Four-Fluid Axisymmetric Plasma Equilibrium Model Including Relativistic Electrons and Computational Method

A non-relativistic multi-fluid plasma axisymmetric equilibrium model1 was developed recently to account for the presence of an energetic electron fluid. The equilibrium formulation of a multi-fluid plasma with relativistic energetic electrons is developed and reported in this paper. The formulation is then applied to a four-fluid plasma composed of a relativistic energetic electron fluid, one thermal electron fluid, and fluids of two thermal ion species. This equilibrium model is relevant to the analysis of a toroidal hydrogenic plasma with a dominant impurity species (e.g., carbon) and an energetic electron component (e.g., runaway-like electrons).


INTRODUCTION
We adopt in this paper a fluid model to describe a plasma. If velocities of the fluid elements within small volumes are comparable to the speed of light, we must take into account of the wellknown relativistic effect. 2 In fluid motion, this effect also changes number density of the fluid element due to Lorentz contraction. 3 As the fluid element is macroscopic, it contains large number of constituting particles. A second relativistic effect is due to the random motion of particles with large velocities. This effect becomes significant when the species fluid temperature is comparable with or more than their rest mass energies. 3,4 Until recently we have described these electrons using nonrelativistic fluid model. 1 Here we develop axisymmetric equilibrium formulation of relativistic multifluid plasma model and apply the formulation to four-fluid equilibrium.
Strong relativistic electron components such as runaway electrons have been observed in many toroidal experiments. 5 Runaway-dominated discharges are qualitatively different from the normal discharges of tokamak operation, MHD activity levels often being lower than for normal discharge. 5 Energy of runaway electrons in recent tokamaks can reach O (10 1-2 ) MeV. 6 These relativistic runaway electrons have a great potential to carry a large fraction of plasma current and to cause severe damage on the plasma facing components of fusion reactors, being a matter of serious concern for all large tokamaks including ITER. 7 Relativistic electron component plays key role in non-inductive current ramp-up in spherical torus plasmas. 8,9 In these plasmas, almost all plasma current is carried by the energetic electron component. The hard X-ray spectra show that temperature of the bulk of energetic electron component is about 50 keV and the energetic electron component with energy tails up to 600 keV were detected. 9 With dual frequency ECH powers, hard X-ray temperature of an energetic electron component up to about 500 keV was measured. 10 Although the toroidal equilibria of relativistic electron beam plasmas have been studied in the large-aspect ratio limit, 11 our present model is appropriate for relativistic multi-fluid plasmas. As far as we know, the formulation of this type of multi-fluid plasma equilibrium is original. This paper is organized as follows. Section II describes equations of relativistic magnetofluid dynamics in three-dimensional form. It is shown that the generalized momentum of relativistic magnetofluid plays a fundamental role in a plasma fluid model. Section III describes axisymmetric equilibrium formulation for relativistic magnetofluid. The computational method for four-fluid plasma model is described in Section IV. Relevance of this equilibrium model to the analysis of today's toroidal plasmas containing a relativistic electron component is discussed in Section V, which also concludes the paper. Appendix provides a detailed formulation of the thermodynamic property of relativistic ideal fluids.

