ENHANCED MAGNETIC COMPRESSIBILITY AND ISOTROPIC SCALE INVARIANCE AT SUB-ION LARMOR SCALES IN SOLAR WIND TURBULENCE

Theories of plasma turbulence which advocate that a turbulent energy cascade within the dissipation range is mediated by the various linear wave modes of a plasma (Leamon et al. 1999; Schekochihin et al. 2009; Gary et al. 2008, 2012; Chang et al. 2011), suggest predictions for C_∥ (Gary & Smith 2009; Sahraoui et al. 2012; TenBarge et al. 2012). As the wave modes advocated to cascade energy at these scales are dispersive in nature (kinetic Alfvén and Whistler waves), many of the proponents of such theories suggest that what has been dubbed the dissipation range for purely historical reasons, should actually be called the dispersive range (Stawicki et al. 2001; Sahraoui et al. 2012). Within the context of sub-ion Larmor scales, there is a growing body of work showing that the fluctuations at these scales share the characteristic of kinetic Alfvén waves (Leamon et al. 1998b, 1999; Bale et al. 2005; Howes et al. 2008; Sahraoui et al. 2009, 2010; Salem et al. 2012). Indeed, the enhancement of the magnetic compressibility seen above is strongly consistent with the transition, around kρ_i ∼ 1, of purely incompressible shear Alfvén waves into kinetic Alfvén waves, with a strong compressive component (Hollweg 1999) and propagating at highly oblique (near-perpendicular) angles to the background magnetic field.

In Figure 2, we overlay the observational results of C_∥ by predictions of C_∥(kρ_i) from numerical solutions of the linearized Vlasov equation for the kinetic Alfvén wave mode at a highly oblique (85°) angle and at a virtually perpendicular (8999) angle, using the same bulk plasma field and particle parameters of our data. These numerical solutions were plotted using the WHAMP code (Rönnmark 1982) which assumes a strong guide field, i.e., a global background magnetic field. The plot shows that our results are qualitatively consistent with C_∥ predicted for kinetic Alfvén waves. This enhancement of the compressive component in the kinetic Alfvén wave is due to the coupling of the Alfvén mode to the purely compressive ion acoustic mode (Leamon et al. 1999). In the inertial range below kρ_i ∼ 1 both of these modes are decoupled (Howes et al. 2006; Schekochihin et al. 2009). In the context of such theories, the 10% compressible component that we see in the inertial range is most likely due to the presence of magnetosonic compressible modes in the data, all at highly oblique (nearly perpendicular) angles to the background magnetic field. The modes can be in the form of slow modes, fast modes or non-propagating (ω = 0) pressure-balanced structures advected by the flow (mirror modes), all of which have purely parallel (compressible) fluctuations (Sahraoui et al. 2012). Although a recent paper by Howes et al. (2011a) shows that slow modes or pressure-balanced structures constitute the dominant part of the compressible fluctuations in the inertial range, fast magnetosonic modes may also exist. This can be supported by the fact that these fast modes are shown to be considerably undamped (compared to slow modes) in the linear theory at highly oblique angles of propagation and high plasma beta (β_i ≳ 1; Sahraoui et al. 2012), conditions very relevant to the solar wind. In Figure 2, the departure of our results from the theoretical solution at kρ_i < 1 can be explained by the fact that the theoretical curve is only for kinetic Alfvén waves—an addition of purely compressible magnetosonic modes might be able to reduce the difference significantly (as in TenBarge et al. 2012). In addition, the theoretical plot is calculated assuming a strong global background magnetic field. Importantly, the theoretical plot for kinetic Alfvén waves shows that the magnetic compressibility should start to rise before kρ_i ∼ 1 which is also what our results show. Recently, Salem et al. (2012) showed a similar result for magnetic compressibility (their definition of C_∥ not being a squared quantity as in this article) with better agreement with the theoretical C_∥ curve for kinetic Alfvén waves than the results presented here. The differences in our results and those of Salem et al. (2012) could be due to the intervals analyzed: our interval is fast wind with β_i ∼ 1.5, whereas their interval is slow wind with β_i ∼ 0.5; that they used two-fluid linear warm plasma relations, instead of linear Vlasov solutions; or their use of a global background magnetic field instead of a local one. To put both these results in their proper place requires a large data survey with intervals that sample more comprehensive plasma conditions, as well as taking into account details such as the linear polarization property of kinetic Alfvén waves.

