4.4 Two-fluid model of Alfvén modes 1434.4.1 Two-fluid slow (shear) modes
As discussed above these modes haveBz 1 =0,E⊥ 1 =−∇φ 1 ,and∇·uσ⊥ 1 =0.We first
consider the parallel component of the linearized equation of motion, namely
nmσ∂uσz 1
∂t=nqσEz 1 −∂Pσ 1
∂z(4.112)
wherePσ 1 =γσnσ 1 κTσandγ=1if the motion is isothermal andγσ=3if the motion
is adiabatic and the compression is one-dimensional. The isothermal case corresponds to
ω^2 /kz^2 <<κTσ/mσand vice versa for the adiabatic case.
The continuity equation is∂n 1
∂t+∇·(nuσ 1 )=0. (4.113)Because the shear mode is incompressible in the perpendicular direction, the continuity
equation reduces to
∂n 1
∂t
+
∂
∂z
(n 0 uσz 1 )=0. (4.114)Taking the time derivative of Eq.(4.112) gives
∂^2 uσz 1
∂t^2
−γσκTσ
mσ∂^2 uσz 1
∂z^2=
qσ
mσ∂Ez 1
∂t(4.115)
which is similar to electron plasma wave and ion acoustic wave dynamics except it hasnot
been assumed thatEz 1 is electrostatic.
Invoking the assumption that all quantities are of the formf(x,y)exp(ikzz−iωt)
Eq.(4.115) can be solved to give
uσz 1 =iωqσ
mσEz 1
ω^2 −γσkz^2 κTσ/mσ(4.116)
and so the relation between parallel current and parallel electricfield is
μ 0 Jz 1 =iω
c^2Ez 1∑
σω^2 pσ
ω^2 −γσk^2 zκTσ/mσ. (4.117)
Usingzˆ·∇×B 1 =∇·(B 1 ׈z)=∇·(B⊥ 1 ׈z)the parallel component of Ampere’s
law becomes for the shear wave
∇⊥·(B⊥ 1 ×zˆ)=iω
c^2Ez 1∑
σω^2 pσ
ω^2 −γσkz^2 κTσ/mσ. (4.118)
Ion acoustic wave physics is embedded in Eq.(4.118) as well as shear Alfvén physics.
The ion acoustic mode can be retrieved by assuming that the electric field is electrostatic
in which caseB⊥ 1 vanishes. For the special case where the electric field is just in the
zdirection, and assuming thatκTi/mi<< ω^2 /k^2 z<< κTe/methe right hand side of
Eq.(4.118) becomes (
ω^2 pi
ω^2
−
1
k^2 zλ^2 De)
Ez 1 =0 (4.119)