Effects of Solar Activity on Taylor Scale and Correlation Scale in Solar Wind Magnetic Fluctuations

The correlation scale and the Taylor scale are evaluated for interplanetary magnetic field fluctuations from two-point, single time correlation function using the Advanced Composition Explorer (ACE), Wind, and Cluster spacecraft data during the time period from 2001 to 2017, which covers over an entire solar cycle. The correlation scale and the Taylor scale are respectively compared with the sunspot number to investigate the effects of solar activity on the structure of the plasma turbulence. Our studies show that the Taylor scale increases with the increasing sunspot number, which indicates that the Taylor scale is positively correlated with the energy cascade rate, and the correlation coefficient between the sunspot number and the Taylor scale is 0.92. However, these results are not consistent with the traditional knowledge in hydrodynamic dissipation theories. One possible explanation is that in the solar wind, the fluid approximation fails at the spatial scales near the dissipation ranges. Therefore, the traditional hydrodynamic turbulence theory is incomplete for describing the physical nature of the solar wind turbulence, especially at the spatial scales near the kinetic dissipation scales.


Introduction
Solar wind turbulence has received considerable attention for several decades within the community (e.g., Kraichnan 1965;Belcher 1971;Matthaeus & Goldstein 1982a,b;Tu & Marsch 1995), and recently there has been an upsurge in interest in this topic. Much of the interest is driven by the fact that the solar wind can provide a perfect natural laboratory for the study of plasma turbulence in space at low-frequency magneto-hydrodynamic (MHD) scales, which is essential for the studies of solar wind generation, plasma heating, energetic particle acceleration, cosmic ray propagation, and space weather (Jokipii 1968a,b;Coleman 1968;Barnes 1979;Matthaeus et al. 1984).
During the past decades, the solar wind fluctuation properties and equations of motion have been studied in great detail (Belcher 1971;Matthaeus & Goldstein 1982a;Tu et al. 1984Tu et al. , 1989. However, even till now, it is still impossible to make accurate quantitative predictions. Most of the earlier studies used the single-spacecraft time-lagged data to infer solar wind spatial properties based on the well-known "frozen-in flow" approximation (Taylor 1938). Provided that the solar wind flow speed V sw is much greater than the local Alfvén speed V A , thus the solar wind fluctuations which pass a detector are convected in a short time compared to all relevant characteristic dynamical timescales, so the time lags ∆t are equivalent to spatial separations ∆tV 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 remains valid is not fully established (Matthaeus et al. 2005;Weygand et al. 2013), and the correct way to establish the spatial structure is to make use of the simultaneous two-point single time 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 measurements of the magnetic fields. By using this technique, both the correlation length scale and the Taylor scale can be determined, and the values of these two scales were given to be 186 Re (Earth radius, 1 Re = 6378 km) and 0.39 ± 0.11 Re, respectively. Weygand et al. (2007) used the same method and obtained the Taylor scale values from the data of magnetic field fluctuations in plasma sheet and solar wind based on the Richardson extrapolation method. They estimated the Taylor scale in the solar wind as 2400 ± 100 km, which agrees with the value given by Matthaeus et al. (2005). Based on the two-point correlation measurement technique, Weygand et al. (2009) also obtained the values of the correlation scale and the Taylor scale by analyzing the magnetic field data measured from the magnetospheric plasma sheet and the solar wind. Their results showed that in the solar wind, the correlation scale along the magnetic field is longer than that along the perpendicular direction, and the Taylor scale is independent of the magnetic field directions.
These conclusions indicated that the turbulence is anisotropic. Later, Matthaeus et al. (2010) provided a method for estimating the two-time correlation function and the associated Eulerian decorrelation timescale in the turbulence. This method can compare the two-point correlation measurements with the single-point measurements at corresponding spatial separations, and can be used for studying the temporal decorrelation of magnetic field fluctuations in space. Recently, by using the direct two-point measurements and the "frozen-in flow" assumption, Chasapis et al. (2017) examined the second-order and fourth-order structure functions of the magnetic turbulence at very small scales in the solar wind. Their analysis extended several familiar statistical results, including the spectral distribution of energy and the scale-dependent kurtosis, down to the spatial scale near 6 km, and also clarified the earlier results.
