Tom Chang and Sunny W.Y. Tam,     Center for Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

Cheng-chin Wu,     Department of Physics and Astronomy, University of California, Los Angeles, CA 90005 USA

Abstract

The phenomenon of complexity related to intermittent turbulence in space plasmas is described. The ideas are based on the stochastic behavior of coherent structures that arise from plasma resonances. The concept of topological magnetic reconfiguration due to coarse-grained dissipation is also introduced.

Keywords

Complexity

Intermittent Turbulence

Magnetic Reconfiguration

1. Introduction

Complexity has become a hot topic in nearly every field of science. Space plasmas are no exception. When the word “complexity” is mentioned, a number of questions naturally arise in one’s mind. What is complexity? Does one look up the meaning of it from the Webster dictionary, or does it have a specific scientific meaning? What physical phenomena are results of complexity in plasmas? What are the available tools that one might employ to solve problems of complexity related to the physics of space plasmas? It is the purpose of this short discourse to provide some answers to such questions.

To avoid distraction from the flow of presentation of the main ideas, we will provide a narrative description of complexity in space plasmas with the insertion of only a minimum number of references. Most of the ideas described below appear in a recently published paper in Physics of Plasmas authored by Chang et al. [2004]. Readers are encouraged to consult this paper and its reference list for further study of this subject (web site: space.mit.edu/geocosmo).

2. Plasma Resonances and Coherent Structures

Data from in-situ space observations generally fluctuate in time (in the spacecraft frame) with varied intensities. When encountering a regime of noticeable fluctuations of such measured quantities, the standard procedure is to perform a fast Fourier transform of that section of the time series and obtain a so-called Fourier power spectrum in frequency. If one notices that there is a hump in the spectrum near certain characteristic frequency, one pronounces that the fluctuations have a strong signal within that range of frequency. Being indoctrinated by the concepts of waves in standard textbooks when discussing fluctuations, one generally goes one step further and surmises that there are strong signals of “waves” within that range of frequencies in the Fourier power spectrum. Actually, there is generally very little in-situ wave number information that is observed using the current available measuring instruments. And, the fluctuations are not really a nice superposition of plane waves. In fact if one represents the section of the fluctuations in terms of a complete set of localized, scale-dependent functions (a so-called wavelet transform), these fluctuations are generally seen localized, with different scales and amplitudes, Fig. 1. The question is then: “What are these fluctuations comprised of?” We shall try to answer this question below based on a physical point of view.

Let us consider a magnetized plasma. It is known that there are generally various types of linearized waves that may propagate along the magnetic field lines. One is then tempted to express a bundle of fluctuations in terms of a sum of such waves within the observed frequency ranges. We, on the other hand, shall ask a subtler question: “Are there fluctuations that do not propagate as field-aligned plane waves?” In a set of field equations that characterizes the dynamics of the magnetized plasma, there are generally time operators, ∂/∂t, and field-aligned propagation operators, B·∇. It is then obvious that at locations where the field-aligned propagation operator vanishes, the fluctuations cannot propagate as waves. In Fourier space, this means that the condition for non-propagation (or resonance) is when k·B = k|| = 0. In a general three-dimensional field of fluctuations, k, and a three-dimensional magnetic field, B, one visualizes the existence of multitudes of singular space curves that may satisfy this (resonance or non-propagation) condition.

We now ask a more physical question: “How would the fluctuations behave near these singularities?” The fluctuations will try to propagate away from the singularities as waves according to the underlying governing field equations. However, the background magnetic field and plasma medium will generally try to hold these fluctuations back close to the resonance curves. Thus, one visualizes that there are regions (resonance regions) close to the singular curves within which the fluctuations will move more or less coherently together and do not propagate as waves. Such seemingly coherent entities (to be called hereafter coherent structures) are actually bundled fluctuations of all scales that move more or less in unison (due to the nonlinear interactions). They are sporadically generated spatiotemporal structures that may meander, move, and even interact. Some of these structures may move in certain directions at uniform speeds. These, then, could be the so-called solitary waves or they could be simply convective structures.

3. Interactions of Coherent Structures and Dynamical Complexity

In an MHD plasma embedded in a dominant background magnetic field, magnetized coherent structures are usually in the form of field-aligned flux tubes, Fig. 2. When such coherent magnetic flux tubes with the same polarity migrate toward each other, strong local magnetic shears are created, Fig. 3. It has been demonstrated by Wu and Chang [2000] that existing sporadic nonpropagating fluctuations will generally migrate toward the strong local shear region. Eventually the mean local energies of the coherent structures will be dissipated into these concentrated fluctuations in the coarse-grained sense and induce reconfigurations of the magnetic field geometry.

Such enhanced intermittency at the intersection regions has been observed by Bruno et al. [2001] in the solar wind and Consolini et al. [2004] in the plasma sheet. As mentioned above, the coarse-grained dissipation will then initiate “fluctuation-induced nonlinear instabilities” and, thereby reconfigure the topologies of the coherent structures of the same polarity into a combined lower local energetic state, eventually allowing the coherent structures to merge locally. And, this merging process may repeat over and over again among the coherent structures. Such phenomenon is analogous to the classical avalanching process of sand piles. On the other hand, when coherent structures of opposite polarities approach each other due to the forcing of the surrounding plasma, they might repel each other, scatter, or induce magnetically quiescent localized regions. Under any of the conditions of the above interaction scenarios, new fluctuations will be generated. And, these new fluctuations can provide new resonance sites; thereby nucleating new coherent structures of varied sizes, a phenomenon that is distinctly different from the classical concept of the avalanching process of sand piles.

All such interactions can occur at any location of a flux tube along its field-aligned direction, and the phenomenon is fully three-dimensional. The result is the generation of multitudes of coherent structures of all sizes (each being much larger than the particle sizes of the plasma medium). The nonlinear stochastic behavior of the spreading and interactions of the multitudes of the coherent structures is vastly different from that of the laminar motion of the plasma medium.