[2] The solar wind is a prime example of plasma turbulence at low frequency magnetohydrodynamic scales as evident from its power-law energy distribution [*Zhou et al.*, 2004], magnetic field correlations [*Matthaeus et al.*, 2005] and intermittent dynamics [*Burlaga*, 2001]. Aside from the challenge of understanding the nature of this type of turbulence, no less important are the couplings between the solar wind and the Earth's magnetosphere on the one hand [*Baker*, 2000], and solar activity and the solar wind on the other [*Webb*, 1995].

[3] In trying to identify the features of the couplings in such a complex system, it has proved useful to analyze the intermittent dynamics in terms of “bursts”, which are typically defined as periods of activity above a threshold. Guided by ideas based on self-organized criticality [*Bak*, 1996; *Jensen*, 1998], it was found that different burst statistics frequently follow scaling laws summarizing the underlying statistical self-similarity of the intermittent dynamics [e.g., *Aschwanden et al.*, 2000; *Freeman et al.*, 2000a, 2000b; *Uritsky et al.*, 2001; *Watkins*, 2002; *Baiesi et al.*, 2006; *Uritsky et al.*, 2006; *Wanliss and Weygand*, 2007; *Wanliss and Uritsky*, 2010]. More importantly, the comparison between observed burst size and duration time distributions in solar wind and magnetospheric quantities help elucidate statistical and other properties of the solar wind-magnetosphere interaction [*Freeman et al.*, 2000a; *Uritsky et al.*, 2001; *Watkins*, 2002; *Wanliss and Weygand*, 2007]. In particular, *Wanliss and Weygand* [2007] suggest that the solar wind is unlikely to act as a direct driver for the scaling of the *SYM* − *H* index, a global marker of low-latitude geomagnetic fluctuations.

[4] Recent studies show that there is a clear signature of the coupling to the solar activity in the solar wind [*Kiyani et al.*, 2007; *Chapman et al.*, 2008]. For example, during solar maximum the self-similar behavior of the magnetic energy density *B*^{2} measured *in situ* is related to coronal structure and dynamics rather than to local magnetohydrodynamic turbulence. At solar maximum, solar flares dominate the solar activity. At solar minimum, however, other types of solar activity become more important such that one should rather consider the full-disk EUV/XUV solar irradiance [*Greenhough et al.*, 2003] as the driver of the solar wind. While this suggests that the Akasofu ε parameter [*Perreault and Akasofu*, 1978], which is related to *B*^{2} and also a solar-wind proxy for the power input into the Earth's magnetosphere, should be dominated by the solar flare activity during solar maximum, this does not seem to be consistent with what has been observed for the corresponding burst statistics. In particular, the burst duration or lifetime distribution for the solar wind as quantified by the ε parameter follows a power law at small values with a threshold-dependent exponential cutoff at large values [*Freeman et al.*, 2000a; *Wanliss and Weygand*, 2007], while the same distribution for solar flares crosses over from a power law at large times to a constant regime at short times [*Baiesi et al.*, 2006]. The power-law exponents in the two cases are very different.

[5] Here, we show that many of the qualitative and quantitative differences can be resolved by considering sufficiently high thresholds to define bursts in the ε time series during solar maximum. Not only do we find that the burst duration time distribution becomes threshold independent in this regime, but also that its form is well described by that proposed for solar flares, decaying asymptotically as a power law. As for solar flares, we also find that the number of bursts as a function of threshold decays as a power law in this regime, which in turn can be used to determine the relevant threshold values. Moreover, we show that the distribution of burst sizes follows a pure power law independent of the threshold and with an exponent comparable to the one observed for solar flares [*Aschwanden et al.*, 2000]. In contrast to these universal features, we find that the distribution of times between bursts in the solar wind depends markedly on the threshold, with a characteristic time scale of approximately 8 hours for high thresholds beyond which the distribution develops a shoulder.

[6] The Akasofu ε parameter is defined in SI units as

where *v* is the solar wind velocity, *B* the magnetic field, *μ*_{0} = 4*π* × 10^{−7} the permeability of free space, ℓ_{0} ≈ 7 *R*_{E}, and θ = arctan(∣*B*_{y}∣/*B*_{z}) — see *Koskinen and Tanskanen* [2002] for a recent discussion. Geocentric solar magnetospheric (GSM) coordinates are used. For the years 01.01.2000–31.12.2003, i.e., during the last solar maximum, we extracted the magnitude of the *x*-component of the solar wind, and the *y* and *z* components of the magnetic fields, as seen respectively by the SWEPAM and MAG instruments (level 2 data) of the ACE spacecraft [*Stone et al.*, 1998] (http://cdaweb.gsfc.nasa.gov). For the most part, measurements are available every 64 and 16 s for the wind velocity and magnetic fields, respectively. We calculated the ε parameter every 64 s, giving rise to 1 972 350 points over 4 years. Since the wind velocity and magnetic field measurements are not synchronized, we linearly interpolated the magnetic field measurements towards the time of the nearest wind velocity measurement. Approximately 9 per cent of the points comprising the ε series are missing. Just as *Baiesi et al.* [2006], we set missing data points to the value of the last valid recording (irrespective of the size of the data gap). This minimises artefacts associated with points missing at experiment-specific frequencies.

