Reynolds Number and Intermittency in the Expanding Solar Wind: Predictions Based on Voyager Observations

TL;DR

This paper predicts how turbulence intermittency in the solar wind varies with distance from the Sun, using Voyager data and theoretical scaling, and forecasts increased intermittency near the Sun's critical regions.

Contribution

It introduces a novel approach linking Reynolds number variations to turbulence intermittency in the expanding solar wind, validated with Voyager data and predicting future Parker Solar Probe observations.

Findings

Reynolds number decreases with heliocentric distance.

Magnetic field kurtosis decreases with distance, indicating less intermittency.

Predicted increased intermittency near the Sun's critical regions.

Abstract

The large scale features of the solar wind are examined in order to predict small scale features of turbulence in unexplored regions of the heliosphere. The strategy is to examine how system size, or effective Reynolds number, varies, and then how this quantity influences observable statistical properties, including intermittency properties of solar wind turbulence. The expectation based on similar hydrodynamics scalings, is that the kurtosis, of the small scale magnetic field increments, will increase with increasing Reynolds number. Simple theoretical arguments as well as Voyager observations indicate that effective interplanetary turbulence Reynolds number decreases with increasing heliocentric distance. The decrease of scale-dependent magnetic increment kurtosis with increasing heliocentric distance, is verified using a newly refined Voyager magnetic field dataset. We argue that these scalings continue to much smaller heliocentric distances approaching the Alfven critical region, motivating a prediction that the Parker Solar Probe spacecraft will observe increased magnetic field intermittency, stronger current sheets, and more localized dissipation, as its perihelion approaches the critical regions. Similar arguments should be applicable to turbulence in other expanding astrophysical plasmas.

Equations

$$C_R(\tau )=\frac{<\mathbf{b}(t)\cdot \mathbf{b}(t+\tau )>}{<|\mathbf{b}(t){|}^2>},$$

Full text

I Introduction

How can scaling of Voyager observations of interplanetary turbulence inform predictions about plasma turbulence in a broader context? The answer to this question is important not only to anticipate what will be observed by spacecraft such as Parker Solar Probe as well as Solar Orbiter, but also for observation of expanding plasmas in diverse astrophysical situations. We explore this possibility here, with specific predictions for Parker Solar ProbeFox et al. (2016), by employing Voyager magnetic field observationsMatthaeus and Goldstein (1982); Ness and Burlaga (2001). The data allow examination of the behavior of effective turbulence Reynolds number and the kurtosis of magnetic field fluctuations over a wide range of heliocentric distances. We find that the Reynolds number decreases and the kurtosis at a fixed physical scale decreases with increasing distance, with associated expectations concerning the roughness of the magnetic field. Extrapolating the observed scaling to lower heliocentric distances motivates the prediction that the spatial concentration of coherent structures increases approaching the Alfvén critical region from outside, an effect that Parker Solar Probe should soon measure. This letter provides theoretical and observation details that motivate this prediction, and discusses further applications.

Theories of turbulence, in particular Kolmogorov theory Kolmogorov (1941, 1941, 1962); Frisch (1996) and its many variations, are frequently applied to understanding in situ spacecraft observations in the interplanetary plasmaBruno and Carbone (2013); Horbury et al. (2008); Chen (2016). In most instances these theories are invoked, either explicitly or implicitly, in a form appropriate the regime of universality conjectured to obtain in the limit of infinite Reynolds number Kolmogorov (1941, 1941, 1962). However it is well known based on experimental data, especially in hydrodynamicsPope (2000), that the dimensionless Reynolds number, at attainable finite values, controls scaling of numerous statistical quantities, including the approach to an asymptotic dissipation rate and the scaling of higher order moments or increments. There is substantial evidence mainly based on simulations, that magnetohydrodynamics (MHD) and other plasma models exhibit analogous systematic variations with Reynolds-like numbers Linkmann et al. (2015); Bandyopadhyay et al. (2018). Here we will examine turbulence in the solar wind, quantifying variation of a specific intermittency parameter, the kurtosis, as an effective Reynolds number is varied.

In hydrodynamics the value of kinematic viscosity $\nu$ enters into the definition of Reynolds number $R=uL/\nu$, for turbulence speed $u$, and energy containing (outer) scale $L$. For the weakly collisional plasma found in the solar wind, $\nu$ is meaningless, but one may adapt the notion of Reynolds number by exploiting its relationship to “system size”, meaning the extent of the inertial range. In Kolmogorov theory, Reynolds number also relates the outer (correlation) scale $L$ and inner (dissipation) scale $\eta$, as $Re=(L/\eta)^{4/3}$. For a weakly collisional plasma, the Kolmogorov dissipation scale $\eta$ maybe reasonably replaced by the ion inertial scale $d_{i}$ (or the thermal gyroradius if plasma beta becomes large), given that the observed inertial range at MHD scales terminates at the largest proton kinetic scale encountered by the direct energy cascade Leamon et al. (1998); Chen et al. (2014). Accordingly we adopt a definition of effective Reynolds number in terms of the system size $L/d_{i}$, or size of the inertial range, as $Re=(L/d_{i})^{4/3}$.

Below we will employ this definition of effective Reynolds number and examine its behavior between 1 AU and 10 AU in Voyager magnetic field data. The observational finding, backed by an elementary theoretical assessment, is that $Re$ decreases with increasing heliocentric distance in the solar wind. Thus, even as the wind expands to fill the available volume, and in this sense becomes larger, the “system size” from the perspective of turbulence, is decreasing.

Based on the determination of the radial behavior of Reynolds number, the next step will be an assessment of the behavior of a normalized fourth order moment of the magnetic fluctuations, and its scaling with the Reynolds umber. Kurtosis, the normalized fourth moment, measures the roughness of the magnetic field and appearance of coherent structures. The kurtosis of increments of a magnetic field cartesian component time-series $b(t)$ with time lag $\tau$ is defined as $\kappa(\tau)=<\Delta b(\tau)^{4}>/<\Delta b(\tau)^{2}>^{2}$ where $\Delta b(\tau)=b(t+\tau)-b(t)$ is the increment at a scale $\tau$. Following typical practice in solar wind studies, we will exploit the highly supersonic and super-Alfvénic flow at speed $V_{sw}$ to interpret statistical properties at time lags $\tau$ with spatial lags $r=-V_{sw}\tau$, the so-called Taylor hypothesis. The use of Taylor’s hypothesis has been shown to work well for first and second order statistics down to the ion inertial scale and smallerChhiber et al. (2018).

Some expectations based on theory and observations. The expected variation of Reynolds number with heliocentric distance may be anticipated using simple arguments as follows: A von Karman-Howarth phenomenology has been shown to work well for explaining radial variations of turbulence and plasma properties of the solar wind, including correlation length and proton temperature Zank et al. (1996); Breech et al. (2008). Temporarily ignoring expansion, a relevant pair of equationsMatthaeus et al. (1996) is $\frac{dZ^{2}}{dt}=-\frac{1}{2}\frac{Z^{3}}{L}$ and $\frac{dL}{dt}=Z$, where $Z$ is turbulence amplitude and $L$ the correlation scale. The solution behaves as $L(t)\sim\sqrt{t}$, which when employing the Taylor hypothesis becomes $L(R)\sim R^{1/2}$ where here $R$ is heliocentric distance. Meanwhile to a reasonable approximation the proton number density in the solar wind falls off as $n(R)\sim R^{-2}$, and by definition the proton inertial scale in the expanding solar wind behaves as $d_{i}(n)\sim n^{-1/2}\sim R$. Therefore the Reynolds number may behave approximately as $Re=(L/d_{i})^{4/3}\sim(R^{1/2}/R)^{4/3}\sim R^{-2/3}$.

On the observational side, it is well established that the correlation scale varies with heliocentric distanceKlein et al. (1992); Zank et al. (1996). This variation is foundRuiz et al. (2014) to be approximately $\sim R^{0.44}$ in a mixed latitude ensemble, although the results did not have strong dependence on plasma beta.

Turning to the kurtosis at small spatial lags, we find very little in the MHD or plasma literature concerning expectations for its behavior as Reynolds number is varied. There are studies at fixed (or, uncontrolled) Reynolds numbers, of the scale dependent kurtosis of primitive variables in simulations, and in solar wind and magnetosheath observations Sorriso-Valvo et al. (1999); Macek et al. (2017); Chhiber et al. (2018). Some insight into scaling of kurtosis at kinetic scales has been obtained in relatively low Reynolds number kinetic simulations Parashar et al. (2015). There are also studies of multi-fractal scalings, but again, as far as we are aware, always without regard for Reynolds number or its variation. In hydrodynamics the situation is more advanced both in experiments and in theory. The small scale kurtosis of longitudinal increments should at zero separation approach the kurtosis of longitudinal spatial derivatives. Below we will carry out an analysis of increments in the smaller inertial range scales of the solar wind, arguably small enough that the kurtosis approaches that of the magnetic field derivatives. In this regard, we take note of a collection of well studied hydrodynamic experimentsVan Atta and Antonia (1980). The best fits of these data to the variation of $\kappa\sim Re^{\gamma}$ is $\gamma\approx 0.16$ to $0.2$. This nicely brackets an analytical estimate, based on phenomenological estimates as well as extensions of Kolmogorov-Obukhov treatments based on log-normal distributions of increments Oboukhov (1962). These theoretical estimates are discussed in the same referenceVan Atta and Antonia (1980) and lead to, for example, $\gamma=3\mu/4=3/16$ when the log-normal intermittency parameter $\mu$ takes the value 0.25. These hydrodynamic expectations anticipate the similar behavior we find below for solar wind turbulence.

Voyager Data. The Voyager magnetic field datasets are an excellent choice for the present study due the wide range of distances spanned: the associated variation of correlation scale and density are found to give a systematic variation of Reynolds number. Therefore we use 1.92s cadence magnetic field vector data from the Voyager 1 spacecraft. For the present study, we use a refined dataset, explained below, covering the range from 1 AU up to $\sim 10AU$.

The publicly available Voyager magnetic field dataset, at the time of this writing, requires additional work to make it useful for a statistical study. After the cleanup procedure we employed this improved subset of Voyager 1 MAG data will be made available for other purposes.

One problem with the previously available data is due to the large data gaps associated with regular lack of telemetry. The first task we perform is to make the time series uniform by filling in the missing data points with NaNs (not-a-numbers). The final time series has $\sim 60$ million data points.

The data available from the NASA-NSSDC also have many intervals containing “calibration rolls” as well as unexplained “noise” that need to be removed before any reliable turbulence statistics can be computed. We clean the data using a series of techniques. These include application of obvious cutoffs (-50nT,50nT), a Hampel filter where the outliers are replaced with NaNs instead of a median value, and finally, visual inspection of high kurtosis regions to identify and filter out remaining bad data. We describe the technique in detail in a longer paper Cuesta et al. (2019). The final clean time series has $\sim 23.27$ million data points and $\sim 37.74$ million NaNs representing missing data or discarded bad data.

After the cleaning procedure, we bin the data into 450 bins, each of size 0.02AU. A significant number of these bins have no physical data points, and some have very few physical data points. For statistical significance, we exclude bins with less than 10000 physical data points from our analysis. The remaining useful dataset comprises 328 bins, each having $>$10000 points. For data quality purposes, we then compute the kurtosis at a small lag for the data in each bin. This helps further identify bins with potential bad data points as $\kappa$ would be anomalously high for large unphysical discontinuities. Some retained bins contain what appears to be upstream waves Smith et al. (1983) or similar plasma activity that cause $\kappa$ to attain anomalously small values. We recall that using single spacecraft data, we employ increments at very small time lags, and the Taylor hypothesis, to estimate radial magnetic field derivatives.

Analysis procedure: Reynolds number. The two-time vector autocorrelation function of magnetic field is defined as

[TABLE]

where ${\bf b}={\bf B}-\langle{\bf B}\rangle$. The outer scale is computed, using the Taylor frozen-in flow hypothesis, as the scale where the autocorrelation drops to $1/e$. In particular, we estimate the correlation scale as $L=V_{sw}\tau_{o}$ where $\tau_{o}$ is the lag at which the two-time correlation of the magnetic field decreases to $1/e$. With these definitions, the effective Reynolds number can be computed as $Re=\left(\frac{L}{d_{i}}\right)^{4/3}$ where $d_{i}$ is the ion inertial length $c/\omega_{pi}$, chosen to represent the scale at which the inertial range terminates. The ion (proton) inertial length is computed in each bin by appropriate averages of the number density recorded by the Voyager plasma (PLS) plasma instrument. Note that choosing the electron inertial length as the inner scale would simply shift the Reynolds number by a constant value.

