2.1 Phenomenological models of Alfvénic turbulence

Alfvénic turbulence is thought to be captured by the incompressible magnetohydrodynamics (MHD) equations, which in fluctuating Elsasser (Reference Elsasser1950) form (omitting forcing and dissipation terms) are

(2.1) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}\boldsymbol{z}^{\pm }}{\unicode[STIX]{x2202}t}\mp v_{A}\unicode[STIX]{x1D735}_{\Vert }\unicode[STIX]{x1D6FF}\boldsymbol{z}^{\pm }+\unicode[STIX]{x1D6FF}\boldsymbol{z}^{\mp }\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FF}\boldsymbol{z}^{\pm }=-\unicode[STIX]{x1D735}P, & \displaystyle\end{eqnarray}$$

(2.2) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D735}\boldsymbol{\cdot }\unicode[STIX]{x1D6FF}\boldsymbol{z}^{\pm }=0, & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D6FF}\boldsymbol{z}^{\pm }=\unicode[STIX]{x1D6FF}\boldsymbol{v}\pm \unicode[STIX]{x1D6FF}\boldsymbol{b}$ are the fluctuating Elsasser variables, $\boldsymbol{v}$ is the bulk velocity, $\boldsymbol{b}=\boldsymbol{B}/\sqrt{\unicode[STIX]{x1D707}_{0}\unicode[STIX]{x1D70C}}$ is the magnetic field in Alfvén units, $\unicode[STIX]{x1D70C}$ is the mass density, $v_{A}=B_{0}/\sqrt{\unicode[STIX]{x1D707}_{0}\unicode[STIX]{x1D70C}}$ is the Alfvén speed, $P$ contains the total (thermal plus magnetic) pressure and $\unicode[STIX]{x1D735}_{\Vert }$ is the gradient in the direction of the mean magnetic field $\boldsymbol{B}_{0}$ . An early model of Alfvénic turbulence, based on these equations, was developed by Iroshnikov (Reference Iroshnikov1963) and Kraichnan (Reference Kraichnan1965). From the form of the nonlinear terms ( $\unicode[STIX]{x1D6FF}\boldsymbol{z}^{\mp }\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FF}\boldsymbol{z}^{\pm }$ ) it was realised that nonlinear interactions occur via oppositely propagating Alfvén wave packets ( $\unicode[STIX]{x1D6FF}\boldsymbol{z}^{+}$ and $\unicode[STIX]{x1D6FF}\boldsymbol{z}^{-}$ ). The turbulence was assumed to be isotropic so that the presence of a strong mean magnetic field would lead to weak interactions (the linear terms dominating in (2.1)), requiring many such interactions to transfer the energy to a smaller scale. When a constant energy flux through scales is assumed, scaling arguments based on the cascade time lead to an inertial range total energy spectrum

(2.3) $$\begin{eqnarray}E(k)\propto k^{-3/2},\end{eqnarray}$$

where $k$ is the wavenumber of the fluctuations.

Weak Alfvénic turbulence, however, was later shown to develop strong anisotropy, with a spectrum perpendicular to mean magnetic field of $E(k_{\bot })\propto k_{\bot }^{-2}$ and no transfer in the parallel direction (Montgomery & Turner Reference Montgomery and Turner1981; Shebalin, Matthaeus & Montgomery Reference Shebalin, Matthaeus and Montgomery1983; Goldreich & Sridhar Reference Goldreich and Sridhar1997; Galtier et al. Reference Galtier, Nazarenko, Newell and Pouquet2000). This leads to a violation of both the isotropic and weak assumptions as the cascade proceeds to smaller scales, since the degree of nonlinearity increases with wavenumber. Goldreich & Sridhar (Reference Goldreich and Sridhar1995) proposed that a state of critical balance is reached in which the linear and nonlinear terms become (and remain) comparable, and the transfer to smaller scales occurs within a single interaction. From (2.1), the linear Alfvén time is $\unicode[STIX]{x1D70F}_{A}\sim l_{\Vert }/v_{A}$ (where $l_{\Vert }$ is the scale parallel to the mean magnetic field) and the nonlinear time was estimated as $\unicode[STIX]{x1D70F}_{nl}\sim l_{\bot }/\unicode[STIX]{x1D6FF}z$ (where $l_{\bot }$ is the perpendicular scale), which, when equated, lead to a scaling $k_{\Vert }\propto k_{\bot }^{2/3}$  Footnote ² . This results in an anisotropy $k_{\bot }\gg k_{\Vert }$ at small scales and energy spectra

(2.4a,b ) $$\begin{eqnarray}E(k_{\bot })\propto k_{\bot }^{-5/3},\quad E(k_{\Vert })\propto k_{\Vert }^{-2}.\end{eqnarray}$$

Since the nonlinear transfer occurs within one interaction, turbulence in critical balance is considered to be strong at all scales, even when $\unicode[STIX]{x1D6FF}B/B_{0}\ll 1$ .

Boldyrev (Reference Boldyrev2006) later noted that the nonlinear time should contain an extra factor of the alignment angle $\unicode[STIX]{x1D703}$ between the fluctuations $\unicode[STIX]{x1D70F}_{nl}\sim l_{\bot }/(\unicode[STIX]{x1D6FF}z\sin \unicode[STIX]{x1D703})$ . This is because both the fluctuations and their gradients are mostly in the plane perpendicular to the mean field, and within this plane the gradients are perpendicular to the fluctuations due to solenoidality, leading to a greater reduction of the nonlinear terms if the fluctuations are more aligned. It was proposed that $\unicode[STIX]{x1D6FF}\boldsymbol{v}$ and $\unicode[STIX]{x1D6FF}\boldsymbol{b}$ align to the maximum amount permitted at each scale, leading to $\unicode[STIX]{x1D703}\propto k_{\bot }^{-1/4}$ , and three-dimensionally anisotropic eddies at small scales $l_{\Vert }\gg \unicode[STIX]{x1D709}\gg \unicode[STIX]{x1D706}$ , where $\unicode[STIX]{x1D709}$ is the characteristic length in the fluctuation direction and $\unicode[STIX]{x1D706}$ in the direction perpendicular to $l_{\Vert }$ and $\unicode[STIX]{x1D709}$ . The resulting spectra are

(2.5a-c ) $$\begin{eqnarray}E(k_{\unicode[STIX]{x1D706}})\propto k_{\unicode[STIX]{x1D706}}^{-3/2},\quad E(k_{\unicode[STIX]{x1D709}})\propto k_{\unicode[STIX]{x1D709}}^{-5/3},\quad E(k_{\Vert })\propto k_{\Vert }^{-2}.\end{eqnarray}$$

In more recent years, further additions to these models have been proposed to allow for the imbalance of the oppositely directed Alfvénic fluxes (Lithwick, Goldreich & Sridhar Reference Lithwick, Goldreich and Sridhar2007; Beresnyak & Lazarian Reference Beresnyak and Lazarian2008; Chandran Reference Chandran2008; Perez & Boldyrev Reference Perez and Boldyrev2009; Podesta & Bhattacharjee Reference Podesta and Bhattacharjee2010) and intermittency (Chandran, Schekochihin & Mallet Reference Chandran, Schekochihin and Mallet2015; Mallet & Schekochihin Reference Mallet and Schekochihin2016). While models such as these are at the phenomenological level, they provide scaling predictions that can be compared to observations to distinguish the physical processes taking place.

2.2 Anisotropy and critical balance

It is well known that Alfvénic turbulence in the solar wind is anisotropic (see reviews by Horbury, Wicks & Chen Reference Horbury, Wicks and Chen2012; Oughton et al. Reference Oughton, Matthaeus, Wan and Osman2015). Correlation lengths are typically observed to be much longer in the direction parallel to the mean magnetic field than perpendicularFootnote ³ , consistent with the anisotropy $k_{\bot }\gg k_{\Vert }$ implied by the above strong turbulence models. More recently, however, new techniques have been developed to measure the anisotropic scaling, as a more direct test. In the above models, the parallel length scale is associated with the linear term, corresponding to the propagation of Alfvén wave packets, which are sensitive to the mean magnetic field at their location and scale, i.e. the local mean field. The need to use such a mean field to test the critical balance predictions was identified in simulations (Cho & Vishniac Reference Cho and Vishniac2000; Maron & Goldreich Reference Maron and Goldreich2001) and a technique to do this in the solar wind, making use of the wavelet transform, was developed by Horbury, Forman & Oughton (Reference Horbury, Forman and Oughton2008). At each location, and at each scale (i.e. for each wavelet coefficient in the transform), a local mean field can be defined from the average of the magnetic field weighted by the corresponding wavelet envelope. Fluctuations at a range of scales at a particular angle to the local mean field can be gathered to produce the spectrum in that directionFootnote ⁴ . A similar technique can be applied to structure functions (e.g. Luo & Wu Reference Luo and Wu2010; Chen et al. Reference Chen, Mallet, Yousef, Schekochihin and Horbury2011b ), for which the local mean field can be defined as the average between the points of the structure function. In practice, the results are not sensitive to this precise definition, as long as a local (and not global) mean field is used.

