Magnetic well behavior determined by plasma pressure in the torsatron
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2000
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| Cite this: | Magnetic well behavior determined by plasma pressure in the torsatron / Yu.K. Kuznetsov, I.B. Pinos, V.I. Tyupa // Вопросы атомной науки и техники. — 2000. — № 6. — С. 52-54. — Бібліогр.: 10 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859646403743055872 |
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| author | Kuznetsov, Yu.K. Pinos, I.B. Tyupa, V.I. |
| author_facet | Kuznetsov, Yu.K. Pinos, I.B. Tyupa, V.I. |
| citation_txt | Magnetic well behavior determined by plasma pressure in the torsatron / Yu.K. Kuznetsov, I.B. Pinos, V.I. Tyupa // Вопросы атомной науки и техники. — 2000. — № 6. — С. 52-54. — Бібліогр.: 10 назв. — англ. |
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UDC 533.9
52 Problems of Atomic Science and Technology. 2000. № 6. Series: Plasma Physics (6). p. 52-54
MAGNETIC WELL BEHAVIOR DETERMINED BY PLASMA PRESSURE
IN THE TORSATRON
Yu.K. Kuznetsov, I.B. Pinos, V.I. Tyupa
Institute of Plasma Physics, National Science Center “Kharkov Institute of Physics and
Technology”, Kharkov 61108, Ukraine
The magnetic field minimum is an important factor
exerting influence on the plasma stability [1, 2]. It has
been demonstrated in several papers that the growth of
gas-kinetic plasma pressure gives rise to the magnetic
well with the result that for a number of most dangerous
MHD instabilities the restriction on ultimate pressure in
plasma stability [3, 4] is removed. However, the
torsatron magnetic system can have both the vacuum
magnetic hill and the vacuum magnetic well; therefore,
when performing analytical calculations of the mean
magnetic well caused by plasma pressure it is essential
to know the both parameters.
This paper gives the analytical calculations of
the vacuum magnetic hill and vacuum magnetic well for
the l=2 torsatron. The formulae for averaging over
magnetic surfaces were used to calculate analytically the
mean magnetic well for various profiles of angles of
rotational transform in the radius and various plasma
pressure profiles [5].
First we derive the relations to calculate the vacuum
magnetic hill of the l=2 torsatron. It is known that the
longitudinal magnetic field and the equation of magnetic
surfaces for the l ≥ 2 torsatron in the cylindrical
coordinate frame (r, ϑ , zo) have the following forms [6]:
θεε lrllllbcaJloBzB cos)(/2 Ι−+= (1)
θε
εε
θ lr
a
r
lllb
a
r
ca
Jl
oBr cos)('
22
22
)
2
(),( Ι−+=Ψ (2)
where ,,
2/
2/sin
)('
24
z
ol
ol
llK
ca
lJ
lb εϑθ
γ
γ
ε
ε
−== a is
the radius of helical winding, J is the current in the
winding pole, ε = 2πa/L is the tangent of tilting angle of
helical conductor to the cylinder axis, L is the pitch
length of helical conductor, γ0 is the angular width of
helical winding pole, Bo is the longitudinal field excited
by external coils, caJl /2 ε is the longitudinal field
created by the helical winding; Il and Kl are the modified
Bessel functions of order l. As it follows from
expression (1), the variable constituent of the
longitudinal field Bzchanges its sign to the opposite at
angles )...3,2,1,0(/2/ =+= klkl ππθ . This means
that the longitudinal field Bz can be both increasing and
decreasing in the radius, and therefore, the field line
belonging to the magnetic surface can be both in the
region of the increasing magnetic field (well) and in the
region of the falling-off longitudinal magnetic field
(antiwell).
The mean vacuum hill (antiwell) value is given
by the following formula [5]
,1/1)(/))()((/ −><=′′−′= zBoBoVoVrVUUδ (3)
where V′( r1) and V′(o) are the derivatives of specific
volumes on the magnetic surface of average radius r
and on the magnetic axis rc, respectively; B01 is the
longitudinal field on the magnetic axis, <Bz> is the
longitudinal field averaged over magnetic surfaces. For
convenience, the subscript z at Bz will be omitted. The
1/ oz BB >< value is calculated according to [5] as
,
2
0 /
/
2
0 /
)(),(
11
1
11011
1111111 ∫
∂Ψ∂
∫
∂Ψ∂
=
>< π θπ θθθ
r
dr
rB
drrB
B
rB
o (4)
where
)5()2coscos2(
/1
/
1
),(
1
2
111222
22
1
11 θθ
ε
εθ
rrcr
crae
ae
B
rB
o
+−
−
−=
is the longitudinal field associated with the magnetic
axis, e = Kϕ A is the ellipticity of magnetic surfaces,
,/sin)2(2/4
/4
2
32
01
),
,/4/()/4(
oo
o
KAcaJAr
caJBB
c
caJoBcaJK
γγεεε
εεεϕ
′−=−
−+=+=
.
