Partially Ionized Plasma - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 23

Finally, we consider the dispersion equation at strong electron and ion
magnetizations. Leaving only transversal (1.102) and longitudinal (1.105)
conductivities and substituting into (1.119) ε ⊥ = c 2 /c 2 A , ε =0 ε =

[( k 0 λ e ) 2 (1 + iν e /ω )] − 1 , we find that (1.119) reduces to the product of two
terms. Equating one of them to zero, we obtain the equation for the so-called
isotropic or the fast magnetosonic (FMS) wave:

−

ω 2 = c 2 A k 2 .

(1.125)

The second wave mode is called the Alfven wave and the dispersion equa-
tion for it is obtained by equating the second term to zero. Thus, the dis-
persion equation for Alfven waves taking into account the attenuation at the
longitudinal resistivity is

[1 + ( k 2
⊥

λ e ) 2 ] ω 2 + iων e ( k 2 λ e ) 2 k 2 c 2 A =0 .

− (1.126)

⊥

These two wave modes are considered in details in Chapter 4.
For the majority of ULF-wave phenomena in the Earth's magnetosphere
and the upper ionosphere, the role of the ohmic part of the transversal and
longitudinal conductivities is insignificant. The media can be considered as
an anisotropic dielectric with the transversal dielectric permeability ε ⊥ and
zero-loss energy. However, a role of the longitudinal conductivity σ and/or
dielectric permeability ε can dominate for small-scale disturbances. For such
disturbances the dissipation and dispersion become significant at scales

= λ e 1+ i ν e
ω

1 / 2

c
ωε 1 / 2

L ⊥ = . (1.127)

In the ionosphere the electron inertial length λ e ∼ 10 − 100 m. In this topic

our concern is only the large-scale disturbances with L | L ⊥ | for which the

effects of longitudinal resistivity can be ignored.

References

1. Akasofu, S. I. and S. Chapman, Solar-Terrestrial Physics , Clarendon Press,
Oxford, England, 1972.
2. Braginskii, S. I., Transport in a plasma, in Reviews of Plasma Physics , 1 ,(Ed.
by M. A. Leontovich), Consultants Bureau, New York 1965.
3. Chapman, S. and T. G. Cowling, Mathematical Theory of Nonuniform Gases ,
3rd Edn., Cambridge: Cambridge University Press, 1970.
4. Clemmow, P. C. and J. P. Dougherty, Electrodynamics of Particles and Plasmas ,
Addison-Wesley, New York, 1990.
5. Ginzburg, V. L., The Propagation of Electromagnetic Waves in Plasmas ,
Pergamon Press, 2d ed., Oxford, 1970.
6. Ginzburg, V. L. and A. A. Rukhadze, Waves in Magnetoactive Plasma , Hand
Physics, 49 , Springer-New York, 1972.