9.9 Dynamic equilibria:flows 291Onceχ,ψandIhave been determined, it is possible to determine the pressure and
electrostatic potential profiles. This is done by taking the divergence of Eq.(9.78) to obtain
∇^2 g=−∇φ·∇×Q. (9.89)The pressure and electrostatic potential are contained within thegterm, whereas theQ
term involves onlyχ,ψandIor functions ofχ,ψandI(e.g., the velocity). Thus, the right
hand side of Eq.(9.89) is known and so can be considered as the source for a Poisson-like
equation for the left hand side. For the equation of motion
gmotion=ρU^2
2+
μ 0
(2π)^2I^2
2 r^2+P (9.90)
so that
P=gmotion−ρU^2
2−
μ 0
(2π)^2I^2
2 r^2. (9.91)
Regions wheregmotionis constant satisfy an extended form of the Bernoulli theorem,
namely
ρU^2
2+
B^2
2 μ 0+P=const. (9.92)Similarly, taking the divergence of Eq.(9.77) gives an equation for the electrostatic po-
tential (using Coulomb gauge so that∇·A=0). Thus, a complete solution for incompress-
ibleflow is obtained by first solving forχ,ψandIusing prescribed boundary conditions,
and then solving for the pressure and electrostatic potential.
9.9.2 Compressible plasma and applied poloidal field
The previous discussion assumed that the plasma was incompressible and the magnetic
field was purely toroidal (i.e., generated by the prescribed poloidal current). The more
general situation involves having a pre-existing poloidal magnetic field such as would be
generated by external coils and also allowing for plasma compressibility. This situation
will now be discussed qualitatively making reference to the sketch provided in Fig.9.13 (a
more thorough discussion is provided in Bellan (2003)).