25F09

Anisotropic Pressure Eﬀects on Diamagnetic Current
in a Magnetospheric Plasma Equilibrium
磁気圏型プラズマ平衡における反磁性電流に対する非等方圧力効果
Shota Iizuka, Masaru Furukawa
飯塚 将太, 古川勝
Grad. Sch. Frontier Sci., Univ. Tokyo, 5-1-5, Kashiwa-no-ha, Kashiwa-shi, Chiba 277-8561, Japan東京大学大学院 新領域創成科学研究科 先端エネルギー工学専攻 〒277-8561 千葉県柏市柏の葉 5-1-5
We studied eﬀects of pressure anisotropy on magnetospheric equilibria, especially focusing on the
diamagnetic current. The current has two contributions; one is from the pressure anisotropy, and theother remains in isotropic equilibria. Since the anisotropic part depends on the gradient of magnetic
field rather than the pressure, it can be significant in magnetospheric configurations. If the pressure
becomes large, the isotropic part changes magnetic field due to the well-known diamagnetic eﬀect.
This nonlinearity results in the change of magnetic field gradient to further increase the importancefor the anisotropic part.

1. Background and Motivation
Magnetospheric plasma confinement has beenstudied in Ring-Trap 1(RT-1) experiment for example. Electron cyclotron heating(ECH) heats
only the perpendicular direction to magnetic
filed, so pressure anisotropy is important forequilibrium[1]. In the pressent paper, we study
eﬀects of pressure anisotropy on magnetosphericequilibria, epsecially focusing on the diamagnetic
current.
2. Equilibrium Equation and Diamagnetic
Current Including Anisotropic Pressure
Balanced equation of single fluid is
0 = j × B − ∇ · p.

(1)

One devide this pressure into pararrel and perlendicular component to the magnetic field likep = pk bb + p⊥ (I − bb), where b := B/B. So,
one can derive the diamagnetic current including
anisotropic pressure.
j⊥ =1
B 2 − µ0 (pk − p⊥ )()pk − p⊥2× B × ∇p⊥ +B × ∇B
B2(2)Next, we use the Grad-Shafranov equation to determine a equilibrium state[2][3]. Three equations∇·p=j×Bµ0 j = ∇ × B∇·B =0(3)(4)(5)

can reduce in the axisymmetric case.
I ∗ (ψ) = σI(6)p⊥ − pk ∂B∂pk+(7)∂θB∂θµ0 R2 ∂pk1∆∗ ψ = −− 2 I ∗ I ∗ 0 (ψ) (8)σ∂ψ B σ1− ∇ψ · ∇σ(9)σWe assume the following pressure profile by using s := (ψ − ψ1 )/(ψ2 − ψ1 ):()( s ) ( B )1−λtanh Ls α1α2pk =tanh(10)sLBn0=and p⊥ = λpk , where L means the scale length andλ means the anisotropy.
We estimate gradient in the magnetic filed byG :=|∇⊥ B 2 |.
B2(11)

3. Coil Size Dependence
We investigated change of the diamagnetic current in midplane(z = 0) by chaning coil size, orgradient in the magnetic field. This lead to examination of liner eﬀects. Figure 1 shows the radius dependence of diamagnetic current in the midplane. Figure 2 shows gradients G in the magneticfields, and fig.3 shows ratio of the anisotropic term
to the isotropic one. Coil radius become big whensubscript is big in fig.1,2.
When coil size become large, both magnitudeof the magnetic field and gradient of one increase.
Then magnitude of the current decrease, but ratio of the anisotropic term to the isotropic one increase because the gradient become large. Therefore, anisotropic eﬀects arise in the point which thegradient is large.

12

j1
j2
j3

0.5
0

0.14
0.13
ratio

log(G)

-1
-1.5

-1

0.12
0.11

-2

-2

0.1
-3

-2.5
0.49

0.5

0.51

0.52
r

0.53

0.54

0.55

0-4
0.48

0.56

Fig.1 Diamagnetic current(Coilsize dependence)

-0.1

0.49

0.51

0.52
r

0.53

0.54

0.55

-0.2

0.0022

-0.25

0.002

-0.3

0.00180.0016-0.4

0.51 0.52 0.53 0.54 0.55 0.56
r

Fig.4 Anisotropic term of diamagnetic current(Pressure dependence)

0.35

0.00080.48 0.49

0.4

G1G2G3-2-2.5-3-3.5-4-4.5

-6

0.001

-6.5
0.48
0.5

0.51 0.52 0.53 0.54 0.55 0.56
rFig.5 Magnitude of magnetic field

4. Pressure Dependence
Next, we investigated change of the diamagnetic current in midplane by chaning the magnitude of pressure. This lead to examination of nonliner eﬀects. Fig.4 shows the radius dependenceof anisotropic term of diamagnetic current. Fig.5
shows the magnetic fields, and fig.6 shows gradients in the magnetic fields.
When pressure increases, magnetic field localy
decreases inside the plasma. Then gradient in
the magnetic field increase, and anisotropic pressure term becomes big. Thus, we found that theanisotropic pressure eﬀects is major in high pressure.
5. Summery

0.3
rc

-5.5

0.0012
0.5

0.25

-5

0.0014-0.45

0.2

Fig.3 Ratio of the anisotropic
term to the isotropic one

B1
B2
B3

B

-0.15

-0.35

0.08

0.56

-1.5
0.00260.0024-0.5
0.48 0.49

0.5

Fig.2 Coil size dependence ofgradient in the magnetic field

j1-2
j2-2j3-2-0.05

0.09

log(G)-30.48jratio

0.15

0-0.5
j0.16

G1G2
G3

1

0.49

0.5

0.51

0.52
r0.53

0.54

0.55

0.56

Fig.6 Pressure dependence ofgradient in the magnetic field

We studied the anisotropic pressure eﬀects on diamagnetic current in a magnetospheric plasma. We
found that when we change a gradient in the magnetic field, anisotropic pressure eﬀects is major in
the point which the gradient is large. In high pressure, we found that the anisotropic term becomeslarge because of the gradient in the magnetic field.
References[1] V.D.Pustovitov. Anisotropic pressure eﬀects onplasma equilibrium in toroidal systems. Plasma
Phys. Control. Fusion, Vol. 52, No. 6, p. 065001,
2010.
[2] H. Grad. Toroidal containment of a plasma. Phys.
Fluids, Vol. 10, No. 137, 1967.
[3] D. B. Nelson G. O. Spies. Properties of magnetically confined plasma equilibria. Phys. Fluids,Vol. 17, No. 1879, 1974.