All these applications of the two-point correlation measurements validate the excellent performance of this technique for studying the structure of the solar wind turbulence. In the current paper we determine the values of correlation and Taylor scales of the interplanetary magnetic field fluctuations based on two-point single time correlation functions throughout a solar cycle to study the effects of solar activity on the structure of the plasma turbulence in the solar wind. This paper is organized as follows. In Section 2, we give a brief description of some basic concepts of the fluid turbulence, which can also be used in the field of plasma turbulence.
In Section 3, we provide a detailed description of the method and the procedure of the two-point measurements. In Section 4, we calculate the Taylor scale and the correlation scale in different time ranges, and discuss how the solar activity influences the structure of the plasma turbulence. Several significant conclusions will be provided in Section 5.

Fundamental Concepts in Fluid Turbulence
A turbulent flow should satisfy the Navier-Stokes equation, which is the momentum evolution of an element of fluid and can be written as Here u is the velocity, which is a fluctuating quantity in time t and space x, ∇ is the gradient with respect to x, ρ is the density, p is the pressure, and ν is the kinematic viscosity. Note that Equation (1) From Equation (2), we can derive that 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 velocity fluctuations.
Since small-scale motions tend to have small time scales, one can assume that these motions are statistically independent of the relatively slow large-scale turbulence. If this assumption makes sense, the small-scale motion should depend only on the rate at which it is supplied with energy by the large-scale motion and on the kinematic viscosity. In accordance 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 in the turbulent flow is proportional to U 2 ; the rate of transfer of energy is proportional to U/L ∼ 1/T , where T denotes the characteristic transfer time; the kinematic viscosity ν is proportional to U · L; and the dissipation rate per unit mass ε, which should be equal to the supply rate, is proportional to U 2 /T ∼ U 3 /L. With these parameters, we can obtain length, time, and velocity scales as η ≡ (ν 3 /ε) 1/4 , τ ≡ (ν/ε) 1/2 , and v ≡ (νε) 1/4 , respectively.
These scales are known as the Kolmogorov microscales of length, time, and velocity. The Reynolds number formed with ν and v is R η = ηv/ν = 1. The value R η = 1 indicates that the small-scale motion is quite viscous, and that the viscous dissipation adjusts itself to the energy supply by adjusting length scales. From the expression we know that the viscous dissipation of energy could be estimated from the large-scale dynamics that does not involve viscosity. Thus, the dissipation can be seen as a passive process in the sense that it proceeds at a rate dominated by the inviscid inertial behavior of the large eddies. The above expression is one of the cornerstone assumptions of the classical hydrodynamic turbulence. However, we should keep in mind that the large eddies only lose a negligible fraction of their energy compared to direct viscous dissipation effects. Supposing that the time scale of the energy decay is L 2 /ν, then the energy loss proceeds at a rate of νU 2 /L 2 , which is very small compared to U 3 /L if the Reynolds number R = UL/ν = U 3 /L νU 2 /L 2 is very large. Kolmogorov (1941a,b) suggested that the small-scale structure of turbulence is always approximately isotropic when the Reynolds number is large enough. This is the well-known local isotropy theory. In isotropic turbulence, the dissipation rate can be simply expressed as where λ denotes the Taylor scale. Combining Equations (4), (5), and the form of the Kolmogorov microscale, we can obtain Comparing Equation (3) with Equation (6), we can see that the Taylor scale is related to the viscous dissipation. Equation (6) also suggests that for hydrodynamic turbulence with large Reynolds number, the Taylor scale is larger than the Kolmogorov microscale.
