Correlation properties of magnetosheath magnetic field fluctuations

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 114,, doi: /2009ja014173, 2009 Correlation properties of magnetosheath magnetic field fluctuations O. Gutynska, 1 J. Šafránková, 1 and Z. Němeček 1 Received 18 February 2009; revised 31 March 2009; accepted 11 May 2009; published 12 August [1] The magnetosheath is characterized by a variety of low-frequency fluctuations, but their features and sources are different. Taking advantage of multipoint magnetic field measurements of the Cluster spacecraft, we present a statistical study to reveal properties of waves. We compute cross-correlation coefficients of magnetic field strengths as measured by pairs of the Cluster spacecraft and determine the correlation length of magnetosheath waves. We discuss the relationship between the correlation length and upstream parameters as well as its connection with the wave mode and frequency power spectrum. In the frequency interval of Hz, we found that the correlation length of fluctuations increases (1) through intervals of high solar wind speeds, (2) with increasing interplanetary magnetic field (IMF) strength, (3) if the cross-correlation between the IMF and magnetosheath magnetic field is higher, and (4) if the amplitude of fluctuations is larger. Citation: Gutynska, O., J. Šafránková, and Z. Němeček (2009), Correlation properties of magnetosheath magnetic field fluctuations, J. Geophys. Res., 114,, doi: /2009ja Introduction 1 Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic. Copyright 2009 by the American Geophysical Union /09/2009JA [2] The interaction of the supermagnetosonic solar wind flow with the Earth s magnetic field forms a magnetospheric cavity bounded by the magnetopause and a bow shock. The layer in between these two surfaces, known as the magnetosheath, enables the incident solar wind flow, reduced to submagnetosonic speeds by the shock, to be diverted around the magnetosphere. The magnetosheath is a region that contains a variety of wave modes which carry information and redistribute energy and momentum from the bow shock to the magnetopause and vice versa. Since it is the magnetosheath which delivers solar wind material to the magnetosphere, the extent to which the magnetosheath modifies the solar plasma has important implications for the basic solar-terrestrial interaction. [3] The magnetosheath is typically a high-b, anisotropic environment. Particle reflection at the bow shock and the ion foreshock provide upstream sources of turbulence and free energy to drive local instabilities. On the other hand, magnetic field line draping and compression at the magnetopause provide sources of free energy which are able to influence the local plasma and turbulence in the magnetosheath [e.g., Savin et al., 2001, 2002, 2004]. Thus, the result is a turbulent magnetosheath with significant power over a wide range of the low-frequency spectrum (see, e.g., review of Schwartz et al. [1996]). [4] According to, e.g., Schwartz et al. [1996], two wave modes dominate the magnetosheath, and grow there owing to the ion T? > T k anisotropy [Denton, 2000]. The Alfvén/ ion cyclotron (AIC) mode grows under modest b conditions, and is found behind the weaker quasi-perpendicular bow shocks and in the plasma depletion layer. At higher b, for instant, behind strong quasi-perpendicular bow shocks and in the middle magnetosheath, the zero-frequency compressive mirror mode dominates the power spectra at low frequencies. A modified version of the mirror mode, with finite frequency and propagation speed, appears to exist downstream of the slow mode transition close to the subsolar magnetopause. However, several aspects complicate the mode identification: (1) There is usually a mixture, possibly phase coherent, of modes and/or frequencies, rather than an isolated mode. (2) Frequencies are often Doppler shifted by an unknown amount. (3) Wave vectors are often difficult to determine from the one (or few) point measurement available from the spacecraft. (4) The mode eigenstate can depend on the wave vector, plasma b, and temperature anisotropy, as well as on the contributions of multispecies or non-maxwellian kinetic features. The result is that both the background state and fluctuations are not accurately known. [5] Numerous studies have been dedicated to the origin and nature of plasma waves in the magnetosheath and its adjacent regions. Song et al. [1990, 1992a, 1992b] showed evidence of a standing slow mode wave in front of the magnetopause over which higher-frequency mirror modes convected with the magnetosheath flow are superposed. Hubert et al. [1998] suggest that the distance from the bow shock is a key parameter determining the nature of plasma waves. They found compressive and AIC modes from the ramp to the undershoot of an oblique shock, pure AIC waves in the outer magnetosheath, a mixture of AIC and mirror modes close to the shock and in the middle magnetosheath, pure mirror modes in the inner magnetosheath, and distorted mirror modes (observed also by Denton et al. [1995]) close to the magnetopause, while 1of11