In the more general nonlinear setting, this enhancement of magnetic compressible fluctuations can be explained by the increased prominence of the Hall term, j × B, in the generalized Ohm's law, where j is the current density arising from unequal ion and electron fluid velocities. To show this in the dynamical evolution of the magnetic field, we will turn to the Hall-MHD model, as it is the simplest physical model of a plasma with unmagnetized ions (Shaikh & Shukla 2009). Although Hall MHD might not be an accurate representation of the full kinetic physics, especially the linear kinetic physics in hot plasmas (Ito et al. 2004; Howes 2009), it serves as a simpler nonlinear model to explain the rising magnetic compressibility and the isotropy that we observe here. Primarily, we will look at the Hall-MHD induction equation (in SI units):

$$\begin{equation} \frac{\partial \bold{B}}{\partial t} = \nabla \times (\bold{v} \times \bold{B}) + \frac{1}{\mu e n_{e}}\nabla \times (\bold{B}\cdot \nabla \bold{B}) + \eta \nabla ^{2}\bold{B} \, \end{equation} \tag{ 3 }$$

where v is the bulk (ion) velocity, η is the magnetic diffusivity, μ is the permeability, e is the electron charge, and n_e is the electron number density. The first term on the right-hand side of Equation (3) is the convective/dynamo term, the second term is the Hall term, and the last term is the diffusive term. We have neglected the effects of any electron pressure gradients here to simplify the discussion, and as these are $\mathcal {O}(\delta n_{e}/n^{2}_{e})$ their effects are considered small. We have kept the diffusive term—which is normally negligible for these solar wind parameters—in order to retain an energy sink in the equations, but also for the possibility of a renormalized, or turbulent, magnetic diffusivity; however, we will not be making use of this term in the following arguments. Although other terms, such as the electron inertia that are very relevant to high-frequency modes, e.g., Whistler modes, become relevant at scales kλ_e ∼ 1 (where λ_e is the electron inertial length), we stick to the Hall-MHD model above and focus on the effects of the Hall term with respect to the convective term. Importantly, in $\beta \sim \mathcal {O}(1)$ plasmas (λ_i ∼ ρ_i as $\lambda _{i}=\rho _{i}/\sqrt{\beta }$), such as the solar wind at 1AU, as one approaches the ion-Larmor scale ρ_i, the Hall term becomes of the order of the convective term (Goossens 2003). In the inertial range, at scales above the Larmor radius, it is the convective term which dominates and the magnetic field is frozen to the ion flow, such that any flow of the plasma perpendicular to the magnetic field affects the evolution of the magnetic field. Thus, if we assume that much of the power in inertial range magnetic field fluctuations is locally generated from the velocity fluctuations (via the nonlinear evolution of the ion momentum equation) in the form of a dynamo effect (∇ × (v × B) term in Equation (3)), and is not being passively advected in the solar wind from the Sun, then it is natural that most of the power in the inertial range will be in transverse fluctuations—also considered a signature of Alfvénic fluctuations (Belcher & Davis 1971; Horbury et al. 2005). Any residual parallel (compressible) fluctuations arise from fluctuations in the plasma density (which also contribute to the transverse fluctuations).

