Third-moment descriptions of the interplanetary turbulent cascade, intermittency and back transfer

Abstract

We review some aspects of solar wind turbulence with an emphasis on the ability of the turbulence to account for the observed heating of the solar wind. Particular attention is paid to the use of structure functions in computing energy cascade rates and their general agreement with the measured thermal proton heating. We then examine the use of 1 h data samples that are comparable in length to the correlation length for the fluctuations to obtain insights into local inertial range dynamics and find evidence for intermittency in the computed energy cascade rates. When the magnetic energy dominates the kinetic energy, there is evidence of anti-correlation in the cascade of energy associated with the outward- and inward-propagating components that we can only partially explain.

1. Introduction

For most of the space age our view of solar wind fluctuations (magnetic, velocity, density, etc.) has been based on the theory of plasma waves. Attempts to incorporate turbulence concepts into this thinking have often been treated as little more than an afterthought that is either a secondary dynamic or a concept in direct conflict with the wave interpretation. The so-called weak turbulence theory illustrates this point wherein the primary dynamic is wave propagation with the secondary dynamic of waves exchanging energy on a time scale that is long compared with the wave period. Coleman illustrated this conflict with two papers: in the first [1], he argued that transverse magnetic fluctuations in the solar wind resemble low-frequency wave behaviour; whereas in the second [2], he argued that the reproducible spectrum of interplanetary fluctuations more nearly resembles the physics of hydrodynamic (HD) turbulence. If there was debate on these conflicting ideas, the wave viewpoint gained momentum with the observations that the fluctuations are transverse to the mean magnetic field, largely non-compressive, and with correlations suggestive of anti-Sunward propagation [3]. These are the properties of low-frequency Alfvén waves originating at the Alfvén radius.

The idea of the solar wind functioning as a giant wind tunnel for the purpose of studying magnetohydrodynamic (MHD) turbulence seems diminished, with many authors viewing turbulence as weakly interacting waves. There are problems with the wave interpretation and they are greatest when one considers the non-interacting wave viewpoint [4]. This view precludes the dynamics of many turbulent processes such as magnetic reconnection [5,6]. Within the context of solar wind heating, the greatest problem in the non-interacting wave viewpoint is that the wave energy available at scales that resonate with thermal particles is finite and insufficient to provide the observed heating rates [7]. Turbulence, and specifically the energy cascade that exists within well-developed turbulence, replenishes the energy at these scales and permits ongoing heating at a rate in general agreement with the observations [8–31].

Turbulence is the nonlinear evolution of a fluid generally considered to involve a wide range of spatial scales [32]. While it may arise from a single instability, it incorporates a broad range of dynamical processes acting in randomly distributed, coherent dynamics across the volume. It is not one nonlinear process—it is generally many dynamical processes. A fully developed turbulent spectrum is generally regarded to possess three subranges. The largest and most long-lived fluctuations constitute the ‘energy-containing range’. An example of these is the large-scale shear-flow regions in the solar wind. Once injected into the flow, these scales drive the turbulence and provide an energy reservoir for the excitation of smaller scales. The intermediate-scale fluctuations that are created by the slow destruction of the energy-containing range, but that are removed from the scales at which dissipation occurs, are called the ‘inertial range’. The inertial range conserves energy as various instabilities destroy and create coherent fluctuations in the process of moving energy to different spatial scales [33–35]. This analysis focuses on these scales. The small-scale fluctuations form the ‘dissipation range’ where the spectrum steepens and turbulent energy is converted into heat [2,34,36–43]. In the solar wind, there is likely to be a second inertial range at scales smaller than those where dissipation on thermal ions occurs and a second dissipation range where dissipation on thermal electrons occurs [44].

Statistical arguments derived from the fundamental equations without the approximations that lead to a finite set of dynamics or a basis set of oscillations have provided insights and reproducible results that can be confirmed by experiment and simulation. One of those reproducible results is the average transport of energy between scales. In three-dimensional HD and MHD turbulence which is statistically independent of location (homogeneous), energy is transported on average from large- to small-scale fluctuations. This is called a ‘cascade’. The cascade of energy is the central result that explains the heating of the solar wind: when small-scale fluctuations that permit one or another form of energy exchange between collective (fluid) motion are exhausted that energy is replaced by the turbulent cascade so that heating can continue. Its description does not rely on a selection of wave modes or a single dynamic describing their interaction. In this way, the large-scale fluctuations of the solar wind provide the energy that is transported to smaller scales where dissipation can occur. Small scales provide the heating dynamic, whereas the large scales provide the reservoir of energy.

There is a fundamental difficulty in assessing turbulent processes in the solar wind and it is best described by examination of the dynamical equations. The Navier–Stokes (N-S) equation that describes a traditional fluid is given by

1.1

where ρ is the density, v is the velocity fluctuation, P is the pressure and ν is the viscosity. Three-dimensional gradients such as those seen here (∇) cannot be evaluated using a single spacecraft. This prevents direct evaluation of the nonlinear terms.

