Fundamentals of Plasma Physics

2.5 Magnetohydrodynamic equations 49

A formal proof of this frozen-influx property will now be established by direct calcula-

tion of the rate of change of the magneticflux through a surfaceS(t)bounded by a material

lineC(t), i.e., a closed contour which moves with the plasma. This magneticflux is

Φ(t) =

∫

S(t)

B(x,t)·ds (2.78)

and theflux changes with respect to time due to either (i) the explicit time dependenceof

B(t)or (ii) changes in the surfaceS(t)resulting from plasma motion. The rate of change

offlux is thus

DΦ

Dt

= lim

δt→ 0

(∫

S(t+δt)B(x,t+δt)·ds−

∫

S(t)B(x,t)·ds

δt

)

. (2.79)

The displacement of a segmentdlof the bounding contourCisUδtwhereUis the velocity

of this segment. The incremental change in surface area due to this displacement is∆S=

Uδt×dl. The rate of change offlux can thus be expressed as

DΦ

Dt

= lim

δt→ 0

∫

S(t+δt)

(

B(x,t) +δt

∂B

∂t

)

·ds−

∫

S(t)

B(x,t)·ds

δt

= lim

δt→ 0

∫

S(t)

(

B(x,t) +δt

∂B

∂t

)

·ds+

∮

C

B(x,t)·Uδt×dl−

∫

S(t)

B(x,t)·ds

δt

=

∫

S(t)

∂B

∂t

·ds+

∮

C

B(x,t)·U×dl

=

∫

S(t)

[

∂B

∂t

+∇×(B×U)

]

·ds. (2.80)

Thus, if

∂B

∂t

=∇×(U×B) (2.81)

then

DΦ

Dt

= 0 (2.82)

so that the magneticflux linked by any closed material line is constant. Therefore, magnetic

flux is frozen into an ideal plasma because Eq.(2.76) reduces to Eq.(2.81) ifη= 0.Equation

(2.81) is called the ideal MHD induction equation.

2.5.5 MHD equations of state

Double adiabatic laws A procedure analogous to that which led to Eq.(2.35) gives the
MHD adiabatic relation
PMHD
ργ

= const. (2.83)

where againγ = (N+ 2)/N andNis the number of dimensions of the system. It

was shown in the previous section that magneticflux is conserved in the plasma frame.