A review of spectral anisotropy and variance anisotropy for solar wind fluctuations is given, with the discussion covering inertial range and dissipation range scales. For the inertial range, theory, simulations and observations are more or less in accord, in that fluctuation energy is found to be primarily in modes with quasi-perpendicular wavevectors (relative to a suitably defined mean magnetic field), and also that most of the fluctuation energy is in the vector components transverse to the mean field. Energy transfer in the parallel direction and the energy levels in the parallel components are both relatively weak. In the dissipation range, observations indicate that variance anisotropy tends to decrease towards isotropic levels as the electron gyroradius is approached; spectral anisotropy results are mixed. Evidence for and against wave interpretations and turbulence interpretations of these features will be discussed. We also present new simulation results concerning evolution of variance anisotropy for different classes of initial conditions, each with typical background solar wind parameters.
Ample evidence exists for in situ heating of the solar wind. In particular, observations of the proton temperature, Tp(r), as a function of heliocentric radius, r, are well above the level associated with a pure adiabatic expansion of the medium (by factors of 10–100 in the outer heliosphere) . Dissipation at shocks and reconnection zones, turbulent dissipation and various types of wave damping are all processes that may contribute to the heating. These situations typically involve fluctuations in the velocity and magnetic field over wide ranges of length and time scales, and in many of them fluctuation anisotropies may lead to enhanced (or curtailed) dissipation. For instance, if a dynamical process steers energy away from small parallel scales, this will obviously restrict the energy available to dissipation channels active at those scales, e.g. ion cyclotron damping. Here, we review two types of fluctuation anisotropy that are robustly observed in the solar wind: spectral anisotropy and variance anisotropy. A summary of the observational results is presented, along with discussion of candidate explanations for the origin of these anisotropies. Some roles they may play in the dissipation of solar wind fluctuations are also noted.
Our focus is primarily at magnetohydrodynamic (MHD) scales because, at length and time scales larger than those characteristic of proton gyromotion, the dynamics of solar wind fluctuations can often be usefully described using MHD models. The relative simplicity of these fluid models affords some advantages for theoretical and simulation work. Of course, the dynamics near (and below) proton gyroscales involves various plasma processes, so that in those ranges, it is appropriate to employ theoretical frameworks that retain more plasma and particle effects than MHD does. We will include some discussion related to these scales. Following traditional usage in the literature, we refer to frequencies above the proton gyrofrequency as the dissipation range, even though the (plasma) physics at these scales differs significantly from the viscous diffusion characteristic of Navier–Stokes dissipation (see appendix A).
This review is intended to complement existing reviews on solar wind anisotropy and related matters [2–6], and also other papers appearing in this theme issue [7–11]. Thus, we do not attempt a complete review of the literature. Familiarity with the basic features and properties of the solar wind is assumed here, as is a basic understanding of turbulence concepts like inertial range, dissipation range, spectra, correlation lengths and times, and energy cascades [12–14]. See appendix A for a terse summary.
The paper is structured as follows. We begin by defining spectral and variance anisotropies and discussing how these arise from distinct features of a vector field. Separate sections for each type of anisotropy follow, wherein we review relevant observational studies and discuss possible explanations. Some new results from compressible three-dimensional MHD simulations are also included. A short summary concludes the main body of the paper. Appendix A covers some primary points regarding spectra, spectral ranges and related ideas, and appendix B considers issues related to global and local mean magnetic fields.
2. Two types of anisotropy: definitions and distinctions
Let us write the velocity and magnetic fields in terms of mean values plus fluctuations about those means, 2.1and 2.2where 〈⋯〉 denotes an appropriate space (x) and/or time (t) averaging operator, which might be global or local. Throughout this paper, the magnetic field is in Alfvén velocity units, i.e. the actual magnetic field (SI units) is , where ρ is the mass density and μ is the magnetic permeability of the medium. We also make use of the Elsasser variables for the fluctuations, z±(x,t)=v±b, where these variables should not be confused with the Cartesian coordinate z. Usually, Cartesian mean-field-aligned coordinates are employed, with . The circumflex indicates unit vectors, with that parallel to heliocentric radius being , for example. To good approximation beyond approximately 0.1 AU, the mean solar wind speed is . Other standard notation used is Alfvén speed V A, proton plasma beta βp, proton gyroradius ρi, proton gyrofrequency Ωci=2πfci and proton skin depth di≡V A/Ωci. It will sometimes be important to note that the mean field employed is a local one, and we then write in place of B0 (cf. appendix B).
If the Cartesian components of the fluctuating field have unequal energies, then the field is said to exhibit variance (also known as component) anisotropy. For v, this would mean departures from . In the solar wind, one typically has , if the averaging is over many correlation lengths, Lcor, where Lcor∼106 km at 1 AU [15–17].
However, considerable care is needed when interpreting observed variance anisotropy: even for a perfectly isotropic distribution of energy, calculated variances will be anisotropic(!) if, in effect, one only has access to reduced spectra (or, equivalently, to one-dimensional correlation functions). A reminder of Batchelor's , p. 50 original incompressible hydrodynamics demonstration is enlightening. Suppose the energy distribution is isotropic, so that its spectral tensor is Sij(k)=(1/2)(δij−kikj/k2)E(k), where E(k) is the modal spectrum with E(k)=Ck−α in the inertial range; for Kolmogorov scaling (see appendix A). Unfortunately, modal spectra are often not experimentally available, with researchers having to make do with the reduced spectra. (This is also the case for most single spacecraft solar wind datasets.) The reduced spectral tensor relative to the x direction is defined as , and it is straightforward to show that for kx in the (powerlaw) inertial range, the ratio of reduced spectral amplitudes is anisotropic, 2.3(the kx dependence has cancelled out). This is 4/3 for Kolmogorov scaling. Similarly, variances calculated using only contributions from inertial range (and smaller) scales are also misleadingly anisotropic. For instance, if we define , etc., with the parameter kx in the inertial range, then it follows from the same assumptions leading to equation (2.3) that . So, somewhat counter-intuitively, a measured variance anisotropy does not necessarily imply anisotropy of the modal energy spectrum. (Variance isotropy is recovered when kxLcor≪1, i.e. when the integrals include all scales with significant energy.) Similar results can be derived for isotropic MHD and axisymmetric MHD [19–21].
