Effects of Solar Activity on Taylor Scale and Correlation Scale in Solar Wind Magnetic Fluctuations
aa r X i v : . [ phy s i c s . s p ace - ph ] A ug Effects of Solar Activity on Taylor Scale and Correlation Scale inSolar Wind Magnetic Fluctuations
G. Zhou , , H.-Q. He , , , and W. Wan , , Received ; accepted Key Laboratory of Earth and Planetary Physics, Institute of Geology and Geophysics,Chinese Academy of Sciences, Beijing 100029, China; [email protected] College of Earth and Planetary Sciences, University of Chinese Academy of Sciences,Beijing 100049, China Innovation Academy for Earth Science, Chinese Academy of Sciences, Beijing 100029,China Beijing National Observatory of Space Environment, Institute of Geology and Geo-physics, Chinese Academy of Sciences, Beijing 100029, China 2 –
ABSTRACT
The correlation scale and the Taylor scale are evaluated for interplanetarymagnetic field fluctuations from two-point, single time correlation function usingthe Advanced Composition Explorer (ACE), Wind, and Cluster spacecraft dataduring the time period from 2001 to 2017, which covers over an entire solar cy-cle. The correlation scale and the Taylor scale are respectively compared withthe sunspot number to investigate the effects of solar activity on the structureof the plasma turbulence. Our studies show that the Taylor scale increases withthe increasing sunspot number, which indicates that the Taylor scale is positivelycorrelated with the energy cascade rate, and the correlation coefficient betweenthe sunspot number and the Taylor scale is 0.92. However, these results are notconsistent with the traditional knowledge in hydrodynamic dissipation theories.One possible explanation is that in the solar wind, the fluid approximation failsat the spatial scales near the dissipation ranges. Therefore, the traditional hydro-dynamic turbulence theory is incomplete for describing the physical nature of thesolar wind turbulence, especially at the spatial scales near the kinetic dissipationscales.
Subject headings:
Solar wind; Interplanetary turbulence; Interplanetary medium;Solar magnetic fields; Space plasmas; Solar activity
1. Introduction
Solar wind turbulence has received considerable attention for several decadeswithin the community (e.g., Kraichnan 1965; Belcher 1971; Matthaeus & Goldstein1982a,b; Tu & Marsch 1995), and recently there has been an upsurge in interest in thistopic. Much of the interest is driven by the fact that the solar wind can provide aperfect natural laboratory for the study of plasma turbulence in space at low-frequencymagneto-hydrodynamic (MHD) scales, which is essential for the studies of solar windgeneration, plasma heating, energetic particle acceleration, cosmic ray propagation, andspace weather (Jokipii 1968a,b; Coleman 1968; Barnes 1979; Matthaeus et al. 1984).During the past decades, the solar wind fluctuation properties and equations of motionhave been studied in great detail (Belcher 1971; Matthaeus & Goldstein 1982a; Tu et al.1984, 1989; Tu & Marsch 1997; Marsch & Tu 1997). However, even till now, it is stillimpossible to make accurate quantitative predictions. Most of the earlier studies usedthe single-spacecraft time-lagged data to infer solar wind spatial properties based on thewell-known “frozen-in flow” approximation (Taylor 1938). Provided that the solar wind flowspeed V sw is much greater than the local Alfv´en speed V A , thus the solar wind fluctuationswhich pass a detector are convected in a short time compared to all relevant characteristicdynamical timescales, so the time lags ∆ t are equivalent to spatial separations ∆ t V sw .This assumption is relatively reliable under some specific conditions (Paularena et al. 1998;Ridley 2000). However, the time scale over which the “frozen-in flow” assumption remainsvalid is not fully established (Matthaeus et al. 2005; Weygand et al. 2013), and the correctway to establish the spatial structure is to make use of the simultaneous two-point singletime measurements.To overcome the shortcomings of the “frozen-in flow” assumption, Matthaeus et al.(2005) obtained the two-point correlation function making use of simultaneous two-point 4 –measurements of the magnetic fields. By using this technique, both the correlation lengthscale and the Taylor scale can be determined, and the values of these two scales weregiven to be 186 Re (Earth radius, 1 Re = 6378 km) and 0 . ± . Re , respectively.Weygand et al. (2007) used the same method and obtained the Taylor scale values from thedata of magnetic field fluctuations in plasma sheet and solar wind based on the Richardsonextrapolation method. They estimated the Taylor scale in the solar wind as 2400 ±