This problem can be circumvented if one asks the correctly posed statistical question. If one assumes incompressibility, homogeneity, scale separation (there exists scales where energy is conserved that is separate from the scales where energy is dissipated) and addresses the statistics of the ensemble only [33,45–47], then the energy cascade within the inertial range can be described as

1.2

where Δv(L)≡v(x+L)−v(x) is the difference between the velocity measured at two points separated by the vector L and ∇_L is the gradient with respect to L. If one assumes that the statistics are stationary and isotropic and that the lag L resides within the inertial range, then they recover the familiar ‘Kolmogorov's 4/5 Law’ [48]

1.3

where 〈…〉 denotes ensemble average and L is the separation vector. We will refer to this theory as K41a. The separation vector L can now be evaluated in any direction and in the solar wind we choose the anti-Sunward direction of the solar wind flow. Note that a simple scaling of equation (1.3) yields , where v_k is called the ‘characteristic speed’ of an eddy of size L, where k≡2π/L.

The derivation of equations (1.2) and (1.3) is beyond the scope of this paper as it is somewhat lengthy. However, we note that K41a first assumes an expression for an isotropic cascade geometry derived by von Karman & Howarth [33] and derives equation (1.3) from that point. Frisch's textbook [47] also follows this approach. A more modern derivation can be performed that follows the Politano & Pouquet [49,50] derivation for MHD third moments wherein equation (1.2) is obtained without regard for geometry. This approach makes use of simple identities derived from homogeneity and the ability to move derivatives through ensemble averages.

Equations (1.1)–(1.3) require an ensemble average in their derivation. This permits the movement of derivatives from individual terms to the product of terms leading to cancellation and simplification supported by the assumption of homogeneity. As a practical matter, the average over the ensemble is normally taken to be an average over the data using as much data as is available. In order to facilitate error analysis, we normally take 1 h of data to represent a single sample within the ensemble and average over many hours to produce an ensemble average consistent with the above equations. However, it has been suggested that much smaller samples may produce averages for which the above equations are valid [51,52]. We will pursue this idea in §4a and following.

MHD also suffers from the same problem of multi-dimensional gradients in the primitive equations that are not readily analysable using single spacecraft. MHD turbulence can also be addressed in the above manner via third-moment expressions, but the derivation and the resulting expressions are somewhat more complicated. We will pursue them below. Direct assessment of the nonlinearity of solar wind fluctuations generally relies upon statistical arguments such as the average plasma properties at different heliocentric distances [8,9,12–17,19,21,22,24–28] as addressed by transport theory. The use of third-moment structure functions allow you to circumvent the multi-dimensional derivatives, but do require that some assumption regarding the underlying geometry of the turbulence be made. This permits us to replace the multi-dimensional derivatives with derivatives in a single measurable direction and still obtain a quantitative measure of the nonlinear dynamics. While these derivations are statistical in nature, they require only sufficient data to achieve convergence in order to provide a local measure of the nonlinear terms.

In this paper, we will review the use of third-moment expressions to compute estimates for the turbulent cascade and the general agreement of these results with local heating rates. Our recent work has shown that the natural variability of these computed cascade rates can be thought of as a direct measure of intermittency and we will present new results from this effort. Going beyond intermittency, we will show that there generally exists an anti-correlation between the energy cascade of the outward- and inward-propagating components of the solar wind fluctuations that is a fundamental aspect of the turbulence and we will attempt to explain this anti-correlation in terms of the large-scale energy-containing fluctuations. The reader is encouraged to read several particularly relevant papers in this same issue [43,53–57].

2. Heating

The in situ heating of the solar wind is now well established [58–62]. Turbulence, both as a source and a means, is cited from the acceleration [53,63] to the termination region [8,9,12–17,19,21,22,24–28]. If we consider the turbulent cascade as a mechanism for transporting energy from the reservoir of the large-scale, energy-containing range to the small scales where dissipation can occur, we will need a rate. Kolmogorov argued that the central dynamic in the transport of energy within the N-S equation is the interaction of two circulating eddies [34]. He asserts that the lifetime of an eddy will be the time it takes to revolve one time (the ‘turnover time’ L/v_k, where v_k is the characteristic speed of the eddy and L is the size). This sets the rate for energy transport to be the energy contained within the eddy divided by its lifetime, . Dimensional analysis produces a prediction for the energy spectrum in the inertial range (k^−5/3 where k is the wavenumber). The resulting prediction for the power spectrum is P_k=C_Kϵ^2/3k^−5/3, where C_K is determined by experiment to be C_K≃1.6 [64]. We will refer to this theory as K41b.

Iroshnikov [65] and Kraichnan [66] (IK) argue that it is the properties of propagating waves that define the interaction time in MHD turbulence so that they predict a different form for the power spectrum (P_k=A_IK(ϵV_A)^1/2k^−3/2 where V_A is the Alfvén speed) and a different rate of energy cascade in the process. Numerous other predictions have followed [8,67–70]. It is important to realize that the spectrum of magnetic and velocity fluctuations in the solar wind do not agree with either the K41b or IK prediction.

