**4. Fractal model of fast reconnection**

Among the many interesting fast and irreversible processes occurring in plasmas, *magnetic reconnection* is surely one of the most important (see e.g. Biskamp, 2000; Birn and Priest, 2007). The name "magnetic reconnection", originally introduced by Dungey (1953), refers to a process in which a particle acceleration is observed consequently to a change of the magnetic field line topology (*connectivity*). Being associated to a change in the magnetic field line topology, the magnetic reconnection process involves the occurrence of magnetic field line diffusion, disconnection and reconnection and it is also accompanied by plasma heating and particle acceleration, sometimes termed as *dissipation* (actually, in this case dissipation means transfer of energy from the magnetic field to the particle energy, both bulk motion energy, the term *ρV*2/2 in the integrand in (7), and thermal energy, the term *U*(*ρ*,*s*) in the same expression of *H*; in the context of metriplectic dynamics, dissipation is simply the transfer of energy into the addendum *U*(*ρ*,*s*)).

The traditional approach to magnetic reconnection is based on resistive MHD theory. In this framework one of the most famous and first scenarios of magnetic reconnection, able to make some quantitative predictions, was proposed by Parker (1957) and Sweet (1958). The Sweet-Parker model provides a simple 2-dimensional description of steady magnetic reconnection in a non-compressible plasmas (see Figure 1). In this model there are two relevant scales: the global scale *L* of the magnetic field and the thickness Δ of the current sheet (or of the diffusion region). The main result of such a model may be resumed in the very-well known expression for the *Alfvèn Mach number M*A,

Sub-Fluid Models in Dissipative Magneto-Hydrodynamics 51

Although several other models have been proposed (see e.g.: Birn and Priest, 2007), some recent MHD simulation have shown that, when the Hall effect is included, it is possible to obtain fast magnetic reconnection rates, which are independent on the current sheet or reconnection region size. For instance, Huba & Rudakov (2004) obtained a reconnection rate

All the above approaches to magnetic reconnection move from the assumption that plasma media can be viewed as *noncollisional fluid*. This assumption is clearly valid when the

> 1 2 2

*x X* 

<sup>1</sup> 10 ,

*x* = *X* – <*X*>. Conversely, recent observations evidenced that space plasmas are

characterized by an intrinsic stochastic character, and that in many situations *turbulence* is present. This is for instance the case of interplanetary space plasmas, such as the solar wind, and the Earth's magnetotail current sheet, characterized by stochastic and turbulent

Several attempts have been done to include the *stochastic and turbulent nature of the plasma media* and to discuss its effects on the magnetic reconnection process (see e.g. Yankov, 1997; Lazarian & Vishniac, 1999). The common point of such models is the idea that as a consequence of the inherent stochasticity and/or turbulent nature of plasma media, the current sheet and the diffusion region topology cannot be associated with a simple continuous regular medium. Conversely, the current sheet could be imagined like a

Fig. 2. A schematic view of 2-dimensional geometry for the fractal reconnection model, of

size *L* and Δ, with reconnection area *A*rec and reconnection active area Ωrec.

*x* of any local field *X* are negligible with respect to the large scale

(29)

*M*<sup>A</sup> ≤ 0.1 in the case of Hall magnetic reconnection.

fluctuations of the same order of magnitude of the average fields.

filamentary, complex and not space-filling region.

inherent local fluctuations

means,

being 

$$\mathcal{M}\_A = \mathcal{R}\_m^{-1/2} \quad \prime \quad \mathcal{R}\_m = \mu\_0 \mathcal{L} \, V\_A \, \prime \, \eta \,, \tag{26}$$

where *Rm* is the *Lundquist number* (often referred as *magnetic Reynolds number*), *VA* is the Alfvén velocity and is the resistivity.

Fig. 1. A schematic view of 2-dimensional geometry for the Sweet-Parker reconnection scenario

Indeed, being a measure of the electric field normalized by the global electric field, i.e.

$$\mathbf{M}\_A = \mathbf{V} \;/\; V\_A = \mathbf{E} \;/\; V\_A \mathbf{B} \;/\; \tag{27}$$

the Alfvén Mach number *M*A, reported in Eq. (26), provides an estimate of the *reconnection rate*, which is generally expressed in terms of the electric field at the reconnection site.

The typical Lundquist number *Rm* in astrophysical and space plasmas is *Rm* >> 106, implying reconnection rates *MA* << 10-4. These reconnection rates are *too slow to explain the explosive nature of several space processes* associated with the occurrence of reconnection, so that the Sweet-Parker model is considered not suitable to explain reconnection in space plasmas.