2. Fundamental Concepts in Fluid Turbulence
A turbulent flow should satisfy the Navier-Stokes equation, which is the momentumevolution of an element of fluid and can be written as ∂ u ∂t + u · ∇ u = − ρ ∇ p + ν ∇ u . (1)Here u is the velocity, which is a fluctuating quantity in time t and space x , ∇ is thegradient with respect to x , ρ is the density, p is the pressure, and ν is the kinematicviscosity. Note that Equation (1) corresponds to the incompressible case. Furthermore,Equation (1) neglects forces (driving forces, gravity) except pressure. The strength of thenonlinear convective term u · ∇ u against the dissipative term ν ∇ u in Equation (1) canbe measured by the “Reynolds number” which is defined as R = U L/ν , where U and L denote the characteristic flow velocity and the characteristic length scale (or correlationscale in this study), respectively. The turbulent flow is characterized by large Reynoldsnumbers, which requires that the viscous term should be insignificant in this case. However,the boundary conditions or initial conditions may make it impossible to neglect the viscous 6 –term everywhere in the flow field. This can be understood by allowing for the possibilitythat viscous effects may be associated with the small length scales. Under the conditions oflarge Reynolds numbers, to make the dissipative term be the same order of the convectiveterm, the viscous term can survive only by choosing a new length scale l . Thus, U /L = νU/l . (2)From Equation (2), we can derive that lL ∼ (cid:16) νU L (cid:17) / = R − / . (3)The length scale l is called viscous length which represents the width of the boundary layer.Thus, at very small length scales, the viscosity can be effective in smoothing out velocityfluctuations.Since small-scale motions tend to have small time scales, one can assume that thesemotions are statistically independent of the relatively slow large-scale turbulence. If thisassumption makes sense, the small-scale motion should depend only on the rate at whichit is supplied with energy by the large-scale motion and on the kinematic viscosity. Inaccordance with the Kolmogorov’s universal equilibrium theory of the small-scale structure(Kolmogorov 1941a,b), the rate of energy supply should be equal to the rate of dissipation.According to the dimensional analysis, the amount of the kinetic energy per unit mass inthe turbulent flow is proportional to U ; the rate of transfer of energy is proportional to U/L ∼ /T , where T denotes the characteristic transfer time; the kinematic viscosity ν isproportional to U · L ; and the dissipation rate per unit mass ε , which should be equal to thesupply rate, is proportional to U /T ∼ U /L . With these parameters, we can obtain length,time, and velocity scales as η ≡ ( ν /ε ) / , τ ≡ ( ν/ε ) / , and v ≡ ( νε ) / , respectively.These scales are known as the Kolmogorov microscales of length, time, and velocity. TheReynolds number formed with ν and v is R η = ηv/ν = 1. The value R η = 1 indicates that 7 –the small-scale motion is quite viscous, and that the viscous dissipation adjusts itself to theenergy supply by adjusting length scales. From the expression ε ∼ U /L, (4)we know that the viscous dissipation of energy could be estimated from the large-scaledynamics that does not involve viscosity. Thus, the dissipation can be seen as a passiveprocess in the sense that it proceeds at a rate dominated by the inviscid inertial behavior ofthe large eddies. The above expression is one of the cornerstone assumptions of the classicalhydrodynamic turbulence. However, we should keep in mind that the large eddies only losea negligible fraction of their energy compared to direct viscous dissipation effects. Supposingthat the time scale of the energy decay is L /ν , then the energy loss proceeds at a rate of νU /L , which is very small compared to U /L if the Reynolds number R = U L/ν = U /LνU /L is very large. Kolmogorov (1941a,b) suggested that the small-scale structure