The more modern view of the K41b versus IK controversy is based upon the orientation of the wave vectors called ‘geometry’ or ‘spectral anisotropy’ [57]. If the wave vectors are quasi-parallel to the background mean magnetic field, it may be that the IK formalism based on wave dynamics has merit. If the wave vectors are quasi-perpendicular to the mean magnetic field, the wave frequencies for incompressible fluctuations approach zero and the K41b formalism is likely to be more appropriate. Recent results suggest that the energy-containing scales, or at least the correlation scale, is dominated by wave vectors aligned with the mean magnetic field [71]. Earlier studies at still greater spatial scales deep within the energy-containing range suggest wave vectors that are radially aligned and not related to the mean field direction [72]. There is strong evidence that inertial-range fluctuations are dominated by perpendicular wave vectors [39,42,73–77]. This includes observations associated with solar energetic particle (SEP) events [74]. Theory and simulation support this claim [78–81] and several theories of inertial range dynamics build on the idea [55,68,82]. We should note that when the analysis of Bieber et al. [74] is applied to the observations of Belcher & Davis [3] the conclusion is that those fluctuations are also dominated by an abundance of perpendicular wave vectors.

The apparent dominance of perpendicular wave vectors within the inertial range may change in the dissipation scales [39,42,76]. Some authors argue that it does not [83–89] and argue that kinetic Alfvén waves provide the observed steepening, but some of these observations have been brought into question [90]. Dispersion is sometimes invoked as an explanation for the steepening of the power spectrum at ion dissipation scales [55,91–95] and whistler waves are also discussed in the context of forming the dissipation range at ion scales [92,96–98]. The central issue to that question rests with the dissipation processes that remove energy from the turbulence and produce heat. What processes are available depends to a great degree on the underlying geometry, or spectral anisotropy, at the smallest scales of the inertial range. The resultant change in the spectral anisotropy at dissipation scales is a diagnostic of the dissipation processes.

One demonstration of the role of turbulence in the solar wind is the ability of turbulence transport theory to reproduce the solar wind proton temperature as a function of heliocentric distance and time [8,9,12–17,19,21,22,24–28]. Most published applications of this theory have used K41a scaling laws under the tacit assumption that the turbulence is dominated by two-dimensional dynamics, but one analysis used IK theory and reproduced the observations about as well [25]. There is a degree of ambiguity in the theory and this points to the usefulness of a more exact and demanding analysis that may eventually resolve the dilemma.

Turbulence transport theory describes the cascade of energy and assumes that however dissipation occurs it will adapt to the changing cascade rate [41,48,99]. It does not specify the means by which dissipation occurs. In HD, the dissipation scale resides within the fluid approximation. When the energy cascade is greater dissipation moves to smaller scales where the dissipation term ν∇²v is stronger [48]. This does not happen in the solar wind [41]. MHD is a fluid approximation to plasma dynamics. Dissipation sets in at the ion inertial scale [43,54,55,57,100], where the fluid approximation breaks down and kinetic physics takes over. However, there is new evidence to suggest that the spectral break may occur at the proton Larmor radius [101,102] under some conditions. Either way, this results in a spectrum at smaller scales that steepens with increased cascade rate [41] which does not happen in HD.

While transport theory allows us to reproduce the temperatures observed by Helios, Ulysses, Voyager 1 & 2 and Pioneer 10 & 11, long-term measurements at 1 AU permit us to obtain a more exacting comparison. Using the previously published analyses of Helios data, together with Advanced Composition Explorer (ACE) data, Vasquez et al. [61] obtained an expression describing proton heating at 1 AU that varies with wind speed and observed temperature. More recently, these results have been extended to include high latitudes [103,104] and non-zero cross helicity (called ‘imbalanced turbulence’ by some) [62].

3. Third moment theory

The incompressible MHD equations can be written as

3.1

where Z^±≡v±b are the Elsässer variables [105] and we define the magnetic field B in units of the Alfvén speed, . As noted below, equation (1.1) the multidimensional gradients in equation (3.1) prevent direct evaluation of the nonlinear terms when only a single spacecraft is used. The use of multiple spacecraft also poses limitations.

The Elsässer variables Z^± are general representations of MHD dynamics. However, low-frequency waves can also be conveniently described in these terms so that Z⁻ (Z⁺) corresponds to waves propagating parallel (anti-parallel) to the background mean magnetic field. This is often the language of solar wind observations, so we will adopt it as a convenient way to distinguish the two components of the turbulence. However, the reader should note that wave vectors perpendicular to the mean magnetic field are a common point of discussion in solar wind turbulence. Any energy associated with these wave vectors can still be described in terms of Z^± even though some of the associated wave forms are non-propagating. Unless otherwise stated, we do not mean to imply a turbulence theory based on wave interactions alone. We simply use the language as a convenient way of connecting with the observations that whichever of Z^± that is associated with ‘outward’, or anti-Sunward, propagation is generally seen to be energetically dominant.

In the analysis that follows, we employ the term ‘pseudo-energy’ to represent the quantities |Z^±|² so that the total energy is . The energy cascade associated with each pseudo-energy can be computed using these techniques and compared.

If we apply the same considerations used to derive equation (1.2), we can derive an expression governing the cascade of energy in incompressible MHD [49,50]

3.2

where ϵ^± refers to the energy cascade of the pseudo-energies |Z^±|². The expression can be evaluated under various assumed geometries, but the simplest is isotropic which is not especially relevant to the solar wind. However, other attempts at anisotropic geometries we have tried have given similar results. If we assume isotropy, invoke the Taylor frozen-in-flow assumption [106], and employ separation along the solar wind flow, equation (3.2) can be written as

3.3

where V_sw is the solar wind speed, R denotes the radial component away from the Sun, L=−V_swτ and τ is the time separation in the measured data. The total energy cascade is given by ϵ^T=(ϵ⁺+ϵ⁻)/2.

