4 Magn confinement eng - Max Planck Society · PDF fileMagnetic confinement p j B! ! ∇ =...
Transcript of 4 Magn confinement eng - Max Planck Society · PDF fileMagnetic confinement p j B! ! ∇ =...
Magnetic confinement
Bjp!!
×=∇
Pressure gradient is balanced by Lorentz force (currents perpendicular to magn. field)
0=∇⋅ pB!
Pressure along magnetic field lines is constant
0=∇⋅ pj!
Current lines lie surfaces of constant pressure
θ-Pinch
Bz
nT>0
r
ZylindrischePlasmasäulein Bz-Feld
j⊥=jΘ
Θ
z
Diamagnetic current reduces external magnetic field
p(r)
B(r)
0
p(r) B(r)
r 0
if ß is small: almost no change of external magnetic field
Large modification of external field for “high-ß”-case (ß=1 if B=0)
Diamagnetic currents
B
electron netto movement: downwards
T e = const
n e
r
j dia
Pressure gradient generates current perpendicular to B field
B
electron netto movement: downwards
n e = const
T e
r
× × ×
j dia Gyro-Radius ~ T1/2
Coil/external current
B
j ∇ n j ∇ T +
high-ß-plasma causes B-field gradient
p(r)
external current
B ...
j ∇ n j ∇ T +
j Drift new:
Bx ∇ B-drift
- -
j ∇ B
+ new:
× × ×
)()()()( ,, rBrjrBjjjjpeee peDrifteBTne ×=×−++=∇ ∇∇∇∇
high-ß-plasma causes B-field gradient single particle drifts cancel in fluid picture
Bj ×∇=µ 0BBp ×⎟⎟
⎠
⎞⎜⎜⎝
⎛×∇⋅=∇
0
1µ
Magnetic confinement in θ-pinch
ieie ppBjj ∇+∇=×+ )(
Ions and electrons contribute to diamagnetic current:
( )00
2
2 µµBBBp ∇
=⎟⎟⎠
⎞⎜⎜⎝
⎛+∇
Pressure gradient is balanced by
B –field pressure
Field line tension
( )00
2
2 µµBBBp ∇
=⎟⎟⎠
⎞⎜⎜⎝
⎛+∇
In θ-pinch no field line tension (MF constant along MF-lines):
0
20
0
2
22 µµBconstBp ==+Plasma pressure + mf- pressure = const:
Magnetic confinement in θ-pinch
( )00
2
2 µµBBBp ∇
=⎟⎟⎠
⎞⎜⎜⎝
⎛+∇
0
20
0
2
22 µµBconstBp ==+
2
2
020
12/ a
i
BB
Bp
−=≡µ
βNormalised plasma pressure
In θ-pinch no field line tension (MF constant along MF-lines):
Plasma pressure + mf- pressure = const:
Magnetic confinement in θ-pinch
Das Bild k
r
z B Θ
I z Θ
ΘΘ ⋅−⋅=×=∇ BjBjBjp zz!!
Z-Pinch
)(2
)(22
0
0
0 rIr
rjrdrr
B z
r
z ⋅=ʹ′⋅ʹ′⋅ʹ′⋅= ∫ πµ
ππµ
Θ
Θ⋅−=×=∇ BjBjp z
!!Z-Pinch-equilibrium
drId
rI
drdI
rp z
zz )(
)2(2)2(
2
20
20 ⋅
⋅−=⋅⋅−=∇
πµ
πµ
)(2
)(22
0
0
0 rIr
rjrdrr
B z
r
z ⋅=ʹ′⋅ʹ′⋅ʹ′⋅= ∫ πµ
ππµ
Θ
drId
rI
drdI
rp z
zz )(
)2(2)2(
2
20
20 ⋅
⋅−=⋅⋅−=∇
πµ
πµ
πµ8
20
0INkT ⋅=
Θ⋅−=×=∇ BjBjp z
!!
Bennet-condition:
Z-Pinch-equilibrium
Confinement condition Z-pinch:
Screw-Pinch
θ- und Z-pinch are very unstable (detailed analysis later)
Current and und B-field in z- und Θ- direction
ΘΘ ⋅−⋅=×=∇ BjBjBjp zz
!!
)()(rBrB
dzrd
z
θθ=
Pitch of field lines:
Screw-Pinch is (mostly) diamagnetic
Bz is weakened
⊗
⊗
Bz
jz
jθ Bθ
and contributes to the confinement
x Bz jθ
jz x Bθ
-∇p
Confinement is better than in z-pinch, pressure and current profiles can be chosen
“low-ß”
p(r)
0
contribution
B z -field
Bp(r)
Bz(r)
“high-ß”
p(r)
0 contribution
B z -field
Bz(r)
Bp(r)
Screw-Pinch: high and low ß
Only the part of Bz that is generated by the current contributes to the confinement (homogeneous MF influences only stability)
0
20
0
2
22 µµBconstBp ==+
Usually, the plasma is diamagnetic, but at large currents it can be paramagnetic:
p(r)
0
Bz(r)
Bp(r) x Bz jθ
jz x Bθ
-∇p
Confinement then worse than in Z-pinch
Reversed Field Pinch
p(r)
0
“Reversed-Field-Pinch“ - start phase -
Bz(r)
Bp(r) p(r)
0
“Reversed-Field-Pinch“ - final phase -
B z (r) is reversed !
Bp(r)
ΘΘ ⋅−⋅=×=∇ BjBjBjp zz
!!Equilibria with Bz and Bp-field
θ-Pinch:
Z-Pinch: drId
rp z )(
)2(2
2
20 ⋅
⋅−=∇
πµ
02 0
2
=⎟⎟⎠
⎞⎜⎜⎝
⎛+∇
µzBp
Bz and Bp-field: drId
rBp zz )(
)2(22
2
20
0
2
⋅⋅
−=⎟⎟⎠
⎞⎜⎜⎝
⎛+∇
πµ
µ
Z-pinch fusion experiments
Iz-StromZ-Pinch
typ. µs
Iz(t)
t
Bθ-Feld
Z-pinch fusion experiments
θ-pinch fusion experiments
mit Crowbarschalter Iθ(t),Bz(t)
komprimiertePlasmasäule
t
Bz-Feld Iθ-Spule
Θ-Pinchjθ-Plasma
Kompressionsablauf:typ.
einige µs
Zeit
Radius
“schnelle”(Stoßwellen-)Kompression
adiabatischeKompression
B=Maximum
θ-pinch fusion experiments
summary
Bz
nT>0
r
ZylindrischePlasmasäulein Bz-Feld
j⊥=jΘ
Θ
z
θ-pinch
Z-pinch
r
zBΘ
Iz Θ
Z- and θ-pinch are unstable:
summary
Screw-pinch has better stability properties!