of turbulenceis always approximately isotropic when the Reynolds number is large enough. This is thewell-known local isotropy theory. In isotropic turbulence, the dissipation rate can be simplyexpressed as ε ∼ ν U λ , (5)where λ denotes the Taylor scale. Combining Equations (4), (5), and the form of theKolmogorov microscale, we can obtain λL ∼ (cid:16) νU L (cid:17) / = R − / , ηL ∼ (cid:16) νU L (cid:17) / = R − / , λη ∼ (cid:18) U Lν (cid:19) / = R / . (6)Comparing Equation (3) with Equation (6), we can see that the Taylor scale is related tothe viscous dissipation. Equation (6) also suggests that for hydrodynamic turbulence withlarge Reynolds number, the Taylor scale is larger than the Kolmogorov microscale.The length scales L , η and λ mentioned above can characterise the properties ofthe flow with high Reynolds numbers. The correlation scale L , also known as “outer” 8 –scale or “energy containing” scale, is related to the inertial range of the turbulence. Thisparameter represents the size of the largest eddy in the turbulent flow (Batchelor 1953;Tennekes & Lumley 1972; Batchelor 2000). The large eddies perform most of the transportof momentum and contaminants, and the energy input also occurs mainly at large scales.This correlation scale can be measured by classical methods based on the Taylor’s hypothesisand can be associated with the first bend-over point in the power spectrum of the turbulentfluctuations. The Kolmogorov microscale η (“inner” scale or dissipation scale) representsthe smallest length scale in the turbulent flow (Jokipii & Hollweg 1970; Tennekes & Lumley1972) and it is at the end of the inertial range. In the view of traditional hydrodynamics,the viscosity can be effective in smoothing out fluctuations and dissipates small-scaleenergy into heat at very small length scales (note that in the low-collisionality plasmas, thissituation is less clear). A standard method for identifying the Kolmogorov microscale is toassociate it with the breakpoint at the high wave number end of the inertial range abovewhich the spectral index of the power spectral density becomes steeper. The Taylor scale λ is first proposed by Taylor (1935). It can be associated with the curvature of the two-pointmagnetic field correlation function evaluated at zero separation (Tennekes & Lumley 1972;Matthaeus et al. 2005; Weygand et al. 2010, 2011; Chuychai et al. 2014). In contrast to thecorrelation scale and the Kolmogorov microscale, the Taylor scale does not represent anygroup of eddy size, but it can characterise the dissipative effects. Moreover, the Taylor scaleis of the same order of magnitude as the Kolmogorov microscale. Specifically, the latter isoften smaller than the former for hydrodynamic turbulence with large Reynolds numbers.An essential characteristic of turbulence is the transfer of energy across scales. Theenergy resides mainly at large scales, but it can be transferred across scales by nonlinearprocesses, and eventually it arrives at small scales. The dissipation mechanisms at the smallscales would limit the transfer, dissipate the fluid motions, and release the heat (Batchelor1953; Tennekes & Lumley 1972). This is the so-called “cascade” process. When the 9 –associated Reynolds number and magnetic Reynolds number are large compared to unity,this process can be expected in hydrodynamics and in fluid plasma models such as MHD. Inprevious studies, the cascade process is investigated through spectral analysis or structurefunction analysis (Matthaeus & Goldstein 1982a,b; Goldstein et al. 1994; Tu & Marsch1995; Goldstein et al. 1995; Zhou et al. 2004). Many analysis methods describe the inertialrange of scale properties using the well-known power law of Kolmogorov theory for fluids(Kolmogorov 1941a,b) and its variants for plasmas (Kraichnan 1965). In hydrodynamics,the inertial range (or the self-similar range) mentioned above is typically defined extendingfrom the correlation scale (where the turbulence contains most of the energy) down to theKolmogorov microscale. In this work, we use the correlation scale L and the Taylor scale λ , instead of the correlation scale L and the Kolmogorov microscale η , to describe theproperties of the solar wind turbulence since the Taylor scale can be measured relativelyeasily (Tennekes & Lumley 1972; Weygand et al. 2005).