The above-mentioned calculations assume that the energy spectrum is accurately known. When averaging intervals are less than a correlation scale, the system is likely to be statistically/ergodically undersampled so that the estimate for the spectrum may involve random anisotropies owing to non-convergence . Unravelling these stochastic anisotropies from ‘kinematic’ ones like those of equation (2.3) and/or genuine physical anisotropies could be very difficult. Moreover, locally computed anisotropy, even when adequately sampled, may be properly viewed as higher-order statistics .
When the energy distribution at a given scale, say l, is not isotropic, one speaks of spectral (also known as wavevector or correlation) anisotropy. In terms of a modal energy spectrum E(kx,ky,k∥), where k=(kx,ky,k∥) is the Fourier wavevector, spectral anisotropy might take the form E(k,0,0)≫E(0,0,k), for example, where k∼1/l. Equivalently, considering a homogeneous correlation like R(r)=〈b(x)⋅b(x+r)〉, the existence of correlation anisotropy would mean that , for some independent directions , . Naturally, spectral anisotropy is often associated with energy transfer that is also anisotropic. In particular, for incompressible MHD, there is abundant evidence that strong (and even moderate) mean magnetic fields engender significant suppression of energy transfer in the parallel direction, so that the perpendicular energy cascade is much stronger than the parallel cascade. Note that spectral anisotropy usually occurs over a range of scales, rather than at (isolated) individual scales.
It is useful to elucidate the different kinematic origins of spectral anisotropy and variance anisotropy. We make use of the vector potential for the magnetic fluctuations, defined by b=∇×a. Solenoidality of b means that in Fourier space there are only two linearly independent components to a(k), at each wavevector k, where a(k) is the Fourier transform of a(x). Choosing as the reference direction, a toroidal–poloidal decomposition for b(k) is 2.4provided .1 This decomposition is general and in no way assumes that the fluctuations are waves (linear or otherwise). Nonetheless, in cases where linear MHD waves are present, the two potentials are associated rather cleanly with the different types of waves. The toroidal piece of b, proportional to aT(k), is polarized in the same sense as an Alfvén wave's b, whereas magnetosonic waves have their b polarized in the same sense as the aP(k) contributions. In the incompressible limit, v may be decomposed exactly as above, while in the compressible case, a component of v parallel to k must also be allowed for.
Assuming axisymmetry about B0, variance anisotropy is determined by the relative magnitudes of the potentials aT and aP. For instance, if aP(k)=0 for all wavevectors, then bz(x)=0 identically, yielding an extreme case of variance anisotropy (a superposition of Alfvén waves would have this configuration). Spectral anisotropy, on the other hand, is determined by how the potentials vary with the direction of k. For example, how aT(5,0,0) differs from aT(3,0,4), where these both produce a by(k).
Note that the connection between variance anisotropy (related to the minimum variance direction, MVD) and spectral anisotropy is not unique. Hence, variance anisotropy information—on its own—is usually not sufficient to infer the actual spectral anisotropy. A well-known example involving two-dimensional turbulence and slab Alfvén waves illustrates this. Turbulence that is strictly two-dimensional (with respect to the z-axis) has aP(k)=0=bz(x) and a toroidal potential that is independent of k∥: aT(kx,ky). Its MVD is in the z-direction. Slab Alfvén waves also have aP=bz=0 and are polarized in the x–y plane. Thus, a superposition of them will have an MVD in the z-direction—just as two-dimensional turbulence does. However, the wavevector dependence of the slab waves is completely complementary to that of two-dimensional turbulence because they have aT=aT(k∥). Clearly, an MVD analysis cannot discriminate between these physically very different states. Recall, also, that the amplitude of a linear Alfvén wave with wavevector k is in the k×B0 direction, so that the MVD is itself not unique here. That is, all directions in the k–B0 plane serve as MVDs for a non-slab Alfvén wave.
3. Spectral anisotropy
Measurements in laboratory plasma physics devices provided the first indications of the prevalence of spectral anisotropy for systems with an energetically significant mean magnetic field, [24,25]. In these studies, correlation lengths parallel to B0 were found to be much larger than those in directions perpendicular to B0, indicating that at smaller scales energy is mostly associated with perpendicular structure. Physical grounds for this behaviour were presented [26,27], and Shebalin et al.  provided an explanation based on ‘weak MHD turbulence’ and the associated suppression of parallel spectral transfer. Later, Goldreich & Sridhar [29,30] proposed a strong turbulence (‘critical balance’, CB) phenomenology that accounts for the anisotropy, and this and its cousins are widely used in current modelling approaches to MHD turbulence. The derivation of reduced MHD (RMHD) owing to Montgomery [26,27] also makes it clear how this anisotropy arises in a strong turbulence context: the strongest turbulence occurs when nonlinearity operates more quickly than Alfvén wave propagation. This provides the small parameter for deriving RMHD. The theory breaks down if there is too much parallel spectral transfer; however, the argument of Shebalin et al.  shows that such transfer is inhibited. Recently, an analytic explanation that is neither perturbative nor phenomenological was advanced . These explanations are reviewed in §3b.
In the solar wind, an early report of correlation anisotropy was provided by Crooker et al. , using 1 AU magnetic field data from ISEE 1 and ISEE 3. Calculations of a hybrid correlation coefficient between the data from the two spacecraft revealed the presence of anisotropic magnetic features, with parallel scale lengths at least four times the perpendicular ones. Some years later, a two-dimensional correlation function for the magnetic field was constructed, R(r⊥,r∥), using 1 AU data . This too indicated that the magnetic energy distribution was anisotropic, with contours of R(r⊥,r∥) displaying a ‘Maltese cross’-type pattern with power enhancements approximately parallel to and perpendicular to the mean magnetic field. These results suggested that two-component models for the fluctuations might be of use.
The many subsequent observational studies have added quantitative details and are, on the whole, in broad agreement with each other. At MHD scales, results from nonlinear simulations are often in accord with the solar wind observations. Below the proton gyroscales, there is still much debate as to whether the important physics is due to particular wave modes, turbulence and cascades, or some combination of these.