The fluctuations in the solar wind are typically indicative of dominance by the outwardly propagating component. This diagnostic is accomplished via the correlation of the velocity and magnetic field fluctuations. One quantification of this is the cross-helicity [107–110] defined to be , where v and b are the fluctuations of the velocity and magnetic field relative to their means, and N is the number of measurements in the sample. Likewise, the bulk energy of the fluctuations can be defined as . A measure of the significance of the cross-helicity correlation is given by the normalized cross helicity −1≤σ^C≤1, where σ^C≡H_C/E. The cascade of cross helicity can also be expressed in terms of equation (3.3) as ϵ^C≡(ϵ⁺−ϵ⁻)/2.

It is important to stress that the third-moment formalism described above does not rely on a single view of turbulence, a prescription of wave modes or a selection of a specific dynamic. It is entirely general subject to the assumptions listed. It can as accurately describe K41a dynamics as it does IK turbulence, or other theories [8,67–70]. This makes it an objective metric against which diverse theories can be tested.

We have tested the third-moment expressions against the thermal proton heating results [61] under the assumption that whatever energy cascades to small scales is dissipated primarily by heating protons [23,30]. The comparison is strongly favourable in both solar maximum and solar minimum. At the present time, 20–40% of the energy cascade is available to heat heavy ions and electrons when the cascade is strong (wind speed and proton temperature are both high) and less when the cascade is weak [23], which is in agreement with empirical constraints [111].

A simple scaling law can be inferred from equation (3.3) that resembles K41a. Where the HD cascade scales as , MHD exhibits a different dynamic wherein the Z⁺ field cascades the Z⁻ field and vice versa. This leads to a simple scaling of equation (3.3) taking the form [112]

3.4

and

3.5

See related discussions [8,9,113]. We will apply that scaling below.

4. Analysis

In this study, we use magnetic field, velocity and proton density data from the Advanced Composition Explorer (ACE) spacecraft [114–116]. Magnetic field data have been averaged to the same 64 s cadence as the thermal proton data. We use 12 years of measurements from early 1998 through mid-2010. Interplanetary shocks and their driver gases have been removed.

The nominal correlation length for fluctuations in the solar wind is 1 h. Therefore, the statistical means and uncertainties derived from many samples when each is 1 h in duration or longer is determined by simple Gaussian statistics. In this study, we use 1 h samples of 64 s resolution data to compute individual realizations from which statistical quantities are obtained. Measurement errors are insignificant in comparison to statistical variations and are not considered here. In order to obtain meaningful estimates for each 1 h sample, we require that greater than 70% of the data have usable magnetic field, velocity and density measurements. We further require, as an estimate of stationarity within the sample, that the mean magnetic field and average wind speed of the first and last half of every hour of data differ from the average for the hour by no more than 10%. This further removes transients, strong gradients including shear, etc., and thereby imposes homogeneity upon each sample. This avoids the complications associated with third-moment expressions that are applicable to shear regions [117–119]. It also fails to address questions of in situ generation of turbulence in these regions [120–122]. As with our earlier papers, we interpolate the results of each 1 h sample onto a common spatial grid that is equivalent to measurements with a 400 km s⁻¹ wind speed. This is done under the assumption that the in situ dynamics are defined by spatial scale rather than the associated Doppler shifted signal that is dependent on the wind speed. This analysis is based upon data selection for the average energy and cross helicity of a 1 h interval. We will avoid examples of large |σ^C| for reasons described in [123].

We compute third-order structure functions as defined in equation (3.3) and recognize the strong tendency for outward propagating fluctuations to possess greater energy than inward propagating fluctuations as determined from the Z^±. Using the computed mean field for the sample, we associate Z^± with the inward- and outward-propagating components Z^in,out and gather our statistics for accordingly. This permits us to write third-moment forms equivalent to equation (3.3):

4.1

4.2

4.3

The total energy cascade ϵ^T can then be computed from whether we employ Z^± or Z^in,out.

The data spacing is 64 s and the data products derived from equation (4.1)–(4.3) are averages of lagged products. For simplicity, we refer to lags not by the time spacing, but by the data spacing. Therefore, a calculation based on lag=5 means using measurements that are five data points (5×64=320 s) apart.

(a). Intermittency

In most of our past publications on this subject [18,20,23,29,30,119,124,125], we have used relatively long data intervals (typically about 12 h in duration) and many such samples to compute average properties of the third moment and implied cascade. The exception to this is a study performed on 3 h samples [126] that revealed the underlying dynamics of intermittency. Here, we repeat that analysis using a different energy range for selection of our intervals and the shorter 1 h sample size.