3. Methods and Procedures
If the turbulence is homogeneous in space, then the means, variances and correlationvalues of the fluctuations are independent of the choice of origin of the coordinate system(Batchelor 1953; Tennekes & Lumley 1972; Barnes 1979; Batchelor 2000). For a magneticfield B ( x , t ) = B + b , the mean is h B i = B , the fluctuation is b = B − B , and thevariance is σ = (cid:10) | b | (cid:11) . The two-point correlation coefficient is R ( r ) = 1 σ h b ( x ) · b ( x + r ) i . (7)Here r is the separation of two points x and x + r . For homogeneity, R and B areindependent of position x , though they may be weakly dependent on position in reality.The h ... i denotes an ensemble average. In homogeneous medium, the ensemble average isequivalent to a suitably chosen time-averaging procedure. For large | r | , the well-behaved 10 –turbulence becomes uncorrelated and R → L = Z ∞ R ( r ) dr. (8)In addition, the Taylor scale can also be associated with the curvature of R ( r ) at theorigin (see Matthaeus et al. (2005) and Weygand et al. (2007) for more details). Strictlyspeaking the scale defined in Equation (8) is the integral scale which is not necessarilyequal to the bendover scale of the spectrum. However, this will not affect the conclusionsin this work. The discussions of different scales and their relations to each other can befound in Shalchi (2020). A model correlation function with the correct asymptotic behavioris R ( r ) ∼ e − r/L , which has often been used as an approximation tool for estimating L (Matthaeus & Goldstein 1982a). Note that R ( r ) = 1 for r = 0 and R → r → ∞ .From the equations mentioned above, it is clear that the two-point correlationcoefficient R ( r ) plays an important role in determining the correlation scale L and theTaylor scale λ . In the following, we shall focus on the procedures for obtaining the R ( r ), L and λ from multi-spacecraft data.The magnetic field data used in this work was obtained by the instruments on spacecraftACE, Wind, and Cluster during the time period from January, 2001 to December, 2017.Most of the distances between ACE and Wind spacecraft are in the range of 20 − Re ,and the Cluster interspacecraft separations during this period range from about 100 km toover 10000 km. Since the spacecraft ACE and Wind orbit the Lagrangian point L1 which isabout 1.5 million km from the Earth and 148.5 million km from the Sun, the informationof the solar wind can be directly obtained by the spacecraft. The Cluster mission, whichconsists of four identical spacecraft at different positions, can provide the three-dimensionalmeasurements of large-scale and small-scale phenomena in the near-Earth environment 11 –(Escoubet et al. 1997). Note that the four Cluster spacecraft are not always in the solarwind. Occasionally, the Cluster spacecraft are in the Earth’s magnetosphere. Therefore,the data provided by the Cluster mission should be filtered before we use them.The first step for investigating the spatial scales in the solar wind turbulence is toidentify the time intervals during which the spacecraft were immersed in the solar wind.Table 1 shows the typical values of several solar wind parameters near 1 AU. As we can see,the values of the magnetic field magnitude and the plasma parameters drastically changewhen the spacecraft travel in and out of the Earth’s magnetosphere. Generally, in the solarwind the plasma velocity is greater than 200 km/s, and the magnetic field magnitude is ofthe order of several nT . When the spacecraft fly into the Earth’s magnetosphere, however,the plasma velocity decreases rapidly, and the magnetic field magnitude increases to severalhundred nT . Furthermore, both the plasma number density and the proton temperaturealso show typically different values for different cases, namely, in the solar wind and in themagnetosphere.To illustrate the difference in the measurements mentioned above, we take Figure1 as an example. Figure 1 shows the time series of plasma data measured by FluxgateMagnetometer (FGM) and Cluster Ion Spectrometer experiment (CIS) onboard satelliteCluster 1 during the period January 19-31, 2004. In this time period, the satellitetraveled in and out of the Earth’s magnetosphere about 5 times. We can see the relativelyregular variations of the plasma parameters in Figure 1. When the satellite crosses themagnetopause, the plasma number density rapidly increases due to the accumulation ofparticles there. The values of the proton temperature and the magnetic field magnitudealso increase, while the plasma bulk velocity sharply