Geoscience Reference
In-Depth Information
convection is laminar or turbulent. At present the Earth's interior is not adequately
explored to clear up this extremely important question.
The flow character is dependent on the Reynolds number Re
D
lV=, where
l and V are characteristic space scale and velocity of the flow, and is the
kinematical viscosity. So, this parameter is very important for the mechanism of the
Earth's magnetic field generation. The numerical assessment of the flow velocity
for both the laminar and turbulent convection gives approximately the same value
V
D
0.1-0.2 cm=s (Golitsyn
1981
; Stevenson
1983
). Zeldovich and Ruzmaikin
(
1987
) have used the higher estimate V
D
4 cm=s. The characteristic size of
the flow is of the order of the Earth's core radius, that is l
D
10
3
km. The
kinematical viscosity varies within interval of 10
7
<<10
5
m
2
=s, that presents
great difficulties in estimating the Reynolds number. It is usually believed that the
kinematical viscosity possesses the value
D
10
6
m
2
=s, which is close to the
lower limit. The Reynolds number then exceeds 10
6
; that means that the convection
must be turbulent. Nevertheless, the large-scale flow inside the core flow gives no
comprehensive evidence yet of having turbulent structure. In conclusion we note that
the laminar flows have been studied in more detail and numerous dynamo-solutions
for the steady-state velocity fields have been derived (e.g., see Brodsky
1983
;Moffat
1968
; Roberts
1971
; and references therein).
1.1.2
Magnetohydrodynamic (MHD) Equations
To treat the electric and magnetic fields, Maxwell's electrodynamics equations are
required. The first pair of these equations in their full form are given by
r
B
D
0
.
j
C
@
t
D
/;
(1.1)
and
r
E
D
@
t
B
;
(1.2)
where
B
is the induction of magnetic field,
E
is the electric field,
j
is the conduction
current density, @
t
D
is the displacement current density,
0
denotes the magnetic
constant/magnetic permeability of free space, and the symbol @
t
stands for the
partial time-derivative, that is @
t
D
@=@t. In what follows we only consider a non-
magnetic medium so that the magnetic permeability of the medium is equal to unity.
The next pair of Maxwell's equations are given by
r
B
D
0;
(1.3)
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