We limit the average bulk energy of the fluctuations to 300<E<700 km² s⁻² and the average normalized cross helicity to |σ^C|≤0.75. The results that we show here have been seen in every energy and helicity range we have examined to date. This selection results in a sufficient number of samples to produce a clean distribution function and we choose these conditions for this reason only. It does eliminate times of high |σ^C|, which may prove interesting in themselves, but this is where we have chosen to begin this investigation. There are 7075 1 h samples analysed in this study. Functions , and are computed for each 1 h sample. We perform a linear fit to the functions over lags 2–20 (128–1280 s separation) and evaluate the goodness of the fit according to where x_i is the time into the hourly sample (64, 128, 192 s, etc.), and y_i is the associated value of D₃. Figure 1 shows the resulting distribution of R² values. A total of 4787 samples (approx. 68%) had R²≥0.7. We consider these 1 h samples to be well-fit by a line passing through the origin as is required of equation (4.3).

Figure 1.

Analysis of 1 h samples using 12 year of ACE MAG and SWEPAM data showing the distribution of R², the square of the correlation coefficient for versus L, for lags 2–20 corresponding to 128–1280 s. See equations (4.1)–(4.3) for definition of . Analysis limited to mean fluctuation energy 300<E<700 km² s⁻² and mean normalized cross helicity |σ^C|≤0.75.

At the heart of this analysis is the question, ‘What is meant by linear third-moment expressions when applied to a correlation length of data?’ The equations describing the cascade are derived for an ensemble average of many such realizations. It has been suggested that the scaling properties of incremental measurements for the HD cascade show better convergence for smaller data sets, and there is the suggestion that these expressions are applicable to much smaller samples of data than the lengthy ensembles used here [51,52]. A possible explanation for this is that averaging of small-scale fluctuations within a given realization (a sample that we take to have 1 h duration) possesses itself a degree of averaging that accomplishes some of the theoretical goals of the derivation. The implication is that when these third-moment expressions are applied to small samples the result is an estimate for the local cascade rather than the global ensemble average cascade.

In the remainder of this paper, we consider only those 1 h samples that pass the R²≥0.7 test of . Figure 2 shows the distributions of ϵ as computed from , and at a lag of five samples (=320 s). We discuss below why we select this lag value for our statistics. Vertical solid red lines represent the mean of each distribution. Each mean is positive representing an average cascade from large to small scales. Vertical dashed red lines give the computed standard deviation. While the distribution functions are nearly Gaussian in form, they have ‘hot tails’ where the distributions are consistently overpopulated relative to the Gaussian fits for values in excess of the standard deviation.

Figure 2.

Distribution of ϵⁱⁿ as defined in equation (4.1) and computed from for lag 5 (320 s) using 1 h samples. Same for ϵ^out, ϵ^T and ϵ^C. Blue curve is least-squares best fit of Gaussian to the distribution functions. Vertical solid (dashed) lines are the means (standard deviations) of the data. While the best-fit Gaussian does reproduce the centre of the distribution functions, there is a strong abundance of points on the tails of the distributions making the standard deviation of the distributions significantly larger than the standard deviation of the best-fit Gaussians. (Online version in colour.)

While it is difficult to convey from figure 2, the mean of each distribution is positive as is generally expected for the average energy cascade rate. The standard deviations are large when compared with the averages. Both means and standard deviations for each of the four panels are listed in table 1.

Table 1.

Means and standard deviations for figure 2.

	mean	s.d.
PDF(ϵin)	4.08×103	1.29×105
PDF(ϵout)	1.56×104	1.85×105
PDF(ϵT)	9.83×103	6.31×104
PDF(ϵC)	5.75×103	1.47×105

A fundamental property of the third-moment expressions for ϵ (equations (3.3) and (4.1)–(4.3)) is that the functions D₃ are linear in lag. We contend that linearity of the observed hourly estimates is not a coincidence, but results from well-determined functions and a valid assessment of ϵ which should be independent of scale in the inertial range. Two properties of the resulting estimates for ϵⁱⁿ, ϵ^out and ϵ^T are (i) that the standard deviation of each distribution function is significantly larger than the mean and (ii) negative values of ϵ are readily obtained. This points to two important aspects of turbulence in general and solar wind turbulence in particular. First, the predictions of K41a, IK and others address only the mean energy cascade rate for the ensemble. Individual realizations as represented by the statistically independent hourly samples we use here may bear little or no resemblance to the mean value of the distribution. Second, negative cascade rates are possible for individual hourly samples even if it seems impossible that they could persist indefinitely.

If we average the individual estimates for D₃ derived from the 4787 1 h samples with R²≥0.7, we can obtain the underlying forms of these functions relating to the means of the above distributions. Figure 3 shows the resulting average third-moment functions , and . The greater slope of is consistent with the larger average value of ϵ^out when compared with and ϵⁱⁿ. Linearity is far from perfect but is within the computed uncertainty of the estimates shown in the figure.

Figure 3.

Computed average functions , and for the 4787 1 h samples used in this study with R²≥0.7. Third moments are plotted as functions of increments in spatial lag L which is equivalent to a temporal lag τ if the wind speed is 400 km s⁻¹. This represents the fixed grid upon which we interpolate each result to allow for varying wind speed. Example uncertainties as represented by the error-of-the-mean are shown on the figure. (Online version in colour.)

