Proceedings of ITC/ISHW2007

Two-Fluid Flowing Equilibria of Helicity Injected Spherical Toruswith Non-Uniform Density
Takashi KANKI, Loren C. STEINHAUER1) and Masayoshi NAGATA2)
Department of Maritime Science and Technology, Japan Coast Guard Academy, Kure, Hiroshima 737-8512, Japan1)
2)

University of Washington, Redmond Plasma Physics Laboratory, Redmond, Washington 98052, USA

Department of Electrical Engineering and Computer Sciences, University of Hyogo, Himeji, Hyogo 671-2201, Japan

Two-dimensional two-fluid flowing equilibria of helicity-injected spherical torus with non-uniform density andboth toroidal and poloidal flows for each species have been numerically determined by means of the nearby-fluids
procedure. It is found from the numerical results that the equilibrium for the driven λ ( ≡ μ0 j ⋅ B /B2) profile has adiamagnetic toroidal field, high-β (toroidal beta value, βt = 32%), hollow current profile, and centrally broad density.
By contrast, the decaying equilibrium has a paramagnetic toroidal field, low-β (βt = 10%), centrally peaked current
profile, and density with a steep gradient in the outer edge region. In the driven case, the toroidal ion and electron
flows are in the same direction, and two-fluid effects are less important since the E × B drift is dominant. In thedecaying case, the toroidal ion and electron flows are opposite in the outer edge region, and two-fluid effects are
significant locally in the edge due to the ion diamagnetic drift.
Keywords: helicity injection, equilibrium, plasma flow, two-fluid effect, E × B drift, diamagnetic drift

1. Introduction
Many experiments on current drive of sphericaltorus using a coaxial helicity injection (CHI) have been
performed to elucidate the current drive mechanism forCHI. In these experiments, E × B plasma toroidal rotation
with an n=1 kink mode the during CHI has been observed.
It is recognized that the effect of plasma flow is importantin understanding the confinements and relaxed states ofhelicity-injected spherical torus (HI-ST) plasmas.
Especially, in the Helicity Injected Torus-II (HIT-II)
experiments, a rotating n=1 magnetic structure observedat the outer plasma edge is locked to the electron fluids
and not to the ion fluids, suggesting a rotating magnetic
field current drive [1]. Because of this behavior, theequilibrium computation of the HI-ST is required to take
into account two-fluid effects [2,3]. The formalism for
two-fluid flowing equilibria was developed [4], andtwo-dimensional equilibria in helicity-driven systems
using the two-fluid model were previously computed,
showing the existence of an ultra-low-q spherical torus
configuration with diamagnetism and higher beta [5].
However, this computation assumed purely toroidal ion
flow and uniform density. The purpose of this study is to
apply the two-fluid model to the two-dimensional
equilibria of HI-ST with non-uniform density and both
toroidal and poloidal flows for each species by means of
the nearby-fluids procedure [6], and to explore theirproperties. We focus our attention on the equilibria
relevant to the Helicity Injected Spherical Torus (HIST)
author’s e-mail: kanki@jcga.ac.jp

device, which are characterized by either driven or
decaying λ ( ≡ μ0 j ⋅ B /B2) profiles [7]. Here μ0, j, and B
are the permeability of vacuum, current density, magnetic
field. We have qualitatively reproduced the HIST
equilibria in the driven or decaying λ profiles on the basisof the experimental data such as λ, j, and B profiles.

2. Governing Equations
Let us assume axisymmetry about HIST geometricaxis in cylindrical coordinates (r, θ, z). Hereafter all
variables are dimensionless [6]. An axisymmetrictwo-fluid flowing equilibrium is described by a pair ofgeneralized Grad-Shafranov equations for ion surface
variable Y(r, z) and electron surface variable ψ(r, z) [6],⎛ ψ i′ ∇Y ⎞ r
2⎟ = (Bθψ i′ − nuθ ) + nr (H i′ − Ti S i′), (1)2⎝ n r ⎠ ε∇⎛ ψ⎞ rr 2∇ ⋅ ⎜ 2 ⎟ = (Bθψ e′ − nuθ ) − nr 2 (H e′ − Te S e′ ),⎝ r ⎠ ε(2)

