Magnetohydrodynamic equilibria of compressible plasma with mass flow in an axisymmetric tokamak
In the present work the problem of ideal MHD equilibria of compressible plasma with mass flow in axisymmetric tokamak is investigated. The generalized Grad-Shafranov equation is derived for an arbitrary coordinate system. Two equations of equilibrium are derived by means of an expansion with respect...
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2000
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| Zitieren: | Magnetohydrodynamic equilibria of compressible plasma with mass flow in an axisymmetric tokamak / O.K. Cheremnykh // Вопросы атомной науки и техники. — 2000. — № 6. — С. 55-57. — Бібліогр.: 3 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860021992880603136 |
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| author | Cheremnykh, O.K. |
| author_facet | Cheremnykh, O.K. |
| citation_txt | Magnetohydrodynamic equilibria of compressible plasma with mass flow in an axisymmetric tokamak / O.K. Cheremnykh // Вопросы атомной науки и техники. — 2000. — № 6. — С. 55-57. — Бібліогр.: 3 назв. — англ. |
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| container_title | Вопросы атомной науки и техники |
| description | In the present work the problem of ideal MHD equilibria of compressible plasma with mass flow in axisymmetric tokamak is investigated. The generalized Grad-Shafranov equation is derived for an arbitrary coordinate system. Two equations of equilibrium are derived by means of an expansion with respect to the inverse aspect ratio from this equation. It is shown that the Shafranov shift is essentially effected by the toroidal velocity for a conventional tokamak.
|
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Problems of Atomic Science and Technology. 2000. № 6. Series: Plasma Physics (6). p. 55-57 55
UDC 533.9.01
MAGNETOHYDRODYNAMIC EQUILIBRIA OF COMPRESSIBLE
PLASMA WITH MASS FLOW IN AN AXISYMMETRIC TOKAMAK
O.K. Cheremnykh
Space Research Institute, NASU & NSAU, Kyiv, Ukraine
In the present work the problem of ideal MHD equilibria of compressible plasma with mass flow in axisymmetric
tokamak is investigated. The generalized Grad-Shafranov equation is derived for an arbitrary co-ordinate system.
Two equations of equilibrium are derived by means of an expansion with respect to the inverse aspect ratio from this
equation. It is shown that the Shafranov shift is essentially effected by the toroidal velocity for a conventional
tokamak.
1. The reduction of the vector equations
As starting point we consider the following set of
well-known stationary, non-linear single-fluid MHD
equations, which describe the macroscopic dynamics of
ideal compressible plasma with mass flow
( ) ( ) Bjc1p
!!!!
×+∇−=∇⋅ρ vv ,
( )jc4B
!!
π=×∇ , (1)
=×∇ E
!
0B =⋅∇
! ,
where ρ , v
!
, and p are, respectively, plasma mass
density, velocity and pressure, E
!
and B
!
are the electric
and magnetic fields, j
!
is the current density.
In order to close this set of equations in addition one
has to apply ideal Ohm’s law
( ) 0Bc1E =×+
!!!
v , Φ∇−=E
! (2)
together with equation of plasma state
( ) 0S =∇⋅v
! , (3)
where γρ=pS is entropy.
Axially symmetric solutions of the system of Eqs. (1)
- (3) are considered in arbitrary co-ordinates ( )321 x,x,x
[1]. According to our assumption we put 0x3 =∂∂ and
introduce the velocity and magnetic fields in the form
( ) 3
33 ee1 !!!
vv +×ϕ∇
ρ
= , (4)
( ) 3
33 eBeB
!!!
+×ψ∇= ,
where the superscripts and subscripts indicate
contravariant and covariant components of a vector,
respectively.
