Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

Magnetic fields and tokamak plasmas
Alan Wootton
Paramagnetic and diamagnetic flux
Outside the plasma the toroidal field has the form B φe = B φ0 (R 0 /R), with B φ0 the value at a fixed
radius R 0 . This toroidal field, together with the poloidal field, takes part in balancing the plasma
pressure.
We need an equation relating fields to pressure. Substituting ∇ × B = µ 0
j into ∇p = j × B yields
(using B × ( ∇ × B) = ∇B2
2 − ( B • ∇)B
)
⎛
∇⎜
p + B2 ⎞
⎟ = B• ∇
⎝ 2µ 0
⎠
( ) B µ 0
For a straight axially symmetric system (∂/∂z = 0) we obtain
∂ ⎛
∂r
p + B 2 2
⎜ z
+ B θ
⎞
⎟ = − B 2
θ
⎝ 2µ 0
⎠ rµ 0
Multiplying each side by r 2 , letting u = r 2 , du = 2rdr, dv = ∂/∂r(.), v = (..), we obtain by
integrating by parts (∫udv = uv - ∫vdu)
r 2
⎛
⎜
⎝
p + B 2 2
z
+ B θ
2µ 0
⎞
⎟
⎠
a
0
a
⎛
− p + B 2 2
⎜ z
+ B θ
⎞
⎟
∫ 2rdr = − B 2
θ
⎝ 2µ 0
⎠
∫ rdr
µ 0
0
0
a
i.e.,
⎛
⎜
⎝
p + B 2 2
z
+ B θ
2µ 0
⎞
⎟
⎠
r = a
= 1
πa 2
a
∫
0
⎛
⎜
⎝
p + B 2
z
2µ 0
⎞
⎟ 2πrdr
⎠
That is, ignoring curvature and equating B z with B φ , the pressure balance constraint is
2µ 0
p = B θa
2 + B 2 2
φe
− B φ
14.8
where B φ is the toroidal field inside the plasma, B φe is the toroidal field outside the plasma,
means an average over the plasma radius, and we have assumed p = 0 at the boundary (i.e. at r =
a). That is, for a given plasma current I p and pressure , the difference (B φe
2 - ) adjusts
itself to ensure pressure balance. This happens because of a poloidally flowing current, either
diamagnetic or paramagnetic, in the plasma, as we derived in equations 14.4 and 14.17. In a
tokamak we have B φe
2 >> B φe
2 - , so that if the cross section is circular with radius a p ,
105