2 Lacombe et al. [1992] detected Alfvén and mirror modes in the vicinity of the bow shock. [6] Mirror mode waves were often observed in the magnetosheath of the Earth [e.g., Kaufmann et al., 1970; Tsurutani et al., 1982] but also of planets [e.g., Bavassano- Cattaneo et al., 1998; Volwerk et al., 2008], and in the heliosheath [Burlaga et al., 2006]. In the Earth s magnetosheath, Lucek et al. [1999] observed mirror modes in approximately 30% of magnetosheath passes of Equator S throughout the whole magnetosheath under a variety of upstream solar wind conditions. [7] Tatrallyay and Erdos [2002] identified mirror-type fluctuations in magnetic field data from the ISEE 1/2 spacecraft in different regions of the magnetosheath and concluded that these fluctuations do not always originate near the bow shock but that the source may be somewhere else (e.g., at the magnetopause, inside the magnetosheath, or in localized regions of the bow shock). A detailed four-point Cluster study of mirror type magnetic field fluctuations by Tatrallyay et al. [2008] reveals that these fluctuations decrease in the inner regions of the magnetosheath, indicating some saturation in the growth of the waves when proceeding toward the magnetopause. The results suggest that mirror type fluctuations originate from the compression region downstream of the quasi-perpendicular bow shock and that the growth of the fluctuations cannot be described by linear approximations. [8] Blanco-Cano et al. [2006] performed global hybrid simulations and studied foreshock morphology and its influence on the bow shock and magnetosheath. The authors confirmed earlier expectations that downstream from the shock, the magnetosheath is permeated by a variety of waves that result from the convection of upstream waves and also from local wave generation. The wave characteristics are different in the quasi-parallel and quasiperpendicular parts of the magnetosheath. Similar findings follow from a series of papers by Zastenker et al. [2002], Shevyrev et al. [2003], Shevyrev and Zastenker [2005], Shevyrev et al. [2007], and Němeček et al. [2002]. The authors concluded that in the magnetosheath, the plasma flow and magnetic field are turbulent and that a character of the turbulence is strongly controlled by the q BN angle. Behind quasi-parallel shocks, they observed different types of MHD-wave modes and an increase of variations of the ion flux and magnetic field. On the other hand, behind the quasi-perpendicular bow shock, they observed sometimes mirror-mode waves in the middle of the magnetosheath and near the magnetopause. Moreover, Shevyrev and Zastenker [2005] analyzed power spectra of plasma and magnetic field fluctuations upstream and downstream separately for quasiparallel and quasi-perpendicular bow shock conditions and suggested that these fluctuations are not born in the foreshock region but they are generated at the bow shock itself. On the other hand, Constantinescu et al. [2007] found a high concentration of low-frequency wave sources in the electron foreshock and in the cusp region. [9] Sahraoui et al. [2004] investigated the spectrum of the magnetic fluctuations measured by the Cluster satellites in the magnetosheath and confirm the dominance of the mirror mode for frequencies up to 1.4 Hz. Comparing the experimental characteristics of the identified mirror mode with a prediction of the linear theory, the authors have shown that the predicted maximum growth rate is observed in the frequency range Hz. All the rest of the mirror mode, identified for higher frequencies, is more likely to be a nonlinear extension observed in the satellite frame as a temporal spread due to Doppler shift. [10] Narita and Glassmeier [2005] used magnetic field data from Cluster spacecraft to determine the wave vectors across the magnetosheath. The multipoint measurements allowed for Doppler correction and for the determination of the dispersion relation and the wave mode identification. They found a mixture of ion cyclotron and mirror modes close to the shock, then a region where mirror modes were dominating and finally, close to the magnetopause they found distorted mirror modes. [11] Schafer et al. [2005] identified different magnetosheath wave populations and found a multiplicity of standing structures (mirror modes) convected with the plasma flow and a large number of Alfvénic waves. The results confirm previous magnetosheath wave studies [e.g., Denton, 2000] but the authors also discuss a small number of mirror mode-like waves that have propagation speeds up to the local Alfvén velocity, quasi-perpendicular to the magnetic field. [12] The direction of propagation of low-frequency waves (drift mirror and mirror mode waves) in the magnetosheath was studied by Narita and Glassmeier [2006], who found that the antisunward