**2. The dissipation algebrized**

Dissipation is a crucial element of the physical mechanism leading to ISCs in plasmas, and dissipative terms already appear in the smooth deterministic MHD. Moreover, the presence of dissipation, together with non-linearity, is a fundamental mechanism in order for coherent structures to form (Courbage & Prigogine, 1983).

Many fundamental phenomena giving rise to plasma ISCs in nature, such as turbulence, magnetic reconnection or dynamo (Biskamp, 1993), are often described by MHD models containing dissipative terms. For instance, this can account for the finite resistivity of the plasma and/or the action of viscous forces.

Where does dissipation come from? Ultimately, MHD is derived from the Klimontovich equations, describing the dynamics of charged particles interacting with electromagnetic fields (Klimontovich, 1967). This is a Hamiltonian, consequently non-dissipative, system. Nevertheless, dissipative terms appear in some versions of MHD equations as a heritage of averaging and approximations carried out along the derivation procedure and which have spoilt the original Hamiltonian structure of the Klimontovich system. The presence of dissipative terms reflects a transfer of energy from the deterministic macroscopic fluid quantities into the microscopic degrees of freedom of the system, to be treated statistically, which lie outside a macroscopic fluid description. Such transfer of energy, in turn, implies an increase of the entropy of the system.

If dissipative terms are omitted, on the other hand, one expects the resulting MHD system to be Hamiltonian, with a conserved energy (the constant value of the Hamiltonian of the system) and a *conserved entropy*. Indeed, the non-dissipative version of MHD, usually 38 Topics in Magnetohydrodynamics

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

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

Dissipation is a crucial element of the physical mechanism leading to ISCs in plasmas, and dissipative terms already appear in the smooth deterministic MHD. Moreover, the presence of dissipation, together with non-linearity, is a fundamental mechanism in order for

Many fundamental phenomena giving rise to plasma ISCs in nature, such as turbulence, magnetic reconnection or dynamo (Biskamp, 1993), are often described by MHD models containing dissipative terms. For instance, this can account for the finite resistivity of the

Where does dissipation come from? Ultimately, MHD is derived from the Klimontovich equations, describing the dynamics of charged particles interacting with electromagnetic fields (Klimontovich, 1967). This is a Hamiltonian, consequently non-dissipative, system. Nevertheless, dissipative terms appear in some versions of MHD equations as a heritage of averaging and approximations carried out along the derivation procedure and which have spoilt the original Hamiltonian structure of the Klimontovich system. The presence of dissipative terms reflects a transfer of energy from the deterministic macroscopic fluid quantities into the microscopic degrees of freedom of the system, to be treated statistically, which lie outside a macroscopic fluid description. Such transfer of energy, in turn, implies

If dissipative terms are omitted, on the other hand, one expects the resulting MHD system to be Hamiltonian, with a conserved energy (the constant value of the Hamiltonian of the system) and a *conserved entropy*. Indeed, the non-dissipative version of MHD, usually

varies with the magnetic Reynolds number faster than the traditional one.

coherent structures to form (Courbage & Prigogine, 1983).

plasma and/or the action of viscous forces.

an increase of the entropy of the system.

precisely the idea of an ISC.

**2. The dissipation algebrized** 

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 space (Morrison, 1998).

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 treatment of microscopic stochastic collisions).

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 physical models of the form

$$
\partial\_t z^i = F\_H^i(z) + F\_D^i(z) \quad , \quad i = 1, \ldots, N,\tag{1}
$$

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 is Hamiltonian and consequently can be written as

$$\hat{\boldsymbol{\sigma}}\_{t}\boldsymbol{z}^{i} = \boldsymbol{F}\_{\boldsymbol{H}}^{i}\left(\boldsymbol{z}\right) = \left[\boldsymbol{z}^{i}, \boldsymbol{H}\left(\boldsymbol{z}\right)\right]\_{t} \tag{2}$$

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 necessarily a constant of motion.

