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1 Introduction

The whole heliosphere is permeated by the solar wind, a supersonic and super-Alfvénic plasma flow of solar origin which continuously expands into the heliosphere. This medium offers the best opportunity to study directly collisionless plasma phenomena, mainly at low frequencies where high-amplitude fluctuations have been observed. During its expansion, the solar wind develops a strong turbulent character, which evolves towards a state that resembles the well known hydrodynamic turbulence described by Kolmogorov (1941Jump To The Next Citation Point). Because of the presence of a strong magnetic field carried by the wind, low-frequency fluctuations in the solar wind are usually described within a magnetohydrodynamic (MHD, hereafter) benchmark (Kraichnan, 1965Jump To The Next Citation PointBiskamp, 1993Jump To The Next Citation PointTu and Marsch, 1995aJump To The Next Citation PointBiskamp, 2003Jump To The Next Citation Point). However, due to some peculiar characteristics, the solar wind turbulence contains some features hardly classified within a general theoretical framework.

Turbulence in the solar heliosphere plays a relevant role in several aspects of plasma behavior in space, such as solar wind generation, high-energy particles acceleration, plasma heating, and cosmic rays propagation. In the 1970s and 80s, impressive advances have been made in the knowledge of turbulent phenomena in the solar wind. However, at that time, spacecraft observations were limited by a small latitudinal excursion around the solar equator and, in practice, only a thin slice above and below the equatorial plane was accessible, i.e., a sort of 2D heliosphere. A rather exhaustive survey of the most important results based on in-situ observations in the ecliptic plane has been provided in an excellent review by Tu and Marsch (1995aJump To The Next Citation Point) and we invite the reader to refer to that paper. This one, to our knowledge, has been the last large review we find in literature related to turbulence observations in the ecliptic.

In the 1990s, with the launch of the Ulysses spacecraft, investigations have been extended to the high-latitude regions of the heliosphere, allowing us to characterize and study how turbulence evolves in the polar regions. An overview of Ulysses results about polar turbulence can also be found in Horbury and Tsurutani (2001Jump To The Next Citation Point). With this new laboratory, relevant advances have been made. One of the main goals of the present work will be that of reviewing observations and theoretical efforts made to understand the near-equatorial and polar turbulence in order to provide the reader with a rather complete view of the low-frequency turbulence phenomenon in the 3D heliosphere.

New interesting insights in the theory of turbulence derive from the point of view which considers a turbulent flow as a complex system, a sort of benchmark for the theory of dynamical systems. The theory of chaos received the fundamental impulse just through the theory of turbulence developed by Ruelle and Takens (1971Jump To The Next Citation Point) who, criticizing the old theory of Landau and Lifshitz (1971Jump To The Next Citation Point), were able to put the numerical investigation by Lorenz (1963Jump To The Next Citation Point) in a mathematical framework. Gollub and Swinney (1975Jump To The Next Citation Point) set up accurate experiments on rotating fluids confirming the point of view of Ruelle and Takens (1971Jump To The Next Citation Point) who showed that a strange attractor in the phase space of the system is the best model for the birth of turbulence. This gave a strong impulse to the investigation of the phenomenology of turbulence from the point of view of dynamical systems (Bohr et al., 1998Jump To The Next Citation Point). For example, the criticism by Landau leading to the investigation of intermittency in fully developed turbulence was worked out through some phenomenological models for the energy cascade (cf. Frisch, 1995Jump To The Next Citation Point). Recently, turbulence in the solar wind has been used as a big wind tunnel to investigate scaling laws of turbulent fluctuations, multifractals models, etc. The review by Tu and Marsch (1995aJump To The Next Citation Point) contains a brief introduction to this important argument, which was being developed at that time relatively to the solar wind (Burlaga, 1993Carbone, 1993Jump To The Next Citation PointBiskamp, 1993Jump To The Next Citation Point2003Jump To The Next Citation PointBurlaga, 1995Jump To The Next Citation Point). The reader can convince himself that, because of the wide range of scales excited, space plasma can be seen as a very big laboratory where fully developed turbulence can be investigated not only per se, rather as far as basic theoretical aspects are concerned.

Turbulence is perhaps the most beautiful unsolved problem of classical physics, the approaches used so far in understanding, describing, and modeling turbulence are very interesting even from a historic point of view, as it clearly appears when reading, for example, the book by Frisch (1995Jump To The Next Citation Point). History of turbulence in interplanetary space is, perhaps, even more interesting since its knowledge proceeds together with the human conquest of space. Thus, whenever appropriate, we will also introduce some historical references to show the way particular problems related to turbulence have been faced in time, both theoretically and technologically. Finally, since turbulence is a phenomenon visible everywhere in nature, it will be interesting to compare some experimental and theoretical aspects among different turbulent media in order to assess specific features which might be universal, not limited only to turbulence in space plasmas. In particular, we will compare results obtained in interplanetary space with results obtained from ordinary fluid flows on Earth, and from experiments on magnetic turbulence in laboratory plasmas designed for thermonuclear fusion.

 1.1 What does turbulence stand for?
 1.2 Dynamics vs. statistics

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