308 Chapter 10. Stability of static MHD equilibria
∂ρ
∂t
+∇·(ρU)=0 (10.40)
P∼ργ. (10.41)
Ohm’s law, Eq.(10.37), and Faraday’s law, Eq. (10.38), are combined to give theinduction
equation
∂B
∂t
=∇×(U×B) (10.42)
which prescribes the magnetic field evolution and, as discussed in the contextof Eq.(2.82),
shows that the magneticflux is frozen into the magnetofluid. Equation (10.41), the adi-
abatic relation, implies the following relationships for spatial and temporal derivatives of
density and pressure
∇P
P
=γ
∇ρ
ρ
,
1
P
∂P
∂t
=
γ
ρ
∂ρ
∂t
. (10.43)
Combining these relationships with the continuity equation, Eq.(10.40), gives the pressure
evolution equation
∂P
∂t
+U·∇P+γP∇·U=0 (10.44)
which can also be written as
∂P
∂t
+∇·(PU)+(γ−1)P∇·U=0. (10.45)
An expression for the overall energy can be derived by first writing the equation of motion,
Eq.(10.36), as
ρ
∂U
∂t
+ρ∇
(
U^2
2
)
−ρU×∇×U=J×B−∇P (10.46)
and then dotting Eq.(10.46) withUto obtain
ρ
∂
∂t
(
U^2
2
)
+ρU·∇
(
U^2
2
)
=J×B·U−U·∇P. (10.47)
Multiplying the entire continuity equation byU^2 / 2 and adding the result gives
∂
∂t
(
ρU^2
2
)
+∇·
(
ρU^2
2
U
)
=−J·U×B−∇·(PU)+P∇·U. (10.48)
Using Ampere’s law, Eq.(10.39), to eliminateJ, then Ohm’s law, Eq.(10.37), to eliminate
U×B,and the pressure evolution equation, Eq.(10.45), to eliminateP∇·U,this energy
equation becomes
∂
∂t
(
ρU^2
2
)
+∇·
(
ρU^2
2
U
)
=
(∇×B)
μ 0
·E−∇·(PU)−
1
(γ−1)
(
∂P
∂t
+∇·(PU)
)
.
(10.49)
Finally, using the vector identity
∇·(E×B)= B·∇×E−E·∇×B
= −B·
∂B
∂t