Geoscience Reference
In-Depth Information
The implication of this equation, which is similar to the diffusion equation, is that
the magnetic field can diffuse in the conducting media. In order to obtain the order-
of-magnitude estimate of the characteristic time d of the diffusion process one
should give a rough assessment of the derivatives in Eq. ( 1.15 )
B
d m B
l 2 :
(1.16)
Here B is the perturbation of magnetic field and l is the characteristic space scale
of perturbed region. Whence we get
d l 2 = m D 0 l 2 :
(1.17)
This estimate is representative for the diffusion processes since the time scale, d ,
of magnetic field perturbation increases with the scale size l squared.
1.1.3
The Concept of “Frozen-In” Magnetic Field
It is usually the case that the magnetic field falls off with distance from the
source due to Joule dissipation in a conducting medium. However the movement
of conducting fluid may result in the amplification of original/inoculating magnetic
field. The possibility of such phenomena follows from the concept of “frozen-in”
magnetic field lines that have been discovered by Alfvén (e.g., see monograph by
Alfvén 1950 ; Alfvén and Falthammar 1963 , and references therein). The effect
of “frozen-in” magnetic field arises when the temporal scale of magnetic field
perturbations, d , is large as compared to the variation time/period, T , of the velocity
field. In the strict sense, this effect holds in an extreme case of infinite conductivity.
This implies that the Joule dissipation can be neglected. Taking !1 in
Eq. ( 1.14 ) we get
@ t B Dr . V B /:
(1.18)
Expanding the curl operator and taking into account that r B D 0 ,Eq.( 1.18 ) can
also be written
@ t B D . B r / V . V r / B B . r V /:
(1.19)
Now we use the principle of conservation of fluid mass given in Eq. ( 1.11 ).
Eliminating r V from Eqs. ( 1.19 ) and ( 1.11 )gives
B
@ t C . V r / B
B
. V r / D . B r / V :
@ t B
(1.20)
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