Geoscience Reference
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the larger eddies; but
ω
∼
ω(η)
, since virtually all the vorticity lies in the smallest
eddies.
The turbulent vorticity can be surprisingly large. In an atmospheric boundary
layer with energy-containing scales
u
1ms
−
1
and
= 1000 m, so that the
=
10
−
3
m,
ω
is of order 10 s
−
1
.
This is equal to the vorticity in a tornado funnel of core diameter 60 m and winds
of 300 m s
−
1
!
Following
Eq. (2.64)
, the vