Proceedings of ITC/ISHW2007

Axisymmetric equilibria with flow in reduced single-fluid models
Atsushi ITO1) , Jesús J. RAMOS2) and Noriyoshi NAKAJIMA1)
1) National Institute for Fusion Science, 322-6 Oroshi-cho, Toki 509-5292, Japan
2) Plasma Science and Fusion Center, Massachusetts Institute of Technology, Cambridge, Massachusetts, 02139-4307

Reduced magnetohydrodynamic equations for axisymmetric toroidal equilibria of flowing high-β plasmas
are derived with asymptotic expansions in terms of the inverse aspect ratio in order to construct models suitable
for the extension to include hot ion effects that are obtained with asymptotic expansions. Depending on the flow
velocity, different orderings are applied. Singular points at the poloidal flow velocity equal to poloidal sound andpoloidal Alfvén velocity are reproduced. The poloidal sound singularity appears in the higher order equations.
Keywords: magnetohydrodynamics, equilibrium, flow
Plasma flows are suggested to lead transport barriers
and pedestals that show steep profiles in the steady states ofmagnetic confinement. Flowing equilibria have been studied to describe these phenomena in the framework of magnetohydrodynamics (MHD)[1, 2]. However, for such steep
plasma profile features, small-scale effects not included in
the ideal MHD model should be significant. The small
scale effects arising due to the Hall current have been studied with two-fluid or Hall MHD models [6, 7, 8, 9, 10, 11].
However, these models are consistent with kinetic theory
only for cold ions. In order to include the hot ion effects
that are relevant to fusion plasmas, an extension of themodel is necessary. However, a consistent treatment of hotions in a two-fluid framework must include the ion gyroviscosity and other finite ion Larmor radius effects. These effects are obtained by asymptotic expansions in terms of thesmall parameter δ that is the ratio between the ion Larmor
radius and the macroscopic scale length, and are much simplified in the slow dynamics ordering v ∼ δ vth where v andvth are the flow and thermal velocities respectively [12, 13].
In this study, we obtain reduced sets of equations for MHD
equilibria with flow with asymptotic expansions in order toconstruct models suitable for the extension to include hot
ion effects. We shall study two cases of the flow velocity
in the orders of the poloidal sound and poloidal Alfvén velocities. These are the characteristic velocities that bringsingularities in the equilibrium equations [2, 3, 4, 5].
The equilibrium equations for single-fluid MHD are∇ · (ρv) = 0,(1)∇ × E = 0,(2)E = −v × B,(3)ρv · ∇v = j × B − ∇p,(4)µ0 j = ∇ × B(5)v · ∇p + γ p∇ · v = 0,

(6)where ρ is the mass density, v is the flow velocity, E and B
are the electric and magnetic fields, j is the current density,author’s e-mail: atsushi@toki.theory.nifs.ac.jp

and p is the pressure. Here we shall consider the corresponding toroidal axisymmetric equilibria, where the magnetic field B and the current density j can be written as
B = ∇ψ(R, Z) × ∇ϕ + I(R, Z)∇ϕ∗µ0 j = ∇I × ∇ϕ − ∆ ψ∇ϕ,(7)(8)where ψ is the poloidal magnetic flux and ∆∗ ≡ R2 ∇ ·[R−2 ∇]. The asymptotic expansion is defined in terms of
the inverse aspect ratio ε ≡ a/R0  1 where a and R0 are
the characteristic scale length of the minor and major radii
respectively. The following high-β tokamak ordering isapplied,B p ∼ εB0 , p ∼ ε B20 /µ0 .
(9)The variables are expanded asψ = ψ1 + ψ2 + ψ3 + . . . ,I = I0 + I1 + I2 + I3 + . . . ,p = p1 + p2 + p3 + . . . ,ρ = ρ0 + ρ1 + . . . ,R = R0 + x,
where I0 ≡ B0 R0 . The leading order of the force balance
Eq. (4) yieldsI1 +µ0 R0
p1 = const.
B0(10)

