**4. Dissipative system of equations in the approximation of two adiabatic invariants of Chu, Goldberger, Low in the drift approximation.**

In order to obtain a complete system of hydrodynamic equations in the drift approximation, it is necessary to add Maxwell's equations to the equation of motion (19), and to close the system, add two equations of state for the parallel *ρII* and perpendicular *ρ*<sup>⊥</sup> components of the pressure tensor, as is done in the approximation of two adiabatic invariants of the ChGL [14]. If one equation for is a consequence of the applicability of the drift approximation and corresponds to the constancy of the first adiabatic invariant ð Þ¼ *dμ=dt* 0, then the second equation can be obtained on the basis of the energy conservation law in the drift approximation [17]

$$\frac{d\varepsilon}{dt} = \varepsilon \left(\stackrel{\leftarrow}{E} \cdot \stackrel{\leftarrow}{U}\_{dr}\right) + \mu \frac{\partial H}{\partial t},\tag{20}$$

where *<sup>ε</sup>* <sup>¼</sup> *<sup>ε</sup>II* <sup>þ</sup> *<sup>ε</sup>*<sup>⊥</sup> <sup>¼</sup> *<sup>m</sup>ν*<sup>2</sup> *II=*<sup>2</sup> <sup>þ</sup> *<sup>m</sup>ν*<sup>2</sup> <sup>⊥</sup>*=*<sup>2</sup> is particle mean energy, *Udr* is drift velocity. From (20) we obtain

$$\frac{d\varepsilon\_{\mathrm{II}}}{dt} = \varepsilon \left(\overleftarrow{\boldsymbol{E}} \cdot \overleftarrow{\boldsymbol{U}}\_{dr}\right) + \mu \frac{\partial H}{\partial \boldsymbol{t}} - \frac{d\varepsilon\_{\perp}}{dt} = \varepsilon \left(\overleftarrow{\boldsymbol{E}} \cdot \overleftarrow{\boldsymbol{U}}\_{dr}\right) - \mu \left(\overleftarrow{\boldsymbol{U}}\_{dr} \cdot \overleftarrow{\boldsymbol{\nabla}}\right) \mathrm{H}.\tag{21}$$

Since

$$\frac{d(n\varepsilon\_{II})}{dt} = \varepsilon\_{II}\frac{dn}{dt} + n\frac{d\varepsilon\_{II}}{dt}$$

and *pII* ¼ 2*nεII*, *p*<sup>⊥</sup> ¼ *nε*<sup>⊥</sup> and *n*\_ ¼ �*ndiνU dr* are valid, than from (20,21) and the latest expression we obtain

$$\frac{dp\_{\mathrm{II}}}{dt} = 2ne\left(\stackrel{\frown}{E} \cdot \stackrel{\frown}{U}\_{dr}\right) - \frac{2p\_{\perp}}{H}\left(\stackrel{\frown}{U}\_{dr} \cdot \stackrel{\frown}{\nabla}\right)H - p\_{\mathrm{II}}d\stackrel{\frown}{\nu}\stackrel{\frown}{\mathbf{U}}\_{dr}$$

or

$$\frac{dp\_{II}}{dt} + p\_{II}\mathrm{d}\boldsymbol{\nu}\overleftarrow{\boldsymbol{U}}\_{dr} = 2\mathrm{ne}\left(\overleftarrow{\boldsymbol{E}} \cdot \overleftarrow{\boldsymbol{U}}\_{dr}\right) - \frac{2p\_{\perp}}{H}\left(\overleftarrow{\boldsymbol{U}}\_{dr} \cdot \overleftarrow{\boldsymbol{\nabla}}\right)\mathrm{H}.\tag{22}$$