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

Sub-Fluid Models in Dissipative Magneto-Hydrodynamics 41

*j ik k i h i kmn*

can be shown to possess a metriplectic formulation. In (5) we adopted a notation with *SO*(3)-

2 3

*ik ni mk nk mi ik mn ik mn <sup>V</sup>*

is the stress tensor, with *η* and *ν* indicate the viscosity coefficients, *κ* is the thermal conductivity, *T* the plasma temperature and *s* the entropy density per unit mass. In the limit *κ* = *ζ* = *ν* = *σik* = 0, one recovers the ideal MHD system treated by Morrison and Greene

 

 

*s Vs V BB T*

 

 

2

*T T T*

*k i i ik*

 

 

<sup>1</sup> <sup>2</sup> , <sup>2</sup>

*k i i*

 

(8)

 

*s*

**B V** (7)

2

(5)

(6)

with the fields

 

*m n*

2

1 , <sup>2</sup>

*i ki i k k*

,

*B BV BV V B B*

*t j ikh mn*

 

 

This is accomplished first, by identifying the dynamical variables *zi*

plays the role of a continuous 3-index). The Hamiltonian for ideal MHD is then

*V*

.

*s Vs*

*t j j t j*

 

*t k*

,

*B BV BV V B*

,

*i ki i k*

*B B <sup>p</sup> V VV B*

Morrison and Greene (1980) showed that the system (6) can indeed be cast in the form (2).

(**B**(**x**,*t*),**V**(**x**,*t*),*ρ*(**x**,*t*),*s*(**x**,*t*)) (here, the space coordinate **x** labels the dynamical variables and

The three addenda in the integrand correspond to the kinetic, magnetic and internal energy of the system, respectively. *U*(*ρ*,*s*) is related to the plasma pressure and the

<sup>2</sup> , . *U U p T*

The Poisson bracket giving rise to the frictionless (6) through the Hamiltonian (7) is given

2 2 <sup>3</sup> ,,, , . 2 2 *V B H s dx U s* 

*i ii j j j i tjj j j*

*i ii j j j i i*

*B B <sup>p</sup> V VV B*

,

*tjj j j*

indices, which turns out to be practical in this context. We specify that

 

*V*

*t k*

*t j*

 

(1980), reading:

temperature as:

by:

*noncanonical* and no pairs of conjugate variables can be identified. For such brackets, particular invariants, denoted as Casimir invariants, exist. These are quantities *C*(*z*) such that [*C*,*F*] = 0 for every *F*(*z*). Consequently ∂*tC* = [*C*,*H*] = 0 in particular, which shows that Casimir functions are indeed conserved quantities.

Energy conservation and entropy increase in metriplectic systems are "algebrized" via a generalized bracket and a generalized energy functional. More precisely, a metriplectic system is a system of the form

$$\left\{ \hat{\sigma}\_t z^i = \left( z^i, F(z) \right) \right\} = \left[ \left. z^i, F(z) \right\} \right] + \left( z^i, F(z) \right) \; \prime \tag{3}$$

where the metriplectic bracket {\*,\*} = [\*,\*] + (\*,\*) is obtained from a Poisson bracket [\*,\*] and a *metric* bracket (\*,\*). The latter is a bilinear, symmetric and semidefinite (positive or negative) operation, satisfying also the Leibniz property (strictly speaking, a symmetric semi-definite bracket (\*,\*) should be referred to as semi-metric). The metric bracket is also required to be such that (*f*,*H*) = 0, for every function *f*(*z*), with *H* being the Hamiltonian of the system: this means that dissipation does not alter the total energy, since this already includes a part accounting for the energy dissipated.

The function *F* in (3) is denoted as *free energy*, and is given by

$$F = H + \mathcal{X}C\_{\prime} \tag{4}$$

where C is a Casimir of the Poisson bracket, and λ is a constant.

In the cases of interest here, this *C* is chosen as the entropy of the microscopic degrees of freedom of the plasma, involved in the dissipation.

Let us assume the metric bracket be semi-definite negative (the case in which it is positive is completely analogous). The resulting metriplectic system possesses the following important properties:


Metriplectic structures have been identified for different systems as, for instance, Navier-Stokes (Morrison, 1984), free rigid body, Vlasov-Poisson (Morrison, 1986) and, in a looser sense, for Boussinesq fluids (Bihlo, 2008) and constrained mechanical systems (Nguyen & Turski, 2009). An algebraic structure for dissipative systems based on an extension of the Dirac bracket has been proposed by Nguyen and Turski (2001). They have also been used for identifying asymptotic vortex states (Flierl and Morrison, 2011).