Here we consider the the flow velocity in the order ofthe poloidal sound speed,p√v ∼ (B p /B0 ) γ p/ρ ∼ εvAp ,(11)√where vAp ≡ B p / µ0 ρ is the poloidal Alfvén velocity thatis the order of the flow velocity for the usual reduced MHD(RMHD) [15]. Sinceρv2 ∼ ε 2 p ∼ ε 3 B20 /µ0 ,this requires the third-order accuracy of the total energy
like the RMHD for finite aspect ratio tokamaks [15]. From

Proceedings of ITC/ISHW2007the requirements ∇ · v ∼ εv/a to eliminate the fast magnetosonic wave and then v · ∇p ∼ ε 2 v/a, the flow velocity v
can be written as [15]   v ≡ 1 + x R0 ∇U × B B + vk B B≡ v p + vϕ R∇ϕ,

(12)
(13)
(14)

The function U is expanded as
U = U1 +U2 + . . .
In the leading order, the poloidal flow is written in the standard stream function representation,(0)v p = R0 ∇U1 × ∇ϕ,(15)

and the toroidal flow velocity coicides with the parallelflow,(0)vϕ = vk .

(16)

The leading order of the ϕ-component of Ohm’s law (3)yieldsU1 = U1 (ψ1 ) ,ρ0 = ρ0 [U1 (ψ1 )] = ρ0 (ψ1 ),ρ1 − ρ00 ψ2 +

ρ0 vk2x+ ρ0 ≡ ρ∗ (ψ1 ) .
B0 R0U10 R0

(24)
(25)

The projection of the force balance Eq. (4) along B is


vkx I
vp ≡∇ψ + 1 +∇U × ∇ϕ,B
R0 B
Ivkx ∇ψ · ∇Uvϕ R ≡− 1+.
BR0B

Analogously, the continuity equation (1) gives the equations for the zeroth- and first-order density,(17)B · (ρv · ∇v + ∇p) Ivk
ρIx ∇ψ · ∇U=− 2 ∇− 1+× ∇ϕ
RBR0Bvkx I·∇ψ + 1 +∇UBR0 BxρI vk ∗x I ∗− 1+∆ψ+1+∆U
R0 BR2 B
R0 B

 
v x Ik· ∇ψ + ∇ 1 +· ∇U [U, ψ]+∇B
R0 B"
#Ivkρx ∇ψ · ∇U 2− 2− 1+,ψ2RBR0B"
 "2 # #x 2B
ρ 2v + 1+|∇U|2 −· ∇U,ψ+2 k
R0B+ [p, ψ] = 0.

(26)

The first order of Eq. (26) is[ρ0 B0 R0 vk ,U1 ] + [p2 , ψ1 ] + [p1 , ψ2 ] = 0(27)which yields the equation for vk ,and its next order is
R0 ([U2 , ψ1 ] + [U1 , ψ2 ]) = 0,(18)which yields
U2 −U10 ψ2 ≡ U2∗ (ψ1 ) ,(19)where [a, b] ≡ (∇a × ∇b) · ∇ϕ is the Poisson bracket andthe prime denotes the delivative of ψ1 . The first order of∇ · v is obtained from the projection of Faraday’s law (2)along B asvk0(1)(20)+ 2xU1 , ψ1 .
(∇ · v) =B0

B0 R0 ρ0U10 vk + p2 − p01 ψ2 ≡ p3∗ (ψ1 ) ,which is the Bernoulli law in the present system. Equations(23), (25) and (28) show the coupling of vk , p2 and ρ1 dueto the slow magnetosonic (sound) wave, since these aredecoupled in the cold (p1 → 0) or incompressible (γ → ∞)limits, and yieldvk = −(2x/R0 )γ p1 − (p2∗ − p3∗ )MAp vA ,2 )(B2 /µ )(β1 − MAp00p2 =p01 ψ2 +The leading order of the pressure equation (6) yields
p1 = p1 [U1 (ψ1 )] = p1 (ψ1 ),−2β1 − MAp

2β1 − MAp

ρ1 =ρ00 ψ2 + ρ∗ +(1)R0 ([p2 ,U1 ] + [p1 ,U2 ]) = −γ p1 (∇ · v) .