EQUATIONS OF RELATIVISTIC MAGNETOFLUID DYNAMICS
Although we study a multi-fluid plasma model in this paper, we discuss in this section the relativistic forms of the continuity equation and the equation of motion for any single species, neglecting subscript which depicts species for simplicity of notation. The continuity equation is given by .
Here n is the density in the proper frame in which the fluid volume element concerned is at rest, is the fluid velocity and . Note that is the density in the laboratory frame in which the fluid volume element concerned moves with and the factor results from Lorentz contraction. 3 The equation of fluid motion is given by , where m and q are the mass and charge of a particle consisting of the species fluid concerned, p is the pressure, E and B are the electric and magnetic fields, respectively, 3 and is defined by Here it is understood to take sum of j=1, 2 and 3 for the repeated j where , , and . , where is the energy density of the fluid, , 4 T is the temperature of the fluid in energy unit and (z) is the 2 nd kind modified Bessel function defined by .
See Appendix. Note that is the enthalpy density or the heat function per unit volume. 3 The relativistic forms of equation of continuity and equation of fluid motion are used in relativistic large-scale magnetic reconnection. 12 In the four-dimensional form of relativistic equations, the cgs-Gauss unit is convenient because E and B can transfer to each other by Lorentz transformation. As equations (1) and (2) where is the electrostatic potential and A is the magnetic vector potential. Using the continuity equation and the above equation in Eq.
(2) and multiplying 1/n, the resultant equation is given by , (6) where is used. See Appendix. Rewriting the right side of Eq. (6) as, where is the generalized momentum. In order to rewrite the last two terms in the L.H.S. of Eq. (7), we use the following vector identity, where and are arbitrary vectors. First take , and second take . Then, the last two terms in the L.H.S. of Eq. (7) can be written as , where u is the magnitude of the velocity u. Substituting Eq. (8) into Eq. (7) leads to . (9) This is the required form of equation of motion for a "fluid particle". As the "fluid particle" is macroscopic, it contains many microscopic particles. As a result, Eq. (9) describing the behavior of generalized momentum accounts for the effects of pressure gradient, electric field, relativistic macroscopic velocity through , and relativistic velocity of microscopic random motion through . For axisymmetric plasma, it is easily verified that Eq. (9) conserves the z-component of the generalized angular momentum, , i.e. Eq. (10) is satisfied.
This quantity plays a key role in axisymmetric equilibrium formulation.

III. AXISYMMETRIC EQUILIBIUM FORMULATION
For each species fluid, we adopt following two equations from Eq. (1) and Eq. (9). , , where . (13) In this paper we call the modified magnetic field. Maxwell's equations govern the fields where , (14) . (15) We adopt the charge neutrality condition instead of Poisson's equation, .
Hereafter we adopt the right-hand cylindrical coordinates . As the magnetic field , the modified magnetic field and the density flux are divergence-free, these can be expressed using flux or stream functions . , , .
Note that since is defined by Eq. (13), the following relation must be satisfied, , .
This gives for arbitrary R and Z. The above is satisfied when , i.e. the function is arbitrary function of .
We adopt following assumption, .
Validity of this assumption is discussed in Ref. 13. This is natural generalization of for flowing fluid. Key points are as follows: 1) Species poloidal flow velocity is much smaller than the same species thermal velocity, 2) For axisymmetric flowing plasma, is more fundamental than , and 3) Temperature anisotropy is negligible for TST-2 ohmic plasmas. 14,15 This assumption is used successfully for non-relativistic three-fluid equilibrium in Ref. i.e. the function is arbitrary function of .
Next, consider the force balance in -direction. This is given by .
Using Eq. (25), the above can be written as . (27) Hereafter we neglect the second term in Eq. (27). This is equivalent to put in the electric force term in Eq. (12). As the Lorentz force is much larger that the electric force in general, this neglect is not harmful. Define the function F as Within this approximation, Eq. (27) can be satisfied for arbitrary R and Z when , i.e. F must be arbitrary function of Y(R,Z). Finally, consider the force balance in the direction.
Using Eq. (29), the force balance equation (12) can be written as within the approximation Explicit expression of the charge neutrality condition becomes complex as number of fluids constituting a plasma increases. We will discuss its application in the next section.

IV. COMPUTAIONAL METHOD FOR FOUR-FLUID PLASMA MODEL
A plasma considered here consists of two ion species (proton and an arbitrary dominant impurity ion, e.g., carbon) and two electron species (low temperature and high density electron component and high temperature, high speed, and low density electron component). The subscript represent the proton fluid, the impurity ion fluid, the low temperature and high density electron fluid and the high temperature, high speed and low density electron fluid, respectively.
As for the eh-fluid, we adopt the relativistic form of equations derived in section III while as for the other three fluids, we adopt non-relativistic form of equations putting , and for because the non-relativistic limit takes , i.e. speed of light is infinity.