Results:

Figure 1 shows the effective Reynolds number computed from the magnetic field, as a function of heliocentric distance in the 1-10 AU Voyager 1 dataset. Blue stars represent the local value of Reynolds number in each of the 450 bins, each approximately 0.02 AU wide in heliocentric distance. Significant variability is present in the Reynolds number, consistent with variable turbulence conditionsTu and Marsch (1995), such as stream structure, coronal mass ejections, etc. There are some bins in which wave activity may modify the results. (See Supplementary Material.) To get a better view of the average behavior of $Re$, we plot a 15 point running average of the blue stars. A clear trend for $Re$ to drop with heliocentric distance can be seen. The elementary estimate of $Re\sim R^{-2/3}$ based on $L\sim\sqrt{R}$ and $d_{i}\sim R$, is over-plotted as a thick red line. We emphasize the point that this is not a fit of any kind: the average $Re$ follows a trend similar to the simple theoretical prediction. The turbulence “system size” or effective Reynolds number in the expanding solar wind in fact decrease with increasing radial distance.

According to the hydrodynamic analogy, a systematic decrease in $Re$ should be reflected in a decrease in the kurtosis of increments computed at very small lags. This expectation would be obtained whenever the turbulence is fully developed, i.e., it is not too close to its injection or initiation. Having observed a systematic decrease in $Re$, we now examine the small scale magnetic kurtosis in the same dataset as a surrogate for the kurtosis of the longitudinal magnetic derivative. We compute the kurtosis at a lag of $10d_{i}$, where the distance is computed from time lag, using Taylor’s hypothesis, using the average radial plasma velocity for the conversion.

Figure 2 shows the kurtosis of the increment of the radial magnetic field component at the scale 10$d_{i}$ as a function of heliocentric distance. Blue stars show value of kurtosis in each of the 450 bins, the black curve shows a 15 point running average, and the red line shows a power-law fit to the data with an approximate dependence $\kappa\sim R^{-0.86}$. As expected, the kurtosis drops for larger heliocentric distance, along with the decrease of effective Reynolds number with increasing distance. Points that have lower kurtosis appear in the same regions where $Re$ drops. However these anomalously low points do not significantly affect the overall conclusion that both $Re$ and $\kappa$ decrease systematically with increasing heliocentric distance.

Conclusions: The above study, based on analysis of Voyager 1 magnetic field data, shows two main results that are established firmly by the observations. First, (I) the effective Reynolds number, as defined based on the extent of the inertial range, decreases systematically in the interplanetary medium between 1AU and 10 AU. Second (II) the kurtosis of magnetic component increments defined over the same range is also found to decrease with increasing heliocentric distances.

Potentially interesting physics comes in examining the relationship between these results. In fact, the reasoning that connects these two findings may be cast in more than one framework:

First, one might simply assume that the effective Reynolds number is fully equivalent to the ordinary Reynolds number, and then further assert that a relationship such as the hydrodynamic relation $\kappa\sim Re^{\gamma}\sim Re^{2\mu/4}$ also obtains for a weakly collisional plasma. As we pointed out earlier, such a scaling can be obtained on purely empirical grounds. Or, it can be deduced from Refined SimilarityKolmogorov (1962), augmented by Obukhov’s scaling hypothesisOboukhov (1962) that parameterizes anomalous scaling of increments with $L/\eta$ (which here becomes $L/d_{i}$ by a separate argument.) While this reasoning may appeal more strongly to formal turbulence theory, we recall that a Refined Similarity hypothesis has not been firmly established for a collisionless plasma (although there have been preliminary discussions of thisMerrifield et al. (2005); Chandran et al. (2015)). So the linkage in this chain of reasoning, while appealing, is a bit tenuous.