propagation (in the plasma frame) dominates. Narita et al. [2006] found an organization in the wave propagation pattern: the upstream waves propagate toward upstream, while the magnetosheath waves propagate toward the flank region and toward the magnetopause, thus it is concluded that wave properties are different between the upstream and the downstream regions but they are similar between the quasi-parallel and the quasi-perpendicular shock regimes, suggesting that the upstream waves are not transmitted to the downstream region across the shock and that the downstream waves do not depend on the shock angle (between the upstream magnetic field and the shock normal direction). [13] As this brief survey has shown, our knowledge is not sufficient to full understanding of magnetosheath wave modes, their sources and propagation. The present paper contributes to this discussion by a study of correlation length of magnetic field fluctuations. Previous studies found the correlation lengths of magnetosheath ion flux [Gutynska et al., 2007, 2008b] and magnetic field [Gutynska et al., 2008a] fluctuations to be surprisingly low: of the order of 1 R E (in agreement with Lucek et al. [2001], who investigated mirror structures using four-point magnetic field observations by Cluster and showed that these variations occur along the maximum variance direction on scales of km). Gutynska et al. [2008a, 2008b] also found that the correlation length increases with an increasing correlation between the magnetosheath and interplanetary magnetic fields (IMF). However, these papers do not quantify mentioned dependences. [14] On the basis of the Cluster measurements, we continue these magnetosheath studies and perform a statistical investigations of properties of observed fluctuations in the frequency range of Hz. We chose the magnetic field strength because magnetic field measurements are available from all Cluster spacecraft, however, this prefer- 2of11

3 Figure 1. Cross-correlation coefficients of the magnetic field strength measured in two magnetosheath points as a function of their separation. The black line shows the exponential fit, and its parameters (including c 2 = 0.052) are given in the top right corner. The different colors distinguish the groups of measurements analyzed in section 6. ence implies that the analysis is limited to compressible fluctuations. Throughout the paper, we discuss the influence of upstream conditions (the solar wind speed, density, IMF strength, IMF B Z component, distance of a particular magnetosheath point to the magnetopause, etc.) as well as the inner state of the magnetosheath itself on the correlation length. For a rough classification of wave modes, we applied cross-correlation coefficients between the magnetic field strength and density, and the form (slope) of the frequency power spectrum of magnetic fluctuations. 2. Data Selection and Processing [15] Here, we will describe only the basic features of the analyzed data set because the procedure of data selection is broadly discussed by Gutynska et al. [2008a]. The data were collected by four Cluster spacecraft in course of May July and November December 2002 and The reason for such selection was that the spacecraft spent usually several hours at the magnetosheath continuously. Moreover, we have chosen a controversial region in the vicinity of the terminator where the magnetosheath flow is influenced by a presence of the cusp; our set includes both low and high latitudes. Precisely, the measurements cover the magnetosheath in the vicinity of the dawn-dusk meridian ( 7 < X GSE <+7R E ) on both flanks in a broad range of latitudes ( 10 < Z GSE < +10 R E ). We should point out an excellent coverage of the magnetopause region, whereas the bow shock region was visited not so often owing to a low apogee of Cluster. [16] Magnetosheath parts of orbits were broken into 30-min intervals of magnetic field strength measurements and all of them were visually inspected in order to discard those with data gaps or boundary crossings. Altogether we have selected 1080 of those intervals. For each of the selected intervals, we have calculated (1) the maximum of the cross-correlation coefficient between pairs of the spacecraft on the interval of a 20-min duration centered around zero lag (6 values for each interval), (2) averaged upstream parameters (ACE magnetic field [Smith et al., 1998] and plasma [Gold et al., 1998] data lagged on the propagation time are used), and (3) the location of a particular spacecraft with respect to the model magnetopause [Petrinec and Russell, 1996]. The power of the fluctuations is described by a standard deviation and their character by a slope of the frequency spectrum on the interval Hz and by the cross-correlation coefficient between the magnetic field strength and ion density. The analysis uses the spin averaged (4 s) magnetic field strength from four Cluster spacecraft (C1 C4) [Balogh et al., 2001] and the ion density (with 4 s time