We average the longitudinal field B(r1, θ1) over
the magnetic surfaces which are given for the l=2
torsatron by the following equation
)2cos1/( 1
22
1 θerr += (6)
where r is the mean radius of the magnetic surface.
Substituting (5) into (4), writing r1 in terms of θ1 and
using the magnetic-surface equation (6), we obtain after
integration the expression for the averaged magnetic
field
)1)(/1(
/
1
),(
2222
2222
01
11
eacre
are
B
rB
−−
−=
><
ε
εθ
(7)
With a knowledge of the magnetic surface shape, it is
possible to determine the ellipticity e as
)max/min1/()max/min1( 2222 rrrre +−= (8)
where rmin and rmax are, respectively, the minimum and
maximum magnetic-surface radii of the torsatron.
Substituting (7) into formula (3) we obtain the
expression for calculating the magnetic hill value
1)
)1)(/1(
/
1/(1 2222
2222
−
−−
−=
eacre
are
U
U
ε
εδ
(9)
In the toroidal case, the magnetic surface ellipticity
changes in the length of the system. Therefore,
substitution of ellipticity averaged over the length of
helical winding pitch is necessary in formula (9).
We now determine the mean vacuum magnetic
well due to the toroidal magnetic field:
53
)cos)/(1/(1/),( 110111 ϑϑ croRrBrB ±+= (10)
where )/1/()/4(01 oRcrcaJoBB ±+= ε is the toroidal
field on the magnetic axis, Ro is the major radius of the
torus, rc is the magnetic axis displacement. Averaging
the field (10) over magnetic surfaces (6) we obtain the
mean toroidal field on the magnetic surface of average
radius r, for the l=2 torsatron as
),,2,2/3,1(2/11/),( 120111 XFXBrB +=>< ϑ (11)
where 12
2 ,),1/()/( FcroRReRrX ±=±= is the
hypergeometrical Gaussian function. The plus and
minus signs in the expression for X correspond to the
vertical and horizontal location of magnetic surfaces,
respectively. Substituting expression (11) into (3) we
obtain the magnetic well δU+/U and δU-/U values for
horizontally and vertically lying magnetic surfaces,
respectively. The resulting magnetic well δU-/U is
determined as an average of δU+/U and δU-/U:
2/1/ =UUδ )// ( UUUU −++ δδ (12)
In the X<<1 case (small-toroidicity case, ro/R<0.2), the
mean magnetic well can be written in the form
),1/()/(2/1/ 22 eRrUU −−=δ (13)
or, writing in terms of err −= 1/max and
err += 1/min we obtain
,/)minmax(4/1/ 222 RrrUU +−=δ
(14)
Adding together expressions (9) and (12) gives the
magnetic well (or antiwell) value of the torsatron. The
analytical calculations made to determine the mean
vacuum magnetic well and the mean vacuum hill for the
LHD and Uragan-2M torsatrons are in good agreement
with the numerical calculations performed for these
systems [7, 8].
The magnetic well value specified by plasma pressure
profiles P=Po and P=Po ( )(/)(1 1 orr ΨΨ− ) (Ψ is the
averaged function of vacuum magnetic surfaces) is given
by formula [9]
where
)/1/(),/1/()/1( 01111 oRcroBBRrRrd +=+−= is the
longitudinal magnetic field on the magnetic axis, Bo is
the longitudinal field on the axis of the system. The
coefficients a, b, c for the pressure distribution P=Po are
written as
,222
1
2222
1
//)1(,/)1(3
,/)1(2/)1(
orcracoarrcrb
orcrorra
αα
ααα
−=−=
−+−+=
and for the pressure distribution P=Po
( )(/)(1 orr ΨΨ− ) the coefficients a, b, c are
expressed as follows:
22
22
2
2
22
22
2
1
22
22
2
2
2
2
1
/2/31
/12/)1(1
,