If we now approach scales close to ρ_i the Hall term starts to show its effects. This manifests itself in changing the direction of the fluctuations such that if transverse fluctuations dominate, the Hall term will generate parallel fluctuations and thus result in an increase in the magnetic compressibility C_∥. To see this we will focus on the Hall term and assume an oversimplification that (1) it is entirely transverse fluctuations δB_⊥ which dominate the turbulent fluctuations coming from the inertial range and (2) that k_⊥ fluctuations dominate (i.e., perpendicular gradients ∇_⊥)—both of which are supported by our observations and studies of the full three-dimensional k-spectrum (Sahraoui et al. 2010). This results in the Hall term contribution to the induction equation becoming

$$\begin{equation} \left.\frac{\partial \bold{B}_{\parallel }}{\partial t}\right|_{{\rm Hall}}\sim \nabla _{\perp 1}\times (\bold{B}\cdot \nabla {\delta \bold{B}_{\perp 2}}) + \nabla _{\perp 2}\times (\bold{B}\cdot \nabla {\delta \bold{B}_{\perp 1}})\, \end{equation} \tag{ 4 }$$

where we have explicitly separated the two transverse components to make clear that the two directions are not parallel to each other in the vector cross product. Equation (4) clearly shows the origin of the enhancement in the parallel magnetic compressible fluctuations. In k-space the right-hand side of Equation (4) would be in the form of the following convolution:

$$\begin{equation} \int d^{3}\bold{j}\ \bold{k}\cdot \bold{B}(\bold{k}-\bold{j},t)(\bold{k}_{\perp 1} \times \delta \bold{B}_{\perp 2}(\bold{j},t) + \bold{k}_{\perp 2} \times \delta \bold{B}_{\perp 1}(\bold{j},t)), \end{equation} \tag{ 5 }$$

where the wavevector arguments are shown explicitly. More importantly, Equations (4) and (5) also suggest the origin of the isotropy that we observe among the different fluctuation components at scales close to ρ_e, where terms similar to Equations (4) and (5) are dominant and the convective term becomes negligible. At such scales, the Hall term will dominate and will also convert any parallel fluctuations which are generated (or already present) into transverse ones. This will evolve until a steady state is reached between the various terms corresponding to the vector components of the magnetic field fluctuations, i.e., isotropy between all the vector components. Interestingly, this simple observation of the role of the Hall term in the enhancement of magnetic compressibility and the resultant isotropy, also suggests some information on the k-anisotropy, i.e., k_∥/k_⊥. If, say, Equation (5) was dominated by k_∥ instead of k_⊥, then it is clear that an inertial range dominated by transverse fluctuations will transition into something which is also dominated by transverse fluctuations, with little or no parallel fluctuations—this is not what we observe here. This indicates that, in our Hall-MHD formalism, power in k_∥ fluctuations is not dominant. Intriguingly, however, this type of argument also does not rule out the possibility that both k_∥ and k_⊥ fluctuations are significant at sub-Larmor scales (Gary et al. 2010); the only observational study of the full three-dimensional P(k) spectra at such scales by Sahraoui et al. (2010) seems to indicate that this is not the case. This and more detailed calculations of the energy transfer between wavevector triads of fluctuations will be the subject of a future investigation, so we can determine a more precise nature of such steady states. Crucially, measurements of the magnetic compressibility, in the context of a Hall-MHD model, could possibly constrain the measurements of the k vector anisotropy.

A transition range between the inertial and dissipation ranges was suggested by Sahraoui et al. (2010), which was, in the context of dispersive waves, cited as a possible sign of where magnetic field energy is Landau damped into ion heating. This transition range, comprising just under a decade of scales around kρ_i ∼ 1 is also seen in the reduced spectra in Kiyani et al. (2009b) and Chen et al. (2010) and is distinctly different from the power law which follows at smaller scales down to kρ_e ∼ 1 (see Figure 1 in Kiyani et al. 2009b and Figure 2 in Sahraoui et al. 2010). Our arguments above suggest that the competition between the convective and Hall terms could also explain such a transition range. In this case, and from looking at where the compressible fluctuations begin to rise in Figures 1 and 2 (kρ_i ∼ 0.1), it seems that the transition range could actually span a much larger range of scales. It is important to note here that we do not know what the functional form of the transition from the convective-dominated regime to the Hall-dominated regime is. All we know is that in the inertial range, far from the spectral break, one can neglect the effects of the Hall term; well below the ion-Larmor radius and well past the spectral break we can neglect the convective term; and at kρ_i ≃ 1 (for β_i ∼ 1) these terms are of the same order as the other (Goossens 2003). Using the above arguments, and the latter observation about the similar strengths of the convective and Hall terms at kρ_i ≃ 1, we can make the following observation: at kρ_i ≃ 1 the convective term will provide half of the fluctuation power (50%) which we assume, from the arguments and observations above, will be comprised of 10% fluctuations in the parallel direction; the Hall term will contribute the other half of the fluctuation power (50%), and 33% of these fluctuations will be in the parallel direction. This means that the magnetic compressibility at kρ_i ≃ 1 will be