Several useful idealizations or phenomenologies for the observed spectral anisotropy exist, and we now describe two of them. The first is a two-component ‘slab+2D’ model , for which excited wavevector modes have either (the slab piece) or k⋅B0=0 (the two-dimensional piece; figure 1a). The modal energy spectrum is then E(k)=Eslab(k∥)δ(kx)δ(ky)+E2D(kx,ky)δ(k∥). For incompressible fluctuations, the slab fluctuations are strictly transverse to B0, whereas the two-dimensional ones could be so-called 2.5D meaning that they can have an amplitude component parallel to B0. Usually, we will not need to distinguish between slab+2D and slab+2.5D. Further assumptions can be imposed, e.g. axisymmetry with respect to the B0 axis and/or powerlaw inertial ranges. Note that this slab+2D model does not make any inherent assumptions about the specific nature of the fluctuations, only about the wavevector dependence of each piece. That said, Alfvén waves and two-dimensional turbulence are well-motivated candidates for the slab and two-dimensional components, respectively. When dissipation range scales are considered, kinetic Alfvén waves (KAWs) are another candidate for the two-dimensional piece.
Clearly slab+2D is a rather drastic approximation, with excitations restricted to modes either on the k∥ axis or in the k∥=0 plane. Thus, most of k-space is unpopulated and it seems likely this is an oversimplification. Nonetheless, such idealized skeleton models provide attractive analytic tractability and, perhaps surprisingly, have enjoyed some success in matching observations, suggesting that they often capture the important elements of the situation.
A class of inertial range models with more realistic coverage of k-space than slab+2D is the set of phenomenologies based on CB ideas [29,30]. In these, the energy is concentrated in wavevector modes for which the linear wave time scale τA(k) is comparable to the nonlinear time scale τnl(k) (figure 1b). For MHD, the anisotropic nature of τA(k)=1/|k∥B0|, along with the assumption of isotropic forcing at scales of order Lcor (the energy-containing scale for the turbulence) leads to the equal time-scale curve k∥B0∼(u0/Lcor)(kLcor)2/3, where u0 is the RMS turbulence velocity.2 The key point is that modes with ks ‘near’ the equal time-scale curve have important nonlinear physics, whereas modes with ks well outside the curve may have some prominent wavelike physics, because τnl(k)≫τA(k) there. The width of the CB region around the equal time-scale curve is not fully determined by the phenomenology and various models for it have been employed [29,36,37]. Note that CB does not require wavevector modes with k∥≫k⊥ (e.g. slab-like modes) to be unpopulated, although their energy level is assumed to be low. Many implementations of CB treat this component as negligible, despite its importance in applications such as scattering of cosmic rays [38,39]. Around the equal time-scale curve, the Alfvén wave time scale plays a pivotal role. However, any waves at these ks are argued to only last for one or so periods and thus are rather fleeting things . The extent to which wave properties like dispersion relations and eigenstructure are important in turbulence is still under investigation [11,40–42]. Figure 1c indicates typical directions of spectral energy transfer inside and outside the equal time-scale curve (cf. ).
(a) Review of observations
Following the original demonstration of correlation anisotropy , many examples of two-dimensional correlation functions, e.g. R(r⊥,r∥), two-dimensional structure functions, and two-dimensional spectra have been presented. Here, r⊥ and r∥ are relative to the mean field B0, and similarly for k⊥ and k∥. Regrettably, it is not completely straightforward to compare results from these studies, because they may employ distinct techniques, single or multiple spacecraft, and/or pertain to different scale ranges. Table 1 lists results and some relevant parameters from many of these studies, and we discuss a selection of them below.
Using 5 years of near-Earth (ACE) data, Dasso et al.  investigated correlation anisotropy for scales around the middle decade of the inertial range. It was found that fast wind (Usw>500 km s−1) was relatively more dominated by fluctuations with quasi-parallel k, whereas for slow wind (Usw<400 km s−1), the quasi-perpendicular modes were more dominant. As slow wind takes longer to get to 1 AU, this can be interpreted as meaning its turbulence has had more time to evolve, in this case, towards an energy distribution concentrated in quasi-perpendicular ks. Intriguingly, at somewhat smaller scales than Dasso et al. studied, results from the ‘spectrum ratio’ test (see below) indicate very little wind speed or distance (over 0.3–0.9 AU) dependence to the spectral anisotropy [45,46]. Hamilton et al. suggest that at smaller scales the turbulence may already be in approximate steady state by 0.3 AU, whereas at larger scales , the distinct anisotropies seen in fast and slow wind may be a remnant of the solar source of the fluctuations.
Wicks et al.  calculated magnetic energy (wavelet) spectra using Ulysses data. They report that the power is roughly isotropic for scales larger than inertial range ones, . At larger wavenumbers, anisotropy consistent with more power in oblique wavevectors is present and its strength increases as kρi increases, until kρi≈0.5 is reached.
The above results are consistent with the perpendicular energy cascade rate being greater than the parallel one, as is also seen in direct 1 AU measurements of the cascade rate, via third-order moment calculations [8,54]. Simulation results [3,28,36,55,56] and phenomenologies [29,35,57,58] support the notion that small scales are more anisotropic, and achieve their quasi-two-dimensional-type states more rapidly than larger scales.
A related method of investigating spectral anisotropy was proposed by Bieber et al. . When the fluctuations are modelled as slab+2D, the frequency spectrum exhibits a dependence on ψ—the angle between B0 and the mean solar wind speed U—that can be used to estimate the fraction of energy in the slab and two-dimensional components. Assuming powerlaw slopes, α, for each component, one obtains 3.1where C2D and Cslab are the fitting parameters. Using Helios data (0.3–1 AU), they found a best-fit value of 95% two-dimensional, 5% slab, for a frequency band in the inertial range. Bieber et al. also proposed the ‘spectrum ratio’ test, which makes (separate) use of the diagonal components of the spectral matrix, rather than the trace employed in equation (3.1). For this test, the best-fit value was 84% two-dimensional, 16% slab. Both tests strongly support the conclusion that fluctuation energy is mostly in Fourier modes with their wavevectors k perpendicular to the mean field B0.
