# Kinetic construction of the high-beta anisotropic-pressure equilibrium in
the magnetosphere

H. Aibara  Graduate School of Frontier Sciences, The University of Tokyo,
Kashiwa, Chiba 277-8561, Japan  Z. Yoshida  Graduate School of Frontier
Sciences, The University of Tokyo, Kashiwa, Chiba 277-8561, Japan  National
Institute for Fusion Science, Toki, Gifu 509-5292, Japan  K. Shirahata
Graduate School of Frontier Sciences, The University of Tokyo, Kashiwa, Chiba
277-8561, Japan

###### Abstract

A theoretical model of the high-beta equilibrium of magnetospheric plasma was
constructed by consistently connecting the (anisotropic pressure)
Grad–Shafranov equation and the Vlasov equation. The Grad–Shafranov equation
was used to determine the axisymmetric magnetic field for a given
magnetization current corresponding to a pressure tensor. Given a magnetic
field, we determine the distribution function as a specific equilibrium
solution of the Vlasov equation, using which we obtain the pressure tensor. We
need to find an appropriate class of distribution function for these two
equations to be satisfied simultaneously. Here, we consider the distribution
function that maximizes the entropy on the submanifold specified by the
magnetic moment. This is equivalent to the reduction of the canonical Poisson
bracket to the noncanonical one having the Casimir corresponding to the
magnetic moment. The pressure tensor then becomes a function of the magnetic
field (through the cyclotron frequency) and flux function, satisfying the
requirement of the Grad–Shafranov equation.

##  1 Introduction

The magnetosphere is a naturally made system confining a high-beta plasma [1].
A similar system may be created for fusion energy applications [2, 3].
Laboratory magnetospheres, namely, LDX [4] and RT-1 [5], demonstrated a stable
confinement of high-beta ($\sim$ 1) plasmas.

The aim of this study is to formulate a theoretical model of high-beta
equilibrium in the magnetosphere. Magnetized particles in an axisymmetric
magnetic field have three independent first integrals, facilitating an easy
construction of the equilibrium solutions of the drift kinetic equation. The
strong inhomogeneity of the dipole magnetic field is the key to understanding
the localization of magnetized particles to the vicinity of the magnetic
dipole [2]. However, in a high-beta plasma, the magnetic field must be
corrected by considering the diamagnetic current. We observe a significant
expansion of the dipole field due to plasma pressure, which can be used to
estimate the plasma pressure [6, 7]. At the first-order level, we may invoke
the Grad–Shafranov equation [8] to analyze the magnetic field of a finite-beta
plasma. However, this equation falls short of considering the strong
anisotropy of the distribution function, as the kinetic model makes
predictions for the magnetospheric system. The bouncing particles introduces
variations in the velocity distribution function along the field lines. The
anisotropic temperature is also demonstrated experimentally in RT-1 [9].
Appropriate corrections can be made by using the extended Grad–Shafranov
equation, which was developed to model the equilibrium of mirror systems [10,
11]. In fact, each magnetic flux tube in the magnetosphere may be viewed as a
crescent-shaped mirror system. The extended Grad–Shafranov equation, which is
still a macroscopic magneto-fluid model, considers an anisotropic pressure
that is a function of the two-dimensional magnetic coordinates, namely, the
flux function and magnetic field strength. While the functional form of the
pressure tensor remains arbitrary in such a fluid model, parametric studies on
the effect of anisotropic pressure have been performed using numerical
analysis [12, 13]. We have yet to build a consistent relationship between the
kinetic description and the magneto-fluid model, and to provide a physical
reason for selecting an appropriate form of the pressure tensor.

In the present study, we constructed a self-consistent model based on the idea
of the maximum entropy state in a topologically constrained phase space (or a
symplectic leaf foliated by Casimirs) [14]. In the context of magnetospheric
plasma confinement, the adiabatic invariant acts as a topological constraint
(Casimir of the noncanonical Hamiltonian mechanics [15]). Providing the
topological charge (in fact, the Casimir) with a chemical potential, we define
a grand canonical ensemble, on which we consider the Gibbs distribution. As
the magnetic moment is the relevant Casimir, the pair of chemical potential
and Casimir parallels that of the magnetic field and magnetization in the
well-known model of magnetic materials. Such a “thermal equilibrium” yields
the desired pressure tensor to be used in the generalized Grad–Shafranov
equation.