By taking into account the representation (4), the
system of Eqs. (1) - (3) can after some manipulations be
reduced to the following scalar partial differential
equations
{ } { } 0,, =ψφ=ψϕ ,
{ } { }Φ=ψ
π
,g,Bg
4
1 3
33
3
33 v ,
,0,B
d
d 3
3 =
ψ
ρψ
Φ−v { } 0,p =ψργ , (5)
( ) 0,
d
dcg
1
S
2 33
1 =
ψ
ψ
Φ−ρ
−γ
γψ+ −γ 3
2
vv
!
,
( ) −
ϕ∇⋅
ρ
∇+
ρ
ϕ∆ϕ∇π−∇+ψ∇ψ∆ 1
g
14BgB
33
3
33
3
( ) 02P4g4 23
33
3 =ρ∇π−∇π+∇ρπ− vvv
! ,
where the expressions { }",P and ∆ in (5) are given by
2pP 2v
!
ρ+≡ ,
{ }
∂
∂
∂
∂−
∂
∂
∂
∂≡ 1221 x
Y
x
X
x
Y
x
X
g
1Y,X , (6)
+
∂
∂
−
∂
∂
∂
∂≡∆
2
12
1
22
1 x
X
g
g
x
X
g
g
xg
1X
∂
∂−
∂
∂
∂
∂+ 1
12
2
11
2 x
X
g
g
x
X
g
g
xg
1 ,
kig are the coefficients of metric tensor, kigdetg = .
Eqs. (5) describe the general structure of the
magnetic, electric and the velocity fields, and of the
pressure and density profiles for stationary axisymmetric
toroidal dynamics, where the velocity and magnetic
fields are assumed to be in general non-parallel.
2. The modified Grad – Shafranov equation
With the help of Eqs. (5) and (6) one may easily find
the following useful expressions required in the
following investigations
( )( )[ ] ( )( )[ ]2
33
3 dd41ddddc4gIB ψϕρπ−ψϕψΦπ+= ,
(7)
( )( )( ) ( )[ ]ψΦ+ρψϕ= ddc1ddgI 33
3v
( )( )[ ]2dd41 ψϕρπ− ,
where the poloidal current ( )ψ=II is a surface function.
In general, the pressure and mass density are no flux
function.
By taking into account the relations
( ) ψ∇ψϕ=ϕ∇ dd ,
( ) ( ) 222 dddd ψ∇ψϕ+ψ∆ψϕ=ϕ∆ ,
−
∂
ψ∂=∇
2
2
112
xg
g
ø ( 8)
2
2212
1x
ø
g
g
2x
ø
1x
ø
g
g2
∂
∂+
∂
∂
∂
∂−
56
we then infer that Eqs. (5) leads to the modified Grad -
Shafranov equation for compressible plasma with mass
flow in the form
( ) +ψ∇
ψ
ϕ
ψρ
−
ψ
ϕ
ρ
π−
π
ψ∆ 2
2
33
2
33 d
d
d
d
2
g
d
d41
4
g
+
ψπ
+ψ∇⋅
ρ
∇
ψ
ϕ+
d
Id
4
Bg1
d
d 3
33
2
( ) ( ) +
ψ
ϕρ+
ψ
ϕ+ 2
2
32
332
2
332
33
d
dcg
d
dBg vv (9)
0
1
g
d
dS
d
dHg 33
33 =
−γψ
γρ−
ψ
ρ+ ,
where
( )ψ=ρ= γ SpS ,
( ) ( )ψ=
ψ
Φ−ρ
−γ
γψ+≡ −γ H
d
dcg
1
S
2
H 3
33
1
2
vv
!