A. Dimensionless form of equilibrium equations
As dimensionless form is convenient in numerical computation we adopt dimensionless variables hereafter. The primary scales are (1) a reference length of a plasma, (2)

C. Computational method
Estimate of the 2 nd term of in Eq. (40a) using the non-relativistic three-fluid model 1 suggests that it is much less than the 1 st term . Therefore, we treat the 2 nd term as correction term including . The partial differential equation which must be solved is only Eq. (42) and the other equations are algebraic equations. It is assumed that the computational domain is rectangular; the poloidal magnetic flux loops are aligned along its boundary, where the poloidal magnetic flux data, , is experimentally measured. Eddy current is not considered for nearly stationary plasma conditions.
Step 1. Make suitable toroidal current density model, , and solve the equation (52) to find the 0 th order poloidal magnetic flux function which satisfies the boundary condition . .
Put , find the 0 th order densities, compute the toroidal velocities using Eq.(39), update , and compute the current densities using Eq. (41).
Step 3. Update , solving Eq. (42) with the boundary condition for prescribed .
Step 5. Solve Eq. (49) for density ratio iteratively for given other quantities.
Step 6. Update densities for .
Step 7. If convergence is not sufficient, return to Step 3 and iterate until solution with sufficient accuracy is found.

V. SUMMARY AND DISCUSSION
Based on the equations of continuity and motion for relativistic fluid species, we derived the equation for the generalized momentum, Eq. (9). This equation is equivalent to the equation of motion for a "fluid particle". As the "fluid particle" is macroscopic, it contains many microscopic particles.
As a result, Eq. (9) describing the behavior of generalized momentum accounts for the effects of pressure gradient, electric field, relativistic macroscopic velocity through , and relativistic velocity of microscopic random motion through . It is found that the z-component of generalized angular momentum, , defined in Eq. (10b), plays a key role in axisymmetric equilibrium formulation. The formulation is applied to a four-fluid equilibrium including two ion species such as proton and carbon and two electron fluid components. As a solution for the Grad-Shafranov equation requires two profile functions, the present four-fluid model requires twelve profile functions. The equations for densities which are consistent with the electrostatic potential are derived. Finally, we have shown a self-consistent method to calculate four-fluid axisymmetric equilibrium.
Our future work will apply the present model to the solenoid-free RF sustained ST plasmas such as in LATE 8 , QUEST 10 and particularly in the EXL-50 experiment, 16 where an energetic electron component and a prominent impurity species are present. Calculations based on this new model of plasma equilibrium where the energetic electrons become relativistic will also be valuable, in understanding the plasma force balance properties during the current decay phase following the thermal quench of a major disruption in a fusion tokamak such as ITER. 17 Establishing the theoretical equilibrium basis would help develop new ideas for disruption mitigation strategies.

ACKNOWLEDGMENTS
YKMP thanks Dr. Shi Yuejiang ( ) for helpful discussion regarding observations of tokamak plasmas that contain a substantial current component of runaway electrons. AI would like to thank Dr. Shi Yuejiang for discussion on X-ray spectroscopy of energetic electrons. 4 In this Appendix, let us consider energy density and pressure p. The thermal equilibrium can be described only in the proper frame in which the fluid volume element concerned is at rest.

APPENDIX: THERMODYNAMIC PROPERTY OF RELATIVISTIC IDEAL FLUID
Also note that as a particle energy includes a rest-mass energy in relativistic theory, therefore the internal energy density also includes the rest-mass energy.
The distribution function f of thermal equilibrium is given by where p is momentum of a particle, is degree of spin freedom, h is Plank's constant, E is energy of a particle, is chemical potential and T is temperature in energy unit. The -sign is for Bose particle and the + sign for Fermi particle in the denominator. For dilute gas such as plasma fluid considered here, the 1 term can be neglected.

. (A.2)
Since the proper frame is local and instantaneous, the distribution function f depends on time t and coordinate x, we neglect these two for simplicity.
Let us calculate the number density n, Since a particle's momentum and energy are given by where and , we have

DATA AVAILABILITY STATEMENT
Data sharing is not applicable to this article as no new data were created or analyzed in this study.