[7] Bursts in the ε time series are defined as periods during which the values remain above some threshold, as illustrated in the inset of Figure 1. The duration of a burst, *t*_{d}, is the time between a threshold upcrossing and the next threshold downcrossing. Two successive upwards crossings of a threshold give the waiting time, *t*_{w}, between subsequent bursts. Finally, the time spent below threshold is denoted by the quiet time, *t*_{q}, i.e., the time between a threshold downcrossing and the next upcrossing. Thus, *t*_{w} = *t*_{d} + *t*_{q}.

[8] The number of bursts depends on threshold. Figure 1 shows that the variation in the number of bursts is rather small for a wide range of thresholds, but a power law decay sets in beyond ε ≈ 5 × 10^{9}*W*, corresponding to the *q* = 0.9 quantile. This behavior is qualitatively identical to that seen in solar flares [*Baiesi et al.*, 2006], and provides a first explanation for the observed differences in the burst statistics: While previous studies of the solar wind restricted themselves to quantiles in the range *q* = 0.1–0.9 [*Freeman et al.*, 2000a; *Wanliss and Weygand*, 2007], thresholds exclusively in the self-similar regime were considered in the context of solar flares [*Baiesi et al.*, 2006]. Here, we focus on higher quantiles in the self-similar regime. A list of the chosen thresholds is given in Table 1.

Quantile | Threshold | μ | t_{0} | ν | E_{min} | ρ | Symbol |
---|---|---|---|---|---|---|---|

- a
The duration time distribution is fitted with equation (2) using parameters *μ*and*t*_{0}[64 s]. The burst energy distribution is fitted with a decaying power law with exponent*ν*beyond*E*_{min}[10^{11}J]. The burst energy as a function of duration is fitted with a power law with exponent*ρ*. Maximum likelihood estimation is employed for*μ*,*t*_{0}and*ν*, while*ρ*is determined by least-squares fitting.
| |||||||

0.95 | 0.09 | 2.33(5) | 1.5(1) | 1.59(7) | 55.2 | 2.46(2) | plus |

0.97 | 0.13 | 2.31(6) | 1.5(1) | 1.6(1) | 79.6 | 2.46(3) | cross |

0.99 | 0.24 | 2.3(1) | 1.5(3) | 1.53(4) | 12.2 | 2.38(4) | square |

0.993 | 0.30 | 2.4(1) | 2.1(4) | 1.50(6) | 13.0 | 2.39(4) | circle |

0.995 | 0.37 | 2.4(2) | 1.9(5) | 1.48(4) | 4.8 | 2.38(5) | triangle |

[9] Figure 2a displays the probability density functions (PDF) for the duration time, *p*(*t*_{d}). Logarithmic binning is applied to the tails where statistics are poorer. The PDFs appear to be insensitive to the choice of threshold for such high thresholds. Qualitatively similar results have been observed for solar flares [*Baiesi et al.*, 2006], with fitting function:

In Figure 2a, the best fit using maximum likelihood estimation of the parameters *μ* and *t*_{0} is plotted for the curve corresponding to the *q* = 0.993 quantile. A maximum likelihood function reflecting the inherent discreteness of the duration times is required [*Edwards et al.*, 2007]. The *p*-value of the fit is 0.45 using a G test [*Terrell*, 1999] (a *χ*^{2} test gives a similar *p*-value). Table 1 gives the maximum likelihood estimates of the parameters *μ* and *t*_{0} for all thresholds considered.

[10] It is illuminating to consider not just the duration of an event, but also its size, namely the area above threshold, which in this case corresponds to the total energy *E* of the event. Figure 2b shows that the PDFs of the burst energies, *p*(*E*), are roughly independent of the threshold and decay as a power law over many decades

with decay exponent ν ≈ 1.5. See Table 1 for maximum likelihood estimates using Otte's implementation (http://tuvalu.santafe.edu/∼aaronc/powerlaws/) of the method described by *Clauset et al.* [2009]. This method gives a *p*-value of 0.12 for a power-law fit to the *q* = 0.99 quantile data. A power law decay is also seen in the energy distribution of solar flares [*Aschwanden et al.*, 2000], with exponent 1.8.