A second argument for the relationship of results I and II rests on understanding how the nonlinearities in turbulence, the most essential of which are quadratic, give rise dynamically to the formation of coherent structures. Nonlinear spectral transfer forms coherent structures, without direct involvement of dissipation, and progressively at smaller scale Frisch et al. (1983); Wan et al. (2009, 2012). This is evidenced, in both ideal and dissipative MHD, by the monotonic increase in filtered kurtosis (high pass filtered) at scale $\ell$, as the band-pass scale decreases. Given that transfer is mainly local in scaleVerma et al. (2005), it follows that an inertial range with greater bandwidth will incorporate a greater number of octaves of transfer over which the coherent structures may form and intensify. Thus, larger systems, i.e., larger $L/\eta$ or $L/d_{i}$, will have stronger coherent structures and higher small scale kurtosis. But this implies immediately that $\kappa(10d_{i})$ will decrease with effective Reynolds number as we have defined it. It is noteworthy that this second line of reasoning takes no explicit position on the relationship between coherent structures and dissipation, nor does it assume a refined similarity hypothesis, in contrast to the first line of reasoning. For this reason we prefer, at this time, the second track for connecting results I and II, although we do not doubt that the more formal relationships may be established more firmly in the future.

Having completed this excursion into the turbulence theoretic basis for relating I and II, we now may view the behavior of the kurtosis at 10$d_{i}$ as a consequence of the decrease in effective $Re$ at larger heliocentric radial distance. Such a connection has interesting implications beyond magnetic field observed by Voyager in the outer heliosphere.

One major implication is the potential for extrapolating these results to other expanding astrophysical plasmas. We would expect, based on the arguments above that other systems engaged in an approximate spherical expansion, with evolving von Karman turbulence, will also admit a baseline estimate of effective Reynolds number scaling as $Re\sim R^{-2/3}$. Turbulence properties that scale with Reynolds number will systematically respond to this scaling, enabling in principle, a variety of predictions for expanding systems such as galactic winds, supernova remnants, etc. The decrease in kurtosis, or inverse filling factor, of coherent current structures, is only one such prediction.

Closer to home, we are tempted to extrapolate the present results inward, toward the corona, but outside the Alfvén critical region. Such an extrapolation finds some partial support in correlation length scalings in radius from Helios and Ulysses Ruiz et al. (2014) that behave as $L\sim R^{0.43}$ and density scalings McComas et al. (2000) that remains close to $n\sim R^{-2}$ everywhere in the heliosphere on average. Encouraged by this, one may extrapolate that effective Reynolds number increases moving towards the inner interplanetary region where the magnetic control imposed by the corona gives way to the turbulent solar wind DeForest et al. (2016). Approaching this region from outside, we expect that as $Re$ increases, the kurtosis at small scales (multiples of $d_{i}$) will increase. This becomes, in effect a prediction for the recently launched Parker Solar Probe and the upcoming Solar Orbiter spacecraft: as these missions explore the heliosphere approaching the Alfvén critical region, we expect an increase in the frequency and intensity of magnetic discontinuities and current sheets. This should be indicted by higher order statistics such as the kurtosis at small scales. There may be associated implications such as stronger concentrations of dissipation and heating Osman et al. (2012), a possibility we have not pursued here. There are reasons to believe that the turbulence is sustained well below the the Alfvén critical surfce Adhikari et al. (2019) but the nature of turbulence and its driving like differs in that region (the corona), and we do not attempt to extend our prediction beyond this point at present. As a step toward supporting the present prediction, we are currently processing Helios data to confirm this effect down to 0.3AU. Results will be presented in a separate publication, as we anxiously wait the relevant Parker Solar Probe analyses. We may anticipate that there will be further interesting consequences of the systematic variation of interplanetary effective Reynolds number that will he examined in future study.

Acknowledgements.

The authors are thankful to Chuck Smith for useful discussions. This research supported in part by the NASA Heliophysics Guest Investigator Program (80NSSC19K0284, NNX17AB79G) and Supporting Research (80NSSC18K1648) and by the ISOIS Parker Solar Probe Project though Princeton subcontract SUB0000165.