decreases. When the satellite fliesout of the magnetopause, however, these trends are reversed. Based on these behaviors,we can roughly distinguish the data intervals of the solar wind from those of the Earth’s 12 –magnetosphere. In addition, an automated procedure can be adopted to identify the solarwind intervals. In our investigations, the solar wind shocks and other discrete solar windstructures are not removed from the data, since the time period studied is long enough toneglect the impacts of such solar wind structures.The measurements from spacecraft ACE and Wind yield two-point correlationcoefficients at larger separations, and those from spacecraft Cluster provide the correlationcoefficients at smaller separations. For each pair of the spacecraft, we linearly interpolatedthe data to 1 min resolution to simultaneously obtain the field vectors at different spatialpositions, since the sampling rate varied from spacecraft to spacecraft. In order to obtainmeaningful two-point correlation coefficients at larger separations, longer continuousintervals are required for our analysis. Therefore, the ACE-Wind data are investigated witha cadence of 1 min, and the individual correlation estimates are calculated by averagingover contiguous 24 hr periods of data. For Cluster data, the correlation analysis is carriedout with 2 hr sampling.The data used in this study were measured during the time period 2001-2017 thatcovers over an entire solar cycle. The spacecraft can provide us thousands of time intervalsfor studying the effects of solar activity on the correlation scale and the Taylor scale. Theentire data set is divided into a series of 3-year time periods. In each 3-year period, thedata intervals are randomly selected. For each data interval, we calculate the time-averagedtwo-point correlation coefficients of the magnetic field vector. The correlation value isassigned to the time-averaged spacecraft distance in the corresponding interval. Usingthe normalized two-point correlation values calculated from a large number of solar windmeasurements in different divided time periods, we can obtain the two-dimensional,normalized correlation coefficients as functions of the spatial separations and the timeranges. For example, Figure 2 shows the estimates of solar wind correlation coefficients 13 – R ( r ) versus spacecraft separations from ACE-Wind data intervals (left) and Clusterdata intervals (right) during the years 2001-2003. In the left panel of Figure 2, a meancorrelation function with the form R ( r ) ∼ e − r/L is obtained by fitting to the data ofACE-Wind correlation coefficients. Using the definition of the correlation scale L givenby Matthaeus et al. (2005), i.e., R ( r ) = e − = 0 . L can be estimated to be 219 Re during the time period 2001-2003. We canuse the so-called Richardson extrapolation technique (see Weygand et al. (2007) for details)to calculate the Taylor scale. Using the normalized correlation coefficients from Clusterdata in the right panel of Figure 2, we can obtain that the Taylor scale is 4311 . km during2001-2003. Note that the data intervals are selected with a random procedure. Therefore,the correlation scale and the Taylor scale may slightly change their values when we repeatthe calculations of them in the same divided time period. In this work, we use the averagedvalues of these repeated calculations for the correlation scale and the Taylor scale in eachtime period (2001-2003, 2002-2004, . . . , 2014-2016, 2015-2017).We can employ the values in different time periods to investigate the variation trendsof the correlation scale and the Taylor scale. In this work, the sunspot number is chosen tobe the indicator of the solar activity. As we know, the number of sunspots varies with an11-year period, which is called the solar cycle (Parker 1979; Hathaway 2010). Generally,more sunspots indicate that more masses and energies would be released into interplanetaryspace through solar burst activities and events. By means of the data of the sunspotnumber, the correlation scale, and the Taylor scale, we can investigate the effects of thesolar activity on the structure of solar wind turbulence. 14 –
4. Results and Discussion
