Geoscience Reference
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So, the polarization of the Alfvén oscillations differs from that of the poloidal
mode. The electric field of the poloidal mode has only an azimuthal component,
while the magnetic field perturbations and plasma velocity are directed across the
magnetic shell.
6.1.2
Toroidal Mode
The shear Alfvén mode could have an important role to play in the generation
of field line resonances (FLRs hereafter). These kinds of standing waves in the
magnetosphere have frequently been observed onboard the satellites (e.g., see
Glassmeier 1995 ) so that a spatial structure and the source mechanisms of these
waves are of a special interest in geophysical studies.
Using the spherical coordinates, the azimuthal components of Eq. ( 6.2 ) can be
written as
@ t ıB ' D r 1
f @ r Œr . V B 0 / @ Œ. V B 0 / r g :
(6.8)
Substituting the angular and radial components of V B 0 into Eq. ( 6.8 ) yields
@ t ıB ' D r 1 ǚ @ r rV ' B r @ V ' B :
(6.9)
The components B r and B of undisturbed Earth's magnetic field are given
by Eq. ( 1.33 ). Substituting these components into Eq. ( 6.9 ) and rearranging this
equation yields
@ t ıB ' D r sin . B 0 r / V '
r sin
;
(6.10)
where we have introduced the differential operator acting on the functions of
variables r and :
4r 3 2 cos @ r C
@ :
0 M e
sin
r
B 0 rD
(6.11)
Here M e denotes the Earth's magnetic dipole moment.
The azimuthal component of the equation of motion ( 6.5 ) has the form
4r 4 ǚ @ ıB ' sin C 2 cos @ r rıB ' :
M e
@ t V ' D
(6.12)
Combining this equation with Eq. ( 1.33 ) for the components B r and B and
rearranging, we find that
0 r sin @ t V ' D . B 0 r / r sin ıB ' :
(6.13)
 
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