Geology Reference
In-Depth Information
In symbolic notation this is
2 u
λ
(
∇·
u )
+ μ
+ μ
(
∇·
u )
=−
F .
(1.269)
Using the vector identity (A.14),
2 u
=−∇×
(
∇×
u )
+∇
(
∇·
u ),
(1.270)
we obtain the Navier equation for the displacement field,
+
2μ)
(
∇·
u )
μ ∇×
(
∇×
u )
=−
F .
(1.271)
1.4.5 Solutions of the Navier equation
By the Helmholtz separation considered in Section 1.2, the vector displacement
field u can be broken into lamellar and solenoidal parts written as
u
=∇
L
+∇×
A ,
(1.272)
where L is the scalar lamellar potential, and A is the vector potential, giving rise
to the solenoidal part of the vector displacement field, with the gauge
∇·
A
=
0.
Similarly, the vector body force per unit volume can be separated as
F
=∇
L F +∇×
A F ,
(1.273)
with the gauge
0. A further separation derives from the Lamb-Backus
decomposition (1.174) of a vector field into lamellar, poloidal and toroidal or tor-
sional parts. Applied to both u and F ,wehave
∇·
A F
=
u
=
L
+
P
+
T
=∇
L
+∇×
(
∇×
r P )
+∇×
r T ,
(1.274)
and
F
=
L F +
P F +
T F =∇
L F +∇×
(
∇×
r P F )
+∇×
r T F ,
(1.275)
where L , P , T are the scalars for u ,and L F , P F , T F are the scalars for F . Applying
the Helmholtz separation to the Navier equation of equilibrium (1.271) itself yields
2 L
+
2μ)
=−∇
L F ,
(1.276)
μ ∇× ∇×
A ) =−∇×
(
∇×
A F .
(1.277)
Taking the divergence of (1.276) gives
1
λ +
4 L
2 L F ,
=−
(1.278)
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