. (10)
Based on cylindrical co-ordinates
( )ϕ=== 321 x,zx,Rx , when ( )z,Rψ=ψ , we
derive with the help of (7), (8), (9) the Kerner - Tokuda
equation [2] for a compressible plasma with mass flow
+
ψ
ϕ⋅π+
ψ∇
ψ
ϕ
ρ
π−
2
2
2
2
d
dB4
Rd
d41div
!!
v
+
ψ
ψ
ϕ
ρ
π−
ψ
ϕ
ψ
Φπ+
π+
d
dI
d
d41
d
d
d
dcR4I
R
4
2
2
2
+
−γψ
ρπ−
ψ
πρ+ γ
1
1
d
Sd4
d
dH4 (11)
0
d
dc
d
d41
d
dcR
d
dI4
2
2
2
2
=
ψ
Φ
ψ
ϕ
ρ
π−
ϕ
Φ+
ψ
ϕ
ρ
πρ
+
,
where the differential operator
2
2
22 zR
1
RR
1
RR
1
R
div
∂
ψ∂+
∂
ψ∂
∂
∂=
ψ∇=ψ∆ , (12)
appearing in Eq. (11), is well-known Grad-Shafranov
operator.
In the static limit ( )0dddd =ψΦ=ψϕ Eq. (11)
reduce to the Grad-Shafranov equation for the potential
( )z,Rψ .
3. Equilibrium with nested magnetic surfaces
Following [3], we assume that the magnetic flux
surfaces consist of a family of closed nested-in toroidal
surfaces with «on-average» circular cross-section so that
we can introduce a radial co-ordinate a , which
coincides with the average radius of the magnetic
surfaces. We also suppose that magnetic surfaces
wrapped around a single magnetic axis, which is shifted
relatively to the geometric axis by an amount ξ . We
introduce on the magnetic surfaces consta = the
poloidal ( )θ and toroidal φ angles as independent
variables. In these «quasi-toroidal» co-ordinates
( )φθ ,,a a magnetic field can be represented in the form
( )Θ′χ ′
π
= ,,0
g2
1Bi , (13)
where χ and Θ are the poloidal and toroidal magnetic
fluxes, ( ) ( ) a∂∂≡′ "" . Thus the representation (13)
of the magnetic field is formally obtained from the one
of Eq. (4) by transformation
( ) ( )Θ ′π⇒θ g21,aB 3 ,
(14)
( ) ( ) ( ) ( )a21a,a χπ−=ψ⇒θψ .
Since the potential ψ only depends on the radial co-
ordinate a , immediately leading to the following
relations for the physical variables
( ) ( ) ( ) ( ) ( ),aHH,aSS,aII,a,a ===Φ=Φϕ=ϕ
( ) 2211 Bdd,0B ψϕ=== vv ,
∂
∂ψ
∂θ
∂−
∂
∂ψ
∂
∂=ψ∆
ag
g
g
1
ag
g
ag
1 1222 , (15)
2
222
ag
g
∂
ψ∂=ψ∇ .
Stationary toroidal equilibria, described by Eq. (11),
can be investigated by means of an expansion with
respect to the inverse aspect ratio Ra=ε . Following
[3], we obtain with the help of Eq. (11) by neglecting
terms of the order ( )3O ε the non-zero components of
the metric tensor kig in the form:
( ) ,sinaag,cos21g 2
1211 θξ ′−λ ′=θξ ′+=
θλ+= cosa2ag 2
22
( ) ,acos2aRg 2
33 θ−λ+ξ−= (16)
( )( )θ−λ+ξ ′+= cosak1Rag ,
where λ is the parameter of «strightforward» magnetic
force lines (see detailes in [3]).
We employ the first equation (7) to in order to find
an expression for the unknown parameter λ . Under the
conventional tokamak ordering (the superscript ""0
denotes the cylindrical terms)
( ) ( ) ,BB,1q 00 ε≈≈ φθ
(17)
( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) 2
2
020
2
02000 B,B,Bp ε≈ρρ θθφφφ vv
the Eq. (7) gives the following well-known expression
for λ
ak−ξ ′−=λ . (18)
The components of the metric tensor (16) together with
(18) fully describe the geometrical properties of the
equilibrium with mass flow.