The remainder of this paper is organized as follows: In the following section,
we review the Hamiltonian mechanics of magnetized particles and derive the
stationary distribution function that describes the thermal equilibrium under
the topological constraint given by the magnetization. The corresponding
pressure tensor is used to formulate the generalized (anisotropic pressure)
Grad–Shafranov equation in Sec. 3. In Sec. 4, we show examples of numerical
solutions, as well as some relations useful to estimate the anisotropic
pressure effect. Section 5 concludes the paper.

##  2 Particle motion and magnetic field

To construct the high-beta equilibrium of magnetospheric plasma, we combine
two models: one is the kinetic model for calculating the distribution
function, and the other is the macroscopic magneto-fluid magnetohydrodynamics
(MHD) model for calculating the magnetic field. The distribution function is
given as a stationary solution of the Vlasov theory with appropriate coarse
graining. Evaluating the pressure tensor using the distribution function, we
solve the generalized (anisotropic pressure) Grad–Shafranov equation to
determine the magnetic field.

###  2.1 Hamiltonian of magnetized particles in magnetosphere

In an axisymmetric magnetic field (of sufficient strength), the dynamics of a
magnetized particle consist of three different periodic motions: gyro motion,
bounce motion, and drift motion. It is then convenient to span the phase space
by variables

$\bm{z}=(\theta_{g},\,\mu,\,;\,\ell,\,P_{\parallel}\,;\,\theta,\,P_{\theta}),$
(1)

where $\mu:=J_{g}q/m$ is the magnetic moment ($J_{g}$ is the action
corresponding to the gyro motion, $q$ is the particle charge, and $m$ is the
particle mass), $\theta_{g}$ is the gyro angle, $P_{\parallel}$ is the
canonical momentum parallel to the magnetic field, $\ell$ is the parallel
coordinate that constitutes the canonical pair with $P_{\parallel}$,
$P_{\theta}$ is the canonical angular momentum around the geometrical axis,
and $\theta$ is the azimuthal angle. (For convenience, we use
$\ell,\,P_{\parallel}$, instead of the bounce action-angle pair). Neglecting
the kinetic part of the canonical angular momentum, we approximate
$P_{\theta}=q\psi$ ($\psi=rA_{\theta}$ is the flux function in the cylindrical
coordinate system $(r,\theta,z)$, where $A_{\theta}$ is the $\theta$ component
of the vector potential).

Here, we consider two different “reductions” for the Hamiltonian. First, the
gyro angle $\theta_{g}$ is _coarse-grained_ and is eliminated from the
Hamiltonian (such a Hamiltonian only dictates the guiding center motion of the
magnetized particle). Then, we obtain

$\dot{\mu}=\frac{\partial H}{\partial\theta_{g}}=0.$ (2)

Second, the Hamiltonian is independent of the azimuthal angle $\theta$ because
we consider an axisymmetric magnetic field. With the approximation
$P_{\theta}=q\psi$, we obtain

$q\dot{\psi}=\dot{P_{\theta}}=\frac{\partial H}{\partial\theta}=0.$ (3)

The two constants $\mu$ and $\psi$ play an important role in later discussion.

The magnetic field may be written as $\bm{B}=\nabla\psi\times\nabla\theta$
(note that the toroidal magnetic field is absent in the magnetospheric
system). By adding the coordinate $\ell$ that measures the arc length on each
magnetic field line, we define a magnetic coordinate system
$(\ell,\psi,\theta)$. Through the axisymmetry, we can eliminate $\theta$. The
reader is referred to Refs. [16, 17] for the parameterization of magnetized
particles in a dipole magnetic field.