ψ i′r 2∇ ⋅ ⎜and a generalized Bernoulli equations for the density n,γn γ −1 exp[(γ − 1) S i ] +u2+ φE = H i ,2γ −1γn γ −1 exp[(γ − 1) S e ] − φ E = H e .
γ −1(3)(4)

Here u, Tα, ε, φE, and γ are the ion flow velocity, species(α=i, e) temperature, two-fluid parameter, electrostatic
potential, and adiabatic constant, respectively. Note that
ε value is 0.0625 in the HIST experiment, and that ψ(r, z)Proceedings of ITC/ISHW2007
corresponds to the familiar poloidal flux function. The
poloidal flow stream functionψ α , total enthalpy function
Hα and entropy function Sα are arbitrary surface functionsof their respective surface variables. The three arbitrary
functions Λ α (≡ ψ α′ ) , Hα, and Sα for each species can beassumed so as to reflect λ, j, and B profiles on the basisof the experimental data by choosing appropriate function
forms,df,
Λ i ( Y ) = Λ i 0 + ( Λ i1 − Λ i 0 )dx Y −ΔY ;δi1 ,δi 2(5)df,
Λ e (ψ ) = Λ e 0 + (Λ e1 − Λ e 0 )dx ψ −Δψ ;δ e1 ,δ e 2f ( x; δ 1 , δ 2 ) =(6)1
δ1 + δ 2⎧⎫
δ 12 exp[ x / δ 1 ]; x < 0×⎨⎬.
2222;
0δxδδxδδx++−+≥2221⎩ 1⎭(7)Here Λα0, Λα1, δα0, δα1, ΔY, and Δψ are constantparameters. By adjusting these constant parameters, wecan obtain the driven or decaying λ profiles. Otherarbitrary functions Hα and Sα are the same function formas Λα.
Equations (1) and (2) have terms of order 1/ε on theright-hand side, and they cause singularities. We employ
the nearby-fluids procedure to eliminate the singularities[6]. This procedure requires that two arbitrary functions
Λi and Λe must differ only to O(ε). We consider replacingthese two arbitrary functions with a pair of arbitrary
functions F and G expressed as follows:Λ e (ψ ) = F (ψ ), Λ i (Y) = F (Y) + εG ′(Y).

G0 in
G ( Y ) = G0 +4ψ s(r − Rc )( Re − r ),( Re − Rc ) 2i(10)eUnder the above assumptions and boundary conditions,the equilibrium is numerically determined by using a
successive over-relaxation method for updating thepoloidal flux function and a Newton-Raphson method for
updating the density.

3. Numerical Results
We investigate the fundamental properties of theHIST equilibria in the driven or decaying λ profiles. The
radial profiles of the magnetic structure at the midplane areshown in Fig. 1. In the driven λ profile, the toroidal field Bt
has a diamagnetic profile due to the condition GG′ < 0 ,and the toroidal beta value βt is high (βt = 32%). The
condition means Λ e > Λ i and then reflects that theabsolute value of electron flow velocity is larger than thatof ion flow velocity. The poloidal field Bz is much smaller
than Bt and is almost flat in the core region. The toroidalcurrent density jt has a hollow profile and almost zero in
the core region. In the decaying λ profile, Bt has aparamagnetic profile except for the inner edge region, andβt is low (βt = 10%). Also, jt has a centrally peaked profile
with a slightly outward shift.
The radial profiles of the plasma structure at themidplane are shown in Fig. 2. In the driven λ profile, thedensity n is a centrally broad profile. Both ion temperature
Ti and electron temperature Te have flat profiles, and theirmagnitudes are almost same. The electrostatic potential φE
(a)

(8)

Note that G and Λe correspond to the toroidal field
function and the familiar Taylor λ(ψ) function,respectively.
Therefore,
thetoroidal
fieldwith GG ′ > 0 gives a paramagnetic profile, while that withGG′ < 0 gives a diamagnetic one.
Next, we consider the boundary conditions for Eqs.
(1)-(4). No magnetic flux penetrates the flux conserver(FC). Therefore, ψ is fixed at 0 at the FC wall. The bias
flux is given by assigning fixed values of ψ to grid points
corresponding to the entrance port of the FC. These values
are calculated using the formula,ψ bias (r ) =[ Λ (Y)dY − ∫ Λ (ψ )dψ ].
ε ∫1(b)(9)where Rc, Re, and ψs are the radius of the central conductor,
that of the entrance port, and the maximum value of thebias flux, respectively. A toroidal field coil current along
the geometry axis inside the central conductor produces avacuum toroidal field Bt.v. The effect of Bt.v is inserted byFig.1

Radial profiles of the magnetic field
and toroidal current density at themidplane; (a) driven λ profile and (b)decaying λ profile.