In the leading order ( )2O ε Eq. (11) describes the
stationary cylindrical equilibrium
( )
( )( ) ( )
+
π
+
+ φθ
8
BB
p
ad
d 220
0
( )( ) ( )( ) 04B
a4
1 2020 =
πρ−
π
+ θθ v , (19)
57
here
( ) ( ),BBB 20
φφφ += ( ) ,
a2
B 0
π
Θ=φ
( ) ( ) ( )2ak2kakBB 2202 +ξ ′+ξ−= φφ
, (20)
( ) .
ad
d
R2
IB 0 χ
π
=θ
In the next order ( )3ε≈ , Eq.(11) describes the first -
order toroidal correction to the cylindrical equilibrium
(19) leading to
( )( ) ( )
−=ξ ′ θ
θ R
B
Ba
ad
d
a
1 20
20
− ( ) ( )( )2p
ad
d
R
a8 200
φρ+π v , (21)
with the relations
( )
( )
( )
( ) ( )
( )
( ) ( )00
0
0
a0
s0
0
0 B
d
d,
B
EcB
B θθ
θθ
θ
φ ψ
ϕ=+= v
v
v . (22)
If the plasma is surrounded by a perfectly conducting
a rigid wall of radius 0aa = , then we have
( ) 0aa 0 ==ξ . With the help of this boundary condition,
we obtain from Eq.(21) the following simple expression
for the Shafranov shift ξ :
( )
( ) +′′∫ ′′∫
′
′
−=ξ θ
′
θ
adBa
Ba
ad
R
1 20a
0
a
a 20
0
( )
( )
( )
ad
2
pa
Ba
ad
R
8
20
0a
0
2
a
a 20
0
′′
ρ
+∫ ′′∫
′
′π
+ φ′
θ
v . (23)
In the case of 0dd =ψϕ , Eqs. (19) and (21)
describe an equilibrium with purely toroidal rotation and
in the case 0dd =ψΦ - one with purely parallel
rotation.
From Eqs. (17) and (22) one may easily obtained an
estimation for the electric fields required, in order to
sufficiently affect on the plasma equilibrium
( )
( )
( ) ( )0s0
0
A0
a B
c
c~B
c
c
~E θθ
θ
. (24)
In the case of parallel flow the following relation for
the ratio between the poloidal and toroidal velocities is
obtained
( )
( )
( )
( ) 1
a
Rq
B
B
0
0
0
0
>>==
θ
φ
θ
φ
v
v . (25)
On conclusion we note, that from the onditions
( ),aSS = ( ),aHH = it follows, that in the presence of
flow the magnetic surfaces do not coincide with the one
of constant pressure and density
( )( ) ( ) ( )( ) ( ) ,cosa,cospapp 1
a
01
a
0 θρ+ρ=ρθ+=
( ) ( ) ( ) ( ) ( ) ( )
( )0
0
2
S
12
S
100 pc,cp,aSp
ρ
γ=ρ=ρ= γ , (26)
( )
( ) ( ) ( ) ( )
( )
( )
( )
1
1
2
0
0
r
2
0
0
s00
B
cE
2
1
B
B
2
1aH
aS
1
−γ
θθ
θ
+
−
γ
−γ=ρ v ,
( ) ( )
( )
( )
( ) ( )
( )
( ) ( )
( )
2
s
2
0
0
s0
2
0
0
s0
2
0
0
r
01
c
B
B
B
B
B
cE
R
a
−
+
ρ=ρ
θ
θ
θ
θ
θ
v
v
.
Now the presssure and density consist of two parts: the terms
( )ap0 and ( )a0ρ , which are constant on the magnetic
surfaces, and its deviations ( ) ( ) θcosap 1 and ( ) ( ) θρ cosa1
arising from a finite plasma flow.
4. Conclusions
The basic results of this work are as follows:
• The stationary single-fluid MHD vector Eqs. (1) - (3) are
reduced to a set of scalar partial differential Eqs. (5) - (6).
• The modified Grad-Shafranov equation (9) is obtained
from Eqs. (5) for a compressible plasma with non-parallel
flow in axisymmetric toroidal magnetic traps.