Figure 3 displays the evolution of the sunspot number and the correlation scale duringthe time period 2001-2017. The left and right ordinates denote the sunspot number and thecorrelation scale, respectively. Obviously, the variation of the sunspots shows a regular andperiodic trend. As we can see, the correlation scale also shows a weak periodic variationtrend. When the sunspot number is large, the correlation scale becomes large; while whenthe sunspot number is small, the correlation scale becomes relatively small as well. Forexample, during the time periods 2001-2004 and 2011-2014, the sunspot number and thecorrelation scale are larger than those during the time periods 2005-2010 and 2014-2017.As shown in Figure 3, the maximum and the minimum of the correlation scale are 211.6 Reand 152.7 Re, respectively. The averaged value of the correlation scale for all time periodsis 178.12 Re, which is similar to the value 186 Re given in Matthaeus et al. (2005). Thecorrelation coefficient between the sunspot number and the correlation scale is 0.56, whichsuggests a moderate positive correlation between the solar activity and the correlation scale.That is to say, the correlation scale of the solar wind turbulence is modulated by the solaractivity to some extent, but not significantly.Figure 4 depicts the evolution features of the sunspot number and the Taylor scaleduring the time period 2001-2017. The left and right ordinates denote the sunspot numberand the Taylor scale, respectively. As one can see, relative to the correlation scale, theTaylor scale is more significantly related to the sunspot number and the solar activity.The Taylor scale increases or decreases with the increasing or decreasing sunspot number,respectively. As shown in Figure 4, the maximum and the minimum of the Taylor scale are4345.5 km and 1224.5 km, respectively. The averaged value of the Taylor scale for all timeperiods is 2459.3 km (0.39 Re), which agrees well with the value 0 . ± .
11 Re given inMatthaeus et al. (2005) and the value 2400 ±
100 km presented in Weygand et al. (2007). 15 –The correlation coefficient between the sunspot number and the Taylor scale is 0.92, whichindicates a strong positive correlation between the solar activity and the Taylor scale.The high value of the correlation coefficient means that the Taylor scale is significantlymodulated by the solar activity.Based on the Equation (6), we can obtain the form of the effective magnetic Reynoldsnumber R effm as R effm = (cid:18) Lλ (cid:19) . (9)Figure 5 presents the evolution features of the sunspot number and the effective magneticReynolds number calculated with Equation (9) during the time period 2001-2017. Theleft and right ordinates denote the sunspot number and the effective magnetic Reynoldsnumber, respectively. As mentioned above, relative to the correlation scale, the Taylor scaleshows a stronger positive correlation with the sunspot number. The effective magneticReynolds number shows a negative correlation with the sunspot number. The correlationcoefficient between the sunspot number and the effective magnetic Reynolds number is-0.82, which indicates that the turbulence is relatively weak during the time period ofstrong solar activity. This result is somewhat counterintuitive.Generally, the energy output from the Sun varies with the solar activity. The solaractivities include solar flares, coronal mass ejections (CMEs), extreme ultraviolet emissions,and x-ray emissions (Hathaway 2010). Based on the magnetic field data from the spacecraftACE and Wind, we have found that both the magnitude and the standard deviation ofthe magnetic fields increase during the rise phase of the solar cycle, and decrease duringthe declining phase of the solar cycle. Therefore, the magnetic energy B increases withthe increasing solar activity, and decreases with the decreasing solar activity. Taking intoaccount the Equation (4) and replacing the energy with the magnetic energy, we can knowthat if the correlation scale L does not change significantly, the energy dissipation rate ε
16 –will increase with the increasing magnetic energy B during the rise phase of the solar cycle.Combining this result with the Equation (5), we can further derive that in the traditionaltheory of hydrodynamic turbulence, the Taylor scale λ will decrease with the increasing ε during the rise phase of the solar cycle, and the λ will increase with the decreasing ε duringthe declining phase of the solar cycle. Therefore, according to the traditional theory ofhydrodynamic turbulence, there should be negative correlation between the solar activityand the Taylor scale. However, our results show that there is strong positive correlationbetween them.This counterintuitive finding is somewhat identical with the results presented bySmith et al. (2006) and Matthaeus et al. (2008). In the previous studies, the authorsemployed the data sets involving intervals from the magnetic cloud and noncloud situationsin the