The ‘spectrum ratio’ test has since been employed on many other solar wind intervals [43,45–47,51,59] (table 1). Inertial range results are mostly consistent with an energetic predominance for two-dimensional modes, with the two-dimensional fraction ranging from 50% to ≈100%. Moreover, in the inner heliosphere, there is only weak distance or wind speed dependence in the results, at least towards the high-frequency end of the inertial range [45,46].
Slab+2D-type fits similar to equation (3.1) have been widely used [48–50,53,60–62] and these also report high values for the inertial range two-dimensional fraction, e.g. 70–100%. Interestingly, Forman et al.  compared slab+2.5D fits and CB fits for Ulysses inertial range data, finding that both produce good agreement with the observations, although the CB fits are a little better.
He et al.  investigate the radial evolution of inertial range spectral anisotropy using Helios data (0.3–0.87 AU). They report two-dimensional energy spectra that exhibit enhanced power along a ridge at k⊥>k∥, roughly similar to a CB scaling. Another power enhancement exists close to the k∥ axis, suggestive of a slab component. At larger distances, the ridge is more oblique and the slab-like component weaker. These results are obtained using a method, new to the community, that involves fitting of processed data at an intermediate stage. First, the observations are used to calculate a two-dimensional (r⊥,r∥) correlation function, with its arguments relative to local mean fields . Second, a direction θk is chosen, and the two-dimensional function integrated over the coordinate normal to that direction to give a one-dimensional (projected) correlation function. This is done for many different θk. Third, stretched exponentials are fitted to each of these one-dimensional projections (to ensure their Fourier transforms are positive), and then these two-dimensional fits are Fourier transformed and reassembled using their θk to give the three-dimensional spectrum. Further calibration of this procedure on known (synthetic) two-dimensional spectra would be welcome.
We turn now to the dissipation range (frequencies above ∼fci), where spectral anisotropy observations are a little puzzling. When the spectrum ratio test is employed, the usual finding is that slab fluctuations are energetically dominant. See the rows in table 1 with either ‘diss’ or a minimum frequency of 0.3 Hz in the ‘freq. range’ column. Note that in all cases where both inertial range and dissipation range scales were investigated as part of the same study, the two-dimensional fraction for the dissipation range is substantially smaller than that for the inertial range, and usually less than 50%. A possible explanation is that the two-dimensional component is preferentially dissipated and associated with a steepened spectrum at those scales [43,53,64]. Another possibility is that temperature anisotropy instabilities excite whistlers and/or ion cyclotron waves at quasi-parallel wavevectors with k∥di≈1, leading to increased dissipation range energy for the slab-like component [9,65,66].
However, the situation is complicated, as shown, for example, by Podesta . Using STEREO data and a wavelet-based approach (with local mean fields), Podesta determined the fluctuation power at different angles ψ between the solar wind velocity and . Here, we are interested in the power in the angle band near 90° (the quasi-k⊥ power), and that near 0° (the quasi-k∥ power). Their ratio, P( f;ψ≈90°)/P( f;ψ≈0°)=P⊥/P∥, is a measure of the dominance of quasi-k⊥ power over quasi-k∥ power. An example from  is reproduced here as figure 2a. The dissipation range behaviour is interesting: first the spectral anisotropy decreases, then increases to a significant (absolute) maximum for the evaluated scales, at ≈1 Hz, and then decreases sharply to its lowest dissipation range value. At all frequencies shown, the quasi-k⊥ power is dominant, which differs from the typical dissipation range results listed in table 1, although the trend at higher f is towards reduced dominance. Podesta attributes the initial dissipation range decrease of P⊥/P∥ to an enhanced population of nearly parallel-propagating waves at k∥ρi∼1, owing to an unknown source, and the peak to a strong perpendicular cascade at those f. Results from eight other intervals examined by Podesta  are similar.
Other studies, including those employing the multi-spacecraft k-filtering technique [68,69], have also found that the quasi-perpendicular wavevectors remain energetically dominant in the dissipation range [67,70–75], as in the inertial range. In k-filtering, multi-spacecraft observations (e.g. from Cluster) are used to construct the spectral energy density as a function of k=(kx,ky,kz) and spacecraft frame frequency, ωsc. Then, at each ωsc, peaks in the energy density are sought and the associated ks employed, with the Doppler shift k⋅U, to determine the frequencies in the plasma rest frame, and hence points on dispersion relations. Energy not associated with ωsc–k peaks is apparently not usually considered further, or is treated as incoherent noise ([69,74]). The method avoids using the Taylor frozen-flow approximation (appendix A), and thus can provide more direct information on spatial structure.
Figure 2b, reproduced from Sahraoui et al. , presents some k-filtering results from Cluster, finding when , where ω is the plasma frame Fourier frequency. Although this is consistent with linear Maxwell–Vlasov dispersion relations for very oblique KAWs (), with such strong obliquities of the ks the results can also be interpreted as indicating the presence of relatively energetic quasi-two-dimensional turbulence. Indeed, Narita et al. [71,72,76,77] use essentially the same multi-spacecraft technique, over similar wavenumber ranges, and state that they find no evidence of a linear dispersion relation. They favour an explanation in terms of well-developed strong quasi-two-dimensional turbulence.
Recently, two-component and/or CB energy spectrum models have been employed to determine the associated normalized reduced magnetic helicity spectra, σm, as a function of time scale and ψ, the angle between the solar wind and [66,78]. These are compared with observational determinations of σm, in efforts to constrain the free parameters of the energy spectra models. The scales investigated are mostly in the inertial range but, as in many of the studies discussed above, the beginning of the dissipation range is also sampled. The best fits are obtained when the bulk of the energy is in quasi-k⊥ modes.
Summarizing the dissipation range results, those obtained using the ‘spectrum ratio’-type tests usually indicate a preponderance of quasi-k∥ power, whereas most studies employing other techniques find the quasi-k⊥ power is dominant. A resolution of this discrepancy would be welcome. Other recent reviews have further discussion of dissipation range anisotropy [4,6]. Given the disparate results from the various types of studies, it is evident that further work is needed.
Finally, we note that cosmic ray scattering data also provide support for spectral anisotropy. Models for the scattering based on slab waves show poor agreement with observations, whereas slab+2D-type models display much better agreement [19,38,39,79–81].
