J. Plasma
J.
Plasma Fusion
Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 6(2004)(2004)169–172
000–000

Relaxation with High-Speed Plasma Flows andSingularity Analysis in MHD Equilibrium
SHIRAISHI Junya, OHSAKI Shuichi and YOSHIDA Zensho
Grad. Sch. of Frontier Sci., the University of Tokyo, Kashiwa 277-8561, Japan(Received: 9 December 2003 / Accepted: 26 April 2004)

Abstract
Relaxation model that leads to plasma confinement with rigid-rotation is presented. This model applies to Jupiter’smagnetosphere. It is shown that the invariance of canonical angular momentum of electron fluid, which is realized byaxisymmetry through self-organization process, yields plasma confinement. Including poloidal flows in equilibrium
equation makes the problem rather complicated. Singularity due to the poloidal flow is focused on. It is shown thatthe singular equation for equilibrium has the same structure as the equation for linear Alfvén wave. Since the singularsolution for equilibrium equation is physically inadequate, the singularity may be removed by another physical effect.
The Hall-effect is taken into account as a singular perturbation that removes the singularity of equilibrium equation
for ideal magnetohydrodynamics.
Keywords:
relaxation, Jupiter, equilibrium, singularity, singular perturbation

1. Introduction

linear Alfvén wave. It is expected that the singularity isremoved by another physical effect because singular solution
for the equilibrium problem may be physically inadequate. It
has been shown that the Hall-effect can remove the singularity
in the equation for linear Alfvén wave [4]. We will discuss
the possibility of Hall-effect to remove the singularity in theequation for equilibrium.

We construct a relaxation model that leads to plasmaconfinement with rigid-rotation. This research is motivated by
the observation that shows high-β plasma with rigid-rotationin Jupiter’s magnetosphere [1]. Since previous theoreticalmodels for Jupiter’s magnetosphere are based on thestationary solution of the ideal magnetohydrodynamics(MHD) equations, flow and pressure profiles are arbitrary. To
exclude these arbitrariness, we invoke relaxation theory.
Although it is known that a relaxed state with rigid-rotationis achieved by imposing the constraint on total angularmomentum [2], the plasma cannot be confined because of thecentrifugal force. To confine the plasma, we must impose
rather restricted condition: axisymmetry of magnetic field
during relaxation process, which assumes the constancy ofelectron’s canonical angular momentum. With thatassumption, it is shown that the plasma can be confined by
the Lorentz force arising from the difference between angularvelocities of electron and ion fluids.
Since the above equilibrium has only toroidal flow (rigidrotation), the equation has the same structure as the GradShafranov (GS) equation for static equilibrium. Including
poloidal flows requires to generalize the GS equation. It is
well known that the ideal-MHD equilibrium withincompressible flows has a singularity where poloidal flow
coincides with Alfvén speed defined by poloidal magnetic
field [3]. By using one-dimensional model, it is shown that
this singularity has the same structure as the equation for

2. Relaxation with high-speed plasmaflows
We consider a plasma that obeys ideal-MHD equationsin an axisymmetric domain Ω. The ideal-MHD equations,with standard notation in Alfvén units, are∂t B – ∇ × (V × B) = 0,(1)∂t V + (V · ∇)V = (∇ × B) × B – ∇p,(2)where the incompressible flow is assumed, ∇·V = 0. With theboundary condition n·B = 0, n·V = 0 (on ∂Ω), where n is aunit normal vector onto ∂Ω, the basic eqs. (1) and (2) havethe following global invariants; the total energy E = 0.5∫Ω (V 2+ B2)dx, the magnetic helicity H1 = 0.5∫Ω A·Bdx and the crosshelicity H2 = ∫ΩV·Bdx. Moreover, we have an invariant thatstems from the toroidicity of the domain, that is, total angularmomentum. In cylindrical coordinate r-θ-z, the total angularmomentum is defined as follows©2004 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchCorresponding author’s e-mail: shira@ppl.k.u-tokyo.ac.jp1169

Shiraishi J. et al., Relaxation with High-Speed Plasma Flows and Singularity Analysis in MHD Equilibrium∫