Substituting Eq. (20) to Eq. (22), one obtains the equation
for the second order pressure,vk2xp2 − p01 ψ2 + γ p1+≡ p2∗ (ψ1 ) . (23)
B0 R0U10 R0,(30)2(2x/R0 )MAp

ρ02β1 − MApp2∗ − p3∗−ρ ,2 )(B2 /µ ) 0(β1 − MAp00(22)(29)

2 γp(2x/R0 )MAp
12 p −β pMAp2∗
1 3∗(21)and the next order is(28)(31)

whereβ1 (ψ1 ) ≡γ p1
 ,B20 µ0(32)Proceedings of ITC/ISHW2007
2 (ψ ) ≡ µ ρ (R U 0 )2 is the poloidal Alfvén Machand MAp
10 00 1
2 where thenumber. The singularity is found for β1 = MAp
poloidal flow velocity equals to the poloidal sound velocity. The projection of the force balance Eq. (4) along ∇ψyields|∇ψ|2 ∆∗ ψ + I∇ψ · ∇I + µ0 R2 ∇ψ · ∇p+ µ0 R2 ∇ψ · (ρv · ∇v) = 0.

(33)

and the second order is2xB0I3 , ψ1 + [p2 − p01 ψ2 , ψ1 ]p3 +µ0 R0R0I1+[I2 − I10 , ψ1 ] − [g0∗ ψ2 , ψ1 ]µ0 R20ρ0U102 |∇ψ2 |2 /2, ψ1 = 0+
2R0
which yields

The first and second orders of Eq. (33) are ρ0 |∇U1 |2I1
B0 I30+I−Iψ+221µ0 R0 µ0 R202R20 22 γp2MAp1x+− g0∗ ψ2 ≡ E∗ (ψ1 ) .
2R0β1 − MAp|∇ψ1 |2 ∆2 ψ1 + 2µ0 R0 x∇ψ1 · ∇p1 + I1 ∇ψ1 · ∇I1+ µ0 R20 ∇ψ1 · ∇p2 + B0 R0 ∇ψ1 · ∇I2 = 0,p3 +(34)and|∇ψ1 |2 ∆2 ψ2 −1 ∂ ψ1R ∂R

Substituting Eqs. (38) and (40) to Eqs. (34) and (35), we
obtain the expanded Grad-Shafranov equation in the presence of poloidal-sonic flow,"0 #µ0 p21
2 2x 0∆2 ψ1 = −µ0 R0p ++ g∗,(41)R0 1B20 2+ 2µ0 R0 x (∇ψ2 · ∇p1 + ∇ψ1 · ∇p2 )+ ∇ψ1 · ∇ µ0 R20 p3 + R0 B0 I3 + I1 I2− µ0 R20 (∇ψ1 · ∇U1 ) ∆2U1
+ µ0 R20 ∇ψ1 · ∇ |∇U1 |2 /2 = 0,(35)"∆2 ψ2 + µ0 R20

where∆2 ≡∂2∂2+∂ R2 ∂ Z 2

=The projection of the force balance Eq. (4) along ∇ϕ is∇ϕ · ρv × (∇ × v) + µ0−1 (∇ × B) × B) Ivk
ρIx ∇ψ · ∇U=− 2 ∇− 1+× ∇ϕRBR0Bvkx I∇ψ + 1 +∇U·BR0 B− (I/µ0 R2 )[I, ψ] = 0.

[p, ψ] + (I/µ0 R2 )[I, ψ]xρI vk ∗x I ∗− 1+∆ ψ + 1+∆UR0 BR2 BR0 Bv x Ik+∇· ∇ψ + ∇ 1 +· ∇U [U, ψ]
BR0 B"
#


Ivk
ρ
x ∇ψ · ∇U 2− 2− 1+,ψ2RBR0B""
2 # #
ρ 2x 2B2+v + (1 + ) |∇U| −· ∇U,ψ2 kR0B= 0.

(37)|∇ψ1 |21 ∂ ψ122 0+ MAp∆2 ψ1 +MAp
R ∂R "2 2xp01− µ0 R20 E∗0 +R0!0 22 γp2MAp
1x+
2R0β1 − MAp!0 #22x MAp p2∗ − β1 p3∗−.
2R0β1 − MAp(42)

The equation for ψ1 (41) is same as for the static case while
Eq. (42) for ψ2 is modified by the flow and the singularity
appears. In the cylindrical limit x/R0 → 0, the singularity
can be removed whenp2∗ = p3∗ ≡ f∗ (ψ1 ) γ p1 (ψ1 ) ,ρ∗ (ψ1 ) ≡ f∗ (ψ1 ) ρ0 (ψ1 ) ,andf∗ (ψ1 ) = f [U1 (ψ1 )](38)

andf [U1 = 0] = 0.