We contend that the large standard deviation relative to the mean of each distribution function for ϵ is an indication of intermittency. Intermittency has come to mean many things and is most often analysed as a ratio of higher order structure functions representing the filling factor, or natural uniformity, of the turbulence. A complimentary and perhaps more fundamental definition of intermittency is the statistical variation of the turbulent cascade relative to predictions for the mean. In this way, third moments provide a direct measure of the variability of the cascade that we can reasonably presume to vary from one sample to another so long as those samples are separated by a correlation length. It is a measure of the variability of the local dynamics of the turbulence. Here, we say nothing of correlation times. Using the Taylor frozen-in-flow assumption and taking all samples from 1 AU, we have no measure of the duration over which a sample shows the computed cascade dynamics. We assume that back transfer of energy from small to large scales cannot persist unless energy flows from one component to another (exchange of pseudo-energy within the cascade) and we have no reliable measure of this possible dynamic at this time. We can only claim that at a time and a point in space the energy cascade of one of the pseudo-energies may be negative in some instances.

(b). Pseudo-energy scalings

The cross helicity can be rewritten as

4.4

but solar wind studies are better organized using Z^in,out. Therefore, we redefine a modified form of cross helicity according to

4.5

which is equivalent to σ^Cr=−(B_R/|B_R|)σ^C so that σ^Cr>0 represents a dominance of outward-propagating fluctuations. We hereafter drop the superscript ‘r’ and adopt equation (4.5) as our definition of the normalized cross helicity.

Comparison of the distribution functions in figure 2 reveals that the standard deviation for ϵ^T is smaller than for ϵ^in,out. This seems unlikely if the cascades of the two pseudo-energies are independent and uncorrelated. Figure 4 shows scatter plots using the same energy range as in figure 2 for selected ranges of σ^C. Three different ranges of |σ^C| are shown. Top to bottom they are 0≤|σ^C|<0.25, 0.25≤|σ^C|<0.5 and 0.5≤|σ^C|<0.75.

Figure 4.

Scatter plots for ϵ^in,out for three ranges of |σ^C| given above the panels. In each plot, 1 h samples are used to produce estimates of ϵ at lag 5 (320 s). σ^C>0 are represented by black diamonds and samples with σ^C<0 are represented by blue triangles. As expected, there are fewer samples with σ^C<0 than with σ^C>0. Dashed red lines represent ϵⁱⁿ=ϵ^out and ϵⁱⁿ=−ϵ^out. Black and blue lines are derived from equation (4.8) using the average value of σ^C for each population with the additional factor −1 to account for the observed anti-coincidence. We have seen similar results when using 3 h and 12 h samples. (Online version in colour.)

Two clear conclusions can be drawn from figure 4. First, each of the six populations shown in figure 4 appears to scale linearly in ϵⁱⁿ versus ϵ^out. When |σ^C| is small the linear trend of the two populations appears to nearly overlay one another. As |σ^C| increases the linear trend of the two populations appears to separate, or fan, so that the σ^C>0 samples (intervals dominated by outward propagation) show a more shallow slope than the σ^C<0 samples (intervals dominated by inward propagation).

Second, there is an anti-correlation between ϵⁱⁿ and ϵ^out: when one is positive the other is negative. Nothing from the cascade laws we have discussed so far explains this.

Let us first address the observation that the slopes of ϵⁱⁿ versus ϵ^out depend on σ^C. If we adopt equations (3.4) and (3.5) as expressions for the cascade of the pseudo-energies E^in,out, we can rewrite them as

4.6

and

4.7

and from this we can write

4.8

As an example, if σ^C=0 so that E^out=Eⁱⁿ, equation (4.8) predicts that ϵ^out/ϵⁱⁿ=1. When σ^C=0.5 (−0.5) equation (4.8) predicts that ϵ^out/ϵⁱⁿ=3^1/2 (3^−1/2). Using the average value of σ^C for each of the six subsets of σ^C shown in figure 4, we can apply equation (4.8). In the process, we adopt a factor of −1 in equation (4.8) for reasons not yet established. The resulting lines are reproduced on the figure.

The existence of negative values of ϵ in figure 4 would seem to preclude the systematic applicability of equations (4.6)–(4.8). However, this does not mean these expressions fail to describe ϵ when averaged over the entire ensemble. For reasons not yet completely determined, equations (4.6)–(4.7) do reliably describe the ratio ϵ^out/ϵⁱⁿ for each hourly interval if we adopt an additional factor of −1.

(c). Back transfer

We now move to examine the second conclusion from figure 4: the observation that ϵⁱⁿ and ϵ^out are anti-correlated. This will account for the factor −1 above.

Consider the third-moment expressions equations (4.1) and (4.2). Both contain terms like |Z^in,out|² that vary in magnitude, but not sign. It is the terms Z^in,out_R that vary in sign and potentially facilitate the back transfer of energy. Comparison of ϵⁱⁿ/ϵ^out leads us to consider or its inverse, depending on the direction of the mean magnetic field. If |Δv_R(L)|>|Δb_R(L)| then the ratio ϵⁱⁿ/ϵ^out>0. However, if |Δv_R(L)|<|Δb_R(L)| then the ratio ϵⁱⁿ/ϵ^out<0 and this provides some possible insight into the back transfer of one of the pseudo-energies. This suggests that the back transfer of pseudo-energy may be related to the Alfvén ratio R_A=E_v/E_b<1 and this is something we can test.

