4.4 Two-fluid model of Alfvén modes 139The ratio of electron thermal velocity to Alfvén velocity is also ofinterest and isv^2 Te
vA^2=
κTe/me
B^2 /nmiμ 0=
mi
meβe. (4.91)Thus,v^2 Te>> vA^2 whenβe>> me/miandv^2 Te<< v^2 Awhenβe<< me/mi.Shear
Alfvén wave physics is different in theβe>>me/miandβe<<me/miregimes which
therefore must be investigated separately. MHD ignores thisβedependence, an oversim-
plification which leads to the paradoxes.
Both Faraday’s law and the pre-Maxwell Ampere’s law are fundamental to Alfvén wave
dynamics. The system of linearized equations thus is
∇×E 1 = −
∂B 1
∂t(4.92)
∇×B 1 = μ 0 J 1. (4.93)If the dependence ofJ 1 onE 1 can be determined, then the combination of Ampere’s
law and Faraday’s law provides a complete self-consistent description of thecoupled fields
E 1 ,B 1 and hence describes the normal modes. From a mathematical point of view, speci-
fyingJ 1 (E 1 )means that there are as many equations as dependent variables in the pair of
Eqs.(4.92),(4.93). The relationship betweenJ 1 andE 1 is determined by the Lorentz equa-
tion or some generalization thereof (e.g., drift equations, Vlasov equation,fluid equation
of motion). The MHD derivation used the polarization drift to give a relationship between
J 1 ⊥andE 1 ⊥but leaves ambiguous the relationship betweenJ 1 ‖andE 1 ‖.
The two-fluid equations provide a definite description of the relationship betweenJ 1 ‖
andE 1 ‖.At frequencies well below the cyclotron frequency, decoupling of modes also
occurs in the two-fluid description, and this decoupling is more clearly defined and more
symmetric than in MHD. The decoupling in a uniform plasma results because the depen-
dence ofJ 1 onE 1 has the property thatJ 1 z∼E 1 zandJ 1 ⊥∼E 1 ⊥. Thus, forω << ωci
there is a simple linear relation between parallel electric fieldand parallel current and an-
other distinct simple linear relation between perpendicular electric field and perpendicular
current;these two linear relations mean that the tensor relatingJ 1 toE 1 is diagonal (at
higher frequencies this is not the case). The decoupling can be seen by supposing that all
first order quantities have the dependenceexp(ik⊥·x+ikzz)wherek⊥=kxˆx+kyyˆ.
Hereˆk⊥is the unit vector in the direction ofk⊥andˆz׈k⊥is the binormal unit vector so
that the setkˆ⊥,ˆz׈k⊥,zˆform a right-handed coordinate system. Mode decoupling can be
seen by examining the table below which lists the electric and magnetic field components
in this coordinate system:
Ecomponents Bcomponents
ˆk⊥·E 1 ˆk⊥·B 1zˆ×ˆk⊥·E 1 zˆ×kˆ⊥·B 1
zˆ·E 1 ˆz·B 1Because of the property thatJ 1 z∼E 1 zandJ 1 ⊥∼E 1 ⊥the terms in boxes are decoupled
from the terms not in boxes. Hence, one mode consists solely of interrelationships between