solar wind to investigate the spectral properties in the dissipation range. Theyshowed that the spectral form in the dissipation range is not consistent with the predictionsof the hydrodynamic turbulence and its MHD counterparts. For instance, the Taylor scaleis usually larger than the Kolmogorov microscale for the hydrodynamic turbulence withlarge Reynolds number. However, Matthaeus et al. (2008) suggested that under severalconditions, the Taylor scale is smaller than the Kolmogorov microscale even if the magneticReynolds number is large enough. Therefore, the plasma dissipation function is not ofthe familiar viscous-resistive Laplacian form. They suggested that the steeper gradient ofthe dissipation range spectrum is associated with the stronger energy cascade rate. Forweaker cascade rate, the gradient of the dissipation range spectrum is gentler. Here, thesteep dissipation range spectrum indicates that the Taylor scale is large. On the contrary,the gentle dissipation range spectrum means that the Taylor scale is small. This result issimilar to our finding that there exists positive correlation between the Taylor scale andthe energy dissipation rate. This finding highlights the non-hydrodynamic properties of thedissipation process in the solar wind. One possible explanation is that in the solar wind, the 17 –assumption of the fluid approximation fails at the spatial scales near the dissipation range.Therefore, the traditional hydrodynamic turbulence theory is incomplete for describingthe physical nature of the solar wind turbulence, especially at the spatial scales near thekinetic dissipation scales where the particle effects are not negligible. In the solar wind,the dissipation process of the turbulence always results from the breakdown of the fluidapproximation and the domination of the kinetic particle effects such as cyclotron andLandau damping (Smith et al. 2006). Therefore, the dissipation process in the solar windrepresents the coupling of the turbulent fluid cascade and the kinetic dissipation.As the cornerstone assumption of the classical hydrodynamic turbulence, the viscousdissipation rate of energy can be estimated by the supply rate of energy at large-scales.This assumption is described by Equation (4), and can be modified as: ε + ξ = U /L. (10)Here ε is the energy dissipation rate at small scales, U /L denotes the energy cascade rate,and ξ denotes the energy loss occurring when the energy transfers from large scales to smallscales especially near the spatial scales at which the solar wind fluid approximation fails.Equation (10) indicates that the sum of the energy dissipation rate and the total energyloss rate equals to the energy transfer rate. As shown above, the Taylor scale is stronglypositively correlated with the energy cascade rate. Combining this result with Equation(5), we can infer that the Taylor scale λ will increase with the increasing magnetic energy U , which indicates that the energy dissipation rate ε is relatively stable in the solar windturbulence. Therefore, the total energy loss ξ will increase with the increasing energycascade rate U /L . Similarly, the total energy loss ξ will decrease with the decreasingenergy cascade rate U /L . This finding sheds new light on the relationship between theenergy cascade and the dissipation in the low-collisionality plasma turbulence. 18 –
5. Conclusions
In this work, based on the simultaneous measurements (Wind, ACE, and Cluster)of the interplanetary magnetic fields during the time period 2001/01-2017/12, we use thetwo-point, single time correlation function to determine the fundamental parameters of thesolar wind turbulence, such as the correlation scale and the Taylor scale. The data set usedin this study covers an entire solar cycle. It is possible to employ this data set accumulatedover a long time period to study the effects of solar activity on the correlation scale and theTaylor scale. We show that the correlation coefficient between the sunspot number and thecorrelation scale is 0.56, and the correlation coefficient between the sunspot number and theTaylor scale is 0.92. Obviously, the relationship between the Taylor scale and the sunspotnumber is more significant than the relationship between the correlation scale and thesunspot number. Therefore, the effective magnetic Reynolds number is primarily affected bythe Taylor scale. The correlation coefficient between the sunspot number and the effectivemagnetic Reynolds number is -0.82, which indicates that the solar wind turbulence isrelatively weak when the solar activity is strong. This