Figure 1.

(a) Normalised magnetic fluctuation amplitude as a function of parallel and perpendicular wavenumber. (b) Variation of spectral index with angle to the local mean field $\unicode[STIX]{x1D703}_{B}$ . (c) Ratio of linear and nonlinear time scales $\unicode[STIX]{x1D712}$ as a function of scale $k_{\bot }\unicode[STIX]{x1D70C}_{i}$ .

The results of the local structure function technique (Chen et al. Reference Chen, Mallet, Yousef, Schekochihin and Horbury2011b ), applied to the same magnetic field data as in Horbury et al. (Reference Horbury, Forman and Oughton2008), are shown in figure 1 Footnote ⁵ . It can be seen that at large scales ( $k\unicode[STIX]{x1D70C}_{i}\lesssim 10^{-3}$ , where $\unicode[STIX]{x1D70C}_{i}$ is the ion gyroradius) the turbulence is isotropic (Wicks et al. Reference Wicks, Horbury, Chen and Schekochihin2010) with $\unicode[STIX]{x1D6FF}B/B_{0}\sim 1$ , and in the inertial range ( $10^{-3}\lesssim k\unicode[STIX]{x1D70C}_{i}\lesssim 1$ ) the parallel spectrum is steeper than the perpendicular spectrum. The inertial range spectral index as a function of angle to the local mean field, $\unicode[STIX]{x1D703}_{B}$ , is shown in figure 1(b). The perpendicular spectrum is closer to the $k_{\bot }^{-5/3}$ prediction of Goldreich & Sridhar (Reference Goldreich and Sridhar1995) than the $k_{\bot }^{-3/2}$ prediction of Boldyrev (Reference Boldyrev2006), but the parallel spectrum is consistent with the $k_{\Vert }^{-2}$ critical balance prediction of both modelsFootnote ⁶ . Wicks et al. (Reference Wicks, Horbury, Chen and Schekochihin2011) also found a $k_{\Vert }^{-2}$ spectrum for the velocity and dominant Elsasser field (although instrument noise made it difficult to measure the anisotropy of the sub-dominant Elsasser field). It is important to note that the $k_{\Vert }^{-2}$ spectrum is only observed when a scale-dependent local mean field is used; a global mean field cannot be used to test critical balance, since, even for arbitrarily small $\unicode[STIX]{x1D6FF}B/B_{0}$ , the predicted anisotropy would be just large enough that the true local parallel correlation would not be measured (Cho & Vishniac Reference Cho and Vishniac2000; Chen et al. Reference Chen, Mallet, Yousef, Schekochihin and Horbury2011b ).

An alternative way to test the critical balance condition is to estimate the turbulence strength directly from the measured time scales. The parallel and perpendicular lengths of an ‘eddy’ of amplitude $\unicode[STIX]{x1D6FF}B$ can be found using the technique of Chen et al. (Reference Chen, Wicks, Horbury and Schekochihin2010b , Reference Chen, Mallet, Schekochihin, Horbury, Wicks and Bale2012a ). For a particular $\unicode[STIX]{x1D6FF}B$ , for example the horizontal dotted line in figure 1(a), $k_{\Vert }$ is estimated as the wavenumber corresponding to the parallel spectrum (blue vertical dotted line) and $k_{\bot }$ as the wavenumber corresponding to the perpendicular spectrum (red vertical dotted line). From this, the wavevector anisotropy can be determined, e.g. at $k_{\bot }\unicode[STIX]{x1D70C}_{i}=1$ the anisotropy is $k_{\bot }/k_{\Vert }=5.5\pm 0.7$ , corresponding to a wavevector angle $\unicode[STIX]{x1D703}_{kB}=(79.7\pm 1.2)^{\circ }$ . The ratio of linear to nonlinear times (omitting the alignment angle) is then given by $\unicode[STIX]{x1D712}=\unicode[STIX]{x1D70F}_{A}/\unicode[STIX]{x1D70F}_{nl}=(k_{\bot }/k_{\Vert })(\unicode[STIX]{x1D6FF}B/B_{0})$ and is shown in figure 1(c) as a function of $k_{\bot }\unicode[STIX]{x1D70C}_{i}$ , where it can be seen to be close to unity throughout the inertial rangeFootnote ⁷ . Even though the estimate of $\unicode[STIX]{x1D70F}_{nl}$ used here is based on the magnetic fluctuations only, does not include the alignment angle and is only defined to order unity, the critical balance condition ( $\unicode[STIX]{x1D712}\sim 1$ ) appears to be well satisfied in the solar wind.

Figure 2.

(a) Electric field spectrum in the spacecraft frame (blue) and plasma frame (green) (adapted from Chen et al. Reference Chen, Bale, Salem and Mozer2011a ). (b) Magnetic (red), velocity (blue) and residual energy (green) spectra (adapted from Chen et al. Reference Chen, Bale, Salem and Maruca2013a ). Note that the flattening of $E_{v}$ and steepening of $E_{r}$ for $f_{sc}>10^{-2}$ Hz are artificial (due to instrument noise).

2.3 Spectral indices and residual energy

While the results of the previous section would appear to favour the Goldreich & Sridhar (Reference Goldreich and Sridhar1995) model of Alfvénic turbulence, the situation becomes more complicated when also considering the velocity and electric field. The electric field was measured by Bale et al. (Reference Bale, Kellogg, Mozer, Horbury and Reme2005) to follow the magnetic field, having a spectral index of $-5/3$ at MHD scalesFootnote ⁸ . Since the perpendicular plasma velocity in this range should be the $E\times B$ drift, the velocity was expected to display a similar spectrum, however, it has been shown to be shallower, with a value nearer $-3/2$ (e.g. Mangeney et al. Reference Mangeney, Salem, Veltri, Cecconi and Warmbein2001; Podesta, Roberts & Goldstein Reference Podesta, Roberts and Goldstein2007). How can these spectra be consistent with each other? The answer lies in the frame dependence of the electric field (Chen et al. Reference Chen, Bale, Salem and Mozer2011a ). Figure 2 shows the electric field spectrum measured in the spacecraft frame $\boldsymbol{E}_{sc}$ with a spectral index near $-5/3$ , similar to the magnetic field. It also shows the electric field spectrum after Lorentz transforming to the frame of the mean solar wind velocity $\boldsymbol{v}_{0}$ ,

(2.6) $$\begin{eqnarray}\boldsymbol{E}_{sw}=\boldsymbol{E}_{sc}+\boldsymbol{v}_{0}\times \boldsymbol{B}.\end{eqnarray}$$

It can be seen that the solar wind frame spectrum is lower by an order of magnitude and is shallower, having a spectral index close to $-3/2$ . These spectra are, therefore, in fact consistent with the velocity and magnetic field measurements, and are independent confirmation of the difference between them. The $\unicode[STIX]{x1D6FF}\boldsymbol{E}_{sw}$ spectrum matches the velocity spectrum since to leading order it is given by $\unicode[STIX]{x1D6FF}\boldsymbol{E}_{sw}=-\unicode[STIX]{x1D6FF}\boldsymbol{v}\times \boldsymbol{B}_{0}$ , and the $\unicode[STIX]{x1D6FF}\boldsymbol{E}_{sc}$ spectrum matches the magnetic spectrum since to leading order it is $\unicode[STIX]{x1D6FF}\boldsymbol{E}_{sc}=-\boldsymbol{v}_{0}\times \unicode[STIX]{x1D6FF}\boldsymbol{B}$ , i.e. it is dominated by the magnetic fluctuations convected at the mean solar wind velocity.

