Geoscience Reference
In-Depth Information
Equations (6.39) are written for the covariant components
E
k
,b
k
of the
electromagnetic field in an arbitrary coordinate system. Usually, the orthog-
onal coordinate system is used and electric and magnetic fields are expanded
over the local orthonormal basis
E
1
i
1
+
E
2
i
2
+
E
3
i
3
,
b
=
b
1
i
1
+
b
2
i
2
+
b
3
i
3
,
E
=
E
k
=
E
b
k
=
b
·
i
k
,
·
i
k
,
i
k
=
δ
jk
,
E
k
and
b
k
are the physical components of the
E
and
b
fields. The relation between physical and covariant components is
especially simple in the orthogonal coordinates:
where
i
j
=
h
j
e
j
,
i
j
·
E
k
=
E
k
b
k
=
b
k
h
k
.
Since the ULF-frequencies are less than the ion cyclotron frequency, the
Hall conductivity can be neglected in the regions above the
E
-layer. Therefore
(6.39) has the form
h
k
,
1
h
3
∂E
1
∂x
3
−
ik
0
h
1
h
2
b
2
=0
,
(6.40)
1
h
3
∂b
2
∂x
3
−
h
2
h
1
E
1
=
1
h
3
∂b
3
∂x
2
,
ik
0
ε
⊥
(6.41)
ik
0
h
1
h
2
h
3
b
3
=
∂E
2
∂E
1
∂x
2
,
∂x
1
−
(6.42)
1
h
3
∂E
2
∂x
3
+
ik
0
h
2
h
1
b
1
=0
,
(6.43)
1
h
3
∂b
1
∂x
3
+
ik
0
ε
⊥
h
1
h
2
E
2
=
1
h
3
∂b
3
∂x
1
.
(6.44)
If the coupling of the Alfven and FMS modes is weak, then quazi-Alfven
and quazi-FMS-waves can propagate in plasmas. In the quazi-Alfven waves,
the longitudinal electric current is finite but the longitudinal magnetic field is
small. Equation set (6.40)-(6.44) is split into two groups: (6.40)-(6.41) for the
toroidal Alfven mode and (6.43)-(6.44) for the poloidal mode. In quazi-FMS
modes, on the contrary the longitudinal magnetic field is not small but the
field-aligned current is small.
Some Features of the Alfven Waves
Large Transversal Wavenumbers
Coupling of Alfven and FMS-waves is yielding with a decreasing the scale
transversal to the main magnetic field. If the scale approaches zero, MHD-
wave equations split into the separate equations for Alfven and FMS-modes.
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