Also the visco-resistive plasma falls into the category of complete systems. Indeed, the following version of the visco-resistive MHD equations

40 Topics in Magnetohydrodynamics

*noncanonical* and no pairs of conjugate variables can be identified. For such brackets, particular invariants, denoted as Casimir invariants, exist. These are quantities *C*(*z*) such that [*C*,*F*] = 0 for every *F*(*z*). Consequently ∂*tC* = [*C*,*H*] = 0 in particular, which shows that

Energy conservation and entropy increase in metriplectic systems are "algebrized" via a generalized bracket and a generalized energy functional. More precisely, a metriplectic

> , , , , *ii i i <sup>t</sup> z z Fz z Fz z Fz*

where the metriplectic bracket {\*,\*} = [\*,\*] + (\*,\*) is obtained from a Poisson bracket [\*,\*] and a *metric* bracket (\*,\*). The latter is a bilinear, symmetric and semidefinite (positive or negative) operation, satisfying also the Leibniz property (strictly speaking, a symmetric semi-definite bracket (\*,\*) should be referred to as semi-metric). The metric bracket is also required to be such that (*f*,*H*) = 0, for every function *f*(*z*), with *H* being the Hamiltonian of the system: this means that dissipation does not alter the total energy, since this already

> *FH C*

In the cases of interest here, this *C* is chosen as the entropy of the microscopic degrees of

Let us assume the metric bracket be semi-definite negative (the case in which it is positive is completely analogous). The resulting metriplectic system possesses the following important



Metriplectic structures have been identified for different systems as, for instance, Navier-Stokes (Morrison, 1984), free rigid body, Vlasov-Poisson (Morrison, 1986) and, in a looser sense, for Boussinesq fluids (Bihlo, 2008) and constrained mechanical systems (Nguyen & Turski, 2009). An algebraic structure for dissipative systems based on an extension of the Dirac bracket has been proposed by Nguyen and Turski (2001). They have also been used

Also the visco-resistive plasma falls into the category of complete systems. Indeed, the

quantities such as total linear or angular momenta can also be conserved);

(3)

, (4)

Casimir functions are indeed conserved quantities.

includes a part accounting for the energy dissipated.

freedom of the plasma, involved in the dissipation.

with *t* (Courbage & Prigogine, 1983); - isolated minima of *F* are *stable equilibrium points*.

properties:

The function *F* in (3) is denoted as *free energy*, and is given by

where C is a Casimir of the Poisson bracket, and λ is a constant.

for identifying asymptotic vortex states (Flierl and Morrison, 2011).

following version of the visco-resistive MHD equations

system is a system of the form

$$\begin{cases} \begin{aligned} \label{eq:T} \boldsymbol{\partial}\_{t} \boldsymbol{V}^{i} &= -\boldsymbol{V}^{k} \boldsymbol{\partial}\_{k} \boldsymbol{V}^{i} - \frac{1}{2\rho} \boldsymbol{\mathcal{o}}^{i} \boldsymbol{B}^{2} + \frac{B\_{k} \boldsymbol{\mathcal{S}}^{k} \boldsymbol{B}^{i}}{\rho} - \frac{\boldsymbol{\mathcal{o}}^{i} \boldsymbol{p}}{\rho} + \frac{\boldsymbol{\mathcal{o}}\_{k} \boldsymbol{\sigma}^{k}}{\rho}, \\ \begin{aligned} \label{eq:T} \boldsymbol{\mathcal{O}}\_{t} \boldsymbol{B}^{i} &= \boldsymbol{B}^{j} \boldsymbol{\mathcal{O}}\_{j} \boldsymbol{V}^{i} - \boldsymbol{B}^{j} \boldsymbol{\mathcal{O}}\_{j} \boldsymbol{V}^{j} - \boldsymbol{V}^{j} \boldsymbol{\mathcal{O}}\_{j} \boldsymbol{B}^{i} + \boldsymbol{\mathcal{J}} \boldsymbol{\mathcal{O}}^{2} \boldsymbol{B}^{i}, \\ \end{aligned} \end{cases} \end{cases} \tag{5}$$
 
$$\begin{aligned} \label{eq:T} \boldsymbol{\mathcal{O}}\_{t} \boldsymbol{\rho} = -\boldsymbol{\mathcal{O}}\_{j} \left(\rho \boldsymbol{V}^{j}\right) \end{aligned} \tag{6}$$
 