/2/31
)/1)(1(2
2
3
,
/2/31
/2/)1(1
2)1(
orcr
orr
oar
crc
orcr
orcr
oar
crr
b
orcr
orcr
or
cr
or
r
a
−
−−
=
−
−−+
=
−
−−
+−+=
α
αα
α
αα
For P=P0
2))(/)(1( orr ΨΨ− the magnetic well is
given by the expression [10]
,
))(()(
)()3)(()31)((
,
))(()(
)]3(1)[()]3(3)[(
,
))(()(
)]3(1)[()]3()[(
,
)11)(1(2)1(
))(3()3)((
A
111
2
1
11111111
111
2
1
111
2
1
1111
1111
1111111
where
qppqppqq
pqqpqqpppqqq
D
qppqppqq
pqppppqqq
C
qppqppqq
pqqqqqppp
B
qppqppqq
pppqqppqqq
−−+−
−+−−−−−
=
−−+−
−+−−+−−−−
=
−−+−
−+−−+−−−
=
−−+−
−+−−+−−
=
the Al, Bl, Cl, Dl, El values are calculated from the
equations
,1,4)(
)()(
,6)()1(
)()1()(
,1)(,4)1(
)1()(
11111112
111121111
1212
1121112111
112112
121121211
=++=+++
+++++
=++++
+++++++
=++=++
++++++
qDBqEqqqCDqdp
AqBqdpEpqqp
CqdpDpd
AqdpBpdEppqq
CAdECpd
DdApdBdEpp
where
)16(,
]
//
[
2
1
//
2
1
11
111112
11
1
22
22
d
E
qp
qDC
qp
qBAd
qp
qDC
qp
qBA
U
U
+
+
+
+
+
++
+
+
+
+
+
=
δ
1
]
31)1(12
1/1)]1(4[)1(4
2
)1(
[
)1()61()1(
1
/
)15(/
3122)1(12
1/11
22
11
1
2
1
2
1
2
11
2
1
1
1
−
−+−+++
−+++−−+−−
+
+
−
−−+++−
+
−+−+++
−++++
=
cbcbc
bcbcdbcdbcd
d
d
x
dbddcd
d
cbcbc
bcbc
U
Uδ
,2/)48(
,2/)48(
1
2
111
11
2
1
cbybp
cbybp
−+−=
−++=
54
++−=+−+−=−
+−+−=−
−
=
+=+−−−+−=
−+
−
−=
−+
−
+=
cbbdcba
d
bdbcc
oq
cdb
op
opoqE
c
oEoqoEoqy
cby
dyb
yq
cby
dyb
yq
424(,1,
166
164836
,
3612
4
,,
6
,
48
,
48
11
2
11
1
2
1111
2
1
2
1111
32133
1
2
1
11
1
1
2
1
11
!
!!
!!
)),1(3/(2),1/()2(3/2
,
]/)1(4/7/5[/3
/]/)1([
,/)1(],/)1(4/5/)1(4/5
[/3]},/)1(2/9
]/)1(42[/4/)1(2/9
/82[/)1{(/
]},/)1(4/7
/)1(2/9[/5)/)1(2/15
/91(3[/)1(3{/
},/)1(3/]/)1(3
2[3/)1(3/62{
/2/)1((2/)1(
),/1/(1,/)1(
,)2814424(
,/)70106(,/)2814
664422
22
52
1
322
1
22
3
1
244
1
2222
1
44
22
44
1
2222
1
44
22
1
44
1
22
1
22
4422
2222
1
1211
11
111
ααα
αα
αα
ααα
αα
ααα
αα
α
ααα
αα
αα
ααα
ααα
+=++=
−++−
−+
=
−−=−+−+
+−=−+
+−++−+
++−−−=
−+
+−++−+
++−−−=
−+−+
++−++−
−−+−+=
+=++++=
−−−+=
+−=−+
MK
rrrrMrrK
rrrr
N
arrrMNrrrr
arrMNrdrrM
rrMrrrrM
rrMNrrarrc
rrM
rrMrrrrM
rrMNrrrarb
rrMrrrr
MrrMrrM
rNrrrrra
Rrdadcb
adcbd
accad
ocococ
ococ
ocooc
oco
ocooc
ocococ
o
ocooc
ococo
oooc
ococ
ococo
!
!!
!
!!
where α=t(o)/t(ro) is the ratio of the angle of rotational
transform on the magnetic axis to its value at the plasma
boundary of radius r=r0. Using expressions (15) and (16)
we calculate the relative value of the mean magnetic
well caused by the above-considered plasma pressure
distributions at different profiles of vacuum angle of
rotational transform. Fig. shows B01/<B> -1 as a
function of the average radius rl/r0 for the magnetic axis
displacement rc/ro = 0.3. In the plots, α is the parameter
that characterizes the profile of the vacuum angle of
rotational transform (а-α=0; b-α=0,2; с-α=0,4; d-
α=0,6; e-α=0,8; f-α=1,0). It is seen from the figure
that an increasing α the mean magnetic well value
decreases for all plasma pressure profiles considered.
For small α (magnetic systems with a large shear of field
lines), the magnetic well value depends only slightly on
plasma pressure profiles. In the case of α close to unity
(small shear of field lines), the mean magnetic well
determined by uniform plasma distribution in the radius
(P=Po) vanishes. In this case, the magnetic system will
have either a vacuum magnetic hill (formula (9)) or a
vacuum magnetic well (formulae (12), (13)).
