**1. Introduction**

34 Will-be-set-by-IN-TECH

34 Topics in Magnetohydrodynamics

Zheng, L.-J., Chu, M. S. & Chen, L. (1999). Effect of toroidal rotation on the localized modes

Zheng, L. J. & Furukawa, M. (2010). Current-interchange tearing modes: Conversion of interchange-type modes to tearing modes, *Physics of Plasmas* 17(5): 052508.

Zheng, L.-J. & Kotschenreuther, M. (2006). AEGIS: An adaptive ideal-magnetohydrodynamics

Zheng, L. J., Kotschenreuther, M. T. & Van Dam, J. W. (2007). Revisiting linear gyrokinetics

Zheng, L. J., Kotschenreuther, M. T. & Van Dam, J. W. (2010). AEGIS-K code for linear kinetic

Zheng, L.-J. & Tessarotto, M. (1994b). Collisionless kinetic ballooning mode equation in the

Zheng, L.-J. & Tessarotto, M. (1995). Collisional ballooning mode dispersion relation in the

Zheng, L.-J. & Tessarotto, M. (1996). Collisional effect on the magnetohydrodynamic modes

URL: *http://www.sciencedirect.com/science/article/pii/S002199911000032X* Zheng, L.-J. & Tessarotto, M. (1994a). Collisionless kinetic ballooning equations in the

comparable frequency regime, *Physics of Plasmas* 1(9): 2956–2962.

low-frequency regime, *Physics of Plasmas* 1(12): 3928–3935.

URL: *http://www.sciencedirect.com/science/article/pii/S0021999105002950* Zheng, L.-J., Kotschenreuther, M. & Chu, M. S. (2005). Rotational stabilization of resistive wall

modes by the shear Alfvén resonance, *Phys. Rev. Lett.* 95: 255003.

URL: *http://link.aps.org/doi/10.1103/PhysRevLett.95.255003*

shooting code for axisymmetric plasma stability, *Journal of Computational Physics*

to recover ideal magnetohydrodynamics and missing finite Larmor radius effects,

analysis of toroidally axisymmetric plasma stability, *Journal of Computational Physics*

in low beta circular tokamaks, *Physics of Plasmas* 6(4): 1217–1226.

URL: *http://link.aip.org/link/?PHP/6/1217/1*

URL: *http://link.aip.org/link/?PHP/17/052508/1*

211(2): 748 – 766.

229(10): 3605 – 3622.

*Physics of Plasmas* 14(7): 072505.

URL: *http://link.aip.org/link/?PHP/1/2956/1*

URL: *http://link.aip.org/link/?PHP/1/3928/1*

URL: *http://link.aip.org/link/?PHP/2/3071/1*

URL: *http://link.aip.org/link/?PHP/3/1029/1*

banana regime, *Physics of Plasmas* 2(8): 3071–3080.

of low frequency, *Physics of Plasmas* 3(3): 1029–1037.

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

Sub-Fluid Models in Dissipative Magneto-Hydrodynamics 37

field features in the magnetosheath transition region (as described in Retinò et al., 2007; Sundkvist et al., 2007) seem to suggest that the dynamics of such coherent structures can be the origin of a *coherent dissipation mechanism*, a sort of coarse-grained dissipation (Tetrault,

A consistent theory of plasmas in ISC should be a consistent theory of SCSs, valid in a suitable "midland" of the "coupling constant space" (Chang et al., 1978). This "midland of SCSs" should be far from the particle scale (because each SCS involves a large amount of correlated particles), but also some steps under the fluid level (because matter should

Furthermore, this "midland" is not the usual kinetic-fluid transition as described e.g. in Bălescu (1997). In fact, the kinetic description is sensible under some *weak coupling approximation* allowing for a self-consistent Markovian single particle theory to exist, while if mesoscopic coherent structures appear, the correlation length and inter-particle interaction scale are so big that the single particle evolves only together with a large number of its fellows, excluding such weak