Relation (22) is a substantial balance equation in the drift approximation for the pressure tensor component *ρII* with a nonzero right-hand side (the presence of a source). We multiply the left-hand side of (22) by *H*<sup>2</sup> *=ρ*<sup>3</sup> and, taking into account that *ρdiνU dr* ¼ �ð Þ *dρ=dt* , we obtain after transformations

$$\frac{\partial H^2}{\rho^3} \left( \frac{dp\_{\text{II}}}{dt} - \frac{p\_{\text{II}}}{\rho} \frac{d\rho}{dt} \right) = \frac{H^2}{\rho^3} \frac{dp\_{\text{II}}}{dt} + \frac{p\_{\text{II}}H^2}{\rho^2} \frac{d}{dt} \left( \frac{1}{\rho} \right) = \frac{H^2}{\rho^3} \frac{dp\_{\text{II}}}{dt} + p\_{\text{II}} \frac{d}{dt} \left( \frac{H^2}{\rho^3} \right) = \frac{d}{dt} \left( \frac{p\_{\text{II}}H^2}{\rho^3} \right).$$

*Application of Onsager and Prigozhin Variational Principles of Nonequilibrium… DOI: http://dx.doi.org/10.5772/intechopen.103116*

Now, after multiplying the right part (22) by *H*<sup>2</sup> *=ρ*<sup>3</sup> � �, we equate this product to the latest equation. Finally, we obtain

$$\frac{d}{dt}\left(\frac{p\_{\text{II}}H^2}{\rho^3}\right) = \frac{H^2}{\rho^3}\left[2ne\left(\stackrel{\frown}{E}\cdot\stackrel{\frown}{U}\_{dr}\right) - \frac{2p\_\perp}{H}\left(\stackrel{\frown}{U}\_{dr}\cdot\stackrel{\frown}{\nabla}\right)H\right].\tag{23}$$

The condition

$$\frac{d}{dt}\left(\frac{p\_\perp}{\rho H}\right) = 0,\tag{24}$$

equivalent to the condition of the first adiabatic invariant conservation, (since *v*2 <sup>⊥</sup> ≈*p*⊥*=ρ*, where *v*<sup>⊥</sup> is a perpendicular component of particle mean velocity), together with (23) are two condition equations for the parallel *pII* and *p*<sup>⊥</sup> perpendicular components of pressure tensor, which close the dissipative system of equations in drift approximation.

Now, the first part of (23) is under analysis. Since we consider plasma systems in axialsymmetrical magnetic fields with potential electric field equal to zero, than in the stationary case *E* ¼ 0, *U dr* � ∇ � � *<sup>H</sup>* � *<sup>U</sup>φ*ð Þ¼ *<sup>∂</sup>H=∂<sup>φ</sup>* 0 and the right part (23) identically vanish. In variable fields, in our case *E U dr* � � <sup>¼</sup> *<sup>E</sup>φUφ*, since for electric field *<sup>E</sup> <sup>φ</sup>* ¼ � <sup>1</sup> *c ∂A φ <sup>∂</sup><sup>t</sup>* and

$$\left(\overleftarrow{U}\_{dr}\cdot\overleftarrow{\nabla}\right)H \approx \overleftarrow{U}\_{R}\frac{\partial H}{\partial R} = \frac{U\_{R}H}{R\_{cr}} = \frac{U\_{R}H}{R}k,$$

where *k* is a coefficient of proportionality between field line curvature radius *Rcr* and guiding center radius-vector *R* [18, 19], *UR* ¼ *c E*ð Þ *=H* is electric drift velocity. In the result, for the right part of (23) we obtain

$$2\frac{H^3}{\rho^3}n\cdot U\_{\varrho}\cdot U\_R\left(\frac{e}{c}n - \frac{\mu\cdot k}{R\cdot U\_{\varrho}}\right).$$

According to the results of the papers [18, 19], we have

$$\frac{RU\_{\phi}}{k} = \frac{c}{e}\mu + \frac{\nu\_{II}^{2}}{co\iota\_{L}} = \frac{c}{enH}\left(p\_{\perp} + p\_{H}\right).$$

and, finally, for (23)) we may write

$$\frac{d}{dt}\left(\frac{p\_{\text{II}}H^2}{\rho^3}\right) = \left(\frac{p\_{\text{II}}H^2}{\rho^3}\right) \cdot \frac{2nU\_{\text{\textquotedblleft}}}{p\_\perp + p\_{\text{II}}} \cdot eE\_{\text{\textquotedblright}},\tag{25}$$