We compare the presence of a back transfer for one or another pseudo-energies with R_A computed from the fluctuations on the largest scales in the hourly samples (a bulk r.m.s. calculation). We justify using R_A computed from the entirety of a 1 h sample rather than at smaller scales because (i) we are not certain that the critical dynamic resides at the smaller scales and (ii) R_A tends to be approximately preserved across the inertial range scales until the very smallest scales [127]. To determine the presence of back transfer, we compute the quantity R_ϵ≡|(ϵ^out−ϵⁱⁿ)/(ϵ^out+ϵⁱⁿ)|. R_ϵ<1 requires that both ϵⁱⁿ and ϵ^out possess the same sign while R_ϵ>1 requires that ϵⁱⁿ and ϵ^out possess different signs (the anti-correlation we report above).

Figure 5a plots R_A versus R_ϵ for all of the 1 h samples used in this study that had R²>0.7. There is a strong tendency for the data to follow the two axes such that R_ϵ<2 (R_ϵ>2) when R_A>1 (R_A<1). On average, the solar wind at 1 AU possesses [128–130], so anti-correlated pseudo-energy cascades are a common feature in the solar wind.

Figure 5.

(a) Scatter plot for ratio of kinetic to magnetic energy R_A≡E_v/E_b computed from large-scale bulk parameters versus parameter R_ϵ quantifying the anti-correlated cascade of pseudo-energy. (b) R_A computed from the second-order structure functions at same lag as in figures 2 and 4. In both panels, R_ϵ is also computed using same lag as in figures 2 and 4. The vertical dashed lines mark R_ϵ=1. There is a clear tendency for R_ϵ>1 when in both panels. (c) Scatter plot of R_A computed from bulk properties as used in panel (a) and [Δv(τ)]²/[Δb(τ)]² computed for lag=5. (Online version in colour.)

As we stated above, we are not sure at what scale the critical dynamic resides. We can compute the ratio of magnetic to kinetic energy at the same scale where the back transfer is measured (lag=5) by computing the ratio of the second-order structure functions [Δv(τ)]²/[Δb(τ)]². Figure 5b shows the result of this analysis compared with R_ϵ for the same samples used to produce the panel on the left. The two plots are virtually identical.

Figure 5c shows a scatter plot of the two definitions for R_A employed in panels (a,b) of the same figure. There is a strong correlation until R_A>0.7 (approximately) when the two definitions diverge, although not systematically. Regrettably, this obscures any potential answer to the question of which spatial scale possesses the dynamic that results in the anti-correlation of the pseudo-energy cascades.

(d). Consistency across the inertial range

So far we have examined only the lag=5 results. We chose this value because there are approximately 12 such lags within each 1 h sample, and the resulting averaging that occurs within the 1 h sample provides a degree of confidence in the result. However, we have argued that the observed back transfer of energy is the result of having low R_A values (magnetically dominated Z^±). This argument alone does not explain which pseudo-energy back transfers.

There is the question of whether what we have shown here is a consistent dynamic across the inertial range scales or merely a local dynamic that is part of the general ebb and flow of energy that forms the cascade. Figure 6 compares the computed cascade rates at lags of 5 and 15. While agreement is not perfect, due at least in part because the lag=15 values are the result of less averaging than the lag=5 values, the linear trend is obvious.

Figure 6.

We compare the computed cascade rates for lags 5 and 15 for each of the 1 h samples that passed the R²≥0.7 test. Comparison for ϵⁱⁿ, ϵ^out, ϵ^T and ϵ^C. General agreement between the computed ϵ at lags 5 and 15 suggest that the conclusions of the above analysis hold throughout the inertial range, or at least over a wide range of lags within the inertial range. (Online version in colour.)

The conclusion we draw from this is that the anti-correlation in cascade rates ϵ^in,out reported here represent behaviour that spans the inertial range scales. There may be unresolved variations and systematics. We cannot say there is not. What we can assert is that the conclusions drawn here appear to represent the entirety, or significant fraction, of the inertial range scales.

Our previous paper [126] showed that sequential values of ϵ differed by values comparable to the mean. This means that while a given 1 h sample may show back transfer of ϵⁱⁿ it is just as likely that the next 1 h sample will show back transfer in ϵ^out. While this does not require that the cascade change its character on timescales comparable to the correlation time (in the plasma frame), it is suggestive of it. Observations reported here are probably just snapshots in time that change with the evolving dynamics.

(e). Variation of cascade on the correlation scale

We have shown that the apparent rate of energy cascade within 1 h samples of solar wind turbulence at 1 AU exhibit large variability (standard deviation) relative to the mean ensemble values. We have further shown that the cascade associated with pseudo-energies Z^±, or Z^out and Zⁱⁿ, are anti-correlated and exhibit negative cascade rates implying transport of energy from small to large spatial scales. We would like to measure the timescale over which this behaviour changes because a persistent transport of energy from small to large spatial scales would seem impossible to maintain within the normal view of homogeneous turbulence unless there is an unresolved small-scale source driving the cascade. This is not possible using single-spacecraft techniques. However, we can perform an analysis of the variability of the cascade from one 1 h sample to the next and apply the normal turbulence assumption that small-scale structures last for shorter periods of time than large-scale structures.