While the electric, magnetic and velocity spectra are self-consistent, the question remains as to why the velocity and magnetic fluctuations differ. This difference is known as the residual energy, and its spectrum is defined as $E_{r}(\boldsymbol{k})=E_{v}(\boldsymbol{k})-E_{b}(\boldsymbol{k})$ . A measure of the amount of residual energy is the Alfvén ratio $r_{A}=\unicode[STIX]{x1D6FF}\boldsymbol{v}^{2}/\unicode[STIX]{x1D6FF}\boldsymbol{b}^{2}$ , which is $r_{A}=1$ for a pure Alfvén wave, but is measured to be $r_{A}\approx 0.7$ in the solar wind (Chen et al. Reference Chen, Bale, Salem and Maruca2013a ). Statistical arguments have been made to explain this dominance of magnetic fluctuation energy (e.g. Frisch et al. Reference Frisch, Pouquet, Leorat and Mazure1975; Pouquet, Frisch & Leorat Reference Pouquet, Frisch and Leorat1976; Boldyrev, Perez & Zhdankin Reference Boldyrev, Perez and Zhdankin2012b ). The existence of residual energy, both theoretically and observationally, indicates that strong turbulence can produce quantitative differences to the linear wave relationships.

The residual energy has also been described as a balance between the Alfvén effect (linear term) leading to equipartition and a turbulent dynamo (nonlinear term) generating the magnetic excess (Grappin, Leorat & Pouquet Reference Grappin, Leorat and Pouquet1983; Müller & Grappin Reference Müller and Grappin2005). Müller & Grappin (Reference Müller and Grappin2005) used an isotropic closure theory to suggest that $E_{r}$ varies with the total energy spectrum $E_{t}$ , following $k^{-2}$ for a $k^{-3/2}$ total energy spectrum and $k^{-7/3}$ for a $k^{-5/3}$ total energy spectrum. Boldyrev, Perez & Wang (Reference Boldyrev, Perez and Wang2012a ) extended this to an anisotropic model in which the perpendicular spectra are $E_{r}(k_{\bot })\propto k_{\bot }^{-1}$ for weak and $E_{r}(k_{\bot })\propto k_{\bot }^{-2}$ for strong turbulence; in both cases, $E_{r}(k_{\Vert },k_{\bot })$ is concentrated near the $k_{\bot }$ axis, similar to the total energy. In the solar wind, the residual energy has an average spectral index of $-1.9$  Footnote ⁹ (Chen et al. Reference Chen, Bale, Salem and Maruca2013a ), and an example spectrum is shown in figure 2, along with the velocity and magnetic spectraFootnote ¹⁰ . There is also evidence that it is anisotropic and concentrated in the region near $k_{\Vert }\approx 0$ (Yan et al. Reference Yan, He, Zhang, Tu, Marsch, Chen, Wang, Wang and Wicks2016). These observations would be most consistent with the strong turbulence model of Boldyrev et al. (Reference Boldyrev, Perez and Wang2012a ), which provides a possible explanation for the different scaling of the velocity and magnetic fluctuations.

Figure 3.

Three-dimensional magnetic eddy shapes from large (a) to small (c) scales, in which $l$ is in the local mean field direction, $\unicode[STIX]{x1D709}$ the local $\unicode[STIX]{x1D6FF}\boldsymbol{B}_{\bot }$ direction, $\unicode[STIX]{x1D706}$ perpendicular to these and colour represents distance from the origin (from Chen et al. Reference Chen, Mallet, Schekochihin, Horbury, Wicks and Bale2012a ).

2.4 Three-dimensional anisotropy

Alfvénic turbulence is polarised such that magnetic fluctuations are in the plane perpendicular to the mean field, i.e. $\unicode[STIX]{x1D6FF}B_{\bot }\gg \unicode[STIX]{x1D6FF}B_{\Vert }$ . The local $\unicode[STIX]{x1D6FF}\boldsymbol{B}_{\bot }$ direction within this perpendicular plane breaks the symmetry about the mean field, meaning that the turbulence can be three-dimensionally anisotropic, e.g. as in the Boldyrev (Reference Boldyrev2006) model (§ 2.1). The techniques of § 2.2 were recently extended to measure this three-dimensional (3-D) anisotropy in the solar wind (Chen et al. Reference Chen, Mallet, Schekochihin, Horbury, Wicks and Bale2012a ). As well as the local mean field, a local $\unicode[STIX]{x1D6FF}\boldsymbol{B}$ direction can be defined and a 3-D coordinate system constructed, with one axis in the local mean field direction, one in the $\unicode[STIX]{x1D6FF}\boldsymbol{B}_{\bot }$ direction and the third perpendicular to these. By selecting different $\unicode[STIX]{x1D6FF}B$ values, and finding the corresponding lengths in each direction, Chen et al. (Reference Chen, Mallet, Schekochihin, Horbury, Wicks and Bale2012a ) constructed the 3-D eddy shapes of the magnetic fluctuations at several scales, shown in figure 3. They vary from being extended in the $\unicode[STIX]{x1D6FF}\boldsymbol{B}_{\bot }$ direction at large scales, to being three-dimensionally anisotropic in the sense of Boldyrev (Reference Boldyrev2006) at small scales. While the small-scale anisotropy ( $l_{\Vert }>\unicode[STIX]{x1D709}>\unicode[STIX]{x1D706}$ ) and the scaling in the $l_{\Vert }$ and $\unicode[STIX]{x1D709}$ directions are consistent with the Boldyrev (Reference Boldyrev2006) model, the $\unicode[STIX]{x1D706}$ scaling is notFootnote ¹¹ .

There are a few considerations when interpreting these results. Firstly, one might expect the anisotropy with respect to the $\unicode[STIX]{x1D6FF}\boldsymbol{B}$ direction to be a simple result of the solenoidality of the magnetic field. However, it can be shown that solenoidality alone does not determine the anisotropy in the local perpendicular plane (Chen et al. Reference Chen, Mallet, Schekochihin, Horbury, Wicks and Bale2012a ; Mallet et al. Reference Mallet, Schekochihin, Chandran, Chen, Horbury, Wicks and Greenan2016), and extra physics, such as alignment (Boldyrev Reference Boldyrev2006) is involved. Secondly, it has been suggested that the anisotropy in the perpendicular plane at large scales could be a result of the solar wind expansion as it travels away from the Sun. Verdini & Grappin (Reference Verdini and Grappin2015) argued that magnetic flux conservation in the expanding wind causes fluctuations in the radial direction to be smaller, leading to a bias towards lower fluctuation amplitudes in the $\unicode[STIX]{x1D709}$ direction when measurements are made in the radial direction. Some evidence in support of this hypothesis has recently been reported (Vech & Chen Reference Vech and Chen2016). As well as the eddy shapes, this effect could also alter the 3-D scaling, since any effect of the expansion relative to the turbulent dynamics is likely to be scale dependent. Finally, as discussed in § 2.3, the situation is further complicated by the presence of residual energy, which causes the magnetic spectrum to be steeper than the velocity spectrum; how this impacts the 3-D anisotropy is not yet well understood. Therefore, while observations show that Alfvénic turbulence in the solar wind is locally three-dimensionally anisotropic, more work is required to fully understand this.

2.5 Imbalanced turbulence

As shown in § 2.3, the scalings of the velocity and magnetic fluctuations are different, so it is the total energy spectrum $E_{t}(\boldsymbol{k})=E_{v}(\boldsymbol{k})+E_{b}(\boldsymbol{k})$ which should be compared to the models of § 2.1. These models, however, describe balanced Alfvénic turbulence, i.e. turbulence with equal fluxes of Alfvénic fluctuations propagating in each direction along the mean field. In real systems, turbulence is often imbalanced, with unequal fluxes created by localised driving sources, such as supernovae in the interstellar medium or Alfvén waves from the Sun. In the solar wind, the level of imbalance can be quantified with the normalised cross-helicity $\unicode[STIX]{x1D70E}_{c}=2\langle \unicode[STIX]{x1D6FF}\boldsymbol{v}\boldsymbol{\cdot }\unicode[STIX]{x1D6FF}\boldsymbol{b}\rangle /\langle \unicode[STIX]{x1D6FF}\boldsymbol{v}^{2}+\unicode[STIX]{x1D6FF}\boldsymbol{b}^{2}\rangle$ ; $\unicode[STIX]{x1D70E}_{c}\approx 0$ corresponds to balanced turbulence and $\unicode[STIX]{x1D70E}_{c}\approx \pm 1$ to highly imbalanced turbulence. While the solar wind at 1 AU has a typical imbalance of $\unicode[STIX]{x1D70E}_{c}=0.46$ (Chen et al. Reference Chen, Bale, Salem and Maruca2013a ), there is significant variability, enabling the systematic dependence to be studied.

