Geoscience Reference
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cut, it is possible to estimate its influence with the help of contour integrating.
As a result, exponentially damping disturbances are added to the disturbances
damping as 1 /t .
One can consider that these exponentially damping disturbances are small
in comparison with ones damping with the power law. However, it is not so.
First, these exponential disturbances correspond to global oscillations of the
whole cavity. And, on the other hand, their decrement can be small. These
modes can exist for a long time before the energy is transferred from a cav-
ity mode to toroidal oscillations of field-lines each oscillating with its own
frequency.
Temporal Evolution of Poloidal Alfven Waves
Following Mann and Wright [14], we shall describe the time evolution of the
poloidal Alfven waves. At k y
0 , the azimuthal displacement
ξ y is caused by the toroidal Alfven waves, and at k y →∞
0 ,b z
0 x
,b z
0 y
0
the displacement ξ x is caused by the poloidal Alfven waves. Consider the case
of large but finite k y . Suppose that the displacement ξ x ( x, t = 0) is given at
t = 0. The longitudinal magnetic field b
is not disturbed at the initial time
instant, b t =0 = 0. So we can write
t =0
∂ξ x
∂x
+ ik y ξ y | t =0 =0 .
Let the plasma at t = 0 be stationary. The displacements ξ x and ξ y at
the next time instants are shown in Figure 5.8 (left frames). The time scale
τ is obtained from the condition of the equality of the toroidal and poloidal
energies. It is clearly seen from the Fig. 5.8a that the phase mixing exists in
both components. At t =3 τ polarization becomes mostly toroidal. The energy
transfer from the poloidal component (solid line) to the toroidal component
(dashed line) is shown in Fig. 5.8 (right frame).
5.10 Summary
The main purpose of the box model study is to get the qualitative picture
of the penetration of MHD-waves from the magnetospheric boundary into its
inner regions and to study the principal features of global resonance and FLR.
A penetration pattern of the hydromagnetic waves into the magnetospheric
inner regions is contained in (5.7a)-(5.7c).
A scheme of these processes without quantitative details can be summa-
rized as follows. A disturbance propagates as an FMS-wave from the mag-
netospheric boundary to the turning point x t , obtained from the condition
(4.55) with k = k n
k A ( x t )= k y + k n .
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