$$\begin{aligned} \label{eq:T} \boldsymbol{\mathcal{O}}\_{t} \boldsymbol{s} = -\boldsymbol{V}^{j} \boldsymbol{\mathcal{O}}\_{j} \boldsymbol{s} + \frac{\sigma\_{ik}}{\rho \boldsymbol{T}} \boldsymbol{\mathcal{O}}^{k} \boldsymbol{V}^{i} + \frac{\boldsymbol{\mathcal{L}}}{\rho \boldsymbol{T}} \boldsymbol{\varepsilon}\_{ikl} \boldsymbol{s} \end{aligned} \tag{7}$$

can be shown to possess a metriplectic formulation. In (5) we adopted a notation with *SO*(3) indices, which turns out to be practical in this context. We specify that

$$
\sigma\_{ik} = \left[ \eta \left( \mathcal{S}\_{ni} \mathcal{S}\_{mk} + \mathcal{S}\_{nk} \mathcal{S}\_{mi} - \frac{2}{3} \mathcal{S}\_{ik} \mathcal{S}\_{mn} \right) + \nu \mathcal{S}\_{ik} \mathcal{S}\_{mn} \right] \hat{\sigma}^m V^n
$$

is the stress tensor, with *η* and *ν* indicate the viscosity coefficients, *κ* is the thermal conductivity, *T* the plasma temperature and *s* the entropy density per unit mass. In the limit *κ* = *ζ* = *ν* = *σik* = 0, one recovers the ideal MHD system treated by Morrison and Greene (1980), reading:

$$\begin{cases} \begin{aligned} \label{10} \left. \begin{aligned} \mathcal{O}\_{t}V^{i} &= -V^{k}\partial\_{k}V^{i} - \frac{1}{2\rho}\mathcal{O}^{i}B^{2} + \frac{B\_{k}\mathcal{O}^{k}B^{i}}{\rho} - \frac{\mathcal{O}^{i}p}{\rho}, \\ \mathcal{O}\_{t}B^{i} &= B^{j}\mathcal{O}\_{j}V^{i} - B^{i}\mathcal{O}\_{j}V^{j} - V^{j}\mathcal{O}\_{j}B^{i}, \\ \mathcal{O}\_{t}\rho &= -\mathcal{O}\_{j}\left(\rho V^{j}\right), \\ \mathcal{O}\_{t}s &= -V^{j}\mathcal{O}\_{j}s. \end{aligned} \end{cases} \tag{6}$$

Morrison and Greene (1980) showed that the system (6) can indeed be cast in the form (2). This is accomplished first, by identifying the dynamical variables *zi* with the fields (**B**(**x**,*t*),**V**(**x**,*t*),*ρ*(**x**,*t*),*s*(**x**,*t*)) (here, the space coordinate **x** labels the dynamical variables and plays the role of a continuous 3-index). The Hamiltonian for ideal MHD is then

$$H[\mathbf{B}, \mathbf{V}\_{\prime}\rho\_{\prime}s] = \int d^3x \left[\frac{\rho V^2}{2} + \frac{B^2}{2} + \rho \mathcal{U}(\rho\_{\prime}s)\right].\tag{7}$$

The three addenda in the integrand correspond to the kinetic, magnetic and internal energy of the system, respectively. *U*(*ρ*,*s*) is related to the plasma pressure and the temperature as:

$$p = \rho^2 \frac{\partial \mathcal{U}}{\partial \rho}, \quad T = \frac{\partial \mathcal{U}}{\partial \mathbf{s}}.\tag{8}$$

The Poisson bracket giving rise to the frictionless (6) through the Hamiltonian (7) is given by:

Sub-Fluid Models in Dissipative Magneto-Hydrodynamics 43

11 11

*k n*

 

*f fg f TV V*

*i ik m m n*

This metric bracket can be decomposed into two parts. A "fluid" part, corresponding to its first two terms, which was shown to produce the viscous terms of the Navier-Stokes equations (Morrison, 1984), and a "magnetic" part, which accounts for the resistive terms. The proof that the above metric bracket satisfies the properties required by the metriplectic formulation has been given in Materassi & Tassi (2011). The *SO*(3)-tensors needed are

*ikmn ni mk nk mi ik mn ik mn*

2 2 <sup>3</sup> , ,, , 2 2 *V B F s dx Us s* 

Thanks to the metriplectic formulation, it appears evident that the dynamics of the complete visco-resistive MHD takes place on surfaces of constant energy but, unlike Hamiltonian systems, it crosses different surfaces of constant Casimirs. Choosing *C* = *S*, it becomes evident that the fact that the dynamics does not take place at a surface of constant Casimir reflects of course the presence of dissipation in the system, and in particular the increase in

conditions) correspond to equilibria of the system (5) (even if other equilibria are possible). These can be found by setting to zero the first variation of *F* and solving the resulting

