1 INTRODUCTION

One of the characteristic features of turbulence development in plasma systems is turbulent transport, which leads to mixing and equalization of the gradients of the hydrodynamic parameters. Earth’s magnetosphere is a giant plasma laboratory for studying turbulent transport processes in a collisionless plasma at Reynolds numbers exceeding 1010 for Coulomb collisions (Borovsky and Funsten, 2003). In Earth’s magnetotail, various instabilities develop and a turbulent flow regime is established.

A high level of turbulent fluctuations is observed in Earth’s magnetotail, which was noted in early publications (Antonova, 1985; Montgomery, 1987; Angelopoulos et al., 1993, 1999). However, the main focus was on studying particle beams, dipolization of magnetic field lines, and other large-scale phenomena. Consistent study of turbulence in Earth’s magnetotail began with the studies (Borovsky et al., 1997, 1998; Borovsky and Funsten, 2003) based on ISEE-2 satellite data, which focused on magnetic field fluctuations and the plasma velocity. It was shown that the correlation time for velocity fluctuations is ~2 min; for the magnetic field, ~8 min; the correlation length (mixing path length) is ~10 000 km. According to magnetic observations, it was established that turbulence in the plasma sheet has intermittency; i.e., zones with strong fluctuations are spatiotemporally adjacent to quiet zones (Angelopoulos et al., 1999; Vörös et al., 2003; Weygand et al., 2005). The results of (Weygand et al., 2005) showed that the correlation length varies from 4000 to 10 000 km. The relationship between the spectra of magnetic field fluctuations and jet streams bursty bulk flows (BBF) has been studied. It has been established (Angelopoulos et al., 1999; Vörös et al., 2003; Weygand et al., 2005) that in the plasma sheet, turbulence has intermittency, that is, zones with strong fluctuations are adjacent to quiet zones in space and time. Studies of electric field fluctuations in the magnetotail were fraught with certain difficulties and, in fact, began with the launch of a four-satellite Multiscale Magnetosphere Mission (MMS) (Burch et al., 2016; Torbert et al., 2016; Pollock et al., 2016), when it was possible to obtain reliable measurements of three electric field components (see the work by Ovchinnikov et al. (2024) and references therein).

The role of turbulent transport in the dynamics of magnetospheric flows is governed by the eddy diffusion coefficient. The first estimates of this coefficient across the plasma sheet of Earth’s magnetosphere were carried out by Borovsky et al. (1998) based on ISEE-2 satellite data. Measurements on this satellite made it possible to determine plasma velocity fluctuations only in the plane of the plasma sheet (in the direction X, Y of the solar–magnetospheric (SM) coordinate system). Therefore, it was assumed that the level of fluctuations across the sheet coincides with the level of fluctuations along the sheet. The eddy diffusion coefficient was calculated as 2.6 × 105 km2/s. This estimate coincided in order of magnitude with the predictions of the model of a magnetostatically equilibrium turbulent plasma sheet compressed in the Z-direction by the dawn-dusk field (Antonova and Ovchinnikov, 1996; Antonova and Ovchinnikov, 1998). The results of measurements on the Interball/Tail probe satellite (Ermolaev et al., 2000), which determined velocity fluctuations in the direction (Y, Z), confirmed the estimate by Borovsky et al. (1998). During measurements on this satellite, the values of the eddy diffusion coefficient across the sheet were determined during magnetically quiet times and substorms (Ovchinnikov et al., 2000, 2002a, 2002b). Subsequently, the eddy diffusion coefficients were determined from Geotail, Cluster, and THEMIS satellite data (Troshichev et al., 2002; Stepanova et al., 2005, 2009, 2011; Nagata et al., 2008; Wang et al., 2010; Pinto et al., 2011). The studies (Ovchinnikov and Antonova, 2017; Antonova and Stepanova, 2021) review the results obtained. The intermittent nature of plasma sheet turbulence led to eddy diffusion coefficients that differed by more than an order of magnitude (Stepanova et al., 2005, 2009, 2011; Eyelade et al., 2021), which required continued research depending on the solar wind parameters and geomagnetic activity.

The implementation of the MMS project made it possible to determine the characteristics of fluctuations in the plasma sheet parameters with a high reliability and higher time resolution than before. Individual BBF events have been studied in detail in higher resolution mode (see, e.g., (Ergun et al., 2018)). Statistical studies of eddy diffusion coefficients using MMS data have not previously been carried out. In this article, a statistical study of velocity fluctuations has been carried out and the eddy diffusion coefficients are calculated for the project period 2017–2022.

2 MATERIALS AND METHODS

To calculate the eddy diffusion coefficient, we used data from measurements of the hydrodynamic velocity of plasma ions from FPI/DIS instruments of the MMS mission (Pollock et al., 2016) with a time resolution of 1/4.5 s–1. Active fluctuation of the plasma hydrodynamic velocity components in the plasma sheet of the magnetotail was revealed when plotting three-dimensional hodographs of the hydrodynamic velocity. Figure 1 shows an example of the resulting hodograph for the interval 0900–0912 of May 25, 2017.