######  Remark 1 (The adiabatic invariant as a Casimir invariant)

In the view of the Poisson bracket, coarse-graining of the gyro angle is
represented by the modified Poisson matrix [14]

$\mathcal{J}_{\mu}=\begin{pmatrix}0&0&0\\\ 0&J_{c}&0\\\ 0&0&J_{c}\\\
\end{pmatrix},\>J_{c}=\begin{pmatrix}0&1\\\ -1&0\\\ \end{pmatrix}.$ (4)

Here, we consider the variables (1). Then, the modification of the Poisson
matrix makes the Poisson bracket

$\\{F,G\\}_{\mu}:=\langle\partial_{z}F,\,\mathcal{J}_{\mu}\partial_{z}G\rangle$
(5)

noncanonical and the adiabatic invariant $\mu$ is a Casimir invariant. The
foliation of the phase space induced by the Casimir invariant is essential in
our theory.

###  2.2 Kinetic distribution function

####  2.2.1 General form of stationary distribution function

In the Vlasov theory, the stationary distribution function $f$ is given by

$\\{H,f^{*}\\}=0,$ (6)

where $\\{~{},~{}\\}$ is the canonical Poisson bracket, and $f^{*}$ is the
Hodge dual of $f$, i.e., $f=f^{*}\mathrm{vol}$ with the phase-space volume
element $\mathrm{vol}=\mathrm{d}^{3}x\mathrm{d}^{3}v$. When $\\{G_{j},H\\}=0$,
$G_{j}$ is a constant of motion. A scalar function such as
$f^{*}(H,G_{1},\cdots,G_{n})$ gives a stationary distribution
$f=f^{*}\mathrm{vol}$, because

$\\{H,f^{*}(H,G_{1},\cdots,G_{n})\\}=\frac{\partial f^{*}}{\partial
H}\\{H,H\\}+\sum_{j=1}^{m}\frac{\partial f^{*}}{\partial
G_{j}}\\{G_{j},H\\}=0.$

Here, we use the aforementioned $\mu$ and $\psi$ to define a stationary
distribution function using

$f^{*}=g(H,\mu,\psi),$ (7)

where $g$ is an arbitrary function of $H$, $\mu$, and $\psi$.

####  2.2.2 Thermal equilibrium with topological constraints

Although the collisionless kinetic theory leaves infinite freedom in the
stationary distribution function, it provides us with the theoretical basis
for statistical mechanics to define the most probable distribution with the
appropriately defined entropy, that is, the invariant measure is determined by
the guide of the Poisson structure pertinent to the kinetic theory. Here, the
invariant measure is given on the symplectic leaf, defined as the level sets
of the two invariants $\mu$ and $\psi$.

We consider the Hamiltonian

$H=\mu B(\ell,\psi)+\frac{P_{\parallel}^{2}}{2m},$ (8)

where $B$ is the magnetic field. Here, we consider a quasi-neutral plasma
($\phi=0$) and neglect the kinetic energy of the toroidal drift velocity by
approximating $P_{\theta}=q\psi$. The first term $\mu B$ may be regarded as
the potential energy ($\mu$ is constant for each particle) on each contour of
$\psi$ (i.e., magnetic field line).

Leaving only $\psi$ as a free parameter characterizing the thermal
nonequilibrium of the system, we consider the thermal equilibrium (maximum
entropy) distribution function such that (see Appendix A)

$f^{*}=A(\psi)\exp\biggl{(}-\frac{H}{T_{\parallel
0}}\biggr{)}\exp\biggl{(}-\mu B_{0}(\psi)\frac{T_{\parallel 0}-T_{\perp
0}}{T_{\parallel 0}T_{\perp 0}}\biggr{)},$ (9)

where the two constants $T_{\parallel 0}$ and $T_{\perp 0}$ represent the
parallel and perpendicular temperatures at $\ell=0$, respectively. We also
define $B_{0}(\psi):=B(0,\psi)$, where $B(\ell,\,\psi)$ is the magnetic field
strength evaluated as a function of $\ell$ and $\psi$. Then, by introducing
$v_{\parallel}$ and $v_{\perp}$ (which represent the parallel and
perpendicular velocity with respect to the magnetic field) and substituting
(8) and $\displaystyle\mu=\frac{mv_{\perp}^{2}}{2B(\ell,\,\psi)}$ in (9), we
can rewrite (9) as

