Multi-Spacecraft Turbulence Analysis Methods

Abstract

Turbulence is ubiquitous in space plasmas, from the solar wind to supernova remnants, and on scales from the electron gyroradius to interstellar separations. Turbulence is responsible for transporting energy across space and between scales and plays a key role in plasma heating, particle acceleration and thermalisation downstream of shocks. Just as with other plasma processes such as shocks or reconnection, turbulence results in complex, structured and time-varying behaviour which is hard to measure with a single spacecraft. However, turbulence is a particularly hard phenomenon to study because it is usually broadband in nature: it covers many scales simultaneously. One must therefore use techniques to extract information on multiple scales in order to quantify plasma turbulence and its effects.

The Cluster orbit takes the spacecraft through turbulent regions with a range of characteristics: the solar wind, magnetosheath, cusp and magnetosphere. In each, the nature of the turbulence (strongly driven or fully evolved; dominated by kinetic effects or largely on fluid scales), as well as characteristics of the medium (thermalised or not; high or low plasma sub- or super-Alfvenic) mean that particular techniques are better suited to the analysis of Cluster data in different locations. In this chapter, we consider a range of methods and how they are best applied to these different regions.

Perhaps the most studied turbulent space plasma environment is the solar wind, see Bruno and Carbone [2005]; Goldstein et al. [2005] for recent reviews. This is the case for a number of reasons: it is scientifically important for cosmic ray and solar energetic particle scattering and propagation, for example. However, perhaps the most significant motivations for studying solar wind turbulence are pragmatic: large volumes of high quality measurements are available; the stability of the solar wind on the scales of hours makes it possible to identify statistically stationary intervals to analyse; and, most important of all, the solar wind speed, V SW , is much higher than the local MHD wave speeds. This means that a spacecraft time series is essentially a "snapshot" spatial sample of the plasma along the flow direction, so we can consider measurements at a set of times ti to be at a set of locations in the plasma given by xi = VSW. This approximation,known as Taylor's hypothesis, greatly simplifies the analysis of the data. In contrast, in the magnetosheath the flow speed is lower than the wave speed and therefore temporal changes at the spacecraft are due to a complex combination of the plasma moving over the spacecraft and the turbulent fluctuations propagating in the plasma frame. This is also the case for ion and electron kinetic scale turbulence in the solar wind and dramatically complicates the analysis of the data. As a result, the application of multi-spacecraft techniques such as k filtering to Cluster data (see Chapter 5, which make it possible to disentangle the effects of flow and wave propagation, have probably resulted in the greatest increase in our understanding of magnetosheath turbulence rather than in the solar wind.

We can therefore summarise the key advantages for plasma turbulence analysis of multi-spacecraft data sets such as those from Cluster, compared to single spacecraft data. Multiple sampling points allow us to measure how the turbulence varies in many directions, and on a range of scales, simultaneously, enabling the study of anisotropy in ways that have not previously been possible. They also allow us to distinguish between the motion of fluctuations in the plasma and motion of the plasma itself, enabling the study of turbulence in highly disturbed environments such as the magnetosheath. A number of authors have studied turbulence with Cluster data, using different techniques, the choice of which is motivated by the characteristics of the plasma environment in which they are interested. The complexity of both the Cluster data and the problem of turbulence meant that progress early in the mission was rather limited, although in the last few years several key results have been obtained and it is now a rapidly evolving topic.

At this point, it is worth noting briefly the scope of this chapter: we discuss multi- spacecraft Cluster results and methods regarding turbulence at fluid, ion and electron scales, with the emphasis on the methods more than the physical significance of the results, but we do not consider more wave-like phenomena such as those in the foreshock. This is an entirely artificial distinction, both in terms of the physics and the analysis methods. Nevertheless, this chapter is intended to be largely self-contained and we refer the reader to other chapters in this book for more information about these related topics. We also stress that this chapter is not in any way intended to be an introduction to, or overview of, the analysis and theory of space plasma turbulence, or even of Cluster results in general: instead, references to review articles are provided where appropriate. Belmont et al. [2006] discussed the application of k filtering to turbulence studies in much greater depth than is presented here and we refer the reader to that paper for more details. Single space- craft analysis of Cluster data is revealing important information about turbulent anisotropy [e.g., Mangeney et al., 2006; Lacombe et al., 2006], dissipation processes [e.g., Bale et al., 2005] and even evidence for reconnection triggered by turbulence [e.g., Retino et al., 2007] but again, we do not discuss these results further here: our emphasis is on multi-spacecraft analysis methods.

After fifty years of spacecraft measurements of turbulent space plasmas, many significant questions remain unanswered. Perhaps the three most important, both for our fundamental understanding of plasma turbulence as a process and for quantifying its large scale effects, are: anisotropy due to the presence of a background magnetic field; the nature of the dissipation process; and the origin of the spatial inhomogeneity known as intermittency. All three of these issues have been addressed using Cluster data. We discuss each briefly here in order to provide the context for the methods and results presented in later sections.