References
1. Rosenbluth M., Longmire C Ann. of Phys. 1, 120
(1957)
Fig. Mean magnetic well distributions determined by
various plasma pressure profiles R0/r0 = 8.9 (1 - P= P0,
2 - P=P0 (1 - Ψ(r)/Ψ(ro)), 3 - P=P0(1 -Ψ(r)/Ψ(ro))2
2. Kadomtsev B.B., Plasma physics and problems of
controlled fusion reactions (in Russian), Moscow,
AN SSSR publ., 1958, v.4, pp. 16-23.
3. Pyatov V.N., Sebko V.P., Tyupa V.I., Preprint
KFTI 76-25 (in Russian) Kharkov, 1976.
4. Kovrizhnykh L.M., Shchepetov S.V., Fiz. Plazmy
1981, v.7, is.2, pp. 419-427.
5. Solov’yov L.S., Shafranov V.D., Voprosy teorii
plazmy. Atomizdat publ., Moscow, v. 5, 1967,
pp.3-208.
6. Morozov A.I., Solov’yov L.S., Voprosy teorii
plazmy. Atomizdat publ., Moscow, v. 2, 1963,
pp.3-91.
7. Bykov V.E., Georgievsky A.V., Kuznetsov Yu.K.
et al., Vopr. At. Nauki i Tekhn., ser. Termoyad.
sintez, 1988, is. 2, p.17.
8. Iioshi A., Motojima M. at al. Fusion Technology
17, N 1, 169 (1990).
9. Kuznetsov Yu.K., Pinos I.B., Tyupa V.I. IAEA
Techn. Comm. Meeting 8 th Stellarator Workshop,
Kharkov, USSR, 1991, IAEA, Vienna, 317 (1991).
10. Kuznetsov Yu.K., Pinos I.B., Tyupa V.I. 23 rd EPS
Conf. on Controlled Fusion and Plasma Physics,
Kiev (1996) 20C, Part II, p.535.
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0.00
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0.00
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0.00
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0.00
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0.00
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0.00
0.0 0.2 0.4 0.6 0.8 1.0
(a)
0.0 0.2 0.4 0.6 0.8 1.0
(b)
0.0 0.2 0.4 0.6 0.8 1.0
(c)
0.0 0.2 0.4 0.6 0.8 1.0
(d)
0.0 0.2 0.4 0.6 0.8 1.0
(e)
0.0 0.2 0.4 0.6 0.8 1.0
(f)
B /<B(r )>-101 1
B /<B(r )>-101 1
B /<B(r )>-1
B /<B(r )>-1
B /<B(r )>-101 1
01 1
01 1
r /r
r /r
r /r
r /r
r /r
r /r
1 0
1 0
1 0 1 0
1 0
1 0
2
31
1
3 2
1
2
3
1
3
2
1
3
2
1
3
2
B /<B(r )>-101 1
|
| id | nasplib_isofts_kiev_ua-123456789-78496 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T13:28:07Z |
| publishDate | 2000 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Kuznetsov, Yu.K. Pinos, I.B. Tyupa, V.I. 2015-03-18T16:08:34Z 2015-03-18T16:08:34Z 2000 Magnetic well behavior determined by plasma pressure in the torsatron / Yu.K. Kuznetsov, I.B. Pinos, V.I. Tyupa // Вопросы атомной науки и техники. — 2000. — № 6. — С. 52-54. — Бібліогр.: 10 назв. — англ. 1562-6016 https://nasplib.isofts.kiev.ua/handle/123456789/78496 533.9 en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Magnetic confinement Magnetic well behavior determined by plasma pressure in the torsatron Article published earlier |
| spellingShingle | Magnetic well behavior determined by plasma pressure in the torsatron Kuznetsov, Yu.K. Pinos, I.B. Tyupa, V.I. Magnetic confinement |
| title | Magnetic well behavior determined by plasma pressure in the torsatron |
| title_full | Magnetic well behavior determined by plasma pressure in the torsatron |
| title_fullStr | Magnetic well behavior determined by plasma pressure in the torsatron |
| title_full_unstemmed | Magnetic well behavior determined by plasma pressure in the torsatron |
| title_short | Magnetic well behavior determined by plasma pressure in the torsatron |
| title_sort | magnetic well behavior determined by plasma pressure in the torsatron |
| topic | Magnetic confinement |
| topic_facet | Magnetic confinement |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/78496 |
| work_keys_str_mv | AT kuznetsovyuk magneticwellbehaviordeterminedbyplasmapressureinthetorsatron AT pinosib magneticwellbehaviordeterminedbyplasmapressureinthetorsatron AT tyupavi magneticwellbehaviordeterminedbyplasmapressureinthetorsatron |