0, 0, ,

(since it has been obtained as extremal of the free energy functional, this solution is also an equilibrium for ideal MHD). The equilibrium (13) is rather peculiar because it corresponds to a situation in which all the kinetic and magnetic energy have been dissipated and converted into heat. It ascribes a physical meaning to the constant *λ*, that corresponds to the

*T*

constant

*eq eq eq*

*eq eq eq*

*p Ts U*

**V B**

equation in terms of the field variables. These equilibrium solutions are given by

 

*k n*

.

 

*h*

*ikmn ikh mn*

 

The bracket (11) together with the free energy functional

produces the dissipative terms of the system (5).

Free extremal points of *F* in (12) (i.e., configurations at which one has

   

*f fg f TB B*

*k*

1 1

*BT s BT s*

 

*i ik m m n*

 

 

*VT s VT s*

   

> 

(11)

 

 . 

<sup>2</sup> , <sup>3</sup>

 

*F* = 0 regardless other

(13)

 

 **B V** (12)

   

> 

 

1 11 3 2 ,

*ikmn*

 *ikmn*

defined as:

entropy.

*k*

*f g f g dx T Ts Ts*

 

$$\begin{split} \left[f,g\right] &= -\int d^3x \left| \frac{1}{\rho} \mathcal{O}\_i \delta\left(\frac{\delta f}{\delta s} \frac{\delta g}{\delta V\_i} - \frac{\delta g}{\delta s} \frac{\delta f}{\delta V\_i}\right) + \\ &+ \frac{1}{\rho} \frac{\delta f}{\delta V\_i} \mathcal{E}\_{ijk} \varepsilon^{kmn} B^j \mathcal{O}\_m \left(\frac{\delta g}{\delta B^n}\right) + \frac{\delta f}{\delta B\_i} \varepsilon\_{ijk} \mathcal{O}^j \left(\frac{1}{\rho} \varepsilon^{kmn} B\_m \frac{\delta g}{\delta V^n}\right) + \\ &+ \frac{\delta f}{\delta \rho} \mathcal{O}\_i \left(\frac{\delta g}{\delta V\_i}\right) + \frac{\delta g}{\delta \rho} \mathcal{O}\_i \left(\frac{\delta f}{\delta V\_i}\right) - \frac{1}{\rho} \frac{\delta f}{\delta V\_i} \varepsilon\_{ikj} \varepsilon^{jmn} \frac{\delta g}{\delta V\_k} \mathcal{O}\_m V\_n \right). \end{split} \tag{9}$$

This bracket possesses Casimir invariants (e.g. Morrison, 1982, Holm et al., 1985), such as the magnetic helicity; particularly relevant in our context, the total *entropy* is defined as:

$$S = \int \rho \mathbf{s} d^3 \mathbf{x} \; . \tag{10}$$

*S* is conserved along the motion of the non-dissipative system (6).

Some observation should be made here about the role of the plasma entropy as a Casimir. Casimir are invariants that a theory shows because of the singularity of its Poisson bracket, which is not full-rank. Typically this can happen when a Hamiltonian system is obtained by reducing some larger parent one, which possesses some symmetry (see, e.g., Marsden & Ratiu, 1999, Thiffeault & Morrison, 2000). In the case of ideal MHD, the reduction which leads to the Poisson bracket (9), is the map leading from the Lagrangian to the Eulerian representation of the fluid (Morrison, 2009a). When the system of microscopic parcels is approximated as a continuum, its (Lagrangian or Eulerian) fluid variables (as the velocity **V**(**x**,*t*)) pertain to the centre-of-mass of the fluid parcels of size *d*3*x* within which they may be approximated as constants. However, fluids are equipped with some thermodynamic variable, as the entropy *s* per unit mass here, which represent statistically the degrees of freedom relative-to-the-centre-of-mass of the parcels in *d*3*x*. In the Lagrangian description, the value of the entropy per unit mass is attributed to each parcel at the initial time, and remains constant, for each parcel, during the motion. In the Eulerian description, the total entropy appears as a Casimir, after the reduction, and the symmetry involved in this case is the relabelling symmetry, which is related to the freedom in choosing the label of each parcel at the initial time. In this respect, it is worth recalling that this reduction process implies a loss of information (e.g. Morrison, 1986) in the sense that, through the Eulerian description, one can observe properties of the fluid at a given point in space, but cannot identify which parcel is passing at a given point at a given time.