In the course of the time, to overcome such a limitation of the Sweet-Parker model several other models have been proposed. Among these models one of the most successful is the *Petscheck model* (Petscheck, 1964), where the diffusion region (associated with the current sheet) is greatly reduced in length and the energy conversion is associated with the presence of two pairs of standing slow-modes. As a result, the reconnection rate in terms of Alfvénic Mach number is

$$M\_A = \frac{\pi}{8 \ln R\_m} \tag{28}$$

which is for most of the space and laboratory plasma situations of the order of *MA* ≈ 10-1 to 10-2.

50 Topics in Magnetohydrodynamics

1/2 *M R R LV Am m A* / ,

where *Rm* is the *Lundquist number* (often referred as *magnetic Reynolds number*), *VA* is the

Fig. 1. A schematic view of 2-dimensional geometry for the Sweet-Parker reconnection

Indeed, being a measure of the electric field normalized by the global electric field, i.e.

the Alfvén Mach number *M*A, reported in Eq. (26), provides an estimate of the *reconnection rate*, which is generally expressed in terms of the electric field at the reconnection site.

The typical Lundquist number *Rm* in astrophysical and space plasmas is *Rm* >> 106, implying reconnection rates *MA* << 10-4. These reconnection rates are *too slow to explain the explosive nature of several space processes* associated with the occurrence of reconnection, so that the Sweet-Parker model is considered not suitable to explain reconnection in space plasmas.

In the course of the time, to overcome such a limitation of the Sweet-Parker model several other models have been proposed. Among these models one of the most successful is the *Petscheck model* (Petscheck, 1964), where the diffusion region (associated with the current sheet) is greatly reduced in length and the energy conversion is associated with the presence of two pairs of standing slow-modes. As a result, the reconnection rate in terms of Alfvénic

8ln *<sup>A</sup>*

which is for most of the space and laboratory plasma situations of the order of *MA* ≈ 10-1 to 10-2.

*M*

*m*

*R* 

Alfvén velocity and

scenario

Mach number is

is the resistivity.

0

(26)

*MA AA V V E VB* / / , (27)

(28)

Although several other models have been proposed (see e.g.: Birn and Priest, 2007), some recent MHD simulation have shown that, when the Hall effect is included, it is possible to obtain fast magnetic reconnection rates, which are independent on the current sheet or reconnection region size. For instance, Huba & Rudakov (2004) obtained a reconnection rate *M*<sup>A</sup> ≤ 0.1 in the case of Hall magnetic reconnection.

All the above approaches to magnetic reconnection move from the assumption that plasma media can be viewed as *noncollisional fluid*. This assumption is clearly valid when the inherent local fluctuations *x* of any local field *X* are negligible with respect to the large scale means,

$$\frac{\left\langle \delta \mathbf{x}^{2} \right\rangle^{\frac{1}{2}}}{\left\langle \mathbf{x} \right\rangle} << 10^{-1} \,, \tag{29}$$

being *x* = *X* – <*X*>. Conversely, recent observations evidenced that space plasmas are characterized by an intrinsic stochastic character, and that in many situations *turbulence* is present. This is for instance the case of interplanetary space plasmas, such as the solar wind, and the Earth's magnetotail current sheet, characterized by stochastic and turbulent fluctuations of the same order of magnitude of the average fields.

Several attempts have been done to include the *stochastic and turbulent nature of the plasma media* and to discuss its effects on the magnetic reconnection process (see e.g. Yankov, 1997; Lazarian & Vishniac, 1999). The common point of such models is the idea that as a consequence of the inherent stochasticity and/or turbulent nature of plasma media, the current sheet and the diffusion region topology cannot be associated with a simple continuous regular medium. Conversely, the current sheet could be imagined like a filamentary, complex and not space-filling region.

Fig. 2. A schematic view of 2-dimensional geometry for the fractal reconnection model, of size *L* and Δ, with reconnection area *A*rec and reconnection active area Ωrec.

Sub-Fluid Models in Dissipative Magneto-Hydrodynamics 53

Moving from the above result and assuming *k* = 1 and *Din* =*Dout* = *D*, Eq. (33) for the *fractal reconnection rate* can be reduced to a more simple expression in terms of the *Lundquist* 

We note that this expression reduces to the standard Sweet-Parker solution of the reconnection rate in the limit 1 *D* and that the fractal reconnection rate is always higher than the one predicted by the Sweet-Parker model. Furthermore, although in the limit *Rm* the reconnection rate predicted by the Petschek-like model results the more efficient, there exists always a certain range of the Lundquist number *Rm*, depending on the fractal dimension *D*, for which the fractal reconnection model is more efficient than the Petschek-like model. The crucial point of a correct estimation and applicability of the above expression stands in the correct evaluation of *Din* and *Dout*, which depends on the topology