The length scales L, η and λ mentioned above can characterise the properties of the flow with high Reynolds numbers. The correlation scale L, also known as "outer" scale or "energy containing" scale, is related to the inertial range of the turbulence. This parameter 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 transport of 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 hypothesis and can be associated with the first bend-over point in the power spectrum of the turbulent fluctuations. The Kolmogorov microscale η ("inner" scale or dissipation scale) represents the smallest length scale in the turbulent flow (Jokipii & Hollweg 1970;Tennekes & Lumley 1972) 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-scale energy into heat at very small length scales (note that in the low-collisionality plasmas, this situation is less clear). A standard method for identifying the Kolmogorov microscale is to associate it with the breakpoint at the high wave number end of the inertial range above which 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-point magnetic field correlation function evaluated at zero separation (Tennekes & Lumley 1972;Matthaeus et al. 2005;Weygand et al. 2010Weygand et al. , 2011Chuychai et al. 2014). In contrast to the correlation scale and the Kolmogorov microscale, the Taylor scale does not represent any group of eddy size, but it can characterise the dissipative effects. Moreover, the Taylor scale is of the same order of magnitude as the Kolmogorov microscale. Specifically, the latter is often smaller than the former for hydrodynamic turbulence with large Reynolds numbers.
An essential characteristic of turbulence is the transfer of energy across scales. The energy resides mainly at large scales, but it can be transferred across scales by nonlinear processes, and eventually it arrives at small scales. The dissipation mechanisms at the small scales would limit the transfer, dissipate the fluid motions, and release the heat (Batchelor 1953;Tennekes & Lumley 1972). This is the so-called "cascade" process. When the 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. In previous studies, the cascade process is investigated through spectral analysis or structure function analysis (Matthaeus & Goldstein 1982a,b;Goldstein et al. 1994;Tu & Marsch 1995;Goldstein et al. 1995;Zhou et al. 2004). Many analysis methods describe the inertial range 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 extending from the correlation scale (where the turbulence contains most of the energy) down to the Kolmogorov 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 the properties of the solar wind turbulence since the Taylor scale can be measured relatively easily (Tennekes & Lumley 1972;Weygand et al. 2005).

Methods and Procedures
If the turbulence is homogeneous in space, then the means, variances and correlation values 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 magnetic Here r is the separation of two points x and x + r. For homogeneity, R and B 0 are independent of position x, though they may be weakly dependent on position in reality.
The ... denotes an ensemble average. In homogeneous medium, the ensemble average is equivalent to a suitably chosen time-averaging procedure. For large |r|, the well-behaved turbulence becomes uncorrelated and R → 0.
The direction-averaged correlation scale is defined as (Matthaeus & Goldstein 1982a;Matthaeus et al. 2005) In addition, the Taylor scale can also be associated with the curvature of R(r) at the origin (see Matthaeus et al. (2005) and Weygand et al. (2007) for more details). Strictly speaking the scale defined in Equation (8) is the integral scale which is not necessarily equal to the bendover scale of the spectrum. However, this will not affect the conclusions in this work. The discussions of different scales and their relations to each other can be found in Shalchi (2020). A model correlation function with the correct asymptotic behavior is 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 → 0 for r → ∞.  (Escoubet et al. 1997). Note that the four Cluster spacecraft are not always in the solar wind. 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 to identify the time intervals during which the spacecraft were immersed in the solar wind.  To illustrate the difference in the measurements mentioned above, we take Figure   1 as an example. Figure  We can employ the values in different time periods to investigate the variation trends of the correlation scale and the Taylor scale. In this work, the sunspot number is chosen to be the indicator of the solar activity. As we know, the number of sunspots varies with an 11-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 interplanetary space through solar burst activities and events. By means of the data of the sunspot number, the correlation scale, and the Taylor scale, we can investigate the effects of the solar activity on the structure of solar wind turbulence. That is to say, the correlation scale of the solar wind turbulence is modulated by the solar activity to some extent, but not significantly. Generally, the energy output from the Sun varies with the solar activity. The solar activities include solar flares, coronal mass ejections (CMEs), extreme ultraviolet emissions, and x-ray emissions (Hathaway 2010). Based on the magnetic field data from the spacecraft ACE and Wind, we have found that both the magnitude and the standard deviation of the magnetic fields increase during the rise phase of the solar cycle, and decrease during the declining phase of the solar cycle. Therefore, the magnetic energy B 2 increases with the increasing solar activity, and decreases with the decreasing solar activity. Taking into account the Equation (4) and replacing the energy with the magnetic energy, we can know that if the correlation scale L does not change significantly, the energy dissipation rate ε will increase with the increasing magnetic energy B 2 during the rise phase of the solar cycle.