Figure 4 shows the dependence of the spectral indices on the level of imbalance (all quantities are measured in the middle of the inertial range). The total energy spectral index varies from ${\approx}-5/3$ at low imbalance to ${\approx}-3/2$ at high imbalance (as also found by Podesta & Borovsky Reference Podesta and Borovsky2010). $E_{b}$ and $E_{v}$ scale similarly for $|\unicode[STIX]{x1D70E}_{c}|\approx 1$ , which is consistent with the residual energy being mathematically constrained to be small for large imbalance. It is interesting, however, that they take a value close to $-3/2$ , favouring the Boldyrev (Reference Boldyrev2006) model in the absence of residual energy. The residual energy itself has a spectral index of $-2$ (see § 2.3) for low imbalance, which becomes shallower for larger imbalanceFootnote ¹² . Both Elsasser variables (not shown here) have the same scaling as the total energy, to within uncertainties, for all $|\unicode[STIX]{x1D70E}_{c}|$ Footnote ¹³ . The main dependence of the total energy spectrum with $\unicode[STIX]{x1D70E}_{c}$ , however, is not predicted by any of the current models of imbalanced Alfvénic turbulence (e.g. Lithwick et al. Reference Lithwick, Goldreich and Sridhar2007; Beresnyak & Lazarian Reference Beresnyak and Lazarian2008; Chandran Reference Chandran2008; Perez & Boldyrev Reference Perez and Boldyrev2009; Podesta & Bhattacharjee Reference Podesta and Bhattacharjee2010). Therefore, while several aspects of Alfvénic turbulence in the solar wind are beginning to be understood, a complete explanation still remains to be found.

Figure 4.

Variation of spectral indices of magnetic field ( $E_{b}$ ), velocity ( $E_{v}$ ), total energy ( $E_{t}$ ) and residual energy ( $E_{r}$ ) with the level of imbalance $|\unicode[STIX]{x1D70E}_{c}|$ (adapted from Chen et al. Reference Chen, Bale, Salem and Maruca2013a ).

3.1 Spectra and passivity

It was proposed by Higdon (Reference Higdon1984) that the density variations observed in interstellar turbulence are entropy fluctuations, and possibly magnetosonic waves, passively mixed by the Alfvénic turbulence. This concept was developed further by Goldreich & Sridhar (Reference Goldreich and Sridhar1997), who suggested that for MHD turbulence, slow waves (or pseudo-Alfvén waves in the incompressible limit) would be passive to the Alfvénic turbulence as a result of the anisotropy $k_{\bot }\gg k_{\Vert }$ developed by the cascade. An Alfvén wave ( $\unicode[STIX]{x1D6FF}\boldsymbol{z}_{A}^{\pm }$ ) is polarised perpendicular to $\boldsymbol{B}_{0}$ , and distorts a counter-propagating Alfvén wave packet through the term $\unicode[STIX]{x1D6FF}\boldsymbol{z}_{A}^{\pm }\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FF}\boldsymbol{z}_{A}^{\mp }$ , which is of order ${\sim}k_{\bot }\unicode[STIX]{x1D6FF}z_{A}^{2}$ . However, an oblique slow wave ( $\unicode[STIX]{x1D6FF}\boldsymbol{z}_{s}^{\pm }$ ) is polarised mostly parallel to $\boldsymbol{B}_{0}$ , so distorts a counter-propagating Alfvén wave packet through the term $\unicode[STIX]{x1D6FF}\boldsymbol{z}_{s}^{\pm }\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FF}\boldsymbol{z}_{A}^{\mp }$ , which is of order ${\sim}\unicode[STIX]{x1D6FF}z_{s}k_{\Vert }\unicode[STIX]{x1D6FF}z_{A}$ , i.e. a factor of $k_{\Vert }/k_{\bot }$ smaller (assuming equal amplitudes). This would mean that slow wave fluctuations are scattered by the anisotropic Alfvénic turbulence, but do not actively interfere with it, which would also result in negligible energy transfer from the Alfvénic to slow mode component (Goldreich & Sridhar Reference Goldreich and Sridhar1995; Maron & Goldreich Reference Maron and Goldreich2001). Schekochihin et al. (Reference Schekochihin, Cowley, Dorland, Hammett, Howes, Quataert and Tatsuno2009) argued that for weakly collisional plasmas such as the solar wind, the MHD description is not sufficient for the compressive fluctuationsFootnote ¹⁵ , and a kinetic treatment is required. In the gyrokinetic formalism, where $k_{\bot }\gg k_{\Vert }$ is taken as an ordering assumption, the nonlinear fluid equations for the Alfvénic turbulence at $k_{\bot }\unicode[STIX]{x1D70C}_{i}\ll 1$ (Reduced MHD) decouple from the ion kinetic equation. The active Alfvénic turbulence is self-contained, and passively mixes the non-Alfvénic part of the ion distribution, from which the compressive fluctuations (which produce the kinetic counterparts of the slow and entropy modes) are obtained (Schekochihin et al. Reference Schekochihin, Cowley, Dorland, Hammett, Howes, Quataert and Tatsuno2009). Therefore, the passivity of these modes, originally derived in the context of fluid models, is predicted to hold for kinetic plasmas too.

In hydrodynamic turbulence, a passive scalar $\unicode[STIX]{x1D70E}$ shares the same spectrum as the advecting velocity field $\boldsymbol{v}$ (neglecting intermittency). This is because it follows its continuity equation, $\unicode[STIX]{x2202}\unicode[STIX]{x1D70E}/\unicode[STIX]{x2202}t+\boldsymbol{v}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D70E}=0$ , meaning that its nonlinear time ${\sim}l/\unicode[STIX]{x1D6FF}v$ is the same as that of the active turbulence, leading to $\unicode[STIX]{x1D70E}\propto \unicode[STIX]{x1D6FF}v$ . In plasma turbulence, however, the situation is more complicated, since the nonlinear times involve both the velocity and magnetic fields, and their alignment angles (see § 2.1). It is therefore of interest to compare the scaling of the compressive fluctuations in the solar wind to that of the Alfvénic turbulence. While spectra of compressive fluctuations have long been measured in the solar wind (see reviews by Alexandrova et al. Reference Alexandrova, Chen, Sorriso-Valvo, Horbury and Bale2013; Bruno & Carbone Reference Bruno and Carbone2013), we have recently been able to investigate these with greater precision, to enable comparison to the Alfvénic turbulence. Chen et al. (Reference Chen, Bale, Salem and Mozer2011a ) showed that the spectral indices of both $\unicode[STIX]{x1D6FF}n$ and $\unicode[STIX]{x1D6FF}|\boldsymbol{B}|$ are close to $-5/3$ , similar to the magnetic field, rather than the velocity, which has a $-3/2$ spectral index. An example from this data set is shown in figure 5, where the typical features can be seen: compressive fluctuations at a level much lower than the total power, with a spectral slope similar to that of the Alfvénic magnetic fluctuations. This scaling of the compressive fluctuations indicates that they are not passively advected in the hydrodynamic sense. It is still possible, however, that they are passive to the Alfvénic turbulence, in which they are mixed by both the velocity and magnetic fluctuations.

3.2 Nature of the compressive fluctuations

It is also of interest to determine what type of compressive fluctuations are present, i.e. which wave modes they resemble, and whether these require a kinetic description. Klein et al. (Reference Klein, Howes, TenBarge, Bale, Chen and Salem2012) showed that the kinetic fast and slow modes retain qualitative characteristics of their MHD counterparts, but differ quantitatively so that a kinetic description is necessary. In particular, $\unicode[STIX]{x1D6FF}n$ and $\unicode[STIX]{x1D6FF}B_{\Vert }$ remain predominantly anti-correlated for the slow mode and correlated for the fast mode, with a characteristic $\unicode[STIX]{x1D6FD}_{i}$ dependence. Howes et al. (Reference Howes, Bale, Klein, Chen, Salem and TenBarge2012) compared solar wind measurements of this correlation to that produced from a critically balanced spectrum of kinetic slow waves plus an isotropic spectrum of kinetic fast waves, with different fractions of the two. The results are given in figure 6 and show a strong, $\unicode[STIX]{x1D6FD}_{i}$ -dependent anti-correlation, consistent with the curve in which only slow modes, and not fast modes, contribute to the compressive power. The scarcity of fast mode fluctuations has some interesting implications. Firstly, it helps justify the application of low-frequency approximations, such as gyrokinetics (Schekochihin et al. Reference Schekochihin, Cowley, Dorland, Hammett, Howes, Quataert and Tatsuno2009) and kinetic reduced MHD (Kunz et al. Reference Kunz, Schekochihin, Chen, Abel and Cowley2015) to the solar wind. Secondly, it suggests that there can be little transfer of energy to whistler turbulence at sub-ion scales (consistent with observations; § 4.3), which in turn constrains the possible dissipation mechanisms (§ 5.1).

