Fundamentals of Plasma Physics

4.4 Two-fluid model of Alfvén modes 145

which is called the inertial Alfvén wave (IAW). Ifk⊥^2 c^2 /ω^2 peis not too large, thenω/kz
is of the order of the Alfvén velocity and the conditionω^2 >> k^2 zκTe/mecorresponds to
v^2 A>>κTe/meor

βe=

nκTe

B^2 /μ 0

<<

me

mi

. (4.128)

Thus, inertial Alfvén wave shear modes exist only in the ultra-lowβregime whereβe<<
me/mi.
In the situation whereκTi/mi<<ω^2 /kz^2 <<κTe/me,Eq.(4.126) can be recast as

k^2 ⊥

(ω^2 −kz^2 vA^2 )

−

ω^2 pe

c^2

1

k^2 zκTe/me

+

ω^2 pi

c^2

1

ω^2

=0. (4.129)

Becauseω^2 appears in the respective denominators of two distinct terms, Eq.(4.129) is
fourth order inω^2 and so describes two distinct modes. Let us suppose that the mode in
question is much faster than the acoustic velocity, i.e.,ω^2 /kz^2 >>κTe/mi.In this case the
ion term can be dropped and the remaining terms can be re-arranged to give

ω^2 =k^2 zv^2 A

(

1+

k⊥^2

vA^2

κTe

me

c^2

ω^2 pe

)

; (4.130)

this is called the kinetic Alfvén wave (KAW).

ρ^2 s=

1

vA^2

κTe

me

c^2

ω^2 pe

=

1

ω^2 ci

κTe

mi

(4.131)

as a fictitious ion Larmor radius calculated using the electron temperature instead of the
ion temperature, the kinetic Alfvén wave (KAW) dispersion relationcan be expressed more
succinctly as
ω^2 =k^2 zvA^2

(

1+k⊥^2 ρ^2 s

)

. (4.132)

Ifk⊥^2 ρ^2 sis not too large, thenω/kzis again of the order ofvAand so the conditionω^2 <<
kz^2 κTe/mecorresponds to havingβe>> me/mi.The conditionω^2 /kz^2 >> κTe/mi
which was also assumed corresponds to assuming thatβe<< 1 .Thus, the KAW dispersion
relation Eq.(4.132) is valid in the regimeme/mi<<βe<< 1.
Let us now consider the situation whereω^2 /k^2 z<< κTi/mi, κTe/me.In this case
Eq.(4.126) again reduces to

ω^2 =k^2 zvA^2

(

1+k⊥^2 ρ^2 s

)

(4.133)

but this time

ρ^2 s=

κ(Te+Ti)

miω^2 ci

. (4.134)

This situation would describe shear modes in a highβplasma (ion thermal velocity faster
than Alfvén velocity).
To summarize: the shear mode hasBz 1 =0, Ez 1 =0, Jz 1 =0,E⊥ 1 =−∇φ 1
and exists in the form of the inertial Alfvén wave forβe<< me/miand in the form
of the kinetic Alfvén wave forβe>> me/mi.The shear mode involves incompressible
perpendicular motion, i.e.,∇·uσ⊥ 1 =ik⊥·uσ⊥ 1 =0, which means thatk⊥is orthogonal
touσ⊥ 1 .For example, in Cartesian geometry, this means that ifuσ⊥ 1 is in thexdirection,