Fig. 1.
figure 1

Example of a hodograph of plasma velocities in planes xy, xz, yz for interval 0900–0912 UT on May 25, 2017, according to MMS1 data.

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To highlight the time periods when the spacecraft was within the plasma sheet, we used the criterion proposed by Stepanova et al. (2011): the coordinates of the device in the GSM system satisfy the conditions X < −6RE, |Y| < |X|, |Z| < 8RE (where RE is the Earth’s radius), the concentration of plasma ions ni > 0.1 cm–3, the ion temperature Ti > 0.5 keV, and the plasma parameter β > 1, where β is the ratio of the plasma pressure to the magnetic field pressure. Later, it was shown (Antonova et al., 2013, 2014) that measurements at geocentric distances of up to ~10–13RE correspond to the region of the plasma ring surrounding Earth, onto which most of the auroral oval is projected (Antonova et al., 2014, 2015). Below, we verify the validity of this result.

All parameters were averaged over 6-min time intervals. For 2017–2022, 29 000 6-min intervals were identified when at least one MMS mission device was in the plasma sheet and transmitting data. Each of the 6-min intervals was analyzed together with the previous one.

To calculate the eddy diffusion coefficients, the intervals were combined in pairs; i.e., 12-min intervals were used, each containing 160 measured hydrodynamic velocity values.

The eddy diffusion coefficients were estimated from the velocity data in accordance with the methodology (Borovsky et al., 1997, 1998). For the hydrodynamic velocity components of plasma ions Vα, autocorrelation functions were constructed:

$${{A}_{{\alpha \beta }}}\left( \tau \right) = \frac{{\sum \left( {{{V}_{\alpha }}\left( i \right) - \left\langle {{{V}_{\alpha }}} \right\rangle } \right)\left( {{{V}_{\beta }}\left( {i + \tau } \right) - \left\langle {{{V}_{\beta }}} \right\rangle } \right)}}{{\sqrt {\sum {{{\left( {{{V}_{\alpha }}\left( i \right) - \left\langle {{{V}_{\alpha }}} \right\rangle } \right)}}^{2}}} \sqrt {\sum {{{\left( {{{V}_{\beta }}\left( {i + \tau } \right) - \left\langle {{{V}_{\beta }}} \right\rangle } \right)}}^{2}}} }},$$
(1)

where \(V_{{rms,\alpha \beta }}^{2} = \left\langle {\left( {{{V}_{\alpha }}\left( i \right) - \left\langle {{{V}_{\alpha }}} \right\rangle } \right)\left( {{{V}_{\beta }}\left( i \right) - \left\langle {{{V}_{\beta }}} \right\rangle } \right)} \right\rangle \) is the rms velocity and angle brackets denote averaging over all measurements of the selected interval. Indices α, β ∈ {X, Y, Z}. Figure 2 shows examples of the resulting autocorrelation functions. To calculate the autocorrelation time ταβ, the autocorrelation function was approximated by an exponential function using the least squares method Aαβ(τ) = exp(–τ/ταβ). Use of the correlation time procedure following the approach of (Borovsky et al., 1997, 1998) may contain significant errors (see Fig. 2), associated with the intermittency of turbulence. This was taken into account when analyzing the results.

Fig. 2.
figure 2

Examples of autocorrelation functions of plasma velocity components for interval 020–032 UT May 28, 2017 according to MMS1 data: (a) Axx, (b) Ayy, (c) Azz.

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The eddy diffusion coefficients were calculated in accordance with the relation

$${{D}_{{\alpha \beta }}} = {{V_{{rms,\alpha \beta }}^{2}{{\tau }_{{\alpha \beta }}}} \mathord{\left/ {\vphantom {{V_{{rms,\alpha \beta }}^{2}{{\tau }_{{\alpha \beta }}}} {2.}}} \right. \kern-0em} {2.}}$$
(2)

For each 12-min interval, autocorrelation functions (1) were constructed and the autocorrelation times were calculated. In accordance with formula (2), the eddy diffusion components were obtained. For further analysis, only diagonal components Dxx, Dyy, Dzz were used.

The dependence of the eddy diffusion coefficients on the direction of the interplanetary magnetic field was analyzed using measurements of the interplanetary magnetic field in the solar wind at Lagrange point L1 based on data from the OMNI database. Each 12-min interval was added to the sample, provided that throughout the entire interval, the Bz-component of the interplanetary magnetic field (IMF) did not change sign. The eddy diffusion coefficient values for analyzing their dependence on geomagnetic activity where chosen with account for the values of the SuperMAG geomagnetic index SML, calculated similarly to the AL index, but for a larger number of stations. For each 12-min interval, the following conditions were verified: SML > –50 nT for all observed intervals preceding the one under consideration (and including the one under consideration) for 1 h to select intervals with quiet geomagnetic activity; SML < –200 nT for selecting intervals with high geomagnetic activity.