$$\begin{equation} C_{\parallel }\simeq 0.5\times 0.33 +0.5\times 0.1\simeq 0.22. \end{equation} \tag{ 6 }$$

This value is in excellent agreement with the value of C_∥ at kρ_i ≃ 1 extrapolated from the local field curves in Figure 2.

We now compare our results for higher-order statistics in the dissipation range to other studies from both observations and simulations. In the case of the dissipation range, there are only really two of these: the study of Alexandrova et al. (2008) who used normal-mode (25 Hz) Cluster observations and Cho & Lazarian (2009) who conducted 512³ electron MHD (EMHD) simulations. Although EMHD is a high-frequency model for magnetic field dynamics with inertialess ions (the opposite of Hall MHD), within the scales that we are analyzing here, ρ⁻¹_i ≲ k ≲ ρ⁻¹_e, the induction equation for both Hall MHD and EMHD are nearly identical for an order unity β_i plasma. Thus, the authors (Cho & Lazarian 2009) also use EMHD to study the dynamics and statistics of magnetic field fluctuations in solar wind turbulence.

Alexandrova et al. (2008) used a continuous wavelet transform using the Morlet wavelet to calculate both PDFs and flatness (kurtosis) functions. They too found non-Gaussian PDFs within the dissipation range but, in contrast to our monoscaling results, showed that the dissipation range has a kurtosis which increases rapidly with scale τ, i.e., suggestive of multifractal scaling. They also showed a dependence of this flatness function with β_i. The increase of compressible fluctuations coincided with a steepening of the spectral slope and the increase of intermittency, as reflected in the flatness function. This prompted the authors to suggest that another nonlinear energy cascade, akin to the inertial range, takes place in the dissipation range. However, due to the increased role of compressible fluctuations at these scales, driven by plasma density fluctuations, this is a small-scale compressible cascade. Due to the dominance of the Hall term at these scales, Alexandrova et al. (2008) then suggest a simple model based on compressible Hall MHD, which takes into account density fluctuations to describe the deviation of the energy spectrum slope from the −7/3 predicted purely by the induction equation for incompressible EMHD (Biskamp et al. 1996). Without a further study of more data intervals, it is difficult to say why our interval shows quite a different scaling behavior to the interval investigated by Alexandrova et al. (2008). However, considering our results and our earlier arguments regarding the magnetic compressibility and the role of the Hall term, the notion of a compressible cascade is an appealing one. Although specific results were not explicitly shown, we should also mention the work of Sahraoui & Goldstein (2010), where the monoscaling result presented in this article and in Kiyani et al. (2009b) was also independently confirmed with Cluster magnetic field data.

