4 Magn confinement eng - Max Planck Society · PDF fileMagnetic confinement p j B! ! ∇ =...

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Magnetic confinement B j p ! ! × = Pressure gradient is balanced by Lorentz force (currents perpendicular to magn. field) 0 = p B ! Pressure along magnetic field lines is constant 0 = p j ! Current lines lie surfaces of constant pressure

Transcript of 4 Magn confinement eng - Max Planck Society · PDF fileMagnetic confinement p j B! ! ∇ =...

Page 1: 4 Magn confinement eng - Max Planck Society · PDF fileMagnetic confinement p j B! ! ∇ = × Pressure gradient is balanced by Lorentz force (currents perpendicular to magn. field)

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

Page 2: 4 Magn confinement eng - Max Planck Society · PDF fileMagnetic confinement p j B! ! ∇ = × Pressure gradient is balanced by Lorentz force (currents perpendicular to magn. field)

θ-Pinch

Bz

nT>0

r

ZylindrischePlasmasäulein Bz-Feld

j⊥=jΘ

Θ

z

Diamagnetic current reduces external magnetic field

Page 3: 4 Magn confinement eng - Max Planck Society · PDF fileMagnetic confinement p j B! ! ∇ = × Pressure gradient is balanced by Lorentz force (currents perpendicular to magn. 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)

Page 4: 4 Magn confinement eng - Max Planck Society · PDF fileMagnetic confinement p j B! ! ∇ = × Pressure gradient is balanced by Lorentz force (currents perpendicular to magn. field)

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

Page 5: 4 Magn confinement eng - Max Planck Society · PDF fileMagnetic confinement p j B! ! ∇ = × Pressure gradient is balanced by Lorentz force (currents perpendicular to magn. field)

Coil/external current

B

j ∇ n j ∇ T +

high-ß-plasma causes B-field gradient

p(r)

Page 6: 4 Magn confinement eng - Max Planck Society · PDF fileMagnetic confinement p j B! ! ∇ = × Pressure gradient is balanced by Lorentz force (currents perpendicular to magn. field)

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

Page 7: 4 Magn confinement eng - Max Planck Society · PDF fileMagnetic confinement p j B! ! ∇ = × Pressure gradient is balanced by Lorentz force (currents perpendicular to magn. field)

Bj ×∇=µ 0BBp ×⎟⎟

⎞⎜⎜⎝

⎛×∇⋅=∇

0

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

Page 8: 4 Magn confinement eng - Max Planck Society · PDF fileMagnetic confinement p j B! ! ∇ = × Pressure gradient is balanced by Lorentz force (currents perpendicular to magn. field)

( )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

Page 9: 4 Magn confinement eng - Max Planck Society · PDF fileMagnetic confinement p j B! ! ∇ = × Pressure gradient is balanced by Lorentz force (currents perpendicular to magn. field)

( )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

Page 10: 4 Magn confinement eng - Max Planck Society · PDF fileMagnetic confinement p j B! ! ∇ = × Pressure gradient is balanced by Lorentz force (currents perpendicular to magn. field)

Das Bild k

r

z B Θ

I z Θ

ΘΘ ⋅−⋅=×=∇ BjBjBjp zz!!

Z-Pinch

Page 11: 4 Magn confinement eng - Max Planck Society · PDF fileMagnetic confinement p j B! ! ∇ = × Pressure gradient is balanced by Lorentz force (currents perpendicular to magn. field)

)(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

Page 12: 4 Magn confinement eng - Max Planck Society · PDF fileMagnetic confinement p j B! ! ∇ = × Pressure gradient is balanced by Lorentz force (currents perpendicular to magn. field)

)(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:

Page 13: 4 Magn confinement eng - Max Planck Society · PDF fileMagnetic confinement p j B! ! ∇ = × Pressure gradient is balanced by Lorentz force (currents perpendicular to magn. field)

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:

Page 14: 4 Magn confinement eng - Max Planck Society · PDF fileMagnetic confinement p j B! ! ∇ = × Pressure gradient is balanced by Lorentz force (currents perpendicular to magn. field)

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

Page 15: 4 Magn confinement eng - Max Planck Society · PDF fileMagnetic confinement p j B! ! ∇ = × Pressure gradient is balanced by Lorentz force (currents perpendicular to magn. field)

“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 ==+

Page 16: 4 Magn confinement eng - Max Planck Society · PDF fileMagnetic confinement p j B! ! ∇ = × Pressure gradient is balanced by Lorentz force (currents perpendicular to magn. field)

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

Page 17: 4 Magn confinement eng - Max Planck Society · PDF fileMagnetic confinement p j B! ! ∇ = × Pressure gradient is balanced by Lorentz force (currents perpendicular to magn. field)

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)

Page 18: 4 Magn confinement eng - Max Planck Society · PDF fileMagnetic confinement p j B! ! ∇ = × Pressure gradient is balanced by Lorentz force (currents perpendicular to magn. field)

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

Page 19: 4 Magn confinement eng - Max Planck Society · PDF fileMagnetic confinement p j B! ! ∇ = × Pressure gradient is balanced by Lorentz force (currents perpendicular to magn. field)

Z-pinch fusion experiments

Iz-StromZ-Pinch

typ. µs

Iz(t)

t

Bθ-Feld

Page 20: 4 Magn confinement eng - Max Planck Society · PDF fileMagnetic confinement p j B! ! ∇ = × Pressure gradient is balanced by Lorentz force (currents perpendicular to magn. field)

Z-pinch fusion experiments

Page 21: 4 Magn confinement eng - Max Planck Society · PDF fileMagnetic confinement p j B! ! ∇ = × Pressure gradient is balanced by Lorentz force (currents perpendicular to magn. field)

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

Page 22: 4 Magn confinement eng - Max Planck Society · PDF fileMagnetic confinement p j B! ! ∇ = × Pressure gradient is balanced by Lorentz force (currents perpendicular to magn. field)

θ-pinch fusion experiments

Page 23: 4 Magn confinement eng - Max Planck Society · PDF fileMagnetic confinement p j B! ! ∇ = × Pressure gradient is balanced by Lorentz force (currents perpendicular to magn. field)

summary

Bz

nT>0

r

ZylindrischePlasmasäulein Bz-Feld

j⊥=jΘ

Θ

z

θ-pinch

Z-pinch

r

zBΘ

Iz Θ

Page 24: 4 Magn confinement eng - Max Planck Society · PDF fileMagnetic confinement p j B! ! ∇ = × Pressure gradient is balanced by Lorentz force (currents perpendicular to magn. field)

Z- and θ-pinch are unstable:

summary

Screw-pinch has better stability properties!