MHD-Waves in Layered Media - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 212

have there poles. Equation (7.118) is fulfilled for the high conductivity over
the whole k plane except a narrow zone near the branch cuts. Close to the
cuts tan κ g H 1 . In this case the inequality (7.118) in the

wavenumber range (7.110) holds for

≈ 1 and tanh kh ≈

1 / 2

| ε a /ε g | 1 . Use for the magnitude of

tangent the estimation

| Im z =const = 1+exp(

− 2 | Im κ g H | ) 1

Im κ g H ,

max | tan κ g H

) ≈

1+

1 − exp ( − 2 | Im κ g H |

| Im k =const = 1+exp(

− 2 | Im k | h ) 1

Im kh .

max | tanh kh

h ) ≈

1+

− − |

Im k

|

1 exp ( 2

Thus, close to the branch cuts we have

1
Im κ g H ,

1
Im kh

tan κ g H tan kH

and for

σ a
σ g

Im κ g H Im kh

the influence of finite conductivity on ζ 1 /Y ( e )

g is insignificant. The same esti-
mate is obtained for ζ 2 /Y ( e g . A ratio of the atmospheric conductivity to the
ground conductivity is extremely small. For typical values of the atmospheric
σ a =10 − 4 s − 1 and crust σ g =10 7 s − 1 conductivities, we have

∼

σ a
σ g

10 − 4
10 7

=10 − 11 .

Thus, the finite-ground conductivity has no effect on the electric mode. Note,
that this conclusion is well known in the theory of radio wave propagation.
On the other hand, ground conductivity can essentially effect the magnetic
mode. For the magnetic mode, the ground can be replaced with a perfect
conductor only for an horizontal scale larger than the skin depth in the ground.
Equation ( 7.117) is true if

c
(2 πσ g ω ) 1 / 2 .

kd g 1 , g =

Let us now define the range of the wavenumbers for which one can neglect
penetration of the current from the ionosphere to the ground. First, let us
consider the function ∆ A . It follows at once from (7.102) and (7.116) that at

= k 0 ε a

1 / 2

Re k | k ∗ | ,

h X |

sin I | (7.119)

∗

the expression for

X

∆ A = ζ 1

ζ 2 |

sin I

|−

| sin I |