Proceedings of ITC/ISHW2007
is peaked towards the periphery region, and reflects thatthe electric field is applied by the coaxial helicity source(CHS). In the decaying λ profile, n is a centrally peaked
profile. Both Ti and Te have similarly peaked profiles, andtheir magnitudes are almost same. Also, φE has almost flat
profile which reflects that the electric field is hardly
applied by the CHS.
The radial profiles of the flow structure at themidplane are shown in Fig. 3. In the driven λ profile, theelectron toroidal flow velocity uet is larger than the ion one
uit, and uit and uet are in the same direction. Both toroidal
and poloidal electron flow velocities rise up sharply near
the inner edge region and cause the hollow current profile.
In the decaying λ profile, uit is larger than uet except for the
inner edge region, and uit and uet are opposite in the outeredge region.
We investigate the generalized Ohm’s law,
E + (1 / ε )u × B + F2 F = 0,(11)F2 F = −∇pi / n − u ⋅ ∇u.

(a)

(b)

Here E and F2F express the electric field and two-fluideffect, respectively. The terms −∇pi / n and −u ⋅ ∇u cause
the ion diamagnetic and inertial effects, respectively. The
radial profiles of E, (1/ε)u × B, F2F, −∇pi / n, and −u ⋅ ∇u atthe midplane are shown in Fig. 4. All components areradial. In the driven λ profile, the two-fluid effect is not solarge except for both edge regions. The ion diamagneticand inertial effects are relatively large both edge regions,but they are in the opposite direction. In the decaying λprofile, the two-fluid effect is dominant in the outer edge
region due to the ion diamagnetic effect.
(a)Fig.3 Radial profiles of the toroidal andpoloidal flow velocities at the midplane;(a) driven λ profile and (b) decaying λprofile.
(b)

Fig.2

Radial profiles of the density,
temperatures and electrostatic potential
at the midplane; (a) driven λ profile and(b) decaying λ profile.

The radial profiles of the ion drift velocity at themidplane are shown in Fig. 5. In the driven λ profile, theE × B drift velocity is dominant. This velocity is thesame direction as the toroidal ion flow one, but is theopposite direction to the toroidal current density. This
result is consistent with the observation in the HIST andother CHI experiments. In the decaying λ profile, the ion
diamagnetic drift velocity is dominant. This velocity isthe same direction as the toroidal ion flow one, but is theopposite direction to the E × B one.

Proceedings of ITC/ISHW2007
(a)

(a)

(b)

(b)

Fig.4 Radial profiles of E, (1 / ε )u × B,F2F, −∇pi / n, and −u ⋅ ∇u at themidplane; (a) driven λ profile and (b)decaying λ profile. All components areradial.

Fig.5 Radial profiles of the ion drift velocity
at the midplane; (a) driven λ profile and(b) decaying λ profile. All components
are toroidal.

equilibrium has the diamagnetic toroidal field, high-β(βt=32%), hollow current profile, and centrally broaddensity. The toroidal ion and electron flows are in thesame direction, and two-fluid effects are less importantsince the E × B drift is dominant. 2) In the decaying
λ profile, the equilibrium has the paramagnetic toroidal
field, low-β (βt=10%), centrally peaked current profile,and density with the steep gradient in the outer edge
region. The toroidal ion and electron flows are opposite
in the outer edge region, and two-fluid effects are
significantly locally in the edge due to the ion
diamagnetic drift.

[1]
[2]
[3]

3. Summary and Conclusions
We have computed the two-dimensional two-fluid
flowing equilibria with non-constant density and thepoloidal as well as toroidal flows for each species byusing the nearby-fluids procedure. We focus our attention
on the HI-ST equilibria relevant to the HIST device,
which are characterized by either driven or decaying λprofiles, and explore their properties. Conclusions aresummarized as follows. 1) In the driven λ profile, the

[4]
[5]
[6]
[7]

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