(b) Theoretical explanations
Let us sketch the weak turbulence argument for the development of spectral anisotropy and enhanced perpendicular energy transfer in incompressible MHD  (see also [82–85]). The primary assumption is that at leading order the fluctuations are Alfvénic, meaning that at each point in space either v(x,t)=±b(x,t), or there is a superposition of these two states. Equivalently, in Fourier space, for (inertial range) wavevectors k, one has v(k,t)=±b(k,t), or a superposition of these two states. In the absence of nonlinear effects, such fluctuations are propagating (shear) Alfvén waves with frequencies ω(k)=±k⋅B0, provided k⋅B0≠0. The waves with a group velocity (vg=∇kω) parallel to B0 have ω(k)=k⋅B0,3 whereas those with an antiparallel group velocity have ω(k)=−k⋅B0. However, two-dimensional fluctuations—those with k⋅B0=0—are not linear Alfvén waves and they do not propagate along B0. When their nonlinear self-interactions are strong, they are the elements of two-dimensional turbulence. In ‘wave language’, they may be called non-propagating modes or vortex modes. Whatever they are called, they play a crucial role in the argument.
The next step is to use the leading-order Alfvénic fluctuations to calculate the first-order nonlinear corrections. Recall that for incompressible three-dimensional MHD, ∂z−/∂t=−z+⋅∇z−+⋯ , with an analogous equation for z+. This Elsasser form highlights the key point that nonlinear interactions only occur if both z+ and z− are non-zero. Now, consider the interaction of Fourier modes z+(k′) and z−(k′′), with respective frequencies ω(k′)=−k′⋅B0 and ω(k′′)=k′′⋅B0. These will combine to drive z−(k) which has frequency ω(k)=k⋅B0—if the well-known matching (resonance) conditions for the wavevectors and frequencies are satisfied: k=k′+k′′ and ω(k)=ω(k′)+ω(k′′). Dotting the former with B0 and using the equations for the frequencies yields 3.2and 3.3Clearly, these can both be true only if k′∥=0, i.e. z+(k′) is a two-dimensional mode.4 Physically, this can be interpreted as two Alfvén waves with the same k∥ and frequency, propagating through a ‘two-dimensional structure’.5 The two-dimensional structure enables the transfer of energy from one wave to the other, necessarily at constant k∥, so that it is the perpendicular structure of the waves that alters. Assuming approximate locality of the Fourier space interactions, i.e. k⊥′∼k⊥′′, means that typically and the net energy transfer will tend to be towards larger k⊥: a forward cascade in the k⊥ plane. Because, at this order, no energy is transferred in the parallel direction, it is evident that the energy spectrum will become anisotropic. In essence, this is an argument about how fluctuations that retain a strong wave-like character (i.e. are non-two-dimensional) evolve. Details of the formal application of weak turbulence theory are available [84,85], including a derivation of the associated powerlaw slopes for the energy spectra. For the zero cross helicity case, the inertial range energy spectra scale as , with the k∥ dependence set by the driving and/or initial conditions (ICs).
Gary  has presented a similar analysis for modes in linear Vlasov–Maxwellian plasmas with βp=0.01–1. Assuming a two-dimensional (radial) k-space cascade, and the interaction of either three Alfvén modes, three magnetosonic modes (both at MHD scales: kdi<0.5) or three magnetosonic-whistler modes (0.4<kdi<4) he generally finds that the cascade leads to energetic predominance for k⊥>k∥. The exception is the βp=1 magnetosonic mode cascade, which seems to favour k⊥≈k∥. One conclusion is that there appear to be no cases in which an enhanced parallel cascade occurs.
Although this weak turbulence approach—nonlinear corrections to the linear wave dynamics—appears plausible for systems with low levels of nonlinear activity, it is striking, and intriguing, that spectral anisotropy is also a well-observed feature in numerical simulations of strong MHD turbulence, that is, in systems where the nonlinear effects are strong, rather than perturbative [28,36,55,83,87–90]. A strong turbulence explanation is needed.
One possibility is that even in strong turbulence, the fluctuations retain some wavelike characteristics, such as time scale, eigenstructure or polarization for the mode. CB phenomenologies include such assumptions, as does the recently posited quasi-linear premise which states that ‘some properties of magnetized plasma turbulence can be understood by modelling the turbulence as a collection of randomly phased linear waves’ , the idea being that the nonlinear interactions transfer energy between linear wave modes, with the latter in some sense dominant, because they retain (at least some) linear wave characteristics. In particular, it is argued that for ‘a turbulent fluctuation with a given wavevector, the amplitude and phase relationships between different (vector) components of that fluctuation are likely to be related to linear eigenfunctions of the characteristic plasma wave modes’. Numerical and observational evidence for and against this premise is discussed by Klein et al. , and it is not yet clear whether the premise correctly describes the physics of turbulence. Note that as quasi-linear premise proponents acknowledge, third- and higher-order correlations will probably be incorrect, because the random phase assumption makes it difficult to achieve the appropriate phase correlations inherent in these higher-order correlations. Such correlations are of course essential to the existence of coherent structures, including current sheets. Thus, magnetic reconnection—robustly seen in MHD, hybrid and particle-in-cell simulations, and solar wind and magnetospheric observations—may be difficult to reconcile with the quasi-linear premise (cf. ). Note that weak turbulence approaches do not always require random phases [91,92].
We are aware of only one explanation for spectral anisotropy that is neither perturbative nor phenomenological and also allows for strong turbulence [31,93]. It is based on the coupling of third-order (and higher) correlations to the second-order correlations (e.g. the energy spectrum), and we now outline it. Consider the infinite set of two-point correlation functions in homogeneous incompressible MHD, such as and , where the prime (′) is shorthand for evaluation of the field at position x+r, rather than x. Their evolution is governed by the unclosed hierarchy of von Kármán–Howarth (vKH) equations [93–97]. We write the first three of these schematically for a generic fluctuation u, 3.4 3.5 3.6Most terms involving pressure and dissipation (e.g. viscosity, ν) are not shown, and the only spatial derivatives indicated are those involving B0⋅∇=B0j∂j. (The schematic equations are thus not dimensionally correct. See Wan et al.  for the full equations.) The vKH equations hold for all homogeneous flows, whether laminar, weakly turbulent or strongly turbulent.