In passing we note that when the above scenario is applied using typical length scales estimated by *in-situ* observations of magnetic reconnection in space plasmas, one gets the reconnection rates typically observed and in agreement with the estimated Hall reconnection rate *MA* ≈ 0.09 (Huba & Rudakov, 2004) assuming a diffusion region shaped as

The fractal reconnection model described here is not based on first principles, because the

The important work necessary for further development will be to give a dynamical sense to the quantities *Din* and *Dout*, that here might appear just as convenient fitting parameters. Studies have been made to regard irregular filamentary structures in plasmas as descending

The feeling is however that it would be very interesting to deduce the fractal nature of the reconnection region from kinetic or microscopic-statistical theories, rather than extracting it

Dissipation consists of the irreversible transfer of energy from the proper MHD variables to the particle degrees of freedom of the plasma, considered as "microscopic" (and usually treated via Thermodynamics). Depending on the spatial and temporal scales on which dissipation takes place, it may activate some "sub-fluid level" of the theory, which interpolates between the continuous system, representing the traditional MHD, and the discrete one, describing the plasma through the motion of its particles. This "sub-fluid" level should probably consist of mesoscopic coherent structures existing because of dissipative process, and evolving through a stochastic (strongly noisy) dynamics. Consequently, the self-consistent theory describing this intermediate level of plasma description is expected to

In this Chapter, three models to approach this "SCS Theory" have been exposed: metriplectic algebrization of MHD, stochastic field theory and fractal magnetic reconnection.

a filamentary structure mainly aligned to the inflow region (direction *i* in Figure 2).

non-space filling, self-similar nature of the reconnection region is simply assumed.

from calculable fluid-model processes (Zheng & Furukawa, 2010).

from extreme behaviours of the plasma as a fluid.

<sup>1</sup>

*<sup>D</sup> FRM <sup>D</sup> M R A m* . (34)

*number Rm*:

of the current sheet.

**5. Conclusion** 

be a theory of SCSs.

In 2007 Materassi & Consolini proposed a revised version of the historical Sweet-Parker model, in which the diffusion/current sheet region, where the magnetic reconnection takes place, is imagined like a fractal object in the plane. The very basic assumption of such a fractal reconnection model is that the reconnection active sites form a *not space-filling domain* rec contained in the diffusion region of measure *A*rec, and that such a non-space-filling domain is characterized by a Hausdorff dimension *D*H < *E*, being *E* the embedding dimension (here *E = 2*). Figure 2 shows a schematic view of the 2-dimensional geometry of the diffusion region.

Due to the fractal nature of the diffusion region, the constraint of flux conservation can be written as

$$\Phi\_{S\_{out}}^{eff}\begin{bmatrix}\mathbf{V}\end{bmatrix} = \Phi\_{S\_{in}}^{eff}\begin{bmatrix}\mathbf{V}\end{bmatrix} \tag{30}$$

where *Sin* (*Sout*) is the entrance (exit) surface for the plasma passing through the fractal domain rec. Here, the flux over the entrance and exit surfaces is given by the following expression,

$$\Phi\_{\rm S}^{\rm eff}[\mathbf{V}] = \int\_{\Omega\_{\rm S}} \mathbf{V} \cdot \hat{\mathbf{n}} d\mu\_{\Omega\_{\rm S}} \,\tag{31}$$

where *<sup>S</sup> d* is a proper elemental measure for the fractal domain *S*. Thus, the evaluation of such fluxes requires *an integration over a fractal domain*, which can be performed using the definitions by Tarasov (2005, 2006) involving irregular integrals.

According to the results shown in Tarasov (2006), if *f* is a regular function defined in **R***n* to be integrated over a fractal domain characterized by a Harsdorff dimension *D* < *n*, then the integration can be performed by introducing a proper weight function *<sup>D</sup>*, i.e.

$$\int\_{\Omega} f d\mu\_{\Omega} = \int\_{A\_{\Omega}} f \xi\_{\mathcal{D}} dA\_{\prime} \tag{32}$$

where *A* is the regular set of dimension *n* embedding the considered fractal set

When the above integration technique is applied to the condition of flux conservation (30), one gets for the *fractal reconnection rate*

$$M\_A^{FRM} = k \left(\frac{\sqrt{\pi}}{\ell\_0}\right)^{\delta} \frac{D\_{in}}{D\_{out}} \frac{\Gamma\left(\frac{D\_{in}}{2}\right)}{\Gamma\left(\frac{D\_{out}}{2}\right)} \frac{\Delta^{D\_{out}}}{L^{D\_{in}}}\,\mathrm{},\tag{33}$$

where *k* is a positive constant such that *Vout* = *kVA*, and *L* are the thickness (typically of the order of the ion-inertial length) and the length of the diffusion region, respectively, = *Dout - Din* is the difference of the Hausdorff dimensions of the projection of the fractal domain in the direction of the entrance (*Din*) and exit (*Dout*) directions, and finally *ℓ*0 is a *reference microscopic length scale*. Such a reference length scale has to be much smaller than the typical scales at which the medium displays fractal features (Tarasov, 2005).

