The Earth’s Magnetic Field - Ultra and Extremely Low Frequency Electromagnetic Fields - page 8

This equation can be rearranged to the form

B B

r

V ;

d
dt

D

(1.21)

where the total time derivative is defined in Eq. ( 1.12 ).
Consider for the moment the “fluid line,” which is labeled by “fluid particles” on
the line (Landau and Lifshitz 1982 ). The “fluid line” moves together with the “fluid
particles,” which compose this line. Let ı l be the length element of this line. We now
study the temporal variations of this small element. If V is the mass velocity of the
fluid at the end of the “fluid line,” the mass velocity at another end of the “fluid line”
must be V C .ı l r / V . For the small time dt the increment of the length element
achieves the value dt.ı l r / V , so the total time derivative of the length element is
given by

d
dt ı l D .ı l r / V :

(1.22)

As can be seen from Eqs. ( 1.21 ) and ( 1.22 ) the change of the vectors ı l and B =
are determined by the same equation. In this sense, if these vectors are parallel at
initial time, they are still parallel at any other time. Furthermore, the variations of
their lengths are proportional to each other. This means that if two infinitely near
“fluid particles” are situated at the same magnetic field line at initial time, they
are still at the same field line at any other time. Moreover, the value B= changes
proportionally to the distance between these two “fluid particles.”
Notice that this conclusion is valid not only for infinitely near points but also
for the distant points situated at the same field line. This means that the magnetic
field line moves together with the “fluid particles” related to this line. We usually
say that if !1 , the magnetic field is “frozen in” and can be considered to
move with the fluid. The quantity B= varies in direct proportion to a tension of the
“fluid line.” In the case of incompressible flow, the mass density of moving “fluid
particles” is unchanged. Whence it appears that the magnetic field B itself varies in
direct proportion to the tension of the “fluid line.”
When the frozen-in field takes place, the magnetic flux across an arbitrary surface
S moving with the fluid velocity, that is a product of magnetic induction B and S,
keeps a constant value. This may result in an enhancement of the magnetic field due
to deformation or compression of the surface S. On the other hand, the flow may
complicate the magnetic field lines thereby decreasing its scale size, which makes
for the amplification of energy dissipation.
The concept of frozen-in field, which is very useful for the visualization of
complex flow and field pattern, will be dealt with in a future study. As has already
been stated, the concept of frozen-in field is valid for an extreme case of infinite
conductivity. Nonetheless, the scale sizes of cosmic bodies are so large that the
characteristic damping time d of the magnetic field may be enormous, even though
the medium conductivity is small or moderate. To illustrate this, consider the outer