$f^{*}=A(\psi)\exp\biggl{\\{}-\biggl{(}\frac{mv_{\parallel}^{2}}{2T_{\parallel}(\ell,\psi)}+\frac{mv_{\perp}^{2}}{2T_{\perp}(\ell,\psi)}\biggr{)}\biggr{\\}},$
(10)

where

$\displaystyle T_{\parallel}(\ell,\psi)$ $\displaystyle=T_{\parallel 0},$ (11)
$\displaystyle T_{\perp}(\ell,\psi)$ $\displaystyle=\frac{T_{\parallel
0}T_{\perp 0}}{T_{\perp 0}+\frac{B_{0}(\psi)}{B(\ell,\psi)}(T_{\parallel
0}-T_{\perp 0})}.$ (12)

Therefore, our distribution function is a Maxwellian with varying anisotropic
temperatures (11) and (12). Multiplying the phase-space volume form, we obtain

$\displaystyle f$ $\displaystyle=f^{*}\mathrm{d}^{3}x\,\mathrm{d}^{3}v$
$\displaystyle=f^{*}\mathrm{d}^{3}x\,\frac{D(v_{x},v_{y},v_{z})}{D(v_{\parallel},v_{\perp},\theta_{g})}\mathrm{d}v_{\parallel}\mathrm{d}v_{\perp}\mathrm{d}\theta_{g}$
$\displaystyle=v_{\perp}A(\psi)\exp\biggl{\\{}-\biggl{(}\frac{mv_{\parallel}^{2}}{2T_{\parallel}(\ell,\psi)}+\frac{mv_{\perp}^{2}}{2T_{\perp}(\ell,\psi)}\biggr{)}\biggr{\\}}\mathrm{d}^{3}x\,\mathrm{d}v_{\parallel}\mathrm{d}v_{\perp}\mathrm{d}\theta_{g}.$

Transforming variables in the volume form, we rewrite

$\displaystyle f$ $\displaystyle(\ell,\psi,v_{\parallel},v_{\perp})=$
$\displaystyle 2\pi
v_{\perp}A(\psi)\exp\biggl{\\{}-\biggl{(}\frac{mv_{\parallel}^{2}}{2T_{\parallel}(\ell,\psi)}+\frac{mv_{\perp}^{2}}{2T_{\perp}(\ell,\psi)}\biggr{)}\biggr{\\}}\mathrm{d}^{3}x\,\mathrm{d}v_{\parallel}\mathrm{d}v_{\perp}.$

The corresponding configuration space density is

$\displaystyle n(\ell,\psi)$ $\displaystyle=\iint
f(\ell,\psi,v_{\parallel},v_{\perp})\,\mathrm{d}v_{\parallel}\mathrm{d}v_{\perp}$
(18)
$\displaystyle=A(\psi)\biggl{(}\frac{2\pi}{m}\biggr{)}^{\frac{3}{2}}T_{\parallel}^{\frac{1}{2}}(\ell,\psi)\,T_{\perp}(\ell,\psi),$

and the parallel and perpendicular components of the pressure tensor are

$\displaystyle p_{\parallel}(\ell,\psi)$ $\displaystyle=\iint
f(\ell,\psi,v_{\parallel},v_{\perp})\,mv_{\parallel}^{2}\mathrm{d}v_{\parallel}\mathrm{d}v_{\perp}$
(20)
$\displaystyle=A(\psi)\biggl{(}\frac{2\pi}{m}\biggr{)}^{\frac{3}{2}}T_{\parallel}^{\frac{3}{2}}(\ell,\psi)T_{\perp}(\ell,\psi),$
$\displaystyle p_{\perp}(\ell,\psi)$ $\displaystyle=\frac{1}{2}\iint
f(\ell,\psi,v_{\parallel},v_{\perp})\,mv_{\perp}^{2}\mathrm{d}v_{\parallel}\mathrm{d}v_{\perp}$
(22)
$\displaystyle=A(\psi)\biggl{(}\frac{2\pi}{m}\biggr{)}^{\frac{3}{2}}T_{\parallel}^{\frac{1}{2}}(\ell,\psi)T_{\perp}^{2}(\ell,\psi).$