In a sense, this observation renders the metriplectic a sub-fluid description, because those microscopic degrees of freedom interact with the continuum variables through the role of *S* in (10) in the metric part of the evolution.

If the dissipative terms are re-introduced into Eq. (6) and one goes back to Eq. (5), a *complete* system is obtained, in the sense that *H* in (7) doesn't change along the motion (5), while entropy *S* in (10) is increased (Morrison, 2009b).

Let us illustrate the metriplectic formulation for the system (5). The non-dissipative part of the dynamics is algebrized through the Hamiltonian (7) and the Poisson bracket (9). As far as the construction of the free energy *F* in (4) is concerned, the entropy *S* is taken as the Casimir *C*, whereas the *metric bracket* reads:

42 Topics in Magnetohydrodynamics

*kmn j j kmn ijk m n n ijk m*

*f gf <sup>g</sup> B B V B B V*

 

*i i ikj m n i ii k*

This bracket possesses Casimir invariants (e.g. Morrison, 1982, Holm et al., 1985), such as the

<sup>3</sup> *S sd x* 

Some observation should be made here about the role of the plasma entropy as a Casimir. Casimir are invariants that a theory shows because of the singularity of its Poisson bracket, which is not full-rank. Typically this can happen when a Hamiltonian system is obtained by reducing some larger parent one, which possesses some symmetry (see, e.g., Marsden & Ratiu, 1999, Thiffeault & Morrison, 2000). In the case of ideal MHD, the reduction which leads to the Poisson bracket (9), is the map leading from the Lagrangian to the Eulerian representation of the fluid (Morrison, 2009a). When the system of microscopic parcels is approximated as a continuum, its (Lagrangian or Eulerian) fluid variables (as the velocity **V**(**x**,*t*)) pertain to the centre-of-mass of the fluid parcels of size *d*3*x* within which they may be approximated as constants. However, fluids are equipped with some thermodynamic variable, as the entropy *s* per unit mass here, which represent statistically the degrees of freedom relative-to-the-centre-of-mass of the parcels in *d*3*x*. In the Lagrangian description, the value of the entropy per unit mass is attributed to each parcel at the initial time, and remains constant, for each parcel, during the motion. In the Eulerian description, the total entropy appears as a Casimir, after the reduction, and the symmetry involved in this case is the relabelling symmetry, which is related to the freedom in choosing the label of each parcel at the initial time. In this respect, it is worth recalling that this reduction process implies a loss of information (e.g. Morrison, 1986) in the sense that, through the Eulerian description, one can observe properties of the fluid at a given point in space, but cannot

In a sense, this observation renders the metriplectic a sub-fluid description, because those microscopic degrees of freedom interact with the continuum variables through the role of *S*

If the dissipative terms are re-introduced into Eq. (6) and one goes back to Eq. (5), a *complete* system is obtained, in the sense that *H* in (7) doesn't change along the motion (5), while

Let us illustrate the metriplectic formulation for the system (5). The non-dissipative part of the dynamics is algebrized through the Hamiltonian (7) and the Poisson bracket (9). As far as the construction of the free energy *F* in (4) is concerned, the entropy *S* is taken as the

 

*f gg f f g <sup>V</sup> V VV V*

*i i*

 

 

 

 

<sup>1</sup> .

 

  . (10)

(9)

 

*jmn*

 

 

1 1

 

 

magnetic helicity; particularly relevant in our context, the total *entropy* is defined as:

 

*i*

 

*S* is conserved along the motion of the non-dissipative system (6).

identify which parcel is passing at a given point at a given time.

in (10) in the metric part of the evolution.

entropy *S* in (10) is increased (Morrison, 2009b).

