FLR in Plasma Configurations - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 136

ordinary differential equations. This method is similar to the method of mod-
ulated normal modes in the theory of radio-wave propagation.
The interaction of FMS-waves with Alfven and slow magnetosonic modes
in a straight magnetic field has been studied in (see, e.g., [33], [45]) and
in the general case in ([10], [12], [28], [52]). It was shown that within the
framework of linear magnetohydrodynamics with a finite plasma pressure,
MHD-disturbances still remain singular at the field-lines where the resonance
requirements hold.

6.2 2D Inhomogeneous Plasma in a Uniform
Magnetic Field

Basic Equations

Consider linear oscillations of a cold plasma embedded in a box. Let the z -
axis of the rectangular coordinate system be oriented along a homogeneous
magnetic field B 0 . Plasma inhomogeneity in two directions ( ρ 0 = ρ 0 ( x, z ))
is confined in a box with dimensions l x ,l y ,l z . Maxwell's equations and the
linearized system of the ideal MHD-equations lead to the equation system
(4.43)-(4.45). For the Fourier harmonic of displacement

( ξ x ,ξ y )exp( − iωt + ik y y ) ,

the system reduces to

L ξ y = − ik y b, (6.1)

∂b
∂x =

−

L ξ x , (6.2)

∂ξ x
∂x

= b − ik y ξ y . (6.3)

Here

∂ 2
∂z 2 +

ω 2
c 2 A ( x, z )

L = (6.4)

c A ( x, z )= B 0 / 4 πρ 0 ( x, z )isanAlfven velocity, b = b z /B 0 is dimensionless
longitudinal wave magnetic field, and k y is a value of the azimuthal wave
vector. Recall that in the box model the faces z =0 , and z = l z simulate the
southern and northern ionospheres, respectively. Neglecting dissipation in the
ionosphere, we put that tangential components of the electric field be vanish
on these faces. Then (6.1)-(6.3) should be supplemented by the boundary
conditions

ξ x =0 ξ y =0 , at z =0 ,l z . (6.5)