**7. References**

30 Will-be-set-by-IN-TECH

Next, let us discuss resistivity effects. Resistivity usually is small in magnetically confined fusion plasmas. Due to its smallness resistivity effects are only important in the singular layer region. With ideal MHD singular layer theory detailed in Sec. 4.2 one can rederive resistive singular layer equations given in Ref. Glasser et al. (1975). However, it should be pointed out that, when kinetic enhancement of apparent mass effect is taken into account, the ratio of resistivity and inertia layer widths changes. This leads kinetic description of resistive MHD modes to become substantially different from fluid description Zheng & Tessarotto (1996) Zheng & Tessarotto (1995). Kinetic analysis of low frequency resistive MHD modes becomes

The driving force for ideal MHD instabilities is related to pressure gradient. Resistivity can instead cause field line reconnection and induce the so-called tearing modes. It is important to note that if current gradient is taken into account pressure driven modes and tearing modes are coupled to each other. The underlying driving mechanism for pressure driven modes is the release of plasma thermal energy from the interchange of magnetic flux tubes. Actually, interchange-type modes exchange not only thermal and magnetic energies between flux tubes, but also current. In a plasma with a current (or resistivity) gradient, such an interchange can create a current sheet at a mode resonance surface and result in the excitation of current

Instabilities of interchange type have been widely used to explain anomalous transport in tokamaks in terms of the formation of turbulent eddies through nonlinear coupling. However, the explanation for experimental observations that the electron energy transport is much larger than what one would expect from diffusive process due to Coulomb collisions is still unsatisfactory. The electron Larmor radius is much smaller than ion one. Nonetheless, the electron thermal transport often is stronger than ion transport. In Ref. Rechester & Rosenbluth (1978), the broken magnetic surfaces due to formation of magnetic island and stochastic field lines are used to explain the enhanced electron transport. But, how magnetic islands are formed in axisymmetric tokamak plasmas has not been given. CITM theory shows that interchange-type instabilities can directly convert to current interchange tearing modes. This

Another transport issue we need to discuss is the so-called flow shear de-correlation of turbulences. This concept has been widely used for explaining suppression of plasma turbulences. In fact, this picture is not right for systems with magnetic shear. We use Fig. 3 to explain it (L. J. Zheng and M. Tessarotto, private communication). In Fig. 3, the dashed long arrow represents a magnetic field line on a given magnetic surface *ψ*0, and two solid long arrows denote the magnetic field lines respectively at two time sequences *t*<sup>0</sup> and *t*<sup>0</sup> + Δ*t* on an adjacent magnetic surface *ψ*1. Let us examine the correlation pattern in the local frame moving together with equilibrium velocity of the dashed long arrow on surface *ψ*0. The modes are supposed to locate around the point "*O*" initially at *t* = *t*0. After a time interval Δ*t*, the field line on surface *ψ*<sup>1</sup> moves relatively to the dashed long arrow on the surface *ψ*<sup>0</sup> due to flow shear. From Fig. 3 one can see that the fixed pattern has not been de-correlated by flow shear,

time *t* = *t*<sup>0</sup> + Δ*t*. This indicates that flow shear does not de-correlate turbulence eddies. Only flow curvature can result in the de-correlation. This resembles to ballooning mode behavior

" at subsequent

instead the pattern just propagates from point "*O*" at time *t* = *t*<sup>0</sup> to point "*O*

in rotating plasmas with Cooper representation Waelbroeck & Chen (1991).

interchange tearing modes (CITMs) as shown in Fig. 2 Zheng & Furukawa (2010).

helps to clarify the source of electron transport in tokamaks.

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```
URL: http://stacks.iop.org/0029-5515/20/i=11/a=011
```

**2** 

*1,2Italy 3France* 

**Sub-Fluid Models in Dissipative** 

*1Istituto dei Sistemi Complessi ISC-CNR, Sesto Fiorentino, 2INAF-Istituto di Fisica dello Spazio Interplanetario, Roma, 3Centre de Physique Théorique, CPT-CNRS, Marseille,* 

Massimo Materassi1, Giuseppe Consolini2 and Emanuele Tassi3

Magneto-Hydrodynamics (MHD) describes the plasma as a *fluid* coupled with the selfconsistent magnetic field. The regime of validity of the MHD description of a plasma system is generally restricted to the temporal and spatial scales much larger than the characteristic plasma temporal scales (such as those associated with the plasma frequency, the ion and electrons cyclotron frequencies and the collision frequency), or the typical spatial scales (as the ion and electron inertial scale, the ion and electron Larmor radii and the Debye length). On the large scale, the plasma can be successfully described in terms of a single magnetized fluid by means of generally differentiable and smooth functions: this description of plasma media has met a wide success. However, the last decade of the 20th Century has brought to scientists' attention a wide amount of experimental and theoretical results suggesting substantial changes in classical magnetized plasma dynamics with respect to the MHD picture. In particular, two fundamental characteristics of the MHD as a dynamical theory have started to appear questionable: *regularity* and *determinism*. The MHD variables are, indeed, analytically smooth functions of space and time coordinates. Physicists refer to this as *regularity*. Moreover, once the initial conditions are assigned (together with some border conditions), the evolution of the MHD variables is unique: hence MHD is strictly *deterministic*. Instead, in in-field and laboratory studies, more and more examples have been brought to evidence, where *irregularity* and *stochastic processes* appear to play a role in magnetized plasma dynamics. This is particularly true when one approaches intermediate and small scales where the validity conditions for the MHD description, although still valid, are no longer valid in a strict sense, or when we are in the presence of topologically relevant structures, whose evolution cannot be described in terms of smooth functions. From now on, the conditions of the MHD variables apparently violating smoothness and/or determinism will be referred to as *irregular stochastic configurations* (ISC). In the following we remind, in some detail, these experimental and theoretical results pointing towards the

existence of ISCs, in the context of space plasmas and fusion plasmas.

In the framework of space physics, it has been pointed out that both the global, large scale dynamics and some local processes related to plasma transport could be better explained in

**1. Introduction** 

**Magneto-Hydrodynamics** 


URL: *http://www.sciencedirect.com/science/article/pii/S0021999105002950*


URL: *http://www.sciencedirect.com/science/article/pii/S002199911000032X*