We compute the difference between cascade rates from one 1 h sample to the next. Where data are missing or samples failed to pass our selection criteria, we omit pairings and use only available, sequential 1 h samples. We define Δ_1 hϵ(t)≡ϵ(t+1 h)−ϵ(t) and compute the distribution functions for ϵⁱⁿ, ϵ^out and ϵ^T. These are shown in figure 7. The width of the distributions are comparable to those of the ϵ values, themselves, suggesting that there is little correlation between the ϵ values computed for successive 1 h samples. While this does not provide a timescale over which the cascade rate changes for a given sample of plasma, it does suggest that timescale is shorter than it would be if sequential 1 h samples were strongly correlated.

Figure 7.

(a–c) Distribution functions for the difference between sequential 1 h sample values of ϵⁱⁿ, ϵ^out and ϵ^T. Note the width of the distributions are comparable to the width of the distributions for the ϵ values, themselves, suggesting that there is little correlation between sequential hourly values. (d) Scatter plot of Δ_1 hϵⁱⁿ and Δ_1 hϵ^out showing the persistent anti-correlation of these values. (Online version in colour.)

Figure 7 also shows a scatter plot of Δ_1 hϵⁱⁿ and Δ_1 hϵ^out. It comes as no surprise that these 1 h differences are anti-correlated when the values of ϵ are themselves anti-correlated. To maintain that anti-correlation, they must change sign in unison.

5. Summary and conclusion

Third-moment expressions have now been used by numerous authors in simulation and data analysis [18,20,23,29,30,103,117–119,123–126,131–138]. The technique is not without controversy. The role of expansion is not well quantified [117,139,140] and the techniques for addressing large-scale shear are difficult to apply [117,119]. However, the technique appears to possess considerable potential for unlocking the dynamics of interplanetary turbulence and enabling the solar wind to fullfil its potential as a very large wind tunnel for the study of MHD turbulence.

In this analysis, we have excluded transients, such as shocks and their drivers, and regions of shear. We have not taken into account expansion. Simply put, the published expressions for expansion effects do not agree and our efforts to derive expansion terms have not produced results that we are prepared to support. We do not believe that expansion effects can account for the broad distribution and negative values of ϵ in any general sense or the associated back transfer of energy shown here.

We have shown that 1 h samples of the solar wind taken at 1 AU can be used to produce third-moment estimates for the inertial range cascade that possess linear scaling with lag and appear to offer reasonable estimates for the cascade of the pseudo-energies. As before [20,23,30,126], the average over many samples produces a net forward transfer of both pseudo-energies consistent with the observed solar wind heating. The estimates derived from individual 1 h samples show evidence of intermittent dynamics with a high degree of variability relative to the mean cascade rates. We do not believe the reported variability is a measurement error. We do believe it to be a fundamental measurement of the underlying intermittency of the turbulence.

To that end, we have argued that the transfer of pseudo-energies associated with the inward- and outward-propagating components are often, and perhaps generally, anti-correlated at 1 AU. Where one is seen to forward transfer to small scales the other is seen to back transfer energy to larger scales. Which component back transfers energy is not determined by this analysis and it can be either. We believe the best explanation lies with the dominance of the dynamics by the magnetic term as measured by the Alfvén ratio R_A<1. This provides possible insights into the third-moment expressions that facilitates anti-correlation between the two cascades.

Whenever one discusses back transfer or the preferential transfer of one pseudo-energy over another, several terms are often raised. The first term is ‘inverse cascade’ [141,142]. Inverse cascade is generally taken to mean the back transfer of one quantity while another quantity forward-transfers to smaller scales. It is often described according to absolute equilibrium ensemble theory [35], and it represents a systematic movement of two quantities to separate spatial scales. The second term is ‘dynamic alignment’ [143]. Dynamic alignment has been proposed as a mechanism for producing strong correlations between the magnetic field and velocity fluctuations in the solar wind [113,144]. The third term is ‘selective decay’ [145,146]. All three processes represent a systematic transport, accumulation and dissipation of one statistical quantity over another. The anti-correlated spectral transfer we show here shows no such systematic transport. It is a random exchange between the forward- and back transfer of two pseudo-energies without any apparent systematic determination evident.

The correlation length for spacecraft data at 1 AU in the solar wind is approximately 1 h. This is roughly the scale that undergoes one turbulent lifetime in the time it takes for the wind to reach 1 AU from its solar source. The energy contained within the inertial range is roughly equal to the accumulation of energy at the average cascade rate over the time it takes to reach 1 AU. So, the inertial range has been fully depleted approximately one time by the time it reaches 1 AU. The scales used here, 320 and 960 s, represent the first decade within the measured inertial range. The dissipation scale, where the spectrum breaks and steepens and dissipation sets in, is seen at 2–5 s. This means there are two more decades of inertial range that remain unstudied by this analysis. It also means that if the back transfer reported here varies between pseudo-energies on the plasma-frame timescale of the turbulent fluctuations at 320 and 960 s, then it is possible for the back transfer to draw energy from the inertial range without depleting it. We do not propose that the back transfer reported here persists over long periods of time. We have already shown that individual cascade rates change substantially from 1 h of data to the next [126]. Back transfers of energy should be possible so long as they do not persist for long periods of time.

Funding statement

J.T.C. and C.W.S. are supported by Caltech subcontract 44A-1062037 to the University of New Hampshire in support of the ACE/MAG instrument. B.J.V. is supported by NSF/SHINE grant no. AGS1357893.