Figure 6.

Correlation $C$ of density and parallel magnetic fluctuations as a function of $\unicode[STIX]{x1D6FD}_{i}$ , compared to theoretical predictions for a spectrum of kinetic fast and slow mode waves with different fractions $F$ of fast mode to total energy (from Howes et al. Reference Howes, Bale, Klein, Chen, Salem and TenBarge2012).

3.3 Anisotropy and damping

An interesting question is why there appears to be a cascade of compressive fluctuations at all, when they are expected to be strongly damped in a $\unicode[STIX]{x1D6FD}_{i}\sim 1$ plasma (Barnes Reference Barnes1966). In particular, their damping rate is $\unicode[STIX]{x1D6FE}\sim k_{\Vert }v_{A}$ , comparable to the time scale on which the Alfvénic turbulence would cascade them to small scales. The solution to this may lie in their anisotropy: if they are significantly more anisotropic than the Alfvénic turbulence, i.e. their $k_{\Vert }$ remains small, they are not strongly damped. So what do we expect the anisotropy of passively mixed compressive fluctuations to be? Lithwick & Goldreich (Reference Lithwick and Goldreich2001) suggested that the slow modes inherit the anisotropy of the Alfvénic turbulence that mixes them. However, Schekochihin et al. (Reference Schekochihin, Cowley, Dorland, Hammett, Howes, Quataert and Tatsuno2009) argued that the kinetic equation for the compressive fluctuations becomes linear in the Lagrangian frame of the Alfvénic turbulence, and, therefore, that they have no parallel cascade. This would lead to highly elongated, compressive structures, with very small $k_{\Vert }$ , and could explain why they are not strongly damped.

Figure 7.

(a) 3-D eddy shape for the $\unicode[STIX]{x1D6FF}|\boldsymbol{B}|$ component of the turbulence at small scales ( $k_{\bot }\unicode[STIX]{x1D70C}_{i}\approx 0.4$ ) in the same form as figure 3 (from Chen et al. Reference Chen, Mallet, Schekochihin, Horbury, Wicks and Bale2012a ). (b) Comparison between the anisotropy of the Alfvénic and compressive components of the turbulence.

The 3-D shape of the $\unicode[STIX]{x1D6FF}|\boldsymbol{B}|$ fluctuations was measured (using the technique described in § 2.4) by Chen et al. (Reference Chen, Mallet, Schekochihin, Horbury, Wicks and Bale2012a ) and is shown in figure 7, where it can be seen that they are indeed very anisotropic. They are, in fact, more anisotropic than the Alfvénic turbulence, as illustrated in figure 7 where the anisotropy $k_{\bot }/k_{\Vert }$ of each component is shown as a function of scale. The ratio of the anisotropy of the compressive to Alfvénic component is shown in the lower panel and can be seen to take a value between 3 and 4 throughout most of the inertial rangeFootnote ¹⁶ . Due to the angular resolution of the measurements, this is a lower limit: the compressive fluctuations are at least several times more anisotropic than the Alfvénic turbulence. This is consistent with the prediction of Schekochihin et al. (Reference Schekochihin, Cowley, Dorland, Hammett, Howes, Quataert and Tatsuno2009), although there is an alternative interpretation. It is possible that the cascade attempts to generate compressive fluctuations with an anisotropy similar to the Alfvénic turbulence, but these are quickly damped, leaving the highly anisotropic ones to be observed. So the question becomes: are the compressive fluctuations highly anisotropic because the less anisotropic component is damped, or are they generated highly anisotropic to begin with? This remains to be answered, but has some interesting implications, e.g. if damping is taking place, then energy may be being removed from the cascade throughout the inertial range, rather than just at the small scales. However, a recent suggestion by Schekochihin et al. (Reference Schekochihin, Parker, Highcock, Dellar, Dorland and Hammett2016) is that ‘anti-phase-mixing’ may effectively suppress such damping, providing an alternative explanation for the presence of the compressive fluctuations. Further observations will be required to distinguish these possibilities.

4.1 Phenomenological models of kinetic range turbulence

Similarly to Alfvénic turbulence (§ 2.1), phenomenological models have been developed for an energy cascade between ion and electron scales. Early models were based on the fluid equations of electron MHD (EMHD) (e.g. Kingsep, Chukbar & Yankov Reference Kingsep, Chukbar and Yankov1990), which in fluctuating form are

(4.1) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}\boldsymbol{B}}{\unicode[STIX]{x2202}t}=\frac{1}{\unicode[STIX]{x1D707}_{0}en_{e}}\unicode[STIX]{x1D735}\times [\boldsymbol{B}_{0}\times (\unicode[STIX]{x1D735}\times \unicode[STIX]{x1D6FF}\boldsymbol{B})+\unicode[STIX]{x1D6FF}\boldsymbol{B}\times (\unicode[STIX]{x1D735}\times \unicode[STIX]{x1D6FF}\boldsymbol{B})].\end{eqnarray}$$

This is essentially the induction equation with the ions assumed motionless (and electron inertia and dissipation terms also neglected). While this turns out to be a poor assumption for the solar wind kinetic range (see § 4.3), the form of the EMHD equations can be used to illustrate some of the basic principles of the kinetic range cascade.

Equation (4.1) is similar to (2.1): there is a linear term (describing whistler waves) and a nonlinear term responsible for the turbulence. The key difference, however, is the additional spatial gradient in both terms. This results in whistler waves being dispersive and leads to a steeper turbulence spectrum. This was first shown by Vaǐnshteǐn (Reference Vaǐnshteǐn1973), who considered isotropic strong EMHD turbulence. From the form of the nonlinear term in (4.1) it can be seen that the nonlinear time is $\unicode[STIX]{x1D70F}_{nl}\propto l^{2}/\unicode[STIX]{x1D6FF}B$ . Assuming a constant energy cascade rate $\unicode[STIX]{x1D700}\propto \unicode[STIX]{x1D6FF}B^{2}/\unicode[STIX]{x1D70F}_{nl}\propto \unicode[STIX]{x1D6FF}B^{3}/l^{2}$ leads to a magnetic energy spectrum

(4.2) $$\begin{eqnarray}E_{B}(k)\propto k^{-7/3},\end{eqnarray}$$

steeper than any of the predicted spectra for MHD turbulence. Similarly to MHD turbulence, though, the assumption of isotropy is not robust. When a critical balance between the whistler time scale and nonlinear time scale is assumed, the scale-dependent anisotropy $k_{\Vert }\propto k_{\bot }^{1/3}$ is obtained (Cho & Lazarian Reference Cho and Lazarian2004), a stronger anisotropy than for Alfvénic turbulence. This results in anisotropic spectra

(4.3a,b ) $$\begin{eqnarray}E_{B}(k_{\bot })\propto k_{\bot }^{-7/3},\quad E_{B}(k_{\Vert })\propto k_{\Vert }^{-5},\end{eqnarray}$$

as illustrated in figure 8.

Figure 8.

Schematic of parallel and perpendicular energy spectra for critically balanced Alfvénic turbulence ( $k_{\bot }^{-5/3}$ and $k_{\Vert }^{-2}$ ) at $k_{\bot }\unicode[STIX]{x1D70C}_{i}<1$ , and kinetic Alfvén or whistler turbulence ( $k_{\bot }^{-7/3}$ and $k_{\Vert }^{-5}$ ) at $k_{\bot }\unicode[STIX]{x1D70C}_{i}>1$ , without intermittency or other corrections (from Chen et al. Reference Chen, Wicks, Horbury and Schekochihin2010b ).