resolution) from spacecraft C1, C3, and C4 [Reme et al., 2001]. [17] The main motivation of the present study is to answer the question how far the magnetosheath monitor can be from the magnetopause to be sure that its measurements can be reliably used for a study of magnetopause processes. Since the most appropriate measure for this purpose is the correlation length, we have calculated it for given ranges of the analyzed quantity and plotted it as a function of this quantity. Our definition of the correlation length is illustrated in Figure 1 where cross-correlation (CC) coefficients calculated from magnetic field strengths measured by Cluster pairs are plotted as a function of their spatial separation. These data are fitted with an exponential fit: CC ¼ exp ðss=clþ ð1þ where SS stands for the spacecraft separation and CL for the correlation length. The cross-correlation function, CC, was computed according to the formula: PN 1 1 N 1 x i PN 1 x k N y i PN 1 y k N i¼0 k¼0 k¼0 CC ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 PN 1 1 x i PN 1 PN 1 y i PN 1 N 1 i¼0 k¼0 x k N 1 N 1 where x i, y i stands for two N-element magnetic field samples from two Cluster spacecraft. This formula requires the data measured in equidistantly spaced times that are identical for all spacecraft. Since the study is based on spin averaged data and the spin periods of Clusters are different, a unified time scale was applied and the data of all spacecraft were linearly approximated (truncated) to this scale. [18] Although this problem is broadly discussed in section 7, we should point out here that the definition of the correlation length is not easy task. It is partly connected to the fact that the correlation function is defined on infinite time intervals but we are using only 450 data points for calculation and the correlation computed on such short interval cannot fall to zero even for random sequences. Gutynska et al. [2008b] applied linear fits but the comparable results and trends were similar to those obtained from i¼0 k¼0 y k N ð2þ 3of11

4 Figure 2. Projections of measurements onto X Z GSE and Y Z GSE planes. The colors correspond to the groups in Figure 1. Only the measurements with low (red and green points) and high (black and yellow points) cross-correlation coefficients are shown. exponential fits. Since an exponential growth/decay of waves is usually expected, we use the fits according to equation (1). [19] Another method of the data evaluation is based on the fact that analyzed observations are not distributed equally within the range of spacecraft separations but there are two groups of preferential separations: less than 0.1 R E and between 0.4 and 0.8 R E. Within these ranges of separations, we have selected the groups of points with extremely low (red and green points in Figure 1) and high (black and yellow points) correlation coefficients and we analyze them separately in section 6. The groups are marked with different colors in Figure 1 (in which an overview of all cross-correlation coefficients is plotted as a function of s/c separations). The black line shows the exponential fit of the whole data set. The fit provides a value of correlation length of about 0.75 R E. [20] Since a critical issue can be a distribution of analyzed points within the magnetosheath and possible bias connected with it, we are showing the distribution of extreme correlation coefficients in Figure 2. The colors represent different groups of cross-correlation coefficients from Figure 1. One can note that it is not the case because the colors (from Figure 1) are sufficiently dense in all locations. 3. Correlation Length and Amplitude of Magnetosheath Fluctuations [21] In sections 3 and 4, we divide the whole data set into six subsets according to an analyzed parameter. We evaluate the influence of the solar wind speed and density, the IMF strength and IMF B Z component, etc. Criteria for binning into particular subsets were (1) approximately equal number of points in each subset and (2) a sufficiently dense coverage of spacecraft separations in the interval of 0 2 R E. We have computed the correlation length (equation (1)) within each subset. Obtained correlation lengths are then plotted as a function of the analyzed parameter and fitted with linear fits in Figures 3 10, which do not show error bars, but since the points are averages from individual values, the errors are 0.05 and never exceed [22] Gutynska et al. [2008a] suggested that one of the most important factors influencing the correlation length of magnetosheath magnetic field fluctuations is their amplitude described by the standard deviation (SD) computed on 30-min intervals but this suggestion was not quantified in the paper. Figure 3a shows the fits in six standard deviation bins. Figure 3a can serve as an illustration of our method. The results are plotted in Figure 3b and one can see a clear rising trend of the correlation length from 0.6 R E for SD 2nTupto1.2 R E at