Magnetospheric MHD Resonances and ULF Pulsations - Ultra and Extremely Low Frequency Electromagnetic Fields - page 212

6.1.4 Azimuthal Harmonics

From the above analysis it is clear that in the axisymmetric magnetosphere model
the basic equations can be split into two independent sets for the shear Alfvén and
compressional waves, which can propagate independently of each other. According
to the FLR theory, the magnitude of the standing shear Alfvén wave can reach
a peak value in the vicinity of resonance magnetic shells under certain resonant
conditions, whereas the standing compressional wave is associated with variations
of the magnetic field perpendicular to the magnetic shells (Radoski 1967a ; Radoski
and Carovillano 1969 ; Southwood 1974 ; Chen and Hasegawa 1974 ; Krylov and
Fedorov 1976 ; Krylov and Lifshitz 1984 ). In a general way, that is, in an arbitrary
ambient magnetic field, the MHD-wave equations for these modes can be coupled.
For example, if ' dependence of the normal modes is taken into account, all
the functions should be expanded in a series of azimuthal harmonics exp .im'/,
where m D 0;1;2;::: is azimuthal wave number. Ignoring the ' dependence, we
thereby have chosen m D 0 in the above equations. Furthermore, the shear and
compressional Alfvén waves in the magnetospheric plasma can be coupled through
the boundary conditions at the conducting E-layer of the ionosphere due to both the
tensor character of the ionospheric plasma conductivity and finite value of Hall and
Pedersen conductivities.
If m ¤ 0 and m ¤1 , the set of MHD equations for magnetospheric oscillations
does not reduce to independent equations for the toroidal and poloidal modes.
Nevertheless, away from the resonance magnetic shells the coupling between these
modes is weak under the requirement that m 1, and in the first approximation
they can be considered as independent modes. The interactions between the toroidal
and poloidal modes become significant only in a narrow region in the vicinity of the
resonance shell (Leonovich and Mazur 1993 ; Leonovich 2000 ).
In the case of m 1 the coupling between the shear Alfvén and compressional
modes is so strong that they cannot be divided into two individual modes. Both
of these modes manifest themselves as a single MHD mode, which is more likely
to be similar to the Alfvén wave rather than to compressional one (Leonovich and
Mazur 1993 ; Leonovich 2000 ). Nevertheless, such a quasi-Alfvén wave combines
the properties of both modes, i.e., strong localization across the magnetic shell, that
is typical for the shear Alfvén wave, and the presence of a considerable constituent
of the field-aligned magnetic field variations, that is typical for the compressional
wave.
If m !1 , the azimuthal scale across the main magnetic field tends to zero. This
implies that the derivatives over ' in the MHD equations become much greater than
those with respect to other variables. For this special case the MHD-wave equations
can be split into two independent groups in analogy to the case of m D 0. As would
be expected, considering the small value of the transverse scale, the plasma velocity
V ' is small and the plasma movement is mainly concentrated within the meridional
plane (Dungey 1954 , 1963 ).