Geoscience Reference
In-Depth Information
where
F
is the external force,
p
the pressure,
π
m
the momentum flux density
tensor due to material motions, and
π
w
the momentum flux density tensor due
to waves in the medium. Applying Gauss's theorem again, (2.9) becomes
∂(ρ
U
)/∂
t
=−∇
p
+
F
−∇·
π
m
−∇·
π
w
(2.10)
The pressure gradient term should be familiar. The external force
F
can be of
many different types, which will be treated as they arise in the text.
To understand the material momentum tensor, consider a single particle of
mass
m
moving at velocity
v
. The momentum,
m
v
, is carried along by the velocity
v
, resulting in a momentum flux tensor given by
π
m
=
v
m
v
Written as a 3
×
3 matrix, this becomes
(π
m
)
jk
=
m
υ
j
υ
k
An analogous form for a fluid of mass density
ρ
characterized by a mean flow
velocity
U
is given by
(π
m
)
jk
=
ρ
U
j
U
k
(2.11)
This part of the momentum flux tensor describes how momentum is transferred
within a fluid by the motion of the fluid. If there is any net divergence of this
momentum flux, a net force will occur as indicated in (2.10). The divergence of
a tensor is a vector and may be written
∂π
jk
∂
(
∇·
π)
j
=
k
x
k
where
x
k
is the
k
th Cartesian coordinate. Applying this to the tensor in (2.11),
x
k
ρ
U
j
U
k
∂
(
∇·
π
m
)
j
=
k
∂
U
k
∂
x
k
ρ
U
j
+
ρ
U
j
∂
∂
=
k
x
k
(
U
k
)
∂
which in vector form is
∇·
π
m
=
U
·∇
(ρ
U
)
+
ρ
U
(
∇·
U
)
The first term is the advective derivative of the momentum and can be combined
with the partial derivative to form the total time derivative. In this text we follow
the notation
(
u
,
v
,
w
)
=
(east,north,up) for the neutral wind.
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