Well far from trying to give a self-consistent theory of the SCS, here we just discuss some models and scenarios retaining some properties that such a theory should have. The approaches discussed here are exactly the application of the philosophy well described by Bălescu (1997) to dissipative processes in the MHD. Probably, a first principle analytical theory of turbulence is going to be out of reach for decades. However, something useful for applications can be developed in a more advanced framework than "traditional" statistical mechanics by introducing elements of chaos or stochasticity, non-Gaussian or non-Markovian properties, in some "effective" and "sound" models. In this way, one admits a certain "degree of randomness" in the equations, so that the non-Gaussianity of the basic stochastic processes, the role of the non-Markovian equations of evolution, the role of fractal structures and the emergence of "strange transport" are all SCS theoretical features of which

The schemes presented here are models with these properties, trying to interpolate between the macroscopic, smooth, deterministic physics of traditional MHD and the mesoscopic, irregular, stochastic physics of "that something else" which has not been formulated yet.

In this chapter three sub-fluid models are described, the metriplectic dissipative MHD, the

In the first model, the metriplectic dissipative MHD (§ 2), we focus on the relationship between the fluid dynamical variables and the microscopic degrees of freedom of the plasma. The thermodynamic entropy of the plasma microscopic degrees of freedom turns out to play an essential role in the metriplectic formalism, a tool developed in the 1980s encompassing dissipation within an algebra of observables, and here adapted to MHD. It is considered that thermodynamics, i.e. statistics, naturally arises for the description of the microscopic degrees of freedom. Fluid degrees of freedom are endowed with energy, linear and angular momenta, while an entropy function, measuring how undetermined their "mechanical" microscopic configuration is, can be attributed to the microscopic degrees of

stochastic field theory of resistive MHD and the fractal magnetic reconnection.

1992a, b; Chang et al., 2003) due to interactions that result non-local in the *k*-space.

coupling. Then, if SCSs exist, the kinetic level of the theory does not.

Such phenomenological approach is indicated as *sub-fluid*.

appear granular and fields irregular).

one tries to take into account.

freedom.

terms of stochastic processes, low-dimensional chaos, fractal features, intermittent turbulence, complexity and criticality (see e.g. Chang, 1992; Klimas et al., 1996; Chang, 1999; Consolini, 2002; Uritsky et al., 2002; Zelenyi & Milovanov, 2004 and references therein).

A certain confidence exists in stating that the rate of conversion of the magnetic energy into plasma kinetic one observed in events of *magnetic reconnection* is significantly underestimated by the traditional, smooth and deterministic, MHD (see for example Priest & Forbes, 2000 and also Biskamp, 2000). Lazarian et al. (2004) have found an improvement in the calculation of the magnetic reconnection rate by considering stochastic reconnection in a magnetized, partially ionized medium. This process is stochastic due to the field line probabilistic wandering through the turbulent fluid. In a different context, Consolini et al. (2005) showed that stochastic fluctuations play a crucial role in the current disruption of the geomagnetic tail, a magnetospheric process occurring at the onset of magnetic substorm in the Earth's magnetotail (see, e.g., Kelley, 1989; Lui, 1996). Consistently with the relevance of stochastic processes in space plasmas, tools derived from information theory have been recently applied to describe the near-Earth plasma phenomenology (Materassi et al., 2011; De Michelis et al., 2011). On the other hand, turbulence has been shown to play a relevant role in several different space plasma media as the solar wind (Bruno & Carbone, 2005) or the Earth's magnetotail regions (see, e.g., Borovsky & Funsten, 2003), etc.

In fusion plasmas, phenomena important as anomalous diffusion induced by stochastic magnetic fields (Rechester & Rosenbluth, 1978) have been suggested to be caused by the appearance of irregular modes similar to ISCs: those modes have been documented since a rather long time (Goodall, 1982). In tokamak machines ISCs observed are mesoscopic intermittent and filamentary structures: recently, studies have shown how such structures might be generated by reconnecting tearing modes triggered by a primary interchange instability (Zheng & Furukawa 2010).