Finally, we represent the pressures and functions of the magnetic coordinates
$\psi$ and $B$, and using (11) and (12), we obtain

$\displaystyle p_{\parallel}(\psi,B)$
$\displaystyle=A(\psi)\biggl{(}\frac{2\pi}{m}\biggr{)}^{\frac{3}{2}}T_{\parallel
0}^{\,\frac{5}{2}}\,\frac{T_{\perp 0}}{T_{\perp
0}+\frac{B_{0}(\psi)}{B}(T_{\parallel 0}-T_{\perp 0})},$ (23) $\displaystyle
p_{\perp}(\psi,B)$
$\displaystyle=A(\psi)\biggl{(}\frac{2\pi}{m}\biggr{)}^{\frac{3}{2}}T_{\parallel
0}^{\,\frac{5}{2}}\biggl{(}\frac{T_{\perp 0}}{T_{\perp
0}+\frac{B_{0}(\psi)}{B}(T_{\parallel 0}-T_{\perp 0})}\biggr{)}^{2}.$

##  3 Finite-beta equilibrium

The anisotropic pressures (23) and (2.2.2), evaluated for the thermal
equilibrium on the symplectic leaf of the magnetic moment, can now be used in
the generalized Grad–Shafranov equation. The pressure tensor is

$\bm{P}=\bm{b}\bm{b}p_{\parallel}+(\bm{I}-\bm{b}\bm{b})p_{\perp}$ (25)

where $p_{\parallel}$ and $p_{\perp}$ are the parallel and perpendicular
pressures, respectively, and $\bm{b}$ is the unit vector parallel to $\bm{B}$.
The MHD equilibrium equations are

$\displaystyle(\nabla\times\bm{B})\times\bm{B}=\mu_{0}\nabla\cdot\bm{P},$ (26)
$\displaystyle\nabla\cdot\bm{B}=0,$ (27)

where $\mu_{0}$ is the vacuum permeability. In the axisymmetric magnetospheric
system (which has no toroidal magnetic field), we can convert the MHD
equilibrium equation into a generalized (anisotropic pressure) Grad–Shafranov
equation [10, 11]:

$\Delta^{*}\psi=-\mu_{0}r^{2}\left.\frac{\partial
p_{\parallel}}{\partial\psi}\right|_{B}-\frac{1}{\sigma}\nabla\psi\cdot\nabla\sigma$
(28) $\left.\frac{\partial p_{\parallel}}{\partial
B}\right|_{\psi}+\frac{p_{\perp}-p_{\parallel}}{B}=0,$ (29)

where $\Delta^{*}$ denotes the Grad–Shafranov operator

$\Delta^{*}:=r^{2}\nabla\cdot(\frac{1}{r^{2}}\nabla),$ (30)

and $\sigma$ is defined as

$\sigma:=1+\mu_{0}\frac{p_{\perp}-p_{\parallel}}{B^{2}}.$ (31)

We notice that the equilibrium equations (28) and (29) can also be obtained as
equilibrium equations of the CGL-MHD model[18].

Note that (23) and (2.2.2) satisfy the second equation (29). We only need to
solve (28) to determine $\psi$.

##  4 Numerical analysis

In this section, we show the results of the numerical analysis based on a
numerical code RTEQ (Ring Trap EQuilibrium) developed by Furukawa [13].

###  4.1 Setting and calculation model

As we assume axisymmetry, we consider the 2D-plane shown in fig. 1. The
magnetic field lines correspond to the $\psi$ contour, and $l=0$ on the
magnetic field lines corresponds to the point where $z=0$ at the outer side of
the ring current. Here, $\psi_{1}$ and $\psi_{2}$ denote the magnetic field
line that comes in contact with the fixed limiters that provide the boundary
of the plasma. The internal ring current (which is implemented by the
levitated superconducting coil) generates the vacuum magnetic field.