It was proposed (e.g. Leamon et al. Reference Leamon, Smith, Ness, Matthaeus and Wong1998; Howes et al. Reference Howes, Cowley, Dorland, Hammett, Quataert and Schekochihin2008; Schekochihin et al. Reference Schekochihin, Cowley, Dorland, Hammett, Howes, Quataert and Tatsuno2009; Boldyrev & Perez Reference Boldyrev and Perez2012), however, that Alfvénic turbulence at large scales transitions to kinetic Alfv́en turbulence at sub-ion scales, rather than whistler/EMHD turbulence, reflecting the fact that the kinetic Alfvén wave (KAW) is the continuation of the oblique Alfvén mode in this regime. The nonlinear fluid equations that capture kinetic Alfvén turbulence between the ion and electron gyroscales $1/\unicode[STIX]{x1D70C}_{i}\ll k_{\bot }\ll 1/\unicode[STIX]{x1D70C}_{e}$ (e.g. Schekochihin et al. Reference Schekochihin, Cowley, Dorland, Hammett, Howes, Quataert and Tatsuno2009; Boldyrev et al. Reference Boldyrev, Horaites, Xia and Perez2013) take the same mathematical structure as the EMHD equations for a strong mean field and anisotropy $k_{\bot }\gg k_{\Vert }$ (Boldyrev et al. Reference Boldyrev, Horaites, Xia and Perez2013), so the above scaling predictions for the magnetic fluctuations also apply. A key physical difference, however, is that kinetic Alfvén turbulence is low frequency, $\unicode[STIX]{x1D714}\ll k_{\bot }v_{th,i}$ , (whereas whistler turbulence has $\unicode[STIX]{x1D714}\gg k_{\bot }v_{th,i}$ ) so unlike whistler turbulence, density fluctuations are non-negligible (Chen et al. Reference Chen, Boldyrev, Xia and Perez2013b ; Boldyrev et al. Reference Boldyrev, Horaites, Xia and Perez2013). From the nonlinear kinetic Alfvén equations, the scaling of the magnetic, electric and density fluctuations are

(4.4a-c ) $$\begin{eqnarray}E_{B}(k_{\bot })\propto k_{\bot }^{-7/3},\quad E_{E}(k_{\bot })\propto k_{\bot }^{-1/3},\quad E_{n}(k_{\bot })\propto k_{\bot }^{-7/3}.\end{eqnarray}$$

A modification of these predictions was suggested by Boldyrev & Perez (Reference Boldyrev and Perez2012), who observed strong intermittency in their kinetic Alfvén turbulence simulations, with fluctuation energy concentrated in 2-D sheets. Assuming that the cascade is only active within these sheets (so that the active volume fraction is proportional to the scale of the fluctuations), a weaker anisotropy is obtained, $k_{\Vert }\propto k_{\bot }^{2/3}$ and the total energy spectra are

(4.5a,b ) $$\begin{eqnarray}E(k_{\bot })\propto k_{\bot }^{-8/3},\quad E(k_{\Vert })\propto k_{\Vert }^{-7/2}.\end{eqnarray}$$

As for Alfvénic turbulence, these predictions for the spectra, nature and intermittency of the fluctuations can be directly tested with solar wind observations.

4.2 Kinetic range spectra

It has been known for a long time that the spectrum of magnetic fluctuations in the solar wind steepens at ion scales (e.g. Coleman Reference Coleman1968; Russell Reference Russell, Sonett, Coleman and Wilcox1972), but it is only recently that we have been able to more comprehensively diagnose this range, using new high-resolution spacecraft instrumentation. While the shape of the magnetic spectrum close to ion scales and close to electron scales is somewhat variable, the range in between is generally found to be a power law with a typical spectral index around $-2.8$ (e.g. Alexandrova et al. Reference Alexandrova, Saur, Lacombe, Mangeney, Mitchell, Schwartz and Robert2009; Sahraoui et al. Reference Sahraoui, Huang, Belmont, Goldstein, Rétino, Robert and De Patoul2013). More recently, we have also been able to measure the density spectrum in this range (Chen et al. Reference Chen, Salem, Bonnell, Mozer and Bale2012b ; Šafránková et al. Reference Šafránková, Němeček, Přech and Zastenker2013, Reference Šafránková, Němeček, Němec, Přech, Pitňa, Chen and Zastenker2015). Chen et al. (Reference Chen, Salem, Bonnell, Mozer and Bale2012b ) showed that this also takes a power-law form, with a spectral index of $-2.75\pm 0.06$ , similar to the magnetic spectrum. Typical spectra of the density and magnetic fluctuations are shown in figure 9. Electric field spectra have also been reported to flatten at ion scales (Bale et al. Reference Bale, Kellogg, Mozer, Horbury and Reme2005; Sahraoui et al. Reference Sahraoui, Goldstein, Robert and Khotyaintsev2009; Salem et al. Reference Salem, Howes, Sundkvist, Bale, Chaston, Chen and Mozer2012) and ion velocity and temperature spectra to steepen (Šafránková et al. Reference Šafránková, Němeček, Přech and Zastenker2013, Reference Šafránková, Němeček, Němec, Přech, Chen and Zastenker2016).

Figure 9.

Spectra of density and magnetic fluctuations normalised according to (4.6); the vertical dashed lines correspond to the ion and electron gyroradii and inertial length scales under the assumption of the Taylor hypothesis (from Chen et al. Reference Chen, Bale, Salem and Maruca2013a ).

The fact that the density and magnetic fluctuations have the same scaling as each other is consistent with expectations for kinetic Alfvén turbulence, in which these two fields are directly coupled (Schekochihin et al. Reference Schekochihin, Cowley, Dorland, Hammett, Howes, Quataert and Tatsuno2009; Boldyrev et al. Reference Boldyrev, Horaites, Xia and Perez2013), but the spectral index of $-2.8$ is steeper than the pure cascade prediction of $-7/3$ (4.4)Footnote ¹⁸ . Several explanations have been proposed to explain the steeper scaling. For example, Howes et al. (Reference Howes, TenBarge, Dorland, Quataert, Schekochihin, Numata and Tatsuno2011) also observed a $k_{\bot }^{-2.8}$ magnetic spectrum in their gyrokinetic simulations, interpreting the steeper-than- $k_{\bot }^{-7/3}$ result as due to the presence of electron Landau damping. The steeper spectrum, however, has also been observed in simulations that do not contain such damping (Boldyrev & Perez Reference Boldyrev and Perez2012; Franci et al. Reference Franci, Landi, Matteini, Verdini and Hellinger2015). As mentioned in § 4.1, Boldyrev & Perez (Reference Boldyrev and Perez2012) suggested that the accumulation of the cascade into intermittent 2-D structures would lead to a $k_{\bot }^{-8/3}$ spectrum, which is closer to the observed value. Other explanations have also been proposed (e.g. Meyrand & Galtier Reference Meyrand and Galtier2013; Passot & Sulem Reference Passot and Sulem2015). It remains to be determined which combination of these, or other possibilities, are responsible for the steep kinetic range spectra.

4.3 Nature of the fluctuations

To understand how the kinetic-scale cascade operates and leads to plasma heating, it is necessary to determine the nature of the fluctuations which constitute the turbulence. As mentioned in § 2, turbulence at large scales, $k\unicode[STIX]{x1D70C}_{i}\ll 1$ , is predominantly Alfvénic, i.e. the fluctuations are polarised similarly to Alfvén waves (Belcher & Davis Reference Belcher and Davis1971). The plasma modes which exist in the kinetic range, however, are more complex. To narrow down the possibilities, it is first helpful to determine the anisotropy of the turbulence, i.e. whether it is dominated by parallel ( $k_{\Vert }\gg k_{\bot }$ ), isotropic ( $k_{\bot }\sim k_{\Vert }$ ) or perpendicular ( $k_{\bot }\gg k_{\Vert }$ ) fluctuations. Chen et al. (Reference Chen, Horbury, Schekochihin, Wicks, Alexandrova and Mitchell2010a ) measured the anisotropy of the magnetic fluctuations in the kinetic range using a technique similar to that described in § 2.2 but with multiple spacecraft to simultaneously measure different directions. The result of this analysis is given in figure 10, which shows the power in the perpendicular and parallel field components as a function of parallel and perpendicular scale. For both components, the power contours are elongated in the parallel direction, indicating that the turbulence consists of perpendicular fluctuations, $k_{\bot }\gg k_{\Vert }$ . This anisotropy is similar to that of the Alfvénic turbulence at large scales, and is consistent with the predictions of § 4.1 that kinetic turbulence remains anisotropic.

Figure 10.

Fluctuation power in the perpendicular (a) and parallel (b) magnetic field components as a function of perpendicular ( $l_{\bot }$ ) and parallel ( $l_{\Vert }$ ) length scale, for the kinetic range between the ion and electron gyroscales, $1/\unicode[STIX]{x1D70C}_{i}<k<1/\unicode[STIX]{x1D70C}_{e}$ (from Chen et al. Reference Chen, Horbury, Schekochihin, Wicks, Alexandrova and Mitchell2010a ).

