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that the exponential functions can fit the approximate solutions of Eq. (
6.46
)inthis
region. For example, considering for the moment that V
A
is a constant, we get
sinh
n
l
1
; where
n
D
k
n
k
A
1=2
:
A
n
D
d
n
sinh
n
x
(6.52)
It follows from this equation that the boundary perturbations E
y
decay in amplitude
with decreasing the distance x due to the exponential fall off of all the coeffi-
cients A
n
. The interpretation we make is that the FMS waves must attenuate when
propagating from the outer boundary to the inner region of the magnetosphere. This
tendency is valid for the case of space-varying Alfvén speed except for the region
where the FLR occurs.
In the resonance region the coupling of the shear Alfvén and the FMS waves
cannot be ignored. The mode coupling is studied in more detail in Appendix F.
As we have noted above, if k
y
¤
0, then Eqs. (
6.25
)-(
6.29
) cannot be split into
two independent sets of equations. In this case the FMS wave can excite the shear
Alfvén wave and vice versa. The interaction between these two modes may greatly
affect the field amplitude as the resonance condition
k
A
.x/
D
k
n
(6.53)
holds true. If x
D
is a root of Eq. (
6.53
), then at this point the wave frequency
! equals to one of the Alfvén resonance frequencies !
n
.x/ that are given by
Eqs. (
6.39
)-(
6.41
).
As has already been stated in Appendix F, the amplitude the Alfvén mode which
includes the components ıB
y
, E
x
, and V
y
, has a peak of Lorentz form near the
FLR position x
D
. The schematic representation of the amplitude of ıB
y
as a
function of x= is displayed in Fig.
6.3
with solid line 1. According to Eqs. (
6.139
)
and (
6.141
), the amplitude of the components ıB
x
, E
y
, and V
x
has a maximum at
the same resonance point but this maximum is not so distinct as is shown in Fig.
6.3
with dashed line 2. It is worth mentioning that, as shown in Appendix F, the phase of
the resonance components E
x
and ıB
y
, changes by when crossing the maximum.
The next singular point x
D
, can be found from the following equation
k
A
.x/
D
k
n
C
k
y
:
(6.54)
The implication here is that the roots of this equation correspond to turning points
x
D
where solutions change from being oscillatory in nature to characteristically
growing or decaying with coordinate x. At the turning point the wave reflection
occurs. It should be noted that if V
A
.x/ is not a monotonic function there may be
more turning and resonance points.
The MHD box model is based on an idealized field geometry that ignores
the magnetic field line curvature and dip angle but includes the field variations
with radial distance and boundary conditions at the ionosphere and magnetopause.
The MHD box model provides us with a qualitative theory of the FLR in the
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