ABSTRACT
Throughout this chapter, we shall assume that the fluid is nonviscous and has a negligible resistivity; thus, we consider “ideal” MHD. In other words, the magnetic Reynolds number, ReM = Lv/ηe, is very large (L and v are the characteristic length and characteristic velocity, respectively, and ηe is the electric resistivity, Equation (1.145)). Extensions to nonideal situations are discussed by Stone (1999). With these assumptions, the MHD equations (1.148) reduce to
∂ρ
∂t + ∇ · (ρv) = 0 (8.1)
∂ρv
∂t + ∇ · (ρvv) = −∇ P − 1
8π ∇(B2) + 1
4π (B · ∇)B (8.2)
∂e
∂t + ∇ · (ev) = −P∇ · v (8.3)
∂B ∂t
= ∇ × (v × B), (8.4)
where, as in Chapter 6, Section 6.5, the internal energy is used instead of the total energy. We shall assume that the MHD flow is axially symmetric; however, all three
components of velocity and magnetic field will be accounted for (the “2 1/2-D” approximation). As in Chapter 6, Section 6.6.1, we shall use cylindrical coordinates (R, Z , ) in which the set (Equation 8.1 to Equation 8.4) takes the form
∂ρ
∂t + ∇ · (ρv) = 0 (8.5)
∂(ρu) ∂t
+ ∇ · (ρuv) = −∂ P ∂ R
+ A 2
ρR3 + 1
4π BZ
∂ BR ∂ Z
− 1 4π
(
BZ ∂ BZ ∂ R
+ B R
∂(RB) ∂ R
)
(8.6)
∂(ρw) ∂t
+ ∇ · (ρwv) = −∂ P ∂ Z
+ 1 4π
BR ∂ BZ ∂ R
− 1 4π
(
BR ∂ BR ∂ Z
+ B ∂ B ∂ Z
)
(8.7)
∂ A ∂t
+ ∇ · (Av) = 1 4π
(
BR ∂(RB)
∂ R + RBZ ∂ B
∂ Z
)
(8.8)
∂e
∂t + ∇ · (ev) = −P∇ · v (8.9)
∂ BR ∂t
= ∂E ∂ Z
(8.10)
∂ BZ ∂t
= − 1 R
∂(RE) ∂ R
(8.11)
∂ B ∂t
= ∂EZ ∂ R
− ∂ER ∂ Z
, (8.12)
where the electromotive force E ≡ v × B (8.13)
has been introduced as an auxiliary vector field. As before, A is the angular momentum per unit volume. Note that some terms cancel out when the magnetic pressure gradient ∇(B2) and the magnetic tension (B · ∇)B (see Chapter 1, Section 1.7) are explicitly calculated. As a result, the expressions for the R and Z components of the magnetic pressure gradient (terms in brackets on the right-hand side of Equation 8.6 and Equation 8.7) contain only two derivatives instead of three. Note also that because of the assumed symmetry the component of the magnetic pressure gradient is equal to 0.