Then, the equations for vk , p2 , ρ1 and ψ2 are rewritten asvk = −

2xR0



The first order of Eq. (37) yields
B0I2 ≡ g∗ (ψ1 ) ,µ0 R0
00 #2x 00µ0 p21p ++ g∗ψ2R0 1
B20 2(36)The difference between Eqs. (26) and (36) is given by

p2 +(40)

+ 2 (∇ψ1 · ∇ψ2 ) ∆2 ψ1
+ µ0 x2 ∇ψ1 · ∇p1 + ∇ψ2 · ∇ I12 /2(39)p2 = p01 ψ2 −β1 MAp vA,
2β1 − MAp
  2f∗ − 2x R0 MAp− f ∗ β12β1 − MAp(43)γ p1 ,(44)Proceedings of ITC/ISHW2007
  2f∗ − 2x R0 MAp− f∗ β1ρ1 = ρ00 ψ2 −2β1 − MAp"
∆2 ψ2 + µ0 R20=

ρ0 ,(45)

00 #µ0 p212x 00p ++ g∗
ψ2R0 1
B20 2

|∇ψ1 |21 ∂ ψ122 0+ MAp∆2 ψ1 +MAp
R ∂R "2 22xxp01 −( f∗ γ p1 )0− µ0 R20 E∗0 +R0R0!0 # 22 γp2MAp
1x+.
2R0β1 − MAp(46)

In comparson with the analysis of the transonic flow for
low-β tokamaks [5, 2], the singularity at the poloidal flow
velocity equal to poloidal sound velocity in the densityand pressure and its dependence on toroidicity has been
reproduced as higher-order effects and the singularity inthe higher order magnetic structure has been found in thepresent study. However, in order to reproduce the radial
discontinuity of the density and pressure found in Ref. [2],2 ∼ εM 2 will be neca local analysis assuming β1 − MAp
Apessary. Finally we note that the hyperbolic region between
the cusp velocity and the poloidal velocity of the slow magnetosonic wave as pointed out in Ref. [4] may be degenerated because the difference between them goes to higher
order in the present ordering.
Next we consider the case for the poloidal-Alfvénicflow v ∼ vAp where the usual RMHD ordering applies. The
first order of Eq. (33) is|∇ψ1 |2 ∆2 ψ1 + 2µ0 R0 x∇ψ1 · ∇p1 + I1 ∇ψ1 · ∇I1+ µ0 R20 ∇ψ1 · ∇p2 + B0 R0 ∇ψ1 · ∇I22− MAp[|∇ψ1 |2 ∆2 ψ1 − ∇ψ1 · ∇(|∇ψ1 |2 /2)] = 0. (47)

The first order of Eq. (37) yields
p2 +B0I2 + ρ0 (R0U10 )2 |∇ψ1 |2 /2 ≡ g∗ (ψ1 ) . (48)µ0 R0Substituting Eq. (48) to Eq. (47), we obtain the equationfor ψ1 in the following form,|∇ψ1 |222 0(1 − MAp)∆2 ψ1 −MAp2"0 #2
p2xµ01= −µ0 R20
p0 ++ g∗.
R0 1
B20 2(49)The singularity at the poloidal flow velocity equal topoloidal Alfvén velocity arises in the first order of themagnetic structure. This singularity is independent of thetoroidicity [16].
We have shown reduced sets of equations for MHD
equilibria with flow with asymptotic expansions and reproduced the singular points at the poloidal flow velocity
equal to poloidal sound and poloidal Alfvén velocity. They

will be extended to include hot ion effects by setting δ ∼ ε
for the poloidal-sonic flow, and δ 2 ∼ ε, as usual reduced
two-fluid models [17, 18], for the poloidal-Alfvénic flow.
This work was partially supported by the Ministry of Education, Culture, Sports and Technology of Japan, Grant-in-Aid for Young Scientists (B)No. 18740358 and the National Institutes of Natural Sciences, NIFS06NPRX001.
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