SD 6 nt. Since the average magnetosheath magnetic field strength in our data set is between 8 and 20 nt, we can conclude that even if the magnitude of the magnetic field fluctuations is comparable with the mean value; their correlation length only slightly exceeds 1 R E. 4. Correlation Length and Upstream Parameters [23] Conditions for excitation of different wave modes, their growth rates, and their amplitudes would depend on the upstream parameters. However, our results show that the connection between these parameters and the correlation length of magnetosheath fluctuations is surprisingly weak. Figure 4 suggests that the correlation length is a slightly rising function of the solar wind speed. On the other hand, it is not influenced by the upstream density, as it is shown in Figure 5. The correlation length as a function of the IMF strength is plotted in Figure 6 and we can note that longer correlations can be expected if the IMF strength is above average. Since this conclusion is based on one point, we have confirmed it by a plot of the correlation length as a function of the local magnetosheath magnetic field (not shown) that exhibits the same feature. We suggest that its is connected with the processes at the magnetopause because 4of11

5 Figure 4. Correlation length of magnetosheath fluctuations as a function of the upstream speed. for Cluster C1. From a linear dependence, we can note that the values computed for different Clusters agree rather well. It means that the signals from all spacecraft (Clusters and Wind) have a common component. This can be considered as an indirect confirmation of the hypothesis on a penetration of upstream fluctuations into the magnetosheath. The penetrating variations are correlated over longer distances and they increase the correlation length as it can be seen from Figure 9 where the correlation lengths are plotted as a function of cross-correlations between Wind and Cluster C1. Figure 3. (a) Cross-correlation coefficients as a function of the spacecraft separation. The colors stand for bins of the standard deviation (black, SD < 1.7; red, 1.7 < SD < 2.3; yellow, 2.3 < SD < 2.8; green, 2.8 < SD < 3.4; blue, 3.4 < SD < 4.4; purple, 4.4 < SD). (b) Correlation length of magnetosheath fluctuations as a function of their standard deviation. we have found a longer correlation length for large negative values of the IMF B Z component as it is displayed in Figure 7. [24] Furthermore, we have analyzed different combinations of the basic upstream parameters (dynamic pressure, temperature, plasma b, etc.) but none of them orders the correlation lengths better than those shown in Figures 4 7. [25] Since Němeček et al. [2002] suggested that a part of solar wind variations can penetrate from the solar wind into the magnetosheath, we have computed the cross-correlation coefficients between the IMF and magnetosheath magnetic field as a measure of this penetration. The correlation coefficients were computed for all four Clusters using the same upstream monitor (WIND). The values of these coefficients vary from 0 to 0.7 with both mean and median being around about Figure 8 shows the crosscorrelation coefficients computed between a particular Cluster spacecraft (C2 C4) and Wind as a function of those 5. Correlation Length and Downstream Parameters [26] The downstream parameters in a given magnetosheath location are generally considered to be proportional to their upstream values [e.g., Spreiter et al., 1966]. This is probably the reason why plots of the correlation length as functions of the downstream density, magnetic field strength, speed, and ion beta resemble all features of corresponding plots for upstream parameters, thus we are not showing Figure 5. Correlation length of magnetosheath fluctuations versus solar wind density. 5of11

6 Figure 6. Correlation length of magnetosheath fluctuations versus the IMF strength. Figure 8. Cross-correlations between the IMF and magnetosheath magnetic field strength. See text for a detail description. them. However, the proportionality constant is a function of the location in the magnetosheath. We are investigating a narrow magnetosheath slice, so the most important parameter would be a relative location with respect to magnetosheath boundaries that we describe by the radial distance of Cluster 1 from the model [Petrinec and Russell, 1996] magnetopause. The plot of the correlation length versus this distance is shown in Figure 10. Both the magnetopause and bow shock were suggested to be sources of the magnetosheath fluctuations, thus one would expect a better correlation over longer distances near the source. Figure 10 confirms this expectation for the magnetopause but the dependence is only weak. The magnetosheath thickness at the dawn-dusk meridian is about 10 R E and the last point in Figure 10 belongs to the middle of the magnetosheath. An analysis of measurements closer to the bow shock suggests a new rise of the correlation length in this