The Earth’s Magnetic Field - Ultra and Extremely Low Frequency Electromagnetic Fields - page 36

Let ı B and ıP be the small perturbations of the magnetic field B 0 and of the
pressure P 0 , respectively. Similarly, in the first approximation Eq. ( 1.37 )ofthe
conducting fluid motion can be reduced to

1
0 . r ı B / B 0 :

0 @ t V Dr ıP C (1.49)

To simplify the problem, we assume that the fluid conductivity !1 , so that
the magnetic field lines are frozen into the conducting fluid. This means that the
electric field can be derivable from the velocity through Eq. ( 1.35 ), that is E D
B 0 V . In such a case Eq. ( 1.18 ) is reduced to

@ t ı B Dr . V B 0 /: (1.50)

Since the flow is isoentropic, the changes of pressure are related to the changes of
density through

ıP D c s ı; (1.51)

where c s D .@P=@/ S is the squared sound velocity taken at the constant
entropy. We seek for the solution of the set of Eqs. ( 1.48 )-( 1.51 )intheform
of harmonic wave. All the quantities are assumed to vary as exp .i k r i!t/,
where k is the wave vector and ! is the frequency. The following combined set of
dynamic and electrodynamic equations for the conducting fluid remains after these
simplifications:

!ı D 0 k V ; (1.52)

! 0 V D c s ı k C 0 B 0 . k ı B /;

(1.53)

!ı B D k . V B 0 /: (1.54)

In addition, the equation r B D 0 is reduced to k ı B D 0. The former equation
holds automatically since ı B is perpendicular to k as it follows from Eq. ( 1.54 ).

1.4.2 Shear Alfvén Waves

The set of Eqs. ( 1.52 )-( 1.54 ) can be split into two independent sets of variables
(e.g., see Landau and Lifshitz 1982 ). The first one consists of the perpendicular
components of magnetic perturbation ı B ? and the velocity V ? asshowninFig. 1.15
with the arrows parallel to z -axis. Both of these vectors are perpendicular to that
plane in which the undisturbed field B 0 and wave vector k are situated. As is seen
from Eq. ( 1.52 ), this means that ı D 0, i.e., the medium density does not vary.
Combining Eqs. ( 1.53 ) and ( 1.54 ), carrying out the triple cross product

A 1 . A 2 A 3 / D A 2 . A 1 A 3 / A 3 . A 1 A 2 /; (1.55)