Figure 1:  Schematic of the 2D-plane considered in the calculation

Normalized with the typical length $L_{1}$, magnetic field $B_{1}$, pressure
$p_{1}$, and magnetic flux $\Psi_{1}:=B_{1}L_{1}^{2}$, the Grad–Shafranov
equation is expressed as follows:

$\check{\Delta}^{*}\check{\psi}=-\frac{\beta_{o}}{2}\check{r}^{2}\left.\frac{\partial\check{p}_{\parallel}}{\partial\check{\psi}}\right|_{\check{B}}-\frac{1}{\check{\sigma}}\check{\nabla}\check{\psi}\cdot\check{\nabla}\check{\sigma},$
(32)

where we apply (23) and (2.2.2) as the parallel and perpendicular components
of the pressure tensor, respectively, which are rewritten as

$\displaystyle\check{p}_{\parallel}(\psi,\,B)$
$\displaystyle=\bar{p}(\check{\psi})\frac{\lambda_{0}}{\lambda_{0}+\frac{\check{B}_{0}(\psi)}{\check{B}}(1-\lambda_{0})},$
(33) $\displaystyle\check{p}_{\perp}(\psi,\,B)$
$\displaystyle=\bar{p}(\check{\psi})\biggl{(}\frac{\lambda_{0}}{\lambda_{0}+\frac{\check{B}_{0}(\psi)}{\check{B}}(1-\lambda_{0})}\biggr{)}^{2}.$
(34)

The boundary condition is $\psi=0$ at infinity. The equation contains two
parameters: $\displaystyle\beta_{o}:=\frac{2\mu_{0}p_{1}}{B_{1}^{2}}$ and
$\displaystyle\lambda_{0}:=\frac{T_{\perp 0}}{T_{\parallel 0}}$. Here,
$\lambda_{0}$ represents the temperature anisotropy on $l=0$. Moreover,
$\bar{p}(\check{\psi})$ corresponds to the normalized pressure in the
isotropic case, which allows arbitrary functions of $\check{\psi}$, and
$\check{B}_{0}(\psi):=\check{B}(0,\psi)$ is the magnetic field strength at
$\ell=0$ as we defined in Sec. 2.2.2. Here, we assume

$\bar{p}(\check{\psi})\propto-(\check{\psi}-\check{\psi_{1}})^{P}(\check{\psi}-\check{\psi}_{2})^{Q}.$
(35)

Although we can treat $P$ and $Q$ as parameters, we fix $P=1,\,Q=1$ in this
study for simplicity. Then, we obtain

$\bar{p}(\check{\psi}):=-\frac{4(\check{\psi}-\check{\psi}_{1})(\check{\psi}-\check{\psi}_{2})}{(\check{\psi}_{2}-\check{\psi}_{1})^{2}}.$
(36)

###  4.2 Effects of anisotropic temperature on the equilibrium states

First, we analyze the cases where $\lambda_{0}\geq 1$. We show the
distributions of $p_{\parallel}$ and $p_{\perp}$ in fig. 2 and those of
$\psi_{p}$ and $\beta$ in fig. 3, which correspond to the equilibrium states
calculated for $\beta_{o}=4.5\times 10^{-5}$ and (top) $\lambda_{0}=1.0$,
(middle) $\lambda_{0}=1.5$, (bottom) $\lambda_{0}=2.0$. Here,
$\displaystyle\beta:=\frac{2\mu_{0}p}{B^{2}}$ and $\psi_{p}:=\psi-\psi_{v}$,
where $\psi_{v}$ denotes the flux function associated with a vacuum magnetic
field. As shown in fig. 2 when $\lambda_{0}=1$, the distributions of pressure
become isotropic ($\check{p}_{\parallel}=\check{p}_{\perp}=\bar{p(\psi)}$).
Therefore, $p_{\parallel}$ and $p_{\perp}$ are constant along the magnetic
field lines. On the other hand, when $\lambda_{0}>1$, $p_{\parallel}$ and
$p_{\perp}$ are higher at the outer side of the ring current near $z=0$.
Moreover, as $\lambda_{0}$ increases, the distributions are more concentrated
at the outer side of the ring current near $z=0$, and the maximum local values
of $p_{\parallel}$ and $p_{\perp}$ become higher. Even for $\psi_{p}$ and
$\beta$, the distributions are more concentrated at the outer side of the ring
current near $z=0$, and the maximum local values become higher as
$\lambda_{0}$ increases; see fig. 3.