The very informative article by Cho & Lazarian (2009) describes the results of their EMHD turbulence simulations, as well as ERMHD simulations. The UDWT digital filters used in our study are similar (in spirit) to the multipoint structure functions used in the analysis of Cho & Lazarian (2009). The main feature of these simulations which is relevant to our results is that, for the EMHD simulations, the monoscaling behavior found in our results is also seen in the higher-order statistics computed from multipoint structure functions for moment orders up to m = 5. Computing higher order moments greater than 5 becomes very difficult in non-Gaussian heavy-tailed statistics due to the poor sampling in the tails (Dudok De Wit 2004). The scaling exponent H for their ζ(m) = Hm scaling (as they used ζ(p)/ζ(3)) is not explicitly stated in their paper. However, this is easily obtained from the −7/3 spectral slope that they obtain in their energy spectrum (for both EMHD and ERMHD), using the well-known relationship β = −(2H + 1) (Kiyani et al. 2009b), where β is the spectral slope. We can then infer that H = 2/3 ≃ 0.66. This is the scaling for the r_⊥ (k_⊥) variations. The results of Cho & Lazarian (2009) also show moderately non-Gaussian PDFs. The authors then compute two-point flatness functions, as well as wavelet flatness functions using the Morlet wavelet similar to Alexandrova et al. (2008). Interestingly, for r_⊥ they find a nominal dependence of the flatness with separation scale r (equivalent to τ) for the two-point flatness function, but a larger increase with r for the wavelet flatness function. This last part offers slightly conflicting results when considering their multipoint structure function monoscaling for moments of order less than 5. Nevertheless, the results of Cho & Lazarian (2009) offer the closest agreement to our scaling results presented in this article and serve as a good comparison to an appropriate model for dissipation range fluctuations (EMHD induction equation as opposed to ideal MHD). Similarly to Alexandrova et al. (2008), we could speculate that the difference in our exponents, H_⊥ = 0.98 and H_∥ = 0.85 (and also the respective spectral exponents), and the −7/3 from EMHD could be due to the differing role of density fluctuations. However, in recent gyrokinetic simulations of anisotropic plasma turbulence, Howes et al. (2011b) show that spectra steeper than −7/3 are only obtained if the full kinetic physics is considered in terms of retaining a physical damping mechanism, e.g., collisionless Landau damping, transit time damping etc. This is absent in the EMHD, ERMHD, and Hall-MHD descriptions (see also Howes 2009). The gyrokinetic description retains this missing physics and thus Howes et al. (2011b) claim that their simulations successfully recover the steeper scaling seen here and in Kiyani et al. (2009b), Alexandrova et al. (2009), Sahraoui et al. (2010), and Chen et al. (2010) for similar plasma parameters.

There is a small but important data caveat that one should mention with regards to these results. This concerns the noise floor of the STAFF search-coil magnetometer and the signal-to-noise ratio at small scales near the electron gyroradius. Due to the dyadic frequency spacing of the UDWT that we use, frequency resolution is poor. We pay this expense for the faster algorithms and the smoother spectra that we gain, the latter being preferred for broadband spectra where robust estimates of the spectral slope are needed. This gives the impression that the signal-to-noise ratio at the smaller scales is higher than it might actually be. In reality there exist some high power spikes in the noise-floor signal >60 Hz which could contaminate the observations at the electron gyro-scale (see Figure 1 in Kiyani et al. 2009b). These spikes are due to interference from electrical signals of the Digital Wave Processor (DWP) instrument on board Cluster (N. Cornilleau-Wehrlin 2012, private communication). The DWP signals are used to synchronize instrument sampling within the experiments of the Cluster Wave Experiment Consortium (WEC), which STAFF belongs to. These spikes only affect the last data point in Figures 1 and 2 and thus should not change our results and conclusions significantly. Intervals with higher power such as those in the slow solar wind stream would be needed for better estimates at these higher frequencies. In addition, the noise-floor that we use is not the actual noise-floor but a proxy for it, obtained from a very quiet period in the magnetotail lobes where the magnetic field power is very low. The actual noise-floor could be smaller. A discussion on the noise-response of the STAFF search-coil instrument and its sensitivity in different plasma conditions can be found in Sahraoui et al. (2011).

We would also like to note that the PDF for the parallel fluctuations seen in Figure 4 for the inertial range from ACE, show an unusual skew toward positive fluctuations. This is an artifact for this particular interval of ACE magnetic field data due to large spike-drops (discontinuities) in the magnetic field magnitude |B|. As these spikes arise in |B| and not in the components, this reflects itself more in the parallel component of the fluctuations rather than the transverse components as, in many cases, one can use |B| as a proxy for parallel fluctuations, e.g., in Alexandrova et al. (2008). This is also reflected in the large errors for the parallel scaling exponents in the inertial range shown in Figure 3(c). However, as these spikes are primarily in the large rare events in the positive tail of the PDF, we believe that the differences in the PDFs of the transverse and parallel fluctuations within the inertial range still occur and our discussion above remains valid.