The appearance of ISCs should not be expected as an exceptional condition: indeed, timeand space-regular MHD relies on very precise hypotheses, not necessarily holding in real plasmas. As underlined before, it should be considered that MHD is a long time description with respect to the interaction times of particles. In order to expect a smooth deterministic evolution in time, "fast phenomena" should be ignored, and clearly this cannot be done when "fast phenomena" lead to big changes in the MHD variables themselves, on macroscopic scales, as it happens in the *fast magnetic reconnection*.

Space regularity requires the scale at which matter appears as granular to shrink to zero, and this is possible under the hypothesis that such scale is much smaller than the typical scale where the MHD variables do vary. However, in *turbulent regimes* the scale at which the MHD fields vary are so small, that they compare with those scales at which plasma appears as granular.

The phenomenology of plasma ISCs appears to indicate that the role of "fundamental entities" should be played by mesoscopic coherent structures, interacting and stochastically evolving. These *stochastic coherent structures* (SCS) have been observed in several space plasma regions: in solar wind (Bruno et al., 2001) as field-aligned flux tubes, in the Earth's cusp regions (Yordanova et al., 2005), in the geotail plasma sheet as current structures, 2D eddies and so on (see, for instance, Milovanov et al., 2001; Borovsky & Funsten, 2003; Vörös et al., 2004; Kretzschmar & Consolini, 2006). Recent observations of small-scale magnetic 36 Topics in Magnetohydrodynamics

terms of stochastic processes, low-dimensional chaos, fractal features, intermittent turbulence, complexity and criticality (see e.g. Chang, 1992; Klimas et al., 1996; Chang, 1999; Consolini, 2002; Uritsky et al., 2002; Zelenyi & Milovanov, 2004 and references therein).

A certain confidence exists in stating that the rate of conversion of the magnetic energy into plasma kinetic one observed in events of *magnetic reconnection* is significantly underestimated by the traditional, smooth and deterministic, MHD (see for example Priest & Forbes, 2000 and also Biskamp, 2000). Lazarian et al. (2004) have found an improvement in the calculation of the magnetic reconnection rate by considering stochastic reconnection in a magnetized, partially ionized medium. This process is stochastic due to the field line probabilistic wandering through the turbulent fluid. In a different context, Consolini et al. (2005) showed that stochastic fluctuations play a crucial role in the current disruption of the geomagnetic tail, a magnetospheric process occurring at the onset of magnetic substorm in the Earth's magnetotail (see, e.g., Kelley, 1989; Lui, 1996). Consistently with the relevance of stochastic processes in space plasmas, tools derived from information theory have been recently applied to describe the near-Earth plasma phenomenology (Materassi et al., 2011; De Michelis et al., 2011). On the other hand, turbulence has been shown to play a relevant role in several different space plasma media as the solar wind (Bruno & Carbone, 2005) or

In fusion plasmas, phenomena important as anomalous diffusion induced by stochastic magnetic fields (Rechester & Rosenbluth, 1978) have been suggested to be caused by the appearance of irregular modes similar to ISCs: those modes have been documented since a rather long time (Goodall, 1982). In tokamak machines ISCs observed are mesoscopic intermittent and filamentary structures: recently, studies have shown how such structures might be generated by reconnecting tearing modes triggered by a primary interchange

The appearance of ISCs should not be expected as an exceptional condition: indeed, timeand space-regular MHD relies on very precise hypotheses, not necessarily holding in real plasmas. As underlined before, it should be considered that MHD is a long time description with respect to the interaction times of particles. In order to expect a smooth deterministic evolution in time, "fast phenomena" should be ignored, and clearly this cannot be done when "fast phenomena" lead to big changes in the MHD variables themselves, on

Space regularity requires the scale at which matter appears as granular to shrink to zero, and this is possible under the hypothesis that such scale is much smaller than the typical scale where the MHD variables do vary. However, in *turbulent regimes* the scale at which the MHD fields vary are so small, that they compare with those scales at which plasma appears