The two perpendicular electromagnetic waves that can exist in an isotropic $\unicode[STIX]{x1D6FD}\sim 1$ plasma for $1/\unicode[STIX]{x1D70C}_{i}\ll k_{\bot }\ll 1/\unicode[STIX]{x1D70C}_{e}$ are the kinetic Alfvén wave and the oblique whistler wave (e.g. TenBarge et al. Reference TenBarge, Podesta, Klein and Howes2012; Boldyrev et al. Reference Boldyrev, Horaites, Xia and Perez2013) and the models in § 4.1 describe the nonlinear turbulence based on these modes. Chen et al. (Reference Chen, Boldyrev, Xia and Perez2013b ) developed a method to distinguish between these two types of turbulence, based on the relative level of density and magnetic fluctuations. The spectra in figure 9 are of the normalised fluctuations,

(4.6a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}{\tilde{n}}=\sqrt{\frac{\unicode[STIX]{x1D6FD}_{i}}{2}\left(1+\frac{T_{e}}{T_{i}}\right)\left[1+\frac{\unicode[STIX]{x1D6FD}_{i}}{2}\left(1+\frac{T_{e}}{T_{i}}\right)\right]}\frac{\unicode[STIX]{x1D6FF}n}{n_{0}},\quad \unicode[STIX]{x1D6FF}\tilde{\boldsymbol{b}}=\frac{\unicode[STIX]{x1D6FF}\boldsymbol{B}}{B_{0}}.\end{eqnarray}$$

Similarly to the Alfvén ratio (§ 2.3), the kinetic Alfvén ratio can be defined as $r_{KAW}=\unicode[STIX]{x1D6FF}{\tilde{n}}^{2}/\unicode[STIX]{x1D6FF}\tilde{b}_{\bot }^{2}$ , which is $r_{KAW}=1$ for a kinetic Alfvén wave (due to its pressure-balanced nature) and $r_{KAW}\ll 1$ for an oblique whistler wave (due to its high frequency $\unicode[STIX]{x1D714}\gg k_{\bot }v_{th,i}$ ). It can be seen from figure 9 that in the range between ion and electron scales, the density and magnetic fluctuations are of similar amplitude, and $r_{KAW}\sim 1$ . This suggests that the turbulence is predominantly kinetic Alfvén in nature, with whistler fluctuations making up (at most) a small fraction, which is consistent with the transition from Alfvénic turbulence at larger scales. Chen et al. (Reference Chen, Boldyrev, Xia and Perez2013b ) measured the average kinetic Alfvén ratio to be $r_{KAW}=0.75$ in the solar wind and $r_{KAW}=0.79$ in a kinetic Alfvén turbulence simulation, indicating that while the turbulence follows the linear expectation to order unity, the nonlinearities introduce quantitative differences (see also § 2.3).

The kinetic Alfvén nature of the turbulence is consistent with the flattening of the density spectrum seen to occur just before ion scales (e.g. figure 9). The flattening can be explained as the kinetic Alfvén component at small scales taking over from the compressive non-Alfvénic component at larger scales (Harmon & Coles Reference Harmon and Coles2005; Chandran et al. Reference Chandran, Quataert, Howes, Xia and Pongkitiwanichakul2009), a model which is consistent with solar wind observations (Chen et al. Reference Chen, Howes, Bonnell, Mozer, Klein and Bale2013c ). It is also consistent with a significant, rather than negligible, $\unicode[STIX]{x1D6FF}E_{\Vert }$ spectrum (Mozer & Chen Reference Mozer and Chen2013), although the $\unicode[STIX]{x1D6FF}E_{\Vert }/\unicode[STIX]{x1D6FF}E_{\bot }$ ratio measured by Mozer & Chen (Reference Mozer and Chen2013) is larger than the linear predictionFootnote ¹⁹ . Finally, the measured ion-scale spectral break can be compared to the expected dispersive scale for the transition from Alfvénic to kinetic Alfvén turbulence. Chen et al. (Reference Chen, Leung, Boldyrev, Maruca and Bale2014a ) examined periods of extreme $\unicode[STIX]{x1D6FD}_{i}$ so that the various ion scales could be distinguished, and found that for $\unicode[STIX]{x1D6FD}_{i}\gg 1$ the break occurs close to the ion gyroscale ( $k\unicode[STIX]{x1D70C}_{i}\sim 1$ ), which is indeed the dispersive scale for Alfvénic turbulence, but at $\unicode[STIX]{x1D6FD}_{i}\ll 1$ it occurs at the ion inertial length ( $kd_{i}\sim 1$ ). This latter result remains to be explained, but one possibility is the presence of a significant $k_{\Vert }$ component in low- $\unicode[STIX]{x1D6FD}_{i}$ turbulence (Boldyrev et al. Reference Boldyrev, Chen, Xia and Zhdankin2015).

4.4 Intermittency

An important feature of turbulence, alluded to in § 4.1, is intermittency, i.e. the non-Gaussian nature of the fluctuations. Intermittency has been well studied in the solar wind for $k\unicode[STIX]{x1D70C}_{i}<1$ (see reviews by Alexandrova et al. Reference Alexandrova, Chen, Sorriso-Valvo, Horbury and Bale2013; Bruno & Carbone Reference Bruno and Carbone2013), where most quantities are seen to become increasingly non-Gaussian towards smaller scales, a well-known feature of turbulence. This is related to the turbulent energy being concentrated into a smaller fraction of the volume as the cascade proceeds to smaller scales, and results in the formation of energetic structures within the plasma. Understanding this intermittency and structure generation is important for both the cascade process and plasma heating, since the energy dissipation is thought to be focussed at these structures.

As a result of the Alfvénic cascade, the probability density functions (PDFs) of the fluctuations are significantly non-Gaussian by the time ion scales $k\unicode[STIX]{x1D70C}_{i}\sim 1$ are reached. Recently, it has been possible to investigate how the PDF shape continues to change down to electron scales. Intriguingly, Kiyani et al. (Reference Kiyani, Chapman, Khotyaintsev, Dunlop and Sahraoui2009) showed that from ion to electron scales, $1/\unicode[STIX]{x1D70C}_{i}<k<1/\unicode[STIX]{x1D70C}_{e}$ , the rescaled PDFs of magnetic fluctuations do not get significantly more non-Gaussian, but retain a similar shape. Chen et al. (Reference Chen, Sorriso-Valvo, Šafránková and Němeček2014b ) found a similar behaviour for the density fluctuation PDFs, which are shown in figure 11, along with the magnetic fluctuationsFootnote ²⁰ . The similar behaviour of the density and magnetic fluctuations in the kinetic range is expected, since in kinetic Alfvén turbulence these fields are directly coupled (§ 4.1), although the self-similarity of the PDFs is unusual for a turbulent cascade. One possibility is that kinetic range turbulence is inherently mono-fractal; although a recent measurement of the multi-fractal spectrum by Sorriso-Valvo et al. (Reference Sorriso-Valvo, Carbone, Leonardis, Chen, Šafránková and Němeček2016) showed a larger multi-fractality in the the kinetic range. Another possibility is that the kurtosis of the fluctuations is limited by another process, such as the instability of the current sheet structures which may form (Biskamp, Welter & Walter Reference Biskamp, Welter and Walter1990). However, the nature of the intermittency in the kinetic range remains to be fully understood.

Figure 11.

(a) PDFs of magnetic fluctuations from ion to electron scales (using the data interval of Chen et al. Reference Chen, Horbury, Schekochihin, Wicks, Alexandrova and Mitchell2010a ). (b) Same for density fluctuations (from Chen et al. Reference Chen, Sorriso-Valvo, Šafránková and Němeček2014b ). A Gaussian distribution is given by the black dashed line.

An alternative way to examine intermittency is through the distribution of rotation angles of the magnetic field. The rotation angle $\unicode[STIX]{x1D6FC}$ is defined as the angle through which the field rotates over a given spatial scale (or time scale $\unicode[STIX]{x1D70F}$ measured by a spacecraft under the Taylor hypothesis),

(4.7) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}(t,\unicode[STIX]{x1D70F})=\cos ^{-1}\left[\frac{\boldsymbol{B}(t)\boldsymbol{\cdot }\boldsymbol{B}(t+\unicode[STIX]{x1D70F})}{|\boldsymbol{B}(t)||\boldsymbol{B}(t+\unicode[STIX]{x1D70F})|}\right],\end{eqnarray}$$

and has been used to study both the intermittency and the structures that are formed (e.g. Zhdankin, Boldyrev & Mason Reference Zhdankin, Boldyrev and Mason2012, and references therein). Such analysis was recently extended into the kinetic range by Chen et al. (Reference Chen, Matteini, Burgess and Horbury2015), in which the distribution of $\unicode[STIX]{x1D6FC}$ over scales $1/\unicode[STIX]{x1D70C}_{i}<k<1/\unicode[STIX]{x1D70C}_{e}$ was investigated. It was found that, similar to larger scales (Bruno et al. Reference Bruno, Carbone, Primavera, Malara, Sorriso-Valvo, Bavassano and Veltri2004; Zhdankin et al. Reference Zhdankin, Boldyrev and Mason2012), the PDFs of $\unicode[STIX]{x1D6FC}$ are well fit by log-normal distributions, with only small changes in shape from ion to electron scales.