3 RESULTS OF STATISTICAL ANALYSIS

Using the obtained data array for two IMF directions, the distributions of the diagonal components of the eddy diffusion coefficients were constructed as a function of the GSM X- and Y coordinates in the plasma sheet of the magnetosphere; the averaged radial profiles of the diffusion coefficients were determined (Figs. 3, 4), and the average values of the diagonal components of the eddy diffusion coefficient were calculated. The average values for the northern orientation of the IMF were: 6.7 × 104 km2/s, 3.1 × 104 km2/s, 1.1 × 104 km2/s for Dxx, Dyy and Dzz, respectively; for the southern orientation of the IMF, 16.4 × 104 km2/s, 5.9 × 104 km2/s, 1.9 × 104 km2/s for Dxx, Dyy and Dzz, respectively. During averaging, the region of the plasma ring surrounding the Earth was not selected.

Fig. 3.
figure 3

Averaged radial profiles of eddy diffusion coefficients for northern direction of interplanetary magnetic field.

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Fig. 4.
figure 4

Averaged radial profiles of eddy diffusion coefficients for southern direction of interplanetary magnetic field.

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For the selected data sets, the distributions of the diagonal components of the eddy diffusion coefficients were constructed as a function of the GSM X- and Y coordinates in the plasma sheet of Earth’s magnetosphere in a quiet geomagnetic environment for SML > –50 nT and during substorms at SML < –200 nT. The averaged radial profiles of the eddy diffusion coefficients were constructed for quiet times and substorms (Figs. 5, 6). The average components of the eddy diffusion coefficients in a quiet geomagnetic environment were: 5.9 × 104 km2/s, 2.7 × 104 km2/s, 0.9 × 104 km2/s for Dxx, Dyy and Dzz respectively; during substorms, the values were 19.5 × 104 km2/s, 7.6 × 104 km2/s, 2.5 × 104 km2/s for Dxx, Dyy and Dzz, respectively.

Fig. 5.
figure 5

Averaged radial profiles of eddy diffusion coefficients in quiet geomagnetic conditions (SML > –50 nT).

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Fig. 6.
figure 6

Averaged radial profiles of eddy diffusion coefficients in disturbed geomagnetic conditions (SML < –200 nT).

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4 DISCUSSION

The statistical analysis confirmed the permanent existence of high-level plasma velocity fluctuations in Earth’s magnetotail, calculated in the MMS project using the standard method for determining the hydrodynamic parameters of the plasma. It should be noted that the method used for determining the autocorrelation time is not the only possible one (Borovsky et al., 1997) and can lead to underestimation of the calculated eddy diffusion coefficients.

On the whole, as expected, the values of the eddy diffusion coefficients depend both on the direction of the IMF and on geomagnetic activity due to the known statistical dependence of geomagnetic activity on the IMF components.

The results of the statistical analysis of MMS data, in general, confirm the previously obtained patterns and allow us to identify new features. Figures 3 and 4 show that for the southern orientation of the IMF, the average eddy diffusion coefficients are 1.5–2 times greater than for the northern orientation. On average, the eddy diffusion coefficient in the X-direction exceeds the value of the eddy diffusion coefficient in the Y-direction. The values of the eddy diffusion coefficient are minimal cross the plasma sheet. Overall, the average Dxx > Dyy > Dzz. It should be noted that this pattern may not be observed in individual events.

The dependences Dxx > Dyy > Dzz persist for periods of magnetospheric substorms (Fig. 6). During magnetospheric substorms, the eddy diffusion coefficients are several times higher than in quiet times.

The radial profiles of the eddy diffusion components (Figs. 3–6) are characterized by increased values of the coefficients with increasing geocentric distance up to ~14RE, after which plateau is reached. This pattern confirms the conclusions of studies about projection of the auroral oval onto the outer part of the ring current, and not onto the plasma sheet itself, where the turbulence level is constantly high. As is known, at latitudes of the auroral oval in magnetically quiet conditions, nearly stationary vortices can be observed, leading to the formation of inverted-V auroral structures (Antonova and Ovchinnikov, 1998) and stable auroral arcs. In general, the pattern is close to the results of (Stepanova et al., 2009, 2011; Pinto et al., 2011), but was obtained with larger statistics.

5 CONCLUSIONS

The analysis performed using data from the MMS mission confirmed the presence of large fluctuations in plasma velocities in the plasma sheet.

A database was created that made it possible to obtain the first results on the dependence of the eddy diffusion coefficients in the (X, Y, Z) directions on the direction of the IMF and level of geomagnetic activity.

Fluctuations in plasma velocity were analyzed in 12-min intervals in the nighttime sector at X < −6RE, |Y| < |X|, |Z| < 8RE in the region where the plasma parameter exceeds unity, which includes the part of the plasma ring surrounding the Earth and the plasma sheet itself. The values of the diagonal components of the eddy diffusion tensor and their averaged values were obtained.

The dependences of the eddy diffusion tensor components on the IMF direction were studied. It was shown that for a southern orientation of the IMF, the eddy diffusion coefficients are 1.5–2 times greater than for the northern orientation.

The averaged dependences on the level of geomagnetic activity were determined in quiet conditions for SML > –50 nT and in perturbed conditions for SML < –250 nT. It has been established that during magnetic substorms, the eddy diffusion coefficients are several times higher than the values during quiet geomagnetic activity.