The phenomenology of plasma ISCs appears to indicate that the role of "fundamental entities" should be played by mesoscopic coherent structures, interacting and stochastically evolving. These *stochastic coherent structures* (SCS) have been observed in several space plasma regions: in solar wind (Bruno et al., 2001) as field-aligned flux tubes, in the Earth's cusp regions (Yordanova et al., 2005), in the geotail plasma sheet as current structures, 2D eddies and so on (see, for instance, Milovanov et al., 2001; Borovsky & Funsten, 2003; Vörös et al., 2004; Kretzschmar & Consolini, 2006). Recent observations of small-scale magnetic

the Earth's magnetotail regions (see, e.g., Borovsky & Funsten, 2003), etc.

macroscopic scales, as it happens in the *fast magnetic reconnection*.

instability (Zheng & Furukawa 2010).

as granular.

field features in the magnetosheath transition region (as described in Retinò et al., 2007; Sundkvist et al., 2007) seem to suggest that the dynamics of such coherent structures can be the origin of a *coherent dissipation mechanism*, a sort of coarse-grained dissipation (Tetrault, 1992a, b; Chang et al., 2003) due to interactions that result non-local in the *k*-space.

A consistent theory of plasmas in ISC should be a consistent theory of SCSs, valid in a suitable "midland" of the "coupling constant space" (Chang et al., 1978). This "midland of SCSs" should be far from the particle scale (because each SCS involves a large amount of correlated particles), but also some steps under the fluid level (because matter should appear granular and fields irregular).

Furthermore, this "midland" is not the usual kinetic-fluid transition as described e.g. in Bălescu (1997). In fact, the kinetic description is sensible under some *weak coupling approximation* allowing for a self-consistent Markovian single particle theory to exist, while if mesoscopic coherent structures appear, the correlation length and inter-particle interaction scale are so big that the single particle evolves only together with a large number of its fellows, excluding such weak coupling. Then, if SCSs exist, the kinetic level of the theory does not.

Well far from trying to give a self-consistent theory of the SCS, here we just discuss some models and scenarios retaining some properties that such a theory should have. The approaches discussed here are exactly the application of the philosophy well described by Bălescu (1997) to dissipative processes in the MHD. Probably, a first principle analytical theory of turbulence is going to be out of reach for decades. However, something useful for applications can be developed in a more advanced framework than "traditional" statistical mechanics by introducing elements of chaos or stochasticity, non-Gaussian or non-Markovian properties, in some "effective" and "sound" models. In this way, one admits a certain "degree of randomness" in the equations, so that the non-Gaussianity of the basic stochastic processes, the role of the non-Markovian equations of evolution, the role of fractal structures and the emergence of "strange transport" are all SCS theoretical features of which one tries to take into account.

The schemes presented here are models with these properties, trying to interpolate between the macroscopic, smooth, deterministic physics of traditional MHD and the mesoscopic, irregular, stochastic physics of "that something else" which has not been formulated yet. Such phenomenological approach is indicated as *sub-fluid*.

In this chapter three sub-fluid models are described, the metriplectic dissipative MHD, the stochastic field theory of resistive MHD and the fractal magnetic reconnection.

In the first model, the metriplectic dissipative MHD (§ 2), we focus on the relationship between the fluid dynamical variables and the microscopic degrees of freedom of the plasma. The thermodynamic entropy of the plasma microscopic degrees of freedom turns out to play an essential role in the metriplectic formalism, a tool developed in the 1980s encompassing dissipation within an algebra of observables, and here adapted to MHD. It is considered that thermodynamics, i.e. statistics, naturally arises for the description of the microscopic degrees of freedom. Fluid degrees of freedom are endowed with energy, linear and angular momenta, while an entropy function, measuring how undetermined their "mechanical" microscopic configuration is, can be attributed to the microscopic degrees of freedom.