Casimir *C*, whereas the *metric bracket* reads:

*fg gf f g dx s sV sV*

*i i*

<sup>3</sup> <sup>1</sup> ,

 

 

 

$$\begin{split} \mathcal{I}\left(f,g\right) &= \frac{1}{\lambda} \Big[\boldsymbol{d}^{3}\mathbf{x} \Big[\kappa T^{2} \hat{\boldsymbol{\mathcal{O}}}^{k} \Big(\frac{1}{\rho T} \frac{\delta f}{\delta \mathbf{s}}\Big) \hat{\boldsymbol{\mathcal{O}}}\_{k} \Big(\frac{1}{\rho T} \frac{\delta g}{\delta \mathbf{s}}\Big) + \\ &+ T \Lambda\_{ikmn} \Big[\boldsymbol{\hat{\mathcal{O}}}^{i} \Big(\frac{1}{\rho} \frac{\delta f}{\delta V\_{k}}\Big) - \frac{1}{\rho T} \boldsymbol{\hat{\mathcal{O}}}^{i} \boldsymbol{V}^{k} \frac{\delta f}{\delta \mathbf{s}} \Bigg] \Big[\boldsymbol{\hat{\mathcal{O}}}^{m} \Big(\frac{1}{\rho} \frac{\delta g}{\delta V\_{n}}\Big) - \frac{1}{\rho T} \boldsymbol{\hat{\mathcal{O}}}^{m} \boldsymbol{V}^{n} \frac{\delta f}{\delta \mathbf{s}}\Big] + \\ &+ T \Theta\_{ikmn} \Big[\boldsymbol{\hat{\mathcal{O}}}^{i} \Big(\frac{\delta f}{\delta B\_{k}}\Big) - \frac{1}{\rho T} \boldsymbol{\hat{\mathcal{O}}}^{i} \boldsymbol{B}^{k} \frac{\delta f}{\delta \mathbf{s}} \Bigg] \Big[\boldsymbol{\hat{\mathcal{O}}}^{m} \Big(\frac{\delta g}{\delta B\_{n}}\Big) - \frac{1}{\rho T} \boldsymbol{\hat{\mathcal{O}}}^{m} \boldsymbol{B}^{n} \frac{\delta f}{\delta \mathbf{s}}\Big]\Big]. \end{split} \tag{11}$$

This metric bracket can be decomposed into two parts. A "fluid" part, corresponding to its first two terms, which was shown to produce the viscous terms of the Navier-Stokes equations (Morrison, 1984), and a "magnetic" part, which accounts for the resistive terms. The proof that the above metric bracket satisfies the properties required by the metriplectic formulation has been given in Materassi & Tassi (2011). The *SO*(3)-tensors needed are defined as:

$$
\begin{split}
\Lambda\_{ikmn} &= \eta \left( \mathcal{S}\_{ni} \mathcal{S}\_{mk} + \mathcal{S}\_{nk} \mathcal{S}\_{mi} - \frac{2}{3} \mathcal{S}\_{ik} \mathcal{S}\_{mn} \right) + \nu \mathcal{S}\_{ik} \mathcal{S}\_{mm'}, \\
\Theta\_{iknm} &= \mathcal{L}\_{ikh} \varepsilon^h{}\_{mm}.
\end{split}
$$

The bracket (11) together with the free energy functional

$$F[\mathbf{B}, \mathbf{V}, \rho, s] = \int d^3 \mathbf{x} \left[ \frac{\rho V^2}{2} + \frac{B^2}{2} + \rho \mathcal{U}(\rho, s) + \lambda \rho \mathbf{s} \right] \tag{12}$$

produces the dissipative terms of the system (5).

Thanks to the metriplectic formulation, it appears evident that the dynamics of the complete visco-resistive MHD takes place on surfaces of constant energy but, unlike Hamiltonian systems, it crosses different surfaces of constant Casimirs. Choosing *C* = *S*, it becomes evident that the fact that the dynamics does not take place at a surface of constant Casimir reflects of course the presence of dissipation in the system, and in particular the increase in entropy.

Free extremal points of *F* in (12) (i.e., configurations at which one has *F* = 0 regardless other conditions) correspond to equilibria of the system (5) (even if other equilibria are possible). These can be found by setting to zero the first variation of *F* and solving the resulting equation in terms of the field variables. These equilibrium solutions are given by

$$\begin{aligned} \mathbf{V}\_{c\eta} &= \mathbf{0}, \quad \mathbf{B}\_{eq} = \mathbf{0}, \quad T\_{c\eta} = -\mathcal{A}, \\ p\_{eq} &= \rho\_{eq} \left( \mathbf{T} \mathbf{s} - \mathcal{U} \right)\_{eq} = \text{constant} \end{aligned} \tag{13}$$

(since it has been obtained as extremal of the free energy functional, this solution is also an equilibrium for ideal MHD). The equilibrium (13) is rather peculiar because it corresponds to a situation in which all the kinetic and magnetic energy have been dissipated and converted into heat. It ascribes a physical meaning to the constant *λ*, that corresponds to the

