Ionospheric Alfvén Resonator (IAR) - Ultra and Extremely Low Frequency Electromagnetic Fields - page 186

where r ? denotes the perpendicular component of the gradient, that is r ? D

@ x ;@ y , where the symbols @ x D @=@x and @ y D @=@y denote the partial
derivatives with respect to x and y, respectively. It follows from Eq. ( 5.75 ) that
the vector A ?

can be written in the form

A ? Dr ? .‰ O z /; (5.76)

where ‰ is the second scalar potential. Indeed, substituting Eq. ( 5.76 )for A ? into

Eq. ( 5.75 ) gives an identity. Hence we get

A D A O z Cr ? .‰ O z /: (5.77)

Subsisting Eq. ( 5.76 )for A ? into Eqs. ( 5.73 ) and ( 5.74 ) and rearranging yields

2
?

ı B D . r ? A/ O z Cr ? @ z ‰ O z r ‰; (5.78)

and

E Dr ? dž r ? . O z @ t ‰/ O z .@ z dž C @ t A/: (5.79)

Potentials of Shear Alfvén and Compressional Waves in Plasma

The representation of the electromagnetic field via potentials is of frequent use in
plasma waves physics. In specific cases the general wave equations can be split
into two independent sets of equations in such a way that the scalar potentials dž
and A describe the shear Alfvén mode while the potential ‰ corresponds to the
compressional mode.
As the plasma is immersed in the external magnetic field, the plasma conductivity
exhibits anisotropy, which can be described by the tensor of the plasma conductiv-
ity ( 2.5 ) or by the tensor of dielectric permittivity ( 2.18 ). As the field-aligned plasma
permittivity " k , that is, the tensor component parallel to the magnetic field B 0 tends
to infinity, the parallel electric field becomes infinitesimal, that is E z D 0.This
implies that @ z dž C @ t A D 0, so that the component A can be expressed through dž.
The same is true if the field-aligned plasma conductivity k !1 . In particular, if
all perturbed quantities are considered to vary as exp. i!t/, then

i!A D @ z dž: (5.80)

In fact this means that the shear Alfvén and compressional modes can be described
through two scalar potentials, say dž and ‰, instead of three potentials. For example,
the shear Alfvén mode can be represented via only the potential dž