The total equation system of two adiabatic invariant approximations, considering *f dis* in the approximation of ideal conductivity *E<sup>φ</sup>* � *V* , *H* h i , is written as follows:

$$\begin{split} \frac{d\overleftarrow{V}}{dt} &= -\frac{1}{nm} \overset{\textstyle \textstyle H \dot{\nu}}{} \overset{\textstyle \textstyle H \dot{\nu}}{} + \overset{\textstyle \textstyle \textstyle T}{}\_{\text{ext}} + \overset{\textstyle \textstyle \textstyle T}{}\_{\text{div}} \left( \overset{\textstyle \textstyle \textstyle T}{}\_{\text{j}} \overset{\textstyle \textstyle \textstyle T}{}\_{\text{j}} \right), \frac{\partial \textbf{n}}{\text{d}t} = -\overset{\textstyle \textstyle \textstyle T}{} \overset{\textstyle \textstyle \textstyle T}{}\_{\text{j}} \overset{\textstyle \textstyle \textstyle H \dot{\nu}}{} \end{split} \tag{26}$$
 
$$\frac{d}{dt} \frac{d\left(\frac{p\_{\text{II}}H}{\rho^{3}}\right)}{} = \left(\frac{p\_{\text{II}}H^{2}}{\rho^{3}}\right) \cdot \frac{2nU\_{\text{\textquotedblleft}}}{p\_{\text{\textquotedblleft}} + p\_{\text{II}}} \cdot e \overset{\textstyle \textstyle \textstyle H \dot{\}}{}\_{\text{\textquotedblleft}} = \overset{\textstyle \textstyle \textstyle T}{}\_{\text{\textquotedblleft}} \overset{\textstyle \textstyle \neg}{}\_{\text{\textquotedblleft}} \overset{\textstyle \textstyle \neg}{}\_{\text{\textquotedblleft}} \overset{\textstyle \neg}{}\_{\text{\textquotedblleft}} \overset{\textstyle \neg}{}\_{\text{\textquotedblleft}} \overset{\textstyle \neg}{}\_{\text{\textquotedblleft}} \overset{\textstyle \neg}{}\_{\text{\textquotedblleft}} \overset{\textstyle \neg}{}\_{\text{\textquotedblleft}} \overset{\textstyle \neg}{}\_{\text{\textquotedblleft}} \overset{\textstyle \neg}{}\_{\text{\textquotedblright}} \overset{\textstyle \neg}{}\_{\text{\textquotedblleft}} \overset{\textstyle \neg}{}$$

$$\begin{split} \text{where } & \mathbf{-}(\boldsymbol{P})\_{kn} = \boldsymbol{p}\_{\boldsymbol{II}} \overleftarrow{\boldsymbol{e}}\_{k} \overleftarrow{\boldsymbol{e}}\_{n} + \boldsymbol{p}\_{\perp} \left( \boldsymbol{\delta}\_{kn} - \overleftarrow{\boldsymbol{e}}\_{k} \overleftarrow{\boldsymbol{e}}\_{n} \right), \overleftarrow{\boldsymbol{F}}\_{\text{ext}} = \frac{1}{\boldsymbol{H}} \left[ -\boldsymbol{p}\_{\perp} \overleftarrow{\nabla}\_{\boldsymbol{II}} \boldsymbol{H} + (\boldsymbol{p}\_{\boldsymbol{II}} - \boldsymbol{p}\_{\perp}) \overleftarrow{\nabla}\_{\boldsymbol{L}} \boldsymbol{H} \right], \\ \boldsymbol{\overleftarrow{f}}\_{\text{dis}} = \frac{2\boldsymbol{\sqrt{\boldsymbol{V}}}}{\boldsymbol{m}\boldsymbol{H}^{\dagger}} \left[ 2\frac{\boldsymbol{p}\_{\perp}}{\boldsymbol{h}^{2}} \left( \overleftarrow{\nabla} \boldsymbol{H} \left[ \overleftarrow{\boldsymbol{R}} \overleftarrow{\boldsymbol{H}} \overleftarrow{\boldsymbol{H}} \right] \right) - \boldsymbol{p}\_{\boldsymbol{II}} \left( \boldsymbol{m} \overleftarrow{\boldsymbol{H}} \overleftarrow{\boldsymbol{\tau}}\_{\boldsymbol{L}} \overleftarrow{\boldsymbol{H}} \boldsymbol{H} \right) \right]. \end{split}$$