The isotropy that we observe between the power in the different fluctuation components at kρ_e ∼ 1 is by no means a result which would apply to all plasma environments. Indeed, there is some strong observational evidence that the ion/proton plasma β_i also has a strong role to play here. This was shown by Hamilton et al. (2008), who repeated the inertial range analysis of Smith et al. (2006b) within the dissipation range and showed that the variance anisotropy followed the empirical power-law relationship δB²_⊥/δB_∥² ≡ PSD_⊥/PSD_∥ ∼ β^−0.56_p in open field line intervals of the solar wind from the ACE spacecraft. However, these results were limited to 3Hz cadence and thus could not sample as much of the dissipation range as in this article. Also Hamilton et al. (2008) used a global field (mean field over the entire interval) rather than the local scale-dependent field used in this article. This same dependence of the magnetic compressibility on β_i was also indicated in higher cadence (25 Hz) Cluster data in the work by Alexandrova et al. (2008), again suggesting that higher β_i implies higher levels of C_∥ (these authors used δ|B| as a proxy for δB_∥). The effect of varying β_i is not mentioned explicitly in our arguments using the Hall term explained above, as we have neglected density fluctuations to simplify the discussion. However, purely from the induction equation point of view, β_i is taken into account in dissipation range scales if we retain the effects arising from density fluctuations at scales kρ_i > 1 as in the formulation of electron-reduced MHD (ERMHD; Schekochihin et al. 2009; Cho & Lazarian 2009). In ERMHD, the explicit β_i dependence is shown, assuming pressure balance, in the pre-factor of the ∇_⊥ × (B · ∇B_⊥) term in the induction equation for B_∥. Thus, this addition of density fluctuations can generalize our arguments above which were essentially based on the Hall term alone. At higher or lower than unity β_i, this will break the isotropy of the fluctuation components seen in this article at kρ_e ∼ 1, as an additional anisotropic source of magnetic compressibility will be introduced via the density fluctuations (see also Malaspina et al. 2010 for high-cadence measurements of density fluctuations in the solar wind between ρ⁻¹_i < k < ρ_e⁻¹). The explicit β_i dependence for the rising compressibility is also seen in the fluid version of the kinetic Alfvén wave solution of compressible Hall MHD where the compressible component to the linear solution goes as $\delta \bold{B}_{\parallel }\sim k \lambda _{i}(\beta _{i}\,+\,1\,+\,\sin ^{2} \theta _{kB}){/}\sin \theta _{kB}$ (Sahraoui 2003), where λ_i is the ion inertial length and θ_kB is the angle between the wavevector k and the magnetic field vector B. This solution assumes that sin θ_kB ≠ 0, thus it is not valid for purely parallel propagating Alfvén waves. However, the rising compressibility is clearly seen here as k → λ_i.

Note that in most of the previous arguments we have been slightly cavalier in our discussion of field components. Our introduction of the local background magnetic field should not be looked upon lightly. It is clear that in cases where δB ≪ B₀, where B₀ is the global background magnetic field, both global and local field descriptions should have negligible differences. However, as we observed in the section describing higher-order statistics, it is not obvious that this is the case here, as the distribution of δB is highly non-Gaussian with very heavy tails. In our explanation of the rising compressibility in sub-Larmor scales using the Hall-term and also the lower levels of the compressibility in the inertial range using the convective term in the induction equation, we did not mention the spatial (and possibly temporal) gradients which arise due to the now locally varying background magnetic field. In a global DC background field, or very slowly evolving field, these gradients are null or negligible. If there is a large spectral separation between the scales of the background magnetic field and the fluctuations that we are projecting on it, then this conclusion could still be quite valid. However, as our UDWT method ensures that no such large gap exists, the statistics we calculate will include the effects of such gradients. Further investigation and mathematical grounding of a local-field is of crucial importance, as the notion of using a local versus a global background magnetic field is intimately tied with the issue of whether we have a strong background guide field or a weaker more stochastic one (see the discussion by Schekochihin et al. 2009 and references therein for further details and exploration of this topic; and also the recent discussion by Matthaeus et al. 2012 on the stochastic nature of the local background field and its relationship with global background field statistics). This also brings into question the use of linear wave modes, such as the kinetic Alfvén wave, which are normally described as small amplitude perturbations on a static or slowly evolving equilibrium field. Although finite amplitude propagating nonlinear Alfvén waves are an exact (Elsässer) solution to the ideal MHD equations and similar arguments in the context of the ERMHD equations are made by Schekochihin et al. (2009) for finite amplitude kinetic Alfvén waves, this topic also requires further exploration (see also the excellent discussion of this by Dmitruk & Matthaeus 2009, who explore the importance of the strength of the mean background field using direct numerical simulations of the incompressible MHD equations).