M = Ω reθ · Vd x.

magnetic and the flow fields are decoupled. In this case, theBernoulli law reads(3)

If n· eθ = 0 on ∂Ω is satisfied, M is conserved.
Using the above invariants, we apply the selforganization theory. It should be noted that E or H2 cannot be
selected as the target functional (TF) to be minimized because
both are equivalent in their fragility. The correct procedure isto search for the most smooth solution in the norm of thevorticities [5]. The TF should be F = 0.5∫Ω (B2 + |∇ × V |2)dx.
Minimizer of F under the constraint on E, H1, H2 and M isgiven by the variational principleδ (F – µ0 E – µ1H1 – µ2H2 – ω M) = 0,p – r 2ω′ 2/2 = const,

which means the pressure rises toward the plasma edge dueto the centrifugal force.
To confine the plasma, we will consider rather restrictedcase that the magnetic field is axisymmetric throughrelaxation process (allowing the representation B = ∇Ψ (r, z,t) × ∇θ + rBθ (r, z, t)∇θ ). This assumption may be justified
by the observation that the magnetic field of Jupiter’s innermagnetosphere is almost axisymmetric dipole [1]. Then, thecanonical angular momentum of electron fluid (or the total
poloidal flux)(4)

where µ j ( j = 0, 1, 2) and ω are Lagrange multipliers.
Variations of δ A and δ V, under the boundary conditions n ×
δ A = 0 and n × δ V = 0, yield the following Euler-Lagrangeequations(1 – µ0) ∇ × B – µ1B – µ2∇ × V = 0,∇ × (∇ × V) – µ0V – µ2B – rω eθ = 0.

∫3j =12µ 22ω
ωe z – r eθ ,µ 02 µ1µ0

12µ ω
B = ∑ ⎛⎜⎝ λ j2 – µ 0 ⎞⎟⎠ D j G j + 2 ez ,µ 0 µ1j =1 µ 2(15)is conserved because dP/dt = ∫∂Ω n · (Ψ V)ds = 0.
Relaxed state derived by similar scenario reads(5)V = rω′ eθ ,(16)–∆*Ψ – µ′1 2Ψ + r 2ω′1 = 0,(17)

(6)
∂ 2z

where ∆* = r ∂ r r ∂ r +is the usual Grad-Shafranov
operator and ω′1 is a constant related to P. The Bernoulli’s
law takes the form as–1(7)

p – r 2ω′ 2/2 + ω′1Ψ = const .

3(9)where µ′1, µ′2 and ω′ are constants determined by variational
principle (4). Euler-Lagrange equations for the formal
variational principle (9) are∇ × B – µ′1B – µ′2 ∇ × V = 0,(10)V – µ′2B – rω′ eθ = 0,(11)2µ ′2ω ′ez ,µ1′(12)2 µ′2 ω ′ez + rω ′eθ ,µ1′(13)(18)

The third term in the LHS of eq. (18) is the crucial effectcompared with eq. (14). This term stems from theconservation of electron’s canonical angular momentum andrepresents the Lorentz force arising from toroidal current
yielded by the different angular velocities of electron and ion
fluids. Typical solutions for the pressure profiles based on eq.
(14) and eq. (18) are shown in Figs. 1 and 2, respectively.
Figure 1 shows that the pressure rises toward the plasma edge,
while Fig. 2 shows that the pressure falls. From Fig. 2, it canbe said that axisymmetry is needed to confine the plasma andthat this model can apply to the qualitative explanation of thephenomena in Jupiter’s magnetosphere.