region but we have only a limited number of observations there. [27] The introductory survey points out that the bow shock/foreshock and magnetopause processes can significantly contribute to the variations of the magnetosheath magnetic field. Whereas the bow shock can be expected to be responsible for a dawn-dusk asymmetry of the magnetosheath fluctuations due to preferential IMF orientation along the Parker spiral, magnetopause processes would distinguish high and low latitudes. [28] The magnetosheath observations can be easily broken into dawn and dusk subsets (see Figure 2) and we have calculated the mean parameters within these subsets and put them into Table 1. Table 1 shows that the amplitude of dawnside magnetosheath fluctuations are enhanced by a factor of about 1.2 with respect to the dusk ones. This value is similar to that found by Němeček et al. [2003] for ion flux fluctuations. The averaged cross-correlation between Cluster and WIND is nearly the same at dawn and dusk and we can conclude that the foreshock effects expected in front of the dawn magnetosheath do not destroy incoming solar Figure 7. Correlation length of magnetosheath fluctuations as a function of the IMF B Z component. Figure 9. Correlation length of magnetosheath fluctuations as a function of the cross-correlation coefficient between the IMF and magnetosheath magnetic fields. 6of11

7 Figure 10. Correlation length of magnetosheath fluctuations versus the distance to the magnetopause. wind variations. An inspection of Figure 4 shows that a different level of dawn and dusk fluctuations cannot explain the difference in correlation lengths. The reason is probably connected with different wave characteristics of dawn and dusk fluctuations that we will analyze in section 6. [29] We should point out that a similar analysis of the high- and low-latitude magnetosheaths did not reveal any substantial differences between these regions even if a potential influence of the magnetic field B Z component was taken into account. It implies that the presence of cusp does not influence significantly the correlation length, although the magnitude of fluctuations is often very large in the cusp vicinity [Savin et al., 2002, 2004]. 6. Wave Properties of Magnetosheath Fluctuations [30] Although the analysis of the wave properties is not a main purpose of the present paper, it probably can contribute to the answer on the main question: why the correlation length is so short. We have calculated the frequency power spectra along investigated 30-min magnetosheath intervals. Two examples in Figure 11 show that the spectra are a power law type and that they can be well described by their slope. These slopes vary from 0.3 to 2 with a flat maximum around 1;1 as it can be seen from Figure 12. In Figure 12, two histograms show distributions of spectral slopes for dawn (black) and dusk (gray) events. The distributions are very similar but we can note a very slight excess of events with exceptionally low slope (i.e., with an enhanced content of high-frequency components) on the Figure 11. Two examples of power spectra of magnetic field strength fluctuations and computed spectral slopes in the investigated frequency range ( Hz). dawn flank. On the other hand, slopes around 5/3 are rather exceptional and it suggests that the nonlinear turbulence cascades are a dominant process forming the spectra of magnetosheath fluctuations. [31] A tool for distinguishing among wave modes is cross-correlation coefficient of the magnetic field strength and plasma density. It is close to 1 for slow and mirror wave modes, whereas a value +1 characterizes fast wave modes. Our analysis reveals that the magnetosheath is usually occupied with a mixture in which slow modes dominate because the histogram in Figure 13 peaks for a value of 0.4 for both dawn and dusk subsets with a slight preference for larger negative values at the dawn flank. [32] As a first step, we have compared separately two groups of events with larger than average correlation coefficients (black and yellow points in Figure 1) and we did not find any difference. The same was true for a mutual Table 1. Averaged Parameters of the Standard Deviations, Cross- Correlation Coefficients Between Wind and C1, and Correlation Lengths at Dawn and Dusk Magnetosheaths a SD (nt) Cross-Correlation C-W CL (R E ) Y < Y > a SD, standard deviation; CL, correlation length. Figure 12. Distributions of spectral slopes at dawn (black) and dusk (gray) flanks. 7of11

