Magnetohydrodynamic Waves - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 98

in terms of α into the expression for ∆S , we obtain

k y
k 2
||

ω | a ( x 1 ) |

2 .

∆S ≈ | b 0 | (4.66)

8

The amplitude b 0 and location of the FLR-point x 1 depend, in (4.66), on
frequency ω and wavenumber k y . For a complete solution of the problem
either a numerical or an analytical solution of (4.53) or (4.58) are necessary.
The right bottom panel of Fig. 4.3 shows the results of numerical calculations
of the Pointing vector. The wave energy loss is equal to the divergence of the
Pointing vector.
It is important, that the energy flux is finite also behind the turning point.
The tunneling effect supports the energy input to the resonance surface. A
finite energy per unit area equal to the Pointing vector discontinuous jump
at the resonance surface dissipates in a thin resonance layer. Behind the res-
onance surface the energy flux becomes small and it vanishes at σ ⊥ → 0.

Under non-monotonous c A ( x ) distribution more than one resonance surfaces
can exist and the energy flux behind the first resonance surface is finite.

References

1. Alfven, H., Ark. Mat., Astr. Fysik , B29 (2), 1942.
2. Alfven, H., C.-G. Falthammar, Cosmical electrodynamics , 2nd ed., Oxford,
Clarendon Press, 1963.
3. Boyce, W. E. and R. C. DiPrima, Elementary Differential Equations and Bound-
ary Value Problems , 7th edn., John Wiley & Sons, Inc., New York, 2001.
4. Chen, L. and A. Hasegawa, A theory of long-period magnetic pulsations, 1.
Steady state excitation of field line resonance, J. Geophys. Res ., 79 , 1024-1032,
1974.
5. Chandrasekhar,

S. Hydrodynamic and Hydromagnetic Stability , Oxford

University Press,Oxford, England, 1961.
6. Cowling, T. G., Magnetohydrodynamics , 2nd edn., Adam Hilger, London, 1976.
7. Denisov, N. G., On the one feature of the electromagnetic wave propagating in
the inhomogeneous plasma., JETP , 31 , 609, 1956.
8. Forsterling, K, and Wuster, N. O., Uber die Entstehung von Oberwellen in der
Ionospheare, J. Atmos. Terr. Phys ., 2 , 22, 1951.
9. Ginzburg, V. L., The Propagation of Electromagnetic Waves in Plasmas ,
Pergamon, Oxford, 1970.
10. Ginzburg, V. L., Rukhadze A. A., Waves in Magnetoactive Plasma , Hand
Physics, 49 , Springer-Verlag, New York, 1972.
11. Guglielmi, A.V. and V. A. Troitskaya, Geomagnetic Pulsations and Diagnostics
of the Magnetosphere , Nauka, Moscow, 1973.
12. Guglielmi, A.V., Diagnostics of the magnetosphere and interplanetary medium
by means of pulsations, Space Sci. Rev., 16 , 331, 1974.
13. Nishida, A., Geomagnetic Diagnosis of the Magnetosphere , Springer-Verlag, New
York, 1978.