(8)

where ∇ × Gj = λj Gj. These solutions do not even satisfies
the boundary condition (because of the vertical field) nor
represent equilibrium. Next, we consider “adjustmentprocess,” which readsδ (E – µ′1H1 – µ′2H2 – ω′ M) = 0,

∫P = – Ω reθ · Adx = – Ω Ψdx

The general solution is given by a triple Beltrami field withvertical field and with rigid-rotation for VV = ∑ Dj G j –(14)which gives

B = C1 G1 –

1.2
10.8
0.6
0.4
0.2
02

V = µ′2 C1 G1 –54µ1′
1– µ′22

G1 . The state expressed by eqs. (12) andwhere ∇× G1 =(13), which represent an equilibrium state, is a special classof eqs. (7) and (8). Furthermore, to satisfy the boundary
condition, µ′2 = 0 or ω′ = 0 must be required. If ω′ = 0, thesolutions reduce to the well-known single-Beltrami field witha parallel flow. To obtain rigid-rotation, we put µ′2 = 0, whichyields force-free magnetic field and rigid-rotation. The

320Fig. 12170

1
120
3-1
45-2
67-3
89-4-5

A typical pressure profile with the centrifugal force (seeeq. (14)).

Shiraishi J. et al., Relaxation with High-Speed Plasma Flows and Singularity Analysis in MHD Equilibrium

For linear Alfvén wave, it has been shown that the Hall-effectcan change the continuous spectrum due to the singularity[4]. Thus, we consider the Hall-effect as a candidate toremove the singularity.
Hall-MHD is described by eq. (2) and1.2
10.8
0.6
0.4
0.2
001Fig. 223456789 10 -5-4-3-2-101234∂t B – ∇ × [(V – ε∇ × B) × B] = 05with ∇·V = 0 and ∇· B = 0. The coefficient ε is the measureof the ion skin depth li = M /µ 0 ne 2 (M is the ion mass).
Equilibrium model that is based on Hall-MHD and that
reproduces ideal-MHD equilibrium with the limit ε → 0 has
been obtained by Ilgisonis [7]. Ilgisonis’s model has two
characteristic directions that degenerate in the limit ε → 0,e.g., V and Ve = V – ε∇ × B. Writing Ve = ∇Φe(x, y) × ∇z +Vez(x, y)∇z and B* = B + ε∇ × V = ∇Ψ*(x, y) × ∇z + B*z∇z,equilibrium equations become

A typical pressure profile for axisymmetric relaxation(See eq. (18)).

3. Singularity in MHD equilibria
Since the relaxed state obtained in the previous section
has only toroidal flow (rigid-rotation), the governing equation(17) has the same structure as the GS equation. Including
poloidal flow makes the equilibrium problem rathercomplicated. Assuming the incompressible flow, we use V =∇Φ (x, y) × ∇z + Vz(x, y)∇z and B = ∇Ψ (x, y) × ∇z + Bz(x,y)∇z in Cartesian coordinate. The equation for ideal-MHDequilibria can be formulated as(1 – M 2 (Φ )) ∆ Φ – 1 M˙ 2(Φ) | ∇Φ |2 + H˙ (Φ ) = 0,2∆Φ – Ψ′ 2∆Φe – Ψ′ Ψ″ |∇Φe|2 + Ḟ + Vez∂ΦVez – δ φs /ε = 0,(24)

(19)

(25)

Vez = Vz + ε (Ψ′ ∆Φe + Ψ″ |∇Φe|2),(26)(1 – M12(Φ) – ε M12′(Φ)Bz + ε 2(···)) ∆Φ+ ε 2M12(Φ)∆(∆Φ) + ··· = 0,(27)

where M12(Φ) = 1/Ψ′ 2|Φe→Φ is the poloidal Alfvén Mach
number in the limit ε → 0. Although the term with the highest(fourth) derivative contains the coefficient M12, singularity isremoved as seen in eq. (27) because M12 ≠ 0 is assumed in
the present theory.

dΦ 0 d ⎡dφ(1 – Ψ 2(Φ 0 )) ⎤⎥dx dx ⎢⎣dx ⎦(20)

4. Summary

On the other hand, the equation for linear Alfvén wave has aform of

The essential point in the relaxation model presented inSec. 2 is the axisymmetry of the magnetic field throughrelaxation process. Under this restricted condition, thecanonical angular momentum of electron fluid is conserved.
In this case, the equilibrium equation for the relaxed state isslightly modified compared with the case of total angularmomentum constraint. Due to the difference of angularvelocity, the resultant Bernoulli’s law includes Lorentz force,and the plasma confinement is achieved. On the other hand,
the relaxed state of the ion and electron fluids with the sameangular velocity is related to a force-free magnetic field, andthe plasma is not confined because of the centrifugal force.
The physical meaning of the singularity in ideal-MHD
equilibrium equation is examined. By using one-dimensionalmodel, the structure of the singular equation is shown to be
principally identical to the equation for linear Alfvén wave.