Figure 2:  Distributions of $p_{\parallel}$ and $p_{\perp}$, which correspond
to the equilibrium states calculated for $\beta_{o}=4.5\times 10^{-5}$ and
(top) $\lambda_{0}=1.0$, (middle) $\lambda_{0}=1.5$, (bottom)
$\lambda_{0}=2.0$ Figure 3:  Distributions of $\psi_{p}$ and $\beta$, which
correspond to the equilibrium states calculated for $\beta_{o}=4.5\times
10^{-5}$ and (top) $\lambda_{0}=1.0$, (middle) $\lambda_{0}=1.5$, (bottom)
$\lambda_{0}=2.0$

Next, we analyze the cases where $\lambda_{0}<1$. We show the distributions of
$p_{\parallel}$ and $p_{\perp}$ in fig. 4 and those of $\psi_{p}$ and $\beta$
in fig. 5, which correspond to the equilibrium states calculated for
$\beta_{o}=4.5\times 10^{-5}$ and (top) $\lambda_{0}=0.50$, (bottom)
$\lambda_{0}=0.25$. In line with the cases in which $\lambda\geq 1$,
$p_{\parallel}$ and $p_{\perp}$ are higher at the inner side of the ring
current near $z=0$. Moreover, as $\lambda_{0}$ becomes smaller, the
distributions are more concentrated at the inner side of the ring current near
$z=0$, but the maximum local values of $p_{\parallel}$ and $p_{\perp}$ become
lower (see fig. 4). Regarding $\psi_{p}$ and $\beta$, the distributions are
concentrated at the outer side of the ring current, and the maximum local
values become lower as $\lambda_{0}$ becomes smaller; see fig. 5.

Figure 4:  Distributions of $p_{\parallel}$ and $p_{\perp}$, which correspond
to the equilibrium states calculated for $\beta_{o}=4.5\times 10^{-5}$ and
(top) $\lambda_{0}=0.50$, (bottom) $\lambda_{0}=0.25$ Figure 5:  Distributions
of $\psi_{p}$ and $\beta$, which correspond to the equilibrium states
calculated for $\beta_{o}=4.5\times 10^{-5}$ and (top) $\lambda_{0}=0.50$,
(bottom) $\lambda_{0}=0.25$

Finally, we show the basic idea of applying our theoretical model to
experimental studies, which will be future works. When we apply the model to
experiments, we have to determine the value of $\beta_{o}$ and $\lambda_{0}$.
By measuring the magnetic induction on a toroidal flux loop (fig. 6), we can
evaluate $\psi_{p}$ at the corresponding coordinate $r$ and $z$. By fitting
the numerical solutions of the model and experimental $\psi_{p}$ at multiple
points, we can estimate the parameters (see fig. 7-8). Here, we set
$r_{1}=1.01,\,z_{1}=0.35,\,z_{2}=0.20$.

Figure 6:  Schematic of the flux loops  Figure 7:  Relation between
$\lambda_{0}$ and $\psi_{p}$ where $\beta_{o}=1.2\times 10^{-5}$ Figure 8:
Relation between $\beta_{o}$ and $\psi_{p}$ where $\lambda_{0}=2.0$

