Geoscience Reference
In-Depth Information
with
p
the pressure,
g
i
the gravity vector and
σ
ij
the viscous stress tensor. The left
side of
Eq. (1.19)
is density times the total acceleration following the motion, the
sum of local and advective accelerations; the right side is the sum of the pressure-
gradient, gravity, and viscous forces.
Equation (1.19)
requires a nonaccelerating
coordinate system, but our earth-based coordinate system is accelerating because
the earth rotates; as a result
(1.19)
also needs a
Coriolis
term. This can be important
in atmospheric turbulence
(Part II)
but we shall ignore it in
Part I
.
Batchelor
(
1967
) points out that when
ρ
is uniform one can define a pressure
(
p
s
, say) whose gradient exactly balances the gravity force:
∂p
s
∂x
i
+
ρg
i
.
0
=−
(1.20)
It follows that
p
s
=
ρg
j
x
j
+
p
0
, with
p
0
a constant. If we write pressure as
p
s
p
m
,
p
=
+
(1.21)
with
p
m
a
modified pressure
that is due to the fluid motion, then we can write
Eq. (1.19)
as
ρ
∂u
i
∂p
m
∂x
i
+
u
j
∂u
i
∂x
j
∂σ
ij
∂x
j
∂t
+
=−
,
(1.22)
so the gravity term does not appear explicitly. In
Part I
we shall use the form
(1.22)
and drop the superscript m on pressure.
Using
Eq. (1.18)
we can write the momentum
equation (1.22)
as
∂x
j
−
σ
ij
.
ρ
∂u
i
∂p
∂x
i
+
∂
∂t
=−
ρu
i
u
j
+
(1.23)
The final term in
Eq. (1.23)
is in
flux form
. It can be interpreted as the divergence of
the total flux of momentum, the sum of advective and viscous parts. Momentum is
mass times velocity; it is a vector. The momentum flux is the amount of momentum
passing through a unit area per unit time. It is a second-order tensor quantity; it
involves two directions, that of the unit normal to the area and that of themomentum.
Its units, density times velocity squared, are equivalent to (newtons/m
2
), or stress.
Thus, we can also interpret the final term in
Eq. (1.23)
as the divergence of a
generalized stress.
In a incompressible
Newtonian
fluid the viscous stress tensor
σ
ij
is a linear
function of the
strain-rate tensor s
ij
. We write this as
μ
∂u
i
∂u
j
∂x
i
σ
ij
=
∂x
j
+
=
2
μs
ij
,
(1.24)
