294 Chapter 9. MHD equilibria
Integration and using the boundary condition thatP=0atr=agives
P(r)=
λ^2 ψ^20
4 π^2 μ 0 a^2(
1 −
r^2
a^2)
(9.101)
which shows that the on-axis pressure is larger wherea(z)is smaller as indicated in
Fig.9.13(a). The axial component of the equation of motion is thus
Fz = ˆz·(−∇P+Jpol×Btor)= −∂P
∂z+JrBφ= −
λ^2 ψ^20
4 π^2 μ 0∂
∂z(
1
a^2−
r^2
a^4)
−
1
μ 0
Bφ∂Bφ
∂z= −λ^2 ψ^20
π^2 μ 0(
−
1
2 a^3+
r^2
a^5)
∂a
∂z−
1
2 μ 0∂
∂z(
λψ
2 πr) 2
=
λ^2 ψ^20
π^2 a^3 μ 0(
1
2
−
r^2
a^2)
∂a
∂z−
λ^2 ψ^20
8 π^2 r^2 μ 0∂
∂z(r
a) 4
=
μ 0 I 02
2 π^2 a^3(
1 −
r^2
a^2)
∂a
∂z(9.102)
which shows that there is an axial force which peaks on axis. This forceis proportional to
currentflowing along theflux tube and to the axial non-uniformity of theflux tube, i.e., to
∂a/∂zand accelerates plasma away from regions whereais small to regions whereais
large.
If theflow stagnates, i.e., is such that the axial velocity is non-uniform so that∇·Uis
negative, then there will be a net inflow of matter into the stagnation region and hence an
increase of mass density at the locations where∇·Uis negative. Because magneticflux is
frozen to the plasma, there will be a corresponding accumulation of toroidalmagneticflux
at the locations where∇·Uis negative. IfUφis zero (as assumed on the basis of there being
no steady toroidal acceleration and zero initial toroidal velocity), then the accumulation of
toroidalflux will increase the toroidal field. This can be seen by considering the toroidal
component of the induction equation
∂Bφ
∂t=rBpol·∇(
Uφ
r)
−rUpol·∇(
Bφ
r)
−Bφ∇·Upol (9.103)or
DBφ
Dt
=−Bφ∇·Upol (9.104)whereD/DT=∂/∂t+U·∇andUφ=0has been assumed. Thus, it is seen thatBφ
will increase in the frame of the plasma when∇·Upolis negative. However, Ampere’s
law givesμ 0 I=2πrBφand sinceI=I 0 at the outer radius of the current channel, it is
seen that ifBφincreases, then the radius of the current channel has to decrease in order to
keeprBφfixed. The result is that stagnation of theflow tends to collimate theflux tube,
i.e., make it axially uniform as shown in Fig.9.13(b). The plasma accelerated upwards in
Fig.9.12 will then squeeze together theψand thusIsurfaces so that these surfaces will
be become vertical lines;very roughly stagnation of an upward moving plasma in Fig.9.12