52 Topics in Magnetohydrodynamics

In 2007 Materassi & Consolini proposed a revised version of the historical Sweet-Parker model, in which the diffusion/current sheet region, where the magnetic reconnection takes place, is imagined like a fractal object in the plane. The very basic assumption of such a fractal reconnection model is that the reconnection active sites form a *not space-filling domain* rec contained in the diffusion region of measure *A*rec, and that such a non-space-filling domain is characterized by a Hausdorff dimension *D*H < *E*, being *E* the embedding dimension (here *E = 2*). Figure 2 shows a schematic view of the 2-dimensional geometry of

Due to the fractal nature of the diffusion region, the constraint of flux conservation can be

[ ] [ ], *out in*

where *Sin* (*Sout*) is the entrance (exit) surface for the plasma passing through the fractal domain rec. Here, the flux over the entrance and exit surfaces is given by the following

[ ] <sup>ˆ</sup> , *<sup>S</sup> <sup>S</sup>*

is a proper elemental measure for the fractal domain *S*. Thus, the evaluation of

*<sup>S</sup> d*

such fluxes requires *an integration over a fractal domain*, which can be performed using the

According to the results shown in Tarasov (2006), if *f* is a regular function defined in **R***n* to be integrated over a fractal domain characterized by a Harsdorff dimension *D* < *n*, then the

*<sup>D</sup>* , *<sup>A</sup> fd f dA*

When the above integration technique is applied to the condition of flux conservation (30),

<sup>0</sup> <sup>2</sup>

*<sup>A</sup> <sup>D</sup> <sup>D</sup> out*

where *k* is a positive constant such that *Vout* = *kVA*, and *L* are the thickness (typically of the

*Din* is the difference of the Hausdorff dimensions of the projection of the fractal domain in the direction of the entrance (*Din*) and exit (*Dout*) directions, and finally *ℓ*0 is a *reference microscopic length scale*. Such a reference length scale has to be much smaller than the typical

 

*D*

*in*

 2

*D L*

*out in*

where *A* is the regular set of dimension *n* embedding the considered fractal set

*FRM in*

order of the ion-inertial length) and the length of the diffusion region, respectively,

*<sup>D</sup> M k*

scales at which the medium displays fractal features (Tarasov, 2005).

*eff*

definitions by Tarasov (2005, 2006) involving irregular integrals.

one gets for the *fractal reconnection rate*

integration can be performed by introducing a proper weight function

*eff eff S S* **V V** (30)

**V Vn** (31)

(32)

,

(33)

= *Dout -* 

*out*

*D*

*<sup>D</sup>*, i.e.

the diffusion region.

written as

expression,

where *<sup>S</sup> d*

Moving from the above result and assuming *k* = 1 and *Din* =*Dout* = *D*, Eq. (33) for the *fractal reconnection rate* can be reduced to a more simple expression in terms of the *Lundquist number Rm*:

$$M\_A^{FRM} = R\_m^{-\left(\frac{D}{D+1}\right)}.\tag{34}$$

We note that this expression reduces to the standard Sweet-Parker solution of the reconnection rate in the limit 1 *D* and that the fractal reconnection rate is always higher than the one predicted by the Sweet-Parker model. Furthermore, although in the limit *Rm* the reconnection rate predicted by the Petschek-like model results the more efficient, there exists always a certain range of the Lundquist number *Rm*, depending on the fractal dimension *D*, for which the fractal reconnection model is more efficient than the Petschek-like model. The crucial point of a correct estimation and applicability of the above expression stands in the correct evaluation of *Din* and *Dout*, which depends on the topology of the current sheet.

In passing we note that when the above scenario is applied using typical length scales estimated by *in-situ* observations of magnetic reconnection in space plasmas, one gets the reconnection rates typically observed and in agreement with the estimated Hall reconnection rate *MA* ≈ 0.09 (Huba & Rudakov, 2004) assuming a diffusion region shaped as a filamentary structure mainly aligned to the inflow region (direction *i* in Figure 2).

The fractal reconnection model described here is not based on first principles, because the non-space filling, self-similar nature of the reconnection region is simply assumed.

The important work necessary for further development will be to give a dynamical sense to the quantities *Din* and *Dout*, that here might appear just as convenient fitting parameters. Studies have been made to regard irregular filamentary structures in plasmas as descending from calculable fluid-model processes (Zheng & Furukawa, 2010).

The feeling is however that it would be very interesting to deduce the fractal nature of the reconnection region from kinetic or microscopic-statistical theories, rather than extracting it from extreme behaviours of the plasma as a fluid.