One might question a conclusion of a single physical process or type of mediating dynamics (in the dissipation range) by the observation that both transverse and parallel components have slightly different scaling exponents and spectral indices. This can be reconciled by realizing that there is no contradiction in the fact that the two components can have slightly different scaling exponents but have identical PDF functional forms. Take for example the case of fractional Brownian motion (fBm), which is a non-Markovian generalization of Brownian motion. Two fBm stochastic processes with different scaling exponents will still have the same Gaussian functional form for their fluctuations. In fBm the scaling exponents are simply an additional parameter representing the degree of statistical dependence between the fluctuations (Mandelbrot 1983; Samorodnitsky & Taqqu 1994).

If we look from the point of view of linear wave mode characteristics, the distinction between inertial and dissipation range scalings could also be seen to reflect the decoupled and coupled nature of the wave modes in the inertial range and dissipation ranges respectively (Leamon et al. 1999). The decoupling of these modes in the inertial range also results in the decoupling between transverse and compressible magnetic field fluctuations (Schekochihin et al. 2009). In this picture, the scaling in the dissipation range could then be described by nonlinearly interacting kinetic Alfvén waves. In all these cases, as they involve linear wave modes, the nonlinear cascade would then take the form of "critical-balance" type cascades as in the works of Schekochihin et al. (2009) and Howes et al. (2008, 2011b), or if one can show that the nonlinearity is small, a "weak-turbulence" type description (Galtier et al. 2002). Although there is no justification for the latter in the inertial range, there might be a case for it in the dissipation range if the coupling between modes becomes weak and results in small amplitude fluctuations (Rudakov et al. 2011). In hydrodynamics described by the Navier–Stokes equations, this latter assumption can be valid if one uses a local Reynolds number (see Batchelor 1953), which is small in the high-k regime (when the spectral amplitudes become small) well into the hydrodynamic dissipation range. This can then provide a suitably small parameter to base a convergent (or asymptotic) perturbation expansion. Of course, without concrete measurements of nonlinearity, this is all speculation. However, it could possibly be an avenue for a future (maybe fruitful) investigation.

An open question is why such a transition from multifractal scaling to self-similar monoscaling occurs between the inertial and dissipation ranges. Kiyani et al. (2010) briefly discuss this using hydrodynamic analogies (Frisch & Vergassola 1991; Chevillard et al. 2005) of a near-dissipation range where the increased effect of viscosity successively switches off the available scaling exponents of the multifractal field—equivalent to narrowing the "f(α)" spectrum in multifractal models. However, no such apparent analogies with a viscous "fractal-dampening" can happen in a near-collisionless solar wind. Also, unlike the hydrodynamic analogy this transition happens very fast as we cross kρ_i ∼ 1, although a scaling analysis within the "transition range" might yet reveal an intermediate scaling behavior which shows this successive "switching-off" of scaling exponents. Indeed, the steepening of the spectral slope in the transition region is strongly suggestive that such an intermediate scaling range could possibly exist. This is an exciting prospect and will be the subject of a future investigation. This latter investigation is not feasible with our UDWT technique as due to the dyadic spacing, it has poor frequency resolution—a more detailed "fine-toothed comb" technique such as the use of wavelet packets (Walden & Cristan 1998) or the continuous wavelet transform (Horbury et al. 2008; Podesta. 2009) would be needed here.