Inspecting the terms in the vKH equations, it is striking that B0 appears in the third-order and all higher-order equations, but not in the second-order one. And yet, simulations, solar wind observations, and theories and phenomenologies all indicate that the mean field exerts considerable influence on second-order correlations, inducing anisotropy in the energy spectrum, for example. This apparent inconsistency is resolved by considering the coupling of these equations. Equation (3.5) shows that 〈uu′u′〉 has an explicit dependence on B0 (and also an implicit one via the dependence of the fourth-order correlation on B0). In particular, its characteristic time scale will be a function of B0, presumably related to the Alfvén time scale for the lag r. Hence, the evolution of 〈uu′〉, equation (3.4), has a hidden, or implicit, dependence on B0 through the third-order correlation, and this produces anisotropy at second order.
Note that this explanation does not assume anything about the nature of the fluctuations (e.g. wave-like or turbulence), except that the correlations are homogeneous. On the other hand, although the pathway for the development of second-order correlation anisotropy is clear (with the B0 dependence ‘cascading in’ from (all) higher orders), the specifics of the process are not apparent. In this sense, we are still seeking a full strong turbulence explanation.
4. Component (variance) anisotropy
This is conveniently quantified using ratios. There are numerous possible choices, many of which have seen use in the literature; however, provided the same averaging procedure is used in each case, it is straightforward to transform between these variants. Here, we employ a mean-field-aligned coordinate system with B0 along the z-axis, and define the magnetic anisotropy ratio for fluctuations b=(bx,by,b∥) as 4.1where 〈⋯〉 is an appropriate averaging operator, which might be global or local and carried out in either physical space or Fourier space. In addition, whether B0 is computed as a global or local mean is an important issue [23,48,60,61,98–102]; see appendix B. The velocity anisotropy ratio, Av, is defined analogously. A popular alternative ratio is the magnetic compressibility, , although it is sometimes defined as the square root of this. We remind readers that for averaging intervals shorter than a correlation length or correlation time, Ab≠2 does not necessarily imply that the energy distribution is anisotropic. See the discussion around equation (2.3).
Table 2 summarizes the observational results for Ab, and also Av in the few cases where it has been reported. Pertinent features of each study include the wind type sampled (e.g. fast, slow, trailing edge of high-speed stream), heliocentric distance and the time scale (or frequency range) over which the fluctuations were averaged to obtain the anisotropy ratios. In many cases, the mean field was calculated as a global average, i.e. over the whole interval. However, in some structure function- or wavelet-based studies, a local mean field is used, and this has been argued to be a more physically motivated approach, because the fluctuations should respond to the local conditions [98,100,101]. Several studies [101,102] indicate that using the global mean field rather than a typically leads to a larger estimate for , and thus a reduced Ab estimate.
Some trends are evident, although caution should be exercised given the relatively small number of studies listed. First, the dominance of perpendicular power over parallel power is essentially ubiquitous in the inertial range, often by more than a factor of 10. Second, Ab appears to increase with heliocentric distance up to ≈1 AU, and decrease thereafter. The data suggest this is the case both at high latitudes (where the wind is almost always fast), and in the ecliptic. Third, Ab>Av in both fast wind and slow wind, and from 0.3 to 10 AU. Finally, although not particularly clear from the table, above the proton gyrofrequency Ab( f) decreases (often monotonically) with f towards isotropy. In particular, Kiyani et al.  report, using 1 AU data, how Ab goes from anisotropic values at MHD scales to isotropic ones as the electron gyroradius is approached (figure 3). An increasing relevance of KAWs is a candidate explanation, and, more generally, they show how the Hall term in Ohm's law causes the parallel variances to increase at sub-ion scales. Other studies report results consistent with this behaviour [43,45,67,108–111].
Perri et al.  have analysed Cluster data in terms of the eigenstructure of the magnetic variance matrix [112,103] averaged over 13 different time scales from 0.09 to 360 s. They report on the statistics of the ratio of the maximum eigenvalue to the minimum one, Γ, which is related to Ab. Indeed, if the MVD is along the mean field, then Ab≥Γ. This is typically the case at large scales in the solar wind, but is not as likely at smaller scales , and thus Ab can be less than Γ. Because Perri et al. report the statistics for Γ and the MVD angle to , it is not straightforward to determine Ab for their data. They find that Γ>1 almost always, and that the most probable values are large: Γ≈100 for averaging times of 0.2 s and 1.5 s, and Γ≈30 for 46 s averages. Overall, at both inertial range and dissipation range scales, substantial and scale-dependent MVD-type anisotropy is found. The anisotropy increases at f>fci, with the fluctuation amplitude tending to lie in the direction of maximum variance (i.e. be approximately one-dimensional). Note that this is not necessarily inconsistent with the results discussed in the previous paragraph, because as the averaging scale decreases, the MVD tends to lie at a larger angle to the local mean field .
Explanations for variance anisotropy in terms of linear wave properties have a seductive simplicity. Numerous authors have used linear kinetic theory to calculate polarizations and other features associated with specific wave modes (e.g. KAWs, whistlers), and then compared these results with observations. Unfortunately, Ab is often insufficient, at least on its own, to make an unambiguous identification of a mode [42,101]. The point has been made that it is important to use the kinetic wave modes, because there are important differences between them and the MHD wave modes [42,101]. For instance, the collisionless Vlasov–Maxwell Alfvén mode has a component along the mean field (finite Ab), whereas the MHD Alfvén mode is polarized strictly transverse to it (). Reasonable agreement between linear theory and observations is used to conclude in favour of one mode or another. The majority of studies cite KAWs as the most likely candidate, with whistlers typically (but not always ) being a poor fit to measurements [42,43,64,101,108,109,113,114]. Note that linear theory predictions are overwhelmingly determined using a uniform and constant global mean field, whereas recent observational results often employ space (or time)-dependent . However, if the direction, say, of varies on a time scale only somewhat slower than that of the fluctuations of interest, then is not a very stably defined mean quantity. The relevance of the linear theory results to the observational ones is then unclear. Similar comments apply to the length scale over which is defined . Moreover, these explanations all presume that the fluctuations are dominantly wave-like. Although there is some numerical support for the wave-based explanations of variance anisotropy, for example in particle-in-cell simulations  and gyrokinetic simulations , it remains unclear whether strong turbulence has this property, at either MHD scales or sub-ion ones [115–117]; cf. §3b.