Sub-Fluid Models in Dissipative Magneto-Hydrodynamics 45

(the choice of incompressible plasma is done for reasons to be clarified later). *ζ* is the resistivity tensor and *p* is the plasma pressure. The dynamical variables are the fields **V** and **B**. The viscosity *ν* is assumed to be zero. The form of *ζ* and *p*, and of the mathematical relationships among them (necessary to close the system (14)), depend on the microdynamics of the medium. Usually, constitutive hypotheses provide the information on the microscopic nature of the medium (Kelley, 1989). When the (at least local) thermodynamic

being *T* the local temperature field. Then, some heat equation is invoked for *T*, requiring

The aforementioned procedure will only give *ζ* and *p* regular quasi-everywhere. Instead, in the sub-fluid approach presented here, irregularities of *ζ* and *p* are explicitly considered by stating that these local quantities are *stochastic fields*, and by assigning their probability density functions (PDF). The probabilistic nature of the terms *ζ* and *p* will be naturally transferred to **B** and **V** through a suitable SFT. The following vector quantities are defined

these **Ξ**, **Δ**, and **Θ** are considered as *stochastic stirring forces*, and their probability density functional is assigned as some *Q*[**Ξ**,**Δ**,**Θ**]. The resistive MHD equations are then re-written as

> *i i ii j j tj j*

*ii i j jki*

*i hi i ijk*

*j kh*

 *J*

iid

**ΞΔΘ ΞΔΘ**

 iid iid

dyn

mechanics to the fractional kinetics reviewed in Zaslavsky (2002).

 

*t j jk*

*B BV V B V VV B*

,, ,, .

This scheme, clearly, is not self-consistent because the PDF of the noise terms must be assigned *a priori*, as the outcome of a microscopic dynamics not included in this treatment and not predictable by it. Plasma microscopic physics will enter through some PDF *P*dyn[*ζ*,*p*]: as far as *P*dyn[*ζ*,*p*] keeps trace of the plasma complex dynamics, this represents a (rather general) way to provide constitutive hypotheses. Then, the positions (16) are used to

, , ,, ,, . *pP p Q* **ΞΔΘ ΞΔΘ**

A closed form for *Q*[**Ξ**,**Δ**,**Θ**] should be obtained consistently with any microscopic dynamical theory of the ISC plasma, from the very traditional equilibrium statistical

Due to the presence of the stochastic terms **Ξ**, **Δ**, and **Θ** two important things happen: first of all, from each set of initial conditions, *many possible evolutions* of **B** and **V** develop according to

*Q*

the following *Langevin field equations*:

construct mathematically the passage:

,, :

*i i*

,

,

(17)

*J p*

(16)

*T pT* ,... , , 0, (15)

equilibrium is assumed, the constitutive hypotheses read something like:

 

other constitutive hypotheses about the specific heat of the plasma.

opposite of the homogeneous temperature the plasma reaches at the equilibrium. Other equilibria with non trivial magnetic or velocity fields can in principle be obtained by considering Casimir constants other than the entropy, and a different metric bracket, or simply by constraining the condition *F* = 0 onto some manifold of constant value for suitable physical quantities. Moreover, the boundary conditions for the system to work in this way must be such that all the fields behave "suitably" at the space infinity. All the results are obtained for a *visco-resistive isolated plasma*: indeed, all the algebraic relationships invoked hold if **V**, **B**, *ρ* and *s* show suitable boundary conditions, rendering visco-resistive MHD a "complete system".

Such metriplectic formulation conserves, in addition to the energy *H*, also the total linear momentum **P**, the total angular momentum **L** and the generator of Galileo's boosts **G**, which are defined by:

$$\mathbf{P} = \int \rho \mathbf{V} d^3 \mathbf{x}, \quad \mathbf{L} = \int \rho (\mathbf{x} \times \mathbf{V}) d^3 \mathbf{x}, \quad \mathbf{G} = \int \rho (\mathbf{x} - \mathbf{V}t) d^3 \mathbf{x}.$$

About these quantities **P**, **L** and **G**, it should be stressed that, besides modifying the scheme with other quantities conserved in the ideal limit, more interesting equilibria than (13) may be identified by conditioning the extremization of *F* to the initial finite values of the Galilean transformation generators.