8 Figure 13. Histograms of cross-correlation coefficients between the magnetic field strength and ion density at dawn (black) and dusk (gray) flanks. comparison of other two groups (red and green in Figure 1). It means that the analyzed properties are conserved on the distances of the order of 1 R E. For this reason, we have merged two groups of magnetosheath observations with larger correlations (black and yellow) and plotted the probability that an event belongs to the merged group in Figures Figure 14 shows this probability as a function of the cross-correlations of the magnetic field and density. The probability reaches unity when the crosscorrelation is close either +1 or 1, i.e., when the magnetosheath fluctuations are dominated by a particular wave mode. However, these cases are rather rare as can be seen in Figure 13. A more important criterion is probably the spectral slope. Figure 15 shows the probability of larger correlations of the magnetic field measured by two spacecraft as a function of the slope. This probability is nearly zero for slopes from 0 to 1 and rapidly increases for steeper slopes reaching unity for slopes of about 2. Figure 15. The probability of observations of larger than average cross-correlation coefficients as a function the spectral slope. Nevertheless, Figure 12 shows that the number of such cases is again very low. [33] The above histograms have shown that the wave modes present at a particular magnetosheath location influence significantly their correlation over moderate distances (up to 0.8 R E ). The conditions for a local excitation of a particular wave mode or for its propagation from a distant source would depend on the background values of magnetosheath parameters. We have tested the probability that a particular observation belongs to the group of highly correlated events as a function of different parameters including the plasma density, magnetic field strength and direction, ion b, etc. but we have found only the magnetic field strength to exhibit a notable influence on this probability (Figure 16). Large values of the correlation coefficients were preferentially (60%) observed for an averaged local magnetic field below 15 nt, whereas this probability falls dawn to 30% above this value. A similar, but no so Figure 14. The probability of observations of larger than average cross-correlation coefficients as a function the cross-correlation between the magnetic field strength and ion density. Figure 16. The probability of observations of larger than average cross-correlation coefficients as a function the magnetosheath magnetic field strength. 8of11

9 Figure 17. A histogram of cross-correlation coefficients of 300 pairs of random sequences consisting of 450 points. distinct change was observed for ion b (larger correlations belong to the high-b regime). We assume that the reason for a better organization of the cross-correlation coefficients by the magnetic field strength alone is connected with difficulties of a determination of a proper temperature in the anisotropic non-maxwellian magnetosheath plasma. 7. Discussion and Conclusion [34] The paper presents a complex study of correlation properties of low-frequency magnetosheath fluctuations and their correlation length. By contrast to our preliminary statistical study [Gutynska et al., 2008a], we have expected an exponential decay of the cross-correlation coefficient with the spacecraft separation and used exponential fits for the determination of the correlation length because this approach is more common. In spite of a large spread of experimental points for very short separations (see Figure 1), our fits start from unity. We have checked the two-parameter fits but this procedure provided nearly the same correlation length as fits with one parameter because the correlation length is mainly influenced by measurements at larger separations. [35] A second problem connected with the determination of the correlation length is addressed to limited time intervals over which the cross-correlation is calculated. We used 30-min intervals that correspond to 450 measuring points. This limitation results in nonzero correlations even for random signals. In order to quantify this effect, we have generated 300 random sequences and processed them by the same way as the real signals. The distribution of cross-correlation coefficients for these sequences peaks at 0.1 as can be seen in Figure 17 and it represents bias in our data processing. In order to account for this error, we checked if a fit using the equation CC ¼ 0; 9 exp ðss=clþþ0:1 ð3þ provides different values of CL. Since the difference in the correlation lengths determined using equation (1) and equation (3) is of the order of 7% only, we are using the definition according to the equation (1) throughout the paper. [36] If we summarize our investigation, we can conclude that the correlation length of magnetosheath magnetic field fluctuations is generally short (0.7 R E ), however, it is longer under specific upstream conditions: (1) during intervals of the high solar wind speeds (Figure 4); and (2) it slightly increases with higher values of the IMF strength (Figure 6). Similarly, the correlation length is longer (3) if the cross-correlation between the IMF and magnetosheath magnetic field is higher (Figures 8 and 9); and (4) if the amplitude of fluctuations represented by a standard deviation (Figure 3) is larger. However, these effects are not necessarily