• The modified Grad-Shafranov equation is rewritten in
quasi-toroidal coordinates, describing the equilibrium of
toroidal plasma with mass flow and nested-in magnetic
surfaces.
• The metric relations (16), (17), and (18) for quasi-toroidal
coordinates describing the equilibrium of a plasma with
flow in a tokamak are derived. It is found that in a
conventional tokamak only the toroidal velocity (see Eqs.
(20), (21)) can sufficiently affect on the Shafranov shift.
• An estimation (24) for the internal radial electric field,
which can sufficiently affect the plasma equilibrium is
derived.
• It is shown (see (25)) that for a plasma equilibrium with
parallel flow the ratio of the toroidal and poloidal
components of the velocity and the magnetic field are
equal and proportional to the aspect ratio.
• It is shown that due to the plasma flow the constant
pressure and density surfaces are shifted relatively to the
magnetic surfaces, where these deviations are given by
Eq. (26).
Acknowledgments
This work was supported, in part, by the theme
0197U015786 of the NAS and MES of Ukraine.
References
1. O.K. Cheremnykh, J.W. Edenstrasser. Phys. Scripta. 1999,
v. 60, p. 423.
2. W. Kerner, S. Tokuda. Z. Naturforsch. 1987, v. 42a,
p. 1157.
3. V.D. Shafranov. Nucl. Fusion. 1964, v. 4, p. 213.
Space Research Institute, NASU & NSAU, Kyiv, Ukraine
|
| id | nasplib_isofts_kiev_ua-123456789-78497 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T16:48:14Z |
| publishDate | 2000 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Cheremnykh, O.K. 2015-03-18T16:23:32Z 2015-03-18T16:23:32Z 2000 Magnetohydrodynamic equilibria of compressible plasma with mass flow in an axisymmetric tokamak / O.K. Cheremnykh // Вопросы атомной науки и техники. — 2000. — № 6. — С. 55-57. — Бібліогр.: 3 назв. — англ. 1562-6016 https://nasplib.isofts.kiev.ua/handle/123456789/78497 533.9.01 In the present work the problem of ideal MHD equilibria of compressible plasma with mass flow in axisymmetric tokamak is investigated. The generalized Grad-Shafranov equation is derived for an arbitrary coordinate system. Two equations of equilibrium are derived by means of an expansion with respect to the inverse aspect ratio from this equation. It is shown that the Shafranov shift is essentially effected by the toroidal velocity for a conventional tokamak. This work was supported, in part, by the theme 0197U015786 of the NAS and MES of Ukraine. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Magnetic confinement Magnetohydrodynamic equilibria of compressible plasma with mass flow in an axisymmetric tokamak Article published earlier |
| spellingShingle | Magnetohydrodynamic equilibria of compressible plasma with mass flow in an axisymmetric tokamak Cheremnykh, O.K. Magnetic confinement |
| title | Magnetohydrodynamic equilibria of compressible plasma with mass flow in an axisymmetric tokamak |
| title_full | Magnetohydrodynamic equilibria of compressible plasma with mass flow in an axisymmetric tokamak |
| title_fullStr | Magnetohydrodynamic equilibria of compressible plasma with mass flow in an axisymmetric tokamak |
| title_full_unstemmed | Magnetohydrodynamic equilibria of compressible plasma with mass flow in an axisymmetric tokamak |
| title_short | Magnetohydrodynamic equilibria of compressible plasma with mass flow in an axisymmetric tokamak |
| title_sort | magnetohydrodynamic equilibria of compressible plasma with mass flow in an axisymmetric tokamak |
| topic | Magnetic confinement |
| topic_facet | Magnetic confinement |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/78497 |
| work_keys_str_mv | AT cheremnykhok magnetohydrodynamicequilibriaofcompressibleplasmawithmassflowinanaxisymmetrictokamak |