d ⎡⎛ 2 2 2 ⎞ d ⎛ φ ⎞ ⎤2⎛22 2⎞ ⎛ φ ⎞⎢⎜ Ω – k By ⎟⎠ ⎜ ⎟ ⎥ – k ⎜⎝ Ω – k By ⎟⎠ ⎜ ⎟ = 0, (21)dx ⎝ Ω ⎠ ⎦dx ⎣⎝⎝Ω⎠
where φ is the stream function of the fluctuation of the flowand Ω = ω – kVy is the Doppler-shifted frequency [6]. Puttingω̃ = ω /k = 0 (equilibrium) with k ≠ 0, eq. (21) becomesVyΦe = Φ – εBz, Bz = VezΨ′ – φ′s ,where the prime denotes d/dΦe, Vz = Vz(Φ, Bz) and φs is thescalar potential. Here, the notation φs|Φe→Φ means that theargument of function φs(Φe) is formally changed by Φ, andthe notation δ φ s = φ| Ψ→Ψ * – φ ( Ψ ) and F = p + V 2 /2 –φs|Φe→Φ /ε is defined. Assuming ε << 1, we can expand eq.
(24) and the principal part becomes

where the dot denotes derivative with respect to Φ, M 2(Φ) =|∇Φ |2/|∇Ψ |2 = 1/Ψ̇ 2 is the poloidal Alfvén Mach number andH(Φ) = p + |∇Φ |2/2 + B2z /2. Equation (19) becomes singularat the point M 2 = 1. We examine the physical meaning of thesingularity. Assuming one-dimensional system (depending ononly x) and putting Φ = Φ 0 + φ (|φ | << |Φ 0|), we can linearize
eq. (19) as⎧d ⎡d 2Φ 0⎤ ⎫– ⎨ ⎢(1 – Ψ 2 (Φ 0 ))⎬φ = 0 .
dx 2 ⎥⎦ ⎭⎩ dx ⎣(23)d ⎡⎛22 ⎞ dφ ⎤⎥⎢⎜⎝1 – By /Vy ⎟⎠dx ⎦dx ⎣dVy ⎤ ⎫⎧d ⎡2⎛
22⎞– ⎨ ⎢⎛⎜⎝1 – By2 /Vy2 ⎞⎟⎠⎥ ⎬ φ – k ⎜⎝1 – By /Vy ⎟⎠ Vy φ = 0. (22)dx ⎦ ⎭⎩dx ⎣Since we can write dΦ0 /dx = –Vy and Ψ 2(Φ0) = B2y /V 2y ineq. (20), eq. (22) is identical to the principal part of eq. (20).
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Shiraishi J. et al., Relaxation with High-Speed Plasma Flows and Singularity Analysis in MHD Equilibrium[2] R.N. Sudan, Phys. Rev. Lett. 42, 1277 (1979).
[3] H. Tasso and G.N. Throumoulopoulos, Phys. Plasmas 5,2378 (1998).
[4] S. Ohsaki and S.M. Mahajan, Phys. Plasmas 11, 898(2004).
[5] Z. Yoshida and S.M. Mahajan, Phys. Rev. Lett. 88,095001 (2002).
[6] A. Ito et al., Phys. Plasmas 9, 4856 (2002).
[7] V.I. Ilgisonis, Plasma Phys. Control. Fusion 43, 1255
(2001).

The solution may be expected to be singular, which is
physically unrealizable. This singularity is expected to beremoved by another mechanism, e.g., Hall-effect. In this pointof view, the Hall effect as a singular perturbation is discussed.
The Hall-effect imposes rather complicated singular
perturbation, however, regular solutions can be obtained
because all points become regular.

References[1] S.M. Krimigis et al., Science 206, 977 (1979).

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