The large rotations are sometimes identified as the ‘structures’, so it is of interest to investigate their distribution, how much energy they contain and how large the rotations are. Figure 12 shows the fraction of the time that the rotation angles are larger than $\unicode[STIX]{x1D6FC}$ , as a function of $\unicode[STIX]{x1D6FC}$ , for scales $\unicode[STIX]{x1D70F}$ corresponding to the ion and electron gyroscales. Also shown is the fraction of magnetic fluctuation energy $|\unicode[STIX]{x1D6FF}\boldsymbol{B}|^{2}$ contained in these angles. It can be seen that for a given angle, the energy fraction is much larger than the filling fraction, consistent with the non-Gaussian nature of the fluctuations. Similar analyses have been performed for the energy dissipation at current structures in both MHD and kinetic simulations (Wan et al. Reference Wan, Matthaeus, Karimabadi, Roytershteyn, Shay, Wu, Daughton, Loring and Chapman2012, Reference Wan, Matthaeus, Roytershteyn, Parashar, Wu and Karimabadi2016; Zhdankin et al. Reference Zhdankin, Boldyrev, Perez and Tobias2014; Zhdankin, Boldyrev & Uzdensky Reference Zhdankin, Boldyrev and Uzdensky2016b ), with similar results. However, at kinetic scales, the absolute rotation angles are relatively small. The energy fraction in figure 12 falls exponentially, with $e$ -folding angle $9.8^{\circ }$ at ion scales and $0.66^{\circ }$ at electron scales, demonstrating that large angles ${>}30^{\circ }$ do not contain significant energy in the kinetic range. This places important constraints on energy conversion and dissipation mechanisms, such as magnetic reconnection, that have been proposed to be taking place here.

Figure 12.

Fraction of magnetic rotations larger than $\unicode[STIX]{x1D6FC}$ (blue) and magnetic fluctuation energy in those angles (red) at (a) ion and (b) electron scales, along with exponential fits (black dashed) (from Chen et al. Reference Chen, Matteini, Burgess and Horbury2015).

5.1 Dissipation

One basic question is how dissipation is distributed throughout a turbulent plasma, i.e. to what extent it is focussed at the energetic structures generated by intermittency. In fact, measuring the distribution of the dissipation can, in some sense, be considered a more direct way of measuring the intermittency of the cascade, since it is more directly connected to the energy cascade rateFootnote ²¹ . Zhdankin, Boldyrev & Chen (Reference Zhdankin, Boldyrev and Chen2016a ) measured the PDFs of the dissipation, averaged over different MHD range scales, in both MHD simulations and the solar wind (where a proxy based on magnetic field measurements was used). The PDFs were found to be well fitted by log-normal distributions in all casesFootnote ²² , although the higher-order moments of both the solar wind and simulated distributions were better described by log-Poisson, rather than log-normal, scaling predictions. The rate at which these PDFs broaden towards smaller scales indicates the level of intermittency, and this was found to be consistent between the solar wind and simulations and indicative of significant intermittency of the energy dissipation. It is of interest to determine which types of structures this dissipation is focussed at, and whether these correspond to any well-known plasma phenomena. For example, reconnecting current sheets (Retinò et al. Reference Retinò, Sundkvist, Vaivads, Mozer, André and Owen2007; Sundkvist et al. Reference Sundkvist, Retinò, Vaivads and Bale2007), Alfvén vortices (Alexandrova et al. Reference Alexandrova, Mangeney, Maksimovic, Cornilleau-Wehrlin, Bosqued and André2006), double layers (Stawarz, Ergun & Goodrich Reference Stawarz, Ergun and Goodrich2015) and several others have been proposed. It remains to be determined which of these, if any, play a significant role in the dissipation of plasma turbulence.

A more fundamental, and even less well understood, question is the nature of the physical mechanisms which transfer energy from the electromagnetic turbulence to the particles and lead to the irreversible heating of the plasma. Several possibilities have been proposed, including ion cyclotron damping (e.g. Coleman Reference Coleman1968; Smith, Vasquez & Hollweg Reference Smith, Vasquez and Hollweg2012), Landau damping (e.g. Howes et al. Reference Howes, Cowley, Dorland, Hammett, Quataert and Schekochihin2008; TenBarge & Howes Reference TenBarge and Howes2013), stochastic heating (e.g. Chandran et al. Reference Chandran, Li, Rogers, Quataert and Germaschewski2010), entropy cascade (Schekochihin et al. Reference Schekochihin, Cowley, Dorland, Hammett, Howes, Quataert and Tatsuno2009) and reconnection associated mechanisms (e.g. Egedal, Daughton & Le Reference Egedal, Daughton and Le2012; Drake & Swisdak Reference Drake and Swisdak2014). Identifying which combinations of these operate, under which conditions, will likely involve detailed examination of particle distributions. While future missions may be required to fully answer this (§ 5.2), there have been some initial hints in the existing data. For example, He et al. (Reference He, Wang, Tu, Marsch and Zong2015b ) interpreted the contours of measured ion distributions as being consistent with quasilinear expectations for the cyclotron and Landau resonances, and He et al. (Reference He, Tu, Marsch, Chen, Wang, Pei, Zhang, Salem and Bale2015a ) found plateaus in the distributions at the phase speeds of the Alfvén and slow mode waves, suggesting possible resonant damping of these modes. Various other indirect evidence has also been reported for some of the other mechanisms, although it will likely be from more direct techniques (e.g. Klein & Howes Reference Klein and Howes2016) that the answer to this question is finally determined.

5.2 Future missions

In the coming years, we are set to continue understanding more about turbulence and heating in space and astrophysical plasmas, through several new missions which are due to be launched soon, or have been proposed. The Magnetospheric Multiscale (MMS) mission (Burch et al. Reference Burch, Moore, Torbert and Giles2016), launched in 2015, is already producing important new results, in particular from the increased time resolution of the ion and electron measurements (Pollock et al. Reference Pollock, Moore, Jacques, Burch, Gliese, Saito, Omoto, Avanov, Barrie and Coffey2016), in the Earth’s magnetosphere and magnetosheath. In 2017, the apogee of the MMS orbit is due to be raised (Fuselier et al. Reference Fuselier, Lewis, Schiff, Ergun, Burch, Petrinec and Trattner2016), so that for the extended mission, it will spend significant time in the free solar wind, allowing the study of kinetic turbulence there in even greater detail.

The following year (2018) is due to see the launch of Solar Probe Plus (Fox et al. Reference Fox, Velli, Bale, Decker, Driesman, Howard, Kasper, Kinnison, Kusterer and Lario2015) and Solar Orbiter (Müller et al. Reference Müller, Marsden, St. Cyr and Gilbert2013), which will both travel closer to the Sun than any previous spacecraft, and for which turbulence, along with the associated heating, is one of the main science goals. In particular, Solar Probe Plus will travel to within 9 Solar radii of the solar surface, and will have a comprehensive payload measuring the electromagnetic fields (Bale et al. Reference Bale, Goetz, Harvey, Turin, Bonnell, Dudok de Wit, Ergun, MacDowall, Pulupa and Andre2016) and particles (Kasper et al. Reference Kasper, Abiad, Austin, Balat-Pichelin, Bale, Belcher, Berg, Bergner, Berthomier and Bookbinder2015) to explore turbulence and heating within the corona for the first time. At the same time, the Voyager spacecraft are moving further away from the Sun, allowing the first in situ measurements of turbulence in the local interstellar medium (Burlaga, Florinski & Ness Reference Burlaga, Florinski and Ness2015).

Finally, Turbulence Heating ObserverR (THOR) (Vaivads et al. Reference Vaivads, Retino, Soucek, Khotyaintsev, Valentini, Escoubet, Alexandrova, Andre, Bale and Balikhin2016) is a mission concept proposed to the European Space Agency that is currently in study phase, and if selected would be due to launch in 2026. The mission will have the highest-resolution set of instruments to date and will explore the range of near-Earth space environments to understand how turbulent energy is dissipated, how that energy is partitioned and how this operates in different turbulence regimes. THOR would be the first space mission dedicated to the topic of turbulent plasma heating, with broad implications for our understanding of astrophysical plasmas. Therefore, the coming years look very promising for the study of turbulence and heating in a variety of astrophysical environments.