Another way of analysing variance anisotropy is in terms of the power in the toroidal and poloidal components of the fluctuations (and the longitudinal component for v). As stressed in §2, this decomposition is general and does not imply the fluctuations are waves. Nonetheless, the toroidal and poloidal directions do match up cleanly with the polarizations for linearized MHD waves, Alfvén and magnetosonic, respectively. Wicks et al.  performed such an analysis using Ulysses data. A physical interpretation of their results is that (inertial range) solar wind turbulence mostly consists of toroidal fluctuations that have energy concentrated in quasi-perpendicular wavevectors. The poloidal component is less energetic, for a given k, yielding a net elliptical polarization to b(k).
To explore this toroidal/poloidal anisotropy further, we present new results from a set of decaying compressible (polytropic) three-dimensional MHD turbulence simulations (see  for the equations solved). A periodic pseudospectral method with a resolution of 512 Fourier modes in each Cartesian direction was employed. Figure 4 shows some quantities related to variance anisotropy for three particular runs, each with uniform mean field strength B0=2, initial plasma beta βp=1, initial sonic Mach number Ms=0.5, and initially excited wavevectors satisfying 3≤|k|≤9. In the column headings, the angle brackets 〈⋅〉 denote averaging over the whole simulation domain, and perpendicular and parallel are computed with respect to the global mean field . The first column of figure 4 indicates that Ab>Av (as seen in the solar wind), aside from some early-time transients associated with generation of either longitudinal velocity fluctuations (top two rows) or toroidal and poloidal v (bottom row), depending upon the ICs. From the second column, it appears that when there is some toroidal b present initially, the dynamics acts to make this component ≈70–80% of the fluctuation magnetic energy, at least for these particular simulations. The v fluctuations evolve similarly, although their toroidal component is less dominant because an O(M2s) longitudinal velocity component emerges . The final column of figure 4 displays three different v2/b2 ratios: that for all the fluctuations (rA), that for just the toroidal components (rtorA), and that for just the k∥=0 components (r2DA). Although there can be significant fluctuations about the average values, in all cases (i) rtorA≈1, which is not inconsistent with this component being strongly Alfvén wave-like, (ii) r2DA<1 (except at short time), which, as expected, is inconsistent with wave-like features , and (iii) is a little bigger than 1.
The energetic dominance of the toroidal fluctuations in these turbulence simulations is consistent with the Wicks et al.  results; however, solar wind observations also usually report rA<1, corresponding to ‘excess’ magnetic energy rather than the ‘excess’ velocity energy seen here. Part of the explanation may be that the simulations have a two-dimensional energy fraction of ≈10% of the fluctuation energy—which is quite different from the typical ∼80% two-dimensional–20% slab results discussed in §3. A fuller study exploring a range of βp values and two-dimensional energy fractions is underway.
Regarding the velocity and magnetic field fluctuations, one can state with confidence that in the solar wind inertial range, observations indicate that they are predominantly polarized transverse to the mean magnetic field, and that their spectral energy distribution is not isotropic. Indeed, there is multi-pronged evidence for fluctuations with quasi-perpendicular wavevectors being considerably more energetic than those with quasi-parallel wavevectors (of comparable magnitude). These features are consistent with channelling of energy towards small perpendicular scales. It appears to be more difficult to move inertial range energy towards small parallel scales. Thus, dissipation and heating processes that favour large quasi-perpendicular wavenumbers may be of most relevance.
More recently, results extending well into the solar wind dissipation range have been presented and some of these indicate that the spectral and variance anisotropies of the fluctuations may differ from those typically seen in the inertial range. However, for spectral anisotropy the results are not all in accord so that its nature is still uncertain. For example, fits of the slab+2D kind suggest quasi-parallel wavevectors are energetically important, whereas other techniques often report energetic dominance for quasi-k⊥ fluctuations. For variance anisotropy, observational analyses are more consistent and there are good indications that it progressively weakens with scale, towards isotropy at about the electron gyroradius. Note that at kinetic and smaller length scales the fluctuations can still have nonlinear time scales comparable to the typical wave-based time scales, often ∼1/Ωci . Naturally, additional physics comes into play at kinetic scales. Wave–particle effects, distinct cascades and modified and/or new wave modes may all be significant.
At present, wave-based explanations and turbulence-based explanations are both viable candidates for the observed anisotropies and it may well be that this represents the true situation. New observational data—from upcoming missions such as Solar Orbiter and Solar Probe Plus—should assist with uncovering the specifics of anisotropies at small scales in the solar wind, and thereby further our understanding of heating and dissipation in the medium.
This research was supported in part by the UK STFC, US NSF Solar Terrestrial grant nos. AGS-1063439 and SHINE AGS-1156094, the NASA Heliospheric Grand Challenge project NNX14AI63G and Solar probe Plus ISIS project (SWRI subcontract D99031L), and the NASA/EPSCoR Programme (NNX13AB30A).
We thank K. Kiyani, J.J. Podesta and F. Sahraoui for kindly supplying figures from their papers for reproduction here.
Appendix A. Types of spectra and spectral ranges
Turbulence theories and phenomenologies make considerable use of wavevector (k) spectra [13,18]. The most fundamental of these is the modal spectrum E(k), which describes the distribution of energy in three-dimensional wavevector space. The total energy is . Several different integrals of the modal spectrum are useful. The omnidirectional spectrum is obtained by employing spherical polar coordinates in k-space and integrating over the two angular coordinates: , and is particularly relevant for isotropic distributions. The reduced spectrum (relative to the k3 direction) is , where k1,k2,k3 denotes some choice of Cartesian coordinates. For measurements made in systems with a strong mean flow—like the solar wind—it can be approximated using a timeseries obtained at a single point in space, via the Taylor ‘frozen flow’ assumption [12,122]; see below. Except in special high symmetry cases, it is not possible to obtain E(k) from the omnidirectional or reduced forms [18,123]. This is because the reduced spectrum is a projection of the modal spectrum, and hence different modal spectra can yield the same reduced spectrum.