Sub-Fluid Models in Dissipative Magneto-Hydrodynamics 39

referred to as *ideal MHD*, has been shown, long ago, to be a Hamiltonian system (Morrison & Greene, 1980). The elements constituting a Hamiltonian structure are the Poisson bracket, a bilinear operator with algebraic properties, and the Hamiltonian of the system, depending on the dynamical variables: in the case of the MHD, these will be defined in the following (see (7) and (9)). The Hamiltonian formulation of the ideal MHD, apart from facilitating the identification of conserved quantitites, or the stability analysis of the equilibria, renders it evident that the dynamics of the system takes place on *symplectic leaves* that foliate the phase

The inclusion of *dissipative terms* invalidates the Hamiltonian representation: this dissipative breakdown matches the fact that, once dissipation is included, the system becomes "less deterministic" in a certain sense, because there is an interaction with microscopic degrees of freedom that are described in a statistical manner (friction forces *are* a statistically effective

Some dissipative systems possess however an algebraic structure called *metriplectic*, which still permits to formulate the dynamics in terms of a bracket and of an observable, extending the concept of Hamiltonian. Metriplectic structures in general occur in systems which *conserve the energy and increase the entropy*. These are the so called *complete systems*. They are obtained adding friction forces to an originally Hamiltonian system, and then including, in the algebra of observables, the energy and entropy of the microscopic degrees of freedom. The metriplectic formulation permits to reformulate the dynamics of dissipative systems in a geometrical framework, in which information, such as the existence of asymptotically

stable equilibria, may be easily retrieved without even trying to solve the equations.

In order to define what a metriplectic structure is, and apply this concept to the case of MHD, it is convenient to start recalling that, very frequently, one deals with the analysis of

where *z* is the set of the *N* dynamical variables of the system (*N* can be infinite; it is actually a continuous real index for field theories or the MHD) evolving under the action of a vector field *FH*(*z*) + *FD*(*z*). Such vector field is the sum of a non-trivial Hamiltonian component *FH*(*z*) and a component *FD*(*z*) accounting for the dissipative terms. If *FD*(*z*) = 0, the resulting system

where *H*(*z*) is the *Hamiltonian* of the system, and [\*,\*] is the *Poisson bracket*, an antisymmetric bilinear operator, satisfying the Leibniz property and the Jacobi identity (Goldstein, 1980). These properties render the Poisson algebra of group-theoretical nature. An immediate consequence of the antisymmetry of the bracket is that ∂*tH* = [*H*,*H*] = 0, so that *H* is

It is important to point out that, in many circumstances, the Poisson bracket is not of the canonical type. In particular, for Hamiltonian systems describing the motion of continuous media in terms of Eulerian variables, as in the case of ideal MHD, the Poisson bracket is

, 1,..., , *ii i tH D z Fz Fz i N* (1)

, , *ii i t H z F z z Hz* (2)

space (Morrison, 1998).

physical models of the form

necessarily a constant of motion.

treatment of microscopic stochastic collisions).

is Hamiltonian and consequently can be written as

In the second model treated, the stochastic field theory (SFT) (§ 3), the dissipation coefficients appearing in the MHD equations of motion are considered as noise, consistently with the fact that, out of its equilibrium, a medium may be treated statistically. In this way, MHD turns into a set of Langevin field equations. These may be treated through the path integral formalism introduced by Phythian (1977), appearing particularly suitable for non equilibrium statistics. Once the resistive MHD theory is turned into a SFT, transition probabilities between arbitrary field configurations may be calculated via a stochastic action formalism, closely resembling what is usually done for quantum fields. This mimics very precisely the idea of an ISC.

A sub-fluid model of *fast magnetic reconnection* (FMR) is dealt with in § 4. FMR clearly belongs to the class of phenomena in which classical fields apparently undergo quantumlike transitions in considerably short times: when magnetic field lines reconnect, the field topology is changed and a big quantity of magnetic energy, associated to the original configuration, is turned into the kinetic energy of fast jets of particles. In order to mimic a reconnection rate high enough, a successful attempt may be done relaxing the assumption that all the local variables of the plasma and the magnetic field are smooth functions. In particular, in a standard 2-dimensional Sweet-Parker scenario (Parker, 1957, 1963; Sweet, 1958), one assumes that the reconnection region, where finite resistivity exists, is a fractal domain of box-counting dimension smaller than 2. This allows for a reconnection rate that varies with the magnetic Reynolds number faster than the traditional one.