[11] In order to determine whether there is any systematic dependence between the size of an event and its duration, in Figure 3 we plot the scatter of event sizes for a given duration, together with their averages, for the *q* = 0.95 quantile. To a good approximation, the dependence is power-law, *E* ∼ *t _{d}^{ρ}*, with

*ρ*= 2.46(2). In fact, for truly scale-invariant statistics, only two of the three exponents

*μ*,

*ν*,

*ρ*are independent. The transformation of probability densities

gives rise to the scaling relation

with *p*(*t*_{d}) assumed to decay as a pure power law with exponent *μ*. Plugging in the estimated exponents for the *q* = 0.95 quantile, we find *μ*(*ρ*, *ν*) = 2.5(2), where the uncertainty is estimated using standard error propagation. This is in good agreement with *μ* = 2.4(1), estimated by a maximum likelihood fit to a pure power-law decay beyond *t*_{d;min} = 14[64 s]. A successful agreement between exponents was also obtained for low-latitude geomagnetic bursts [*Wanliss and Uritsky*, 2010].

[12] While *p*(*E*) and *p*(*t*_{d}) are roughly insensitive to the choice of threshold in the power-law regime of event numbers and, thus, apparently universal, the same is not true for the PDFs of the quiet times *t*_{q} and waiting times *t*_{w}. As Figure 4 shows, *p*(*t*_{w}) is approximately flat for small *t*_{w} and then decays as a power law for intermediate *t*_{w} up to a characteristic time *t*_{c} ≈ 5 × 10^{2} minutes. For *t* > *t*_{c}, the functional form of *p*(*t*_{w}) is non-universal and develops a shoulder with increasing threshold. *t*_{c} is within the estimated range for the correlation time in solar wind turbulence [*Matthaeus et al.*, 2005], derived from magnetic field fluctuations during solar maximum. Such a characteristic time has also been observed [*Moloney and Davidsen*, 2010], *independent* of the solar cycle. For solar flare waiting times, meanwhile, *p*(*t*_{w}) is of the same form as equation (2), with *t*_{0} depending on threshold [*Baiesi et al.*, 2006]. This suggests that *t*_{c} is an intrinsic property of the solar wind turbulence, unrelated to the solar driver. Indeed, this is confirmed by an analysis of *p*(*t*_{w}) during solar minimum (Moloney and Davidsen, manuscript in preparation, 2011).

[13] *p*(*t*_{q}) shows essentially the same behavior (Moloney and Davidsen, manuscript in preparation, 2011). This is expected since events are relatively localised in time, i.e., 〈*t*_{q}〉 ≫ 〈*t*_{d}〉. Therefore, barring strong correlations between duration and subsequent quiet times, waiting and quiet time distributions should be essentially the same.

[14] To summarize, we have studied the burst statistics of the Akasofu ε parameter during solar maximum. For thresholds above the *q* ≳ 0.9 quantile, we find that the number of events decays as a power law, as in the case of solar flares [*Baiesi et al.*, 2006]. Moreover, the duration time distributions appear to be threshold independent and well described by the same functional form as proposed for solar flares, with the same exponent *μ*. We also observe that the distributions of burst energies decay as a power law, and that the energies scale with duration time as a power law. It is worth noting that alternative observables may give different results, as in the case of the Poynting flux (closely related to the ε parameter) studied by *Freeman et al.* [2000b]. While the former has also been observed for solar flares, the latter has not been investigated for solar flares to our best knowledge. These observations indicate that there is a clear signature of the solar activity in the burst statistics of ε during solar maximum.

[15] Meanwhile, the quiet and waiting time distributions are not asymptotically power law — in contrast to solar flares — and develop pronounced shoulders at around 5 × 10^{2} minutes with increasing threshold. Since this characteristic time does not depend on the solar cycle, it is likely an intrinsic feature of the local magnetohydrodynamic turbulence. This characteristic time is not apparent in the distributions of burst durations since it is comparable with the largest observed burst durations.

[16] All these results are further strengthened by the observation that the bursts statistics of the magnetic energy density *B*^{2} alone (see equation (1)) show the same qualitative behavior as for ε (Moloney and Davidsen, manuscript in preparation, 2011). This is important since *B*^{2} is sometimes thought to be more directly coupled to solar flare activity than ε.

[17] In addition to providing further evidence for the direct influence of the solar activity on the statistical properties of the solar wind, our analysis also raises important questions in the context of the solar wind-magnetosphere interaction. While this coupling has been studied in terms of burst statistics, only thresholds below the *q* = 0.9 quantile have been considered. It will be interesting to see whether any of the recent results [*Freeman et al.*, 2000a; *Uritsky et al.*, 2001; *Watkins*, 2002; *Wanliss and Weygand*, 2007] need to be augmented for higher quantiles.