Let us multiply the first equation in the system (26) scalarly by *V*

$$\left(\overleftarrow{\boldsymbol{V}} \cdot \frac{d\overleftarrow{\boldsymbol{V}}}{dt}\right) = -\frac{1}{nm}\left(\overleftarrow{\boldsymbol{V}} \cdot \boldsymbol{D}\boldsymbol{i}\boldsymbol{v}\overleftarrow{\boldsymbol{P}} + \left(\overleftarrow{\boldsymbol{V}} \cdot \overleftarrow{\boldsymbol{F}\_{\text{ext}}}\right) + \left(\overleftarrow{\boldsymbol{V}} \cdot \overleftarrow{\boldsymbol{f}}\_{\text{dis}}\left(\overleftarrow{\boldsymbol{F}}, \overleftarrow{\boldsymbol{j}}\right)\right)\right)$$

Since we are interested in the influence of the dissipative term on the character of motion of a plasma element with macroscopic velocity *V* , let us assume for simplicity that the scalar product of the first two terms is early to zero, then, given the explicit form *f dis*, we obtain

$$\frac{1}{2}\frac{dV^2}{dt} = \frac{2cV^2}{enH^3}\left[2\frac{P\_\perp}{R^2}\left(\overleftarrow{\nabla}H\left[\overleftarrow{R},\overleftarrow{H}\right]\right) - p\_\text{II}\left(rot\overleftarrow{H}\cdot\overleftarrow{\nabla}\_\perp H\right)\right].$$

The last expression shows that in the case of fluctuations, azimuthal inhomogeneity may appear and, as a consequence, to the coincidence of the direction of the components *F <sup>m</sup>* and *j <sup>c</sup>*, and *j <sup>m</sup>* (see (19) and explanations to it), then, depending on the sign of the term in square brackets, the energy of the plasma element will increase or change.

### **5. Conclusions**

The right parts of the functions *F ext* and *f dis* are expressed through drift current explicit values separating the components of pressure tensor *p*<sup>⊥</sup> and *pII*. In the system (26) the unknown values are *p*⊥, *pII*, *H* , *E <sup>φ</sup>*, *n* and *V* .

Obtaining theoretical models describing the motion of continuous systems is an important branch of continuum mechanics. The construction of these models is based both on the use of experimental data and on the application of the well-known principles of mechanics, thermodynamics, physics, and they are based on the search for additional relationships between the parameters describing the state of the considered continuous medium. It is known that the basic equations of mechanics, electrodynamics, hydrodynamics, and so on are derived on the basis of the variational Lagrange equation. The corresponding analysis shows that with the help of variational principles it is possible to construct any physical models describing both reversible and non-reversible processes. Therefore, the application of the principles of Prigogine and Onsager, combined by Gyarmati, to obtain the equation of motion of a magnetized plasma at the hydrodynamic level of description seems to be quite promising. And here the following should be noted.

In the hydrodynamic approximation, fluctuations are not taken into account, since in continuum mechanics it is assumed to be continuous. The Navier-Stokes equation, in contrast to the Euler equation, already takes into account dissipative phenomena, but does not contain fluctuation interactions (without additional assumptions about the form of the stress tensor that takes into account the molecular structure), which describe Brownian motion. Relying on the concept of continuity, as already noted, it is