##  5 Conclusion

To describe the high-beta equilibrium of a magnetospheric plasma, we need a
consistent relation between the magnetic field (to be modified by the current
in the plasma) and the phase-space distribution function (to be influenced by
the magnetic field). The former is dictated by the field equation, and the
latter is dictated by kinetic theory. We can use the (generalized)
Grad–Shafranov equation as the field equation that determines the magnetic
flux function for a given magnetization (diamagnetic) current. However, the
Grad–Shafranov equation has an additional (in fact, essential) implication,
that is, the internal relation between the magnetic field and the
magnetization current, which is imposed by the magneto-fluid force–balance
relation. Therefore, we need to find a special class of distribution functions
that does not create inconsistencies with the macroscopic magneto-fluid model.
In the present study, we showed that the “thermal equilibrium” on the
topologically constrained phase space (foliated by the adiabatic invariant
$\mu$) is suitable for the generalized Grad–Shafranov equation, which is not
only amenable, but also definitive for the functional form of the pressure
tensor. However, we have left the flux function $\psi$ as a free parameter
that can control the “radial” profile of the pressure tensor. Hasegawa [2]
suggested that $\partial_{\psi}f^{*}=0$, because $P_{\theta}\sim\psi$ is the
most fragile constant influenced by low-frequency ($\sim$ drift frequency)
perturbation. In fact, we observe the “inward diffusion” of particles,
consistent with the relaxation toward $\partial_{\psi}f^{*}=0$ [4, 9].
However, in a real system, the boundaries (located both inside and outside the
confinement domain) deform the distribution from the ideal one. In the present
study, we maintained $\psi$ as an experimental parameter to model the
nonequilibrium property. The theoretical model constructed in the present work
will be useful to analyze the experimental data of dipole systems or satellite
data of planetary magnetospheres. The function $A(\psi)$ will characterize the
non-equilibrium property of the real system in comparison with the simple
model of the relaxed state [2]. The two parameters $\beta_{o}$ (measuring the
beta) and $\lambda_{0}$ (characterizing the temperature anisotropy) can be
determined, in experiments, by measuring the variation of magnetic flux
$\psi_{p}$ using flux loops. We observe a variety of density and pressure
profiles depending on plasma parameters [19]. Detailed experimental analysis
will be discussed in future work.

## Acknowledgment

The authors thank Professor M. Furukawa and Mr. K. Ueda for their support in
the numerical calculations. This study was supported by JSPS KAKENHI (grant
number 17H01177).

##  Appendix A Grand canonical ensemble

The function form of (9) implicitly assumes maximum entropy states under the
appropriate constraints formulated in [14].

When we consider the grand canonical ensemble determined by the Casimir
(magnetic moment) $C=\int\mu f^{*}\mathrm{d}^{n}z$ in addition to the total
particle number $N=\int f^{*}\mathrm{d}^{n}z$ and total energy $E=\int
Hf^{*}\mathrm{d}^{n}z$, the equilibrium state in which entropy $S=-\int
f^{*}\log f^{*}\mathrm{d}^{n}z$ is maximized is calculated as

$\delta(S-\alpha N-\beta E-\gamma C)=0,$ (37)

which yields a Boltzmann distribution

$f^{*}=Z^{-1}\exp(-\beta H-\gamma\mu),$ (38)

where $Z:=\exp(\alpha+1)$ is the normalized factor and
$\alpha,\,\beta,\,\gamma$ are Lagrange multipliers. The Euler-Lagrange
equation of (37) is formally the same as that of a different variational
principle $\delta(E+\alpha_{1}S+\alpha_{2}N+\alpha_{3}C)=0$, which may be
interpreted as the “Energy-Casimir” functional used in the stability theory of
non-canonical Hamiltonian systems (entropy $S$ may be regarded as a Casimir of
the Vlasov Lie-Poisson algebra). Here, we invoke (37) to define the
statistical equilibrium on the “energy shell” that is constrained by $C$.

In our formulation, we also consider $\psi$ as a constant of motion. Then,
$Z,\,\beta$ and $\gamma$ can be the functions of $\psi$. However, in this
study, we consider $\beta$ to be a constant in (9) for simplicity.

When we do not consider the magnetic moment as a constraint, the equilibrium
state obtained via entropy maximization changes drastically. The distribution
function is obtained as

$f^{*}=Z^{-1}\exp(-\beta H),$ (39)

which is equivalent to the distribution function obtained by $T_{\parallel
0}=T_{\perp 0}=T_{0}$ in (9). The distribution function yields the pressure as

$p_{\parallel}=p_{\perp}=A(\psi)\biggl{(}\frac{2\pi}{m}\biggr{)}^{\frac{3}{2}}T_{\parallel
0}^{\,\frac{5}{2}},$ (40)

which is isotropic and the function of $\psi$ (not $\psi$ and $B$). This
simple exercise shows that the factor that causes the nontrivial structure
along the magnetic field at the same time as entropy maximization is the
constraint of the magnetic moment in the view of statistical mechanics.

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