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3.2 Force balance in tokamak plasmas

Parallel force balance

Consider the force balance in the direction of B. Dotting the equilibrium equation (56) by B, we obtain

0 = B ⋅∇P,
(57)

which implies that P is constant along a magnetic field line. Since Ψ is also constant along a magnetic field line, P can be expressed in terms of only Ψ on a single magnetic line. Note that this does not necessarily mean P is a single-valued function of Ψ, (i.e. P = P(Ψ)). This is because P still has the freedom of taking different value on different magnetic field lines with the same value of Ψ while still satisfying B ⋅∇P = 0. This situation can appear when there are saddle points (X points) in Ψ contours (refer to Sec. A.10) and P takes different functions of Ψ in islands of Ψ sepearated by a X point. For pressure within a single island of Ψ, P = P(Ψ) can be safely assumed.

On the other hand, if P = P(Ψ), then we obtain

B ⋅∇P = dP- dΨB ⋅∇Ψ = 0,

i.e., Eq. (57) is satisfied, indicating P = P(Ψ) is a sufficient condition for the force balance in the parallel (to the magnetic field) direction.

Toroidal force balance

Next, consider the ϕ component of Eq. (56), which is written

1∂P JZBR − JRBZ = R--∂ϕ.
(58)

Since P = P(Ψ), which implies ∂P∕∂ϕ = 0, equation (58) reduces to

JZBR − JRBZ = 0
(59)

Using the expressions of the poloidal current density (50) and (51) in the force balance equation (59) yields

∂g ∂g ---BR + ---BZ = 0, ∂R ∂Z
(60)

which can be further written

B ⋅∇g = 0.
(61)

According to the same reasoning for the pressure, we conclude that g = g(Ψ) is a sufficient condition for the toroidal force balance. (The function g defined here is usually called the “poloidal current function” in tokamak literature. The reason for this name is discussed in Sec. A.3.)

Force balance along the major radius: Grad-Shafranov equation

Next, consider the force balance in ˆR direction. The Rˆ component of the force balance equation (56) is written

JϕBZ − JZB ϕ = ∂P ∂R
(62)

Using the expressions of the current density and magnetic field [Eqs. (6), (51), and (54)], equation (62) is written

− 1△ ∗Ψ-1∂Ψ- − 1-∂g-g-= μ0∂P-. R R ∂R R ∂R R ∂R
(63)

Assuming the sufficient condition discussed above, i.e., P and g are a function of only Ψ, i.e., P = P(Ψ) and g = g(Ψ), Eq. (63) is written

− 1△ ∗Ψ 1∂-Ψ − 1-dg-∂Ψ-g-= μ0dP-∂Ψ-, R R∂R R dΨ ∂R R dΨ ∂R
(64)

which can be simplified to

△∗Ψ = − μ R2dP-− -dgg, 0 dΨ dΨ
(65)

i.e.,

∂2Ψ ∂ ( 1 ∂Ψ ) dP dg ∂Z2- +R ∂R- R-∂R- = − μ0R2 dΨ-− dΨg.
(66)

Equation (66) is known as Grad-Shafranov (GS) equation.

[Note that the Z component of the force balance equation is written

JRBϕ − JϕBR = ∂P- ∂Z
⇒−-∂g- ∂Z1- R-g R −-1 R△∗Ψ1- R∂Ψ- ∂Z = μ0dP- dΨ∂Ψ- ∂Z
⇒−-dg dΨ∂Ψ- ∂Z1- Rg- R −1- R△∗Ψ-1 R∂Ψ- ∂Z = μ0dP dΨ∂Ψ- ∂Z
⇒−-dg dΨ1- R-g R −-1 R△∗Ψ1- R = μ0dP dΨ
⇒△∗Ψ = −μ0R2dP dΨ −dg dΨg

which turns out to be identical with the Grad-Shafranov equation. This is not a coincidence. The reason is that the force balance equation has been satisfied in three different directions (namely, ˆ ϕ, ˆ R, and B direction) and thus it must be satisfied in all the directions.]

Using the GS equation, Jϕ can also be expressed as

1 ( dP dg ) Jϕ = ---- μ0R2--- + --g . μ0R dΨ dΨ
(67)

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