result is somewhat counterintuitive.In traditional theory of hydrodynamic turbulence, the dissipation range or the inertialrange can be described by a universal function. The dissipation scale is determined by theenergy cascade rate through the inertial range. Specifically, the stronger cascades generatethe smaller dissipation scales. However, our results suggest that the form of the dissipationprocess in solar wind turbulence is not consistent with the predictions of the hydrodynamicturbulence and its immediate MHD counterparts. Using the solar wind data measured at 1AU, we have shown that the variation of the Taylor scale depends on the energy cascade ratein a manner different from the traditional hydrodynamic case. The Taylor scale increaseswith the increasing sunspot number, and decreases with the decreasing sunspot number.This indicates that the Taylor scale is positively correlated with the energy cascade rate. 19 –One possible explanation is that in the solar wind, the fluid approximation failsat the spatial scales near the dissipation range. Therefore, the traditional theory ofhydrodynamic turbulence is incomplete for describing the physical nature of solar windturbulence, especially at the spatial scales near the kinetic dissipation scale where theparticle effects are not negligible. The dissipation process in the MHD turbulence resultsfrom the breakdown of the fluid approximation and the domination of the kinetic particleeffects such as cyclotron and Landau damping. Therefore, the dissipation process in thesolar wind represents the coupling of the turbulent fluid cascade and the kinetic dissipation.We suggest that the energy dissipation rate ε is relatively stable in solar wind turbulence.The energy cascade rate U /L is positively correlated with the total energy loss ξ .The results presented in this work suggest that solar wind turbulence is influenced bythe solar activity accompanying the solar cycle. In addition, our investigations highlight thenon-hydrodynamic properties of the dissipation process in the solar wind, which providesnew perspectives on the relationship between the energy cascade and the dissipation in thelow-collisionality plasma turbulence. The anisotropy of the solar wind turbulence is anotherimportant subject in the field. In the future work, we will investigate the effects of the solaractivities and the solar cycle on the anisotropy of the solar wind turbulence.This work was supported in part by the B-type Strategic Priority Program of theChinese Academy of Sciences under grant XDB41000000, the National Natural ScienceFoundation of China under grants 41874207, 41621063, 41474154, and 41204130, and theChinese Academy of Sciences under grant KZZD-EW-01-2. H.-Q.H. gratefully acknowledgesthe partial support of the Youth Innovation Promotion Association of the Chinese Academyof Sciences (No. 2017091). We benefited from the data of ACE, Wind, and Cluster providedby NASA/Space Physics Data Facility (SPDF)/CDAWeb. The sunspot data were providedby the World Data Center SILSO, Royal Observatory of Belgium, Brussels. 20 – REFERENCES
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Separation (Re) -0 .4-0.200.20.40.60.81 C o rr . C oe ff. Corr. Coeff.Fitted Curve Separation (km) -0.500.51 C o rr . C oe ff. Fig. 2.— Estimates of solar wind normalized correlation coefficients R ( r ) versus spacecraftseparations from ACE-Wind data intervals (left) and Cluster data intervals (right) duringthe years 2001-2003. The correlation coefficient for the magnetic field vectors decreases withthe increasing spacecraft separation. Fitting to the ACE-Wind data (solid curve) in the leftpanel gives the correlation scale L = 219 Re . 26 – Sunspot Number Correlation Scale (Re)Correlation coefficient r=0.56
Year
Sunspot Number
Correlation Scale (Re)
Fig. 3.— Evolution features of sunspot number (red) and correlation scale (blue) duringtime period 2001-2017. The left and right ordinates denote the sunspot number and thecorrelation scale (Re), respectively. The correlation coefficient between the sunspot numberand the correlation scale is 0.56. 27 –
Sunspot Number Taylor Scale (km)Correlation coefficient r=0.92
Year
Sunspot Number
Taylor Scale (km)
Fig. 4.— Evolution features of sunspot number (red) and Taylor scale (blue) during timeperiod 2001-2017. The left and right ordinates denote the sunspot number and the Taylorscale (km), respectively. The correlation coefficient between the sunspot number and theTaylor scale is 0.92. 28 –
Sunspot Number Effective Magnetic Reynolds NumberCorrelation coefficient r=-0.82
Year