Results and Discussion
Combining this result with the Equation (5), we can further derive that in the traditional theory 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 during the declining phase of the solar cycle. Therefore, according to the traditional theory of hydrodynamic turbulence, there should be negative correlation between the solar activity and the Taylor scale. However, our results show that there is strong positive correlation between them.
This counterintuitive finding is somewhat identical with the results presented by Smith et al. (2006) and Matthaeus et al. (2008). In the previous studies, the authors employed the data sets involving intervals from the magnetic cloud and noncloud situations in the solar wind to investigate the spectral properties in the dissipation range. They showed that the spectral form in the dissipation range is not consistent with the predictions of the hydrodynamic turbulence and its MHD counterparts. For instance, the Taylor scale is usually larger than the Kolmogorov microscale for the hydrodynamic turbulence with large Reynolds number. However, Matthaeus et al. (2008) suggested that under several conditions, the Taylor scale is smaller than the Kolmogorov microscale even if the magnetic Reynolds number is large enough. Therefore, the plasma dissipation function is not of the familiar viscous-resistive Laplacian form. They suggested that the steeper gradient of the dissipation range spectrum is associated with the stronger energy cascade rate. For weaker cascade rate, the gradient of the dissipation range spectrum is gentler. Here, the steep 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 is similar to our finding that there exists positive correlation between the Taylor scale and the energy dissipation rate. This finding highlights the non-hydrodynamic properties of the dissipation process in the solar wind. One possible explanation is that in the solar wind, the assumption of the fluid approximation fails at the spatial scales near the dissipation range.
Therefore, the traditional hydrodynamic turbulence theory is incomplete for describing the physical nature of the solar wind turbulence, especially at the spatial scales near the kinetic 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 fluid approximation and the domination of the kinetic particle effects such as cyclotron and Landau damping (Smith et al. 2006). Therefore, the dissipation process in the solar wind represents the coupling of the turbulent fluid cascade and the kinetic dissipation.
As the cornerstone assumption of the classical hydrodynamic turbulence, the viscous dissipation 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: Here ε is the energy dissipation rate at small scales, U 3 /L denotes the energy cascade rate, and ξ denotes the energy loss occurring when the energy transfers from large scales to small scales 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 energy loss rate equals to the energy transfer rate. As shown above, the Taylor scale is strongly positively 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 2 , which indicates that the energy dissipation rate ε is relatively stable in the solar wind turbulence. Therefore, the total energy loss ξ will increase with the increasing energy cascade rate U 3 /L. Similarly, the total energy loss ξ will decrease with the decreasing energy cascade rate U 3 /L. This finding sheds new light on the relationship between the energy cascade and the dissipation in the low-collisionality plasma turbulence.

Conclusions
In this work, based on the simultaneous measurements ( In traditional theory of hydrodynamic turbulence, the dissipation range or the inertial range can be described by a universal function. The dissipation scale is determined by the energy cascade rate through the inertial range. Specifically, the stronger cascades generate the smaller dissipation scales. However, our results suggest that the form of the dissipation process in solar wind turbulence is not consistent with the predictions of the hydrodynamic turbulence and its immediate MHD counterparts. Using the solar wind data measured at 1 AU, we have shown that the variation of the Taylor scale depends on the energy cascade rate in a manner different from the traditional hydrodynamic case. The Taylor scale increases with 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.
One possible explanation is that in the solar wind, the fluid approximation fails at the spatial scales near the dissipation range. Therefore, the traditional theory of hydrodynamic turbulence is incomplete for describing the physical nature of solar wind turbulence, especially at the spatial scales near the kinetic dissipation scale where the particle effects are not negligible. The dissipation process in the MHD turbulence results from the breakdown of the fluid approximation and the domination of the kinetic particle effects such as cyclotron and Landau damping. Therefore, the dissipation process in the solar 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 3 /L is positively correlated with the total energy loss ξ.