scales and under collisionless conditions. Anti-phase-mixing (Schekochihin et al. 2016) is a potential explanation for this fluid-like behavior. In the turbulent background, nonlinear interactions between fluctuations at different scales can trigger stochastic plasma echoes (Gould et al. 1967; Schekochihin et al. 2016) that may inhibit the transfer of power to higher moments of the velocity distribution. Parker et al. (2016) and Meyrand et al. (2018) find that energy transfer from large to small velocity-space scales nearly cancels due to anti-phase-mixing excited by a stochastic plasma echo. This process leads to an effective low-moment closure, even under collisionless conditions. In KAWs with larger amplitude, the nonlinear trapping of electrons may contribute to the saturation of damping and a more fluid-like behavior (Gershman et al. 2017).

Alternatively, wave–particle interactions can suppress fluctuations in higher moments of the velocity distribution. Verscharen et al. (2016) find that microinstabilities generate fluctuations that scatter protons and thus reduce the anisotropy of the pressure tensor. Wave–particle interactions may then play the role of particle–particle collisions in suppressing fluctuations in higher moments and closing the moment hierarchy at a low order.

Our finding of the fluid-like behavior of KAW turbulence at scales down to the proton inertial length supports the use of fluid models when studying large- and small-scale fluctuations. This discovery will be beneficial to astrophysical modeling since fluid computations are much faster than kinetic computations. More fundamentally, it is of great importance to determine the physics processes that lead to this fluid-like behavior of an otherwise collisionless plasma.

H.W. is supported by the China Scholarship Council for her stay at MSSL. This work is also supported by NSFC under contracts 41474147, 41674171, and 41574168, STFC Ernest Rutherford Fellowships ST/P003826/1 and ST/N003748/2, STFC Consolidated Grant ST/N000722/1, and STFC Solar Orbiter UK Community Support Grant ST/P005489/1. The authors appreciate the valuable conversations with Lloyd Woodham, Owen Roberts, Alexander Pitňa, Petr Hellinger, Olga Alexandrova, and Francesco Valentini. H.W. is grateful for Chuanyi Tu's support during this work.

In the near future, the space missions Parker Solar Probe and Solar Orbiter will carry instruments into the inner heliosphere that will provide us with unprecedented measurements of the fluctuations in both the fields and the particle distributions. Their data will allow us to study kinetic Alfvén turbulence in the solar wind using our methods for the first time.

The solar wind in the inner heliosphere exhibits a broader range of β _p values and typically shows β _p ≈ β _e. We derive predictions for the KAW polarizations under typical solar-wind conditions with θ _kB = 88°, T _∥s  = T _⊥s , β _e/β _p = 1.0, γ _p = 5/3, and γ _e = 1. For completeness, we add the polarizations for electrons. Our kinetic theory applies to protons and electrons likewise. In the two-fluid theory, the calculations for the electron polarizations differ from the calculations for the proton polarization. We find for electrons

$$\begin{eqnarray}&&{\xi }_{{\rm{e}}}={\xi }_{{\rm{p}}},\end{eqnarray} \tag{ 29 }$$

$$\begin{eqnarray}&&{\chi }_{\parallel {\rm{e}}}=\left[-\displaystyle \frac{i{\omega }_{\mathrm{cp}}}{\omega }+\displaystyle \frac{i{\gamma }_{{\rm{e}}}{\beta }_{{\rm{e}}}{\omega }_{\mathrm{cp}}{k}_{\parallel }^{2}{v}_{{\rm{A}}}^{2}}{\omega ({\gamma }_{{\rm{e}}}{\beta }_{{\rm{e}}}{k}_{\parallel }^{2}{v}_{{\rm{A}}}^{2}-2{\omega }^{2}{m}_{{\rm{e}}}/{m}_{{\rm{p}}})}\right]\displaystyle \frac{({m}_{{\rm{p}}}/{m}_{{\rm{e}}})\,\,\,\omega \,\tfrac{\delta {E}_{{\rm{z}}}}{\delta {E}_{{\rm{x}}}}}{{v}_{{\rm{A}}}\left({k}_{\parallel }-{k}_{\perp }\tfrac{\delta {E}_{{\rm{z}}}}{\delta {E}_{{\rm{x}}}}\right)},\end{eqnarray} \tag{ 30 }$$

$$\begin{eqnarray}&&{\chi }_{\perp {\rm{e}}}=\left[-1+\displaystyle \frac{{\gamma }_{{\rm{e}}}{\beta }_{{\rm{e}}}{k}_{\perp }{k}_{\parallel }{v}_{{\rm{A}}}^{2}\tfrac{\delta {E}_{{\rm{z}}}}{\delta {E}_{{\rm{x}}}}}{{\gamma }_{{\rm{e}}}{\beta }_{{\rm{e}}}{k}_{\parallel }^{2}{v}_{{\rm{A}}}^{2}-2{\omega }^{2}{m}_{{\rm{e}}}/{m}_{{\rm{p}}}}\right]\displaystyle \frac{\omega }{{v}_{{\rm{A}}}\left({k}_{\parallel }-{k}_{\perp }\tfrac{\delta {E}_{{\rm{z}}}}{\delta {E}_{{\rm{x}}}}\right)},\end{eqnarray} \tag{ 31 }$$

and

$$\begin{eqnarray}&&{\psi }_{{\rm{e}}}={\gamma }_{{\rm{e}}}{\beta }_{{\rm{e}}}{\xi }_{{\rm{e}}}.\end{eqnarray} \tag{ 32 }$$

Figure 6 shows ξ _s , χ _∥s , χ _⊥s , and ψ _s at k _⊥ ρ _p = 2 as defined in Section 2. Under solar-wind conditions, the two-fluid theory and kinetic theory predict similar ξ _s and ψ _s (in both amplitude and phase) behaviors depending on β _p. For χ _∥s and χ _⊥s , our theoretical results show large differences in both amplitude and phase.

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We expect Parker Solar Probe and Solar Orbiter to test our predictions based on large statistical data sets in the solar wind at different distances from the Sun. This future study will improve our understanding of fluctuations at ion scales and the differences between kinetic and fluid-like behavior in the solar wind.