Spacecraft speeds are usually much slower than those of the supersonic solar wind. Thus, for time intervals that are not too long, the measurements made are a timeseries at approximately the same point in space. These datasets can be used to calculate temporal correlation functions and frequency power spectra, say P(f). Let the mean solar wind speed be , where is the unit vector for the heliocentric radius coordinate. Under the Taylor frozen-flow assumption , namely that Usw greatly exceeds typical wave speeds and fluctuation speeds, the frequency spectrum is equivalent to a reduced wavenumber spectrum: Ered(k3)=UswP(f=k3Usw/2π), where is the component of k parallel to U, and the wavevector components transverse to U have been integrated over . This enables some information about the spatial structure of the turbulence to be obtained from measurements at a single position.
Spectral ranges. For large Reynolds-number isotropic hydrodynamic turbulence, there are three scale (or wavenumber) ranges of particular interest. These are related to two characteristic scales of a steady turbulent flow: the integral scale6 and the Kolmogorov dissipation scale ℓdiss=(ν3/ϵ)1/4, where ν is the kinematic viscosity of the fluid and ϵ the rate of energy injection. At the large scales is the energy-containing range, with scales ℓ∼Lcor, often associated with the scale of the driving. The dissipation range comprises the small scales , and viscous dissipation is the dominant physics there. In between these—and well separated from both of them—is the inertial range, ℓdiss≪ℓ≪Lcor, for which the direct effects of forcing and dissipation are treated as negligible. The physics of the inertial range is then argued to be self-similar, leading to its famous Kolmogorov spectrum ℰ(k)=Cϵ2/3k−5/3, with C the order unity Kolmogorov constant. MHD turbulence is more complicated, in part because of anisotropies and wave effects associated with mean fields . Plasmas, too, have additional richness.
A few words are in order regarding the definition of the inertial range and, particularly, the dissipation range in solar wind data, because plasma effects lead to some differences from the hydrodynamic case. For common 1 AU (ecliptic) conditions the inertial range frequencies are an MHD-scale band from ∼10−4 to ∼10−1 Hz, and approximate powerlaw behaviour is often seen. The high-frequency end is roughly the proton gyrofrequency, fci, and there is usually a fairly sharp transition about there to a steeper powerlaw, with a slope that shows considerable variation .7 If the solar wind were a viscoresistive magnetofluid, this f>fci range would correspond to a fluid dissipation range with approximately exponential falloff and dynamics dominated by viscous and resistive diffusion. Instead, however, the solar wind is a strongly ionized plasma and this range may involve dissipation processes that are more complicated than viscous diffusion. For example, dissipation might occur via wave–particle interactions, wave damping and/or further cascades [7–11]. These processes are expected to become relevant at scales near (and smaller than) the proton gyroradius (ρi) and/or proton inertial length (di=V A/Ωci). Although the physics is not exclusively dissipative, the solar wind literature still commonly refers to this set of scales as the dissipation range, and we follow this usage herein. An alternative designation is the dispersion range [64,124].
Note that when the wavevector energy spectrum is anisotropic, owing to a mean field B0, say, the frequency at which P(f) transitions from the inertial range to the dissipation range can depend on the angle the measurement direction makes with B0 [43,125].
Appendix B. Mean magnetic fields: global and local
An important issue for many types of anisotropy is the scale over which the mean magnetic field is defined. For a local mean field, , it might be the same scale as that of the considered fluctuations, or perhaps 2, 5 or 10 times that. Studies employing either wavelets or structure functions now commonly take this approach. For a global mean field, B0, the averaging occurs over the whole data interval (or simulation domain), so that B0 is uniform. The particular mean field used can lead to significant differences in results [23,100,101]. Possibly the most well-known solar wind case concerns the energy spectrum and how the inertial range slope depends on the angle between the measurement direction and the direction of B0 or . No systematic difference in slopes is seen when a global field is used, but steeper spectral slopes are seen at small angles when local mean fields are employed (see  for a review).
Note that wavelet ‘spectra’ computed relative to local mean fields are, in fact, not spectra in the classical second-order moment sense . Rather, because the direction of is itself a random variable, wavelet ‘spectra’ also involve third- and higher-order correlations.
Recall that the properties of linear wave modes are typically derived assuming a homogeneous background with a mean magnetic field that is uniform and constant . If one wishes to investigate the properties of wave modes relative to a , this is clearly a more complicated situation. In the extreme cases where is defined over the same scale as the fluctuations, it is possible or even probable that changes direction on the same time scale that the fluctuations evolve. It is then difficult to see how features requiring some stableness relative to the direction—for example, k exactly perpendicular to for supposed analogues of ion Bernstein modes—can persist long enough to justify their interpretation. Similar comments apply to exactly two-dimensional turbulence. However, because two-dimensional turbulence does not rely on a dispersion relation or specific modal eigenstructures, the exactness of the two-dimensionality is not as relevant, and quasi-two-dimensional turbulence (i.e. allowing large-scale, but not small-scale, dependence on the parallel coordinate) remains a sound concept.
There are other issues associated with averaging intervals that are shorter than a correlation time or correlation length. In particular, such short averaging intervals may produce unconverged spectral estimates, containing stochastic anisotropies owing to undersampling of the data. Separating such distortions from the true signal is difficult. Thus, observations of variance anisotropy do not necessarily imply actual anisotropy of the energy spectrum! See the discussion around equation (2.3) [18–21].
One contribution of 11 to a theme issue ‘Dissipation and heating in solar wind turbulence’.
↵1 ‘Slab’ modes, i.e. Fourier modes v(k) and b(k) with their k parallel to , need to be handled separately. For example, as .
↵2 Boldyrev [34,35] has shown that if there is also a scale-dependent alignment of v and b fluctuations, the nonlinear time scale is weakened, leading to a different scaling for the equal time-scale curve, e.g. .
↵3 Since then , which has the correct sense of propagation. Note that may be positive or negative.
↵4 The special case where all three of k, k′ and k′′ are perpendicular to B0, giving a purely two-dimensional interaction, is sometimes referred to as the ‘trivially resonant’ case.
- Accepted February 4, 2015.
- © 2015 The Author(s) Published by the Royal Society. All rights reserved.