independent because, for example, fluctuations with larger amplitudes can be excited by high-speed solar wind streams. We have tested this hypothesis by plots of one of aforementioned parameters as a function of another and we did not find any notable functional dependence. [37] On the other hand, Figure 10 shows a slight dependence of the correlation length on the distance of a measuring point from the magnetopause; being longer near the magnetopause. From this follows that the magnetopause plays some role in the wave mode propagation and/or excitation [e.g., Attie et al., 2008; Alexandrova et al., 2008]. This role is probably connected with subsolar reconnection because Figure 7 shows a longer correlation length of fluctuations for a strongly negative IMF B Z component. [38] Previous studies of the magnetosheath waves revealed that the mirror mode dominates especially in high-b plasma regions behind the quasi-perpendicular bow shock and central magnetosheath [e.g., Schwartz et al., 1996; Tatrallyay et al., 2008; Shevyrev and Zastenker, 2005; Shevyrev et al., 2007]. Unfortunately, our statistics cannot distinguish slow and mirror modes but, following Figure 13, one can see a clear preference for negative crosscorrelations of the magnetic field strength and plasma Figure 18. The dependence of the frequency spectrum slope on the distance of a measuring point to the magnetopause. 9of11

10 Figure 19. The dependence of the frequency spectrum slope on the solar wind speed. density that are consistent with the mirror mode. Although the values close to 1 are observed rather rarely (Figure 13), such cases usually exhibit large correlation coefficients (Figure 14). These waves are more frequently observed at the dawn flank (Figure 13) and we assume that their presence can result in a longer correlation length of fluctuations at this region (Table 1). However, the low probability of observations of a single wave mode suggests that a mixture of modes is a typical magnetosheath feature [e.g., Schwartz et al., 1996; Sahraoui et al., 2004]. This is probably a reason why the correlation length is so short in a statistical sense. [39] Shevyrev and Zastenker [2005] carried out a case study of the frequency spectra of magnetosheath magnetic field and ion flux fluctuations and they found a significant difference in their spectral slopes. The authors attributed it to the bow shock type (parallel versus perpendicular). We have used the IMF monitor for a determination of the bow shock type upstream of a particular magnetosheath observation. However, the IMF often changed its direction several times during our 30-min intervals; the mean q BN angle that we computed was often in the range of and thus, it could not be used for a reliable determination of the bow shock type. For this reason, we rely on the preferential IMF orientation and compared dawn and dusk flank magnetosheaths. Our statistics (Figure 12) did not fully confirm Shevyrev and Zastenker s [2005] conclusions because we have found a very similar spread of slopes at both magnetosheath flanks (Figure 12). However, it must be noted that their conclusions were based on case studies that used the calculated q BN angle, whereas our conclusion relies on the average IMF orientation. [40] Figure 15 suggests that the cross-correlations are generally larger when their spectral slope is steeper, i.e., when the spectrum is dominated by low frequencies. Our search for conditions favorable for excitation and propagation of such waves have brought almost negative results because we have found only two parameters exhibiting an influence on the spectral slope. First of them is the distance to the magnetopause as it can be seen from Figure 18 where the spectral slope is plotted as a function of this distance. A preference of steeper slopes at the magnetopause vicinity can probably explain a larger correlation length in this region (Figure 10). On the other hand, steeper slopes are observed during intervals of a low solar wind speed as it can be seen from Figure 19. This dependence is rather complicated but an overall trend is clear and out of error bars. In accord with Figure 15, the steeper spectral slopes would result in longer correlation length of magnetosheath fluctuations but Figure 4 shows an opposite trend. Both dependencies are out of the statistical errors, thus a more complex multifactor study is required for an explanation of this contradiction. [41] As a conclusion, we can note that the correlation length of magnetosheath magnetic field fluctuations varies from 0.5 to 1.5 R E. It means that for a reliable determination of the magnetic field at the magnetopause, the monitor should be as close as 1 R E from the investigated magnetopause point. Since Gutynska et al. [2007, 2008b] have found a similar correlation length of the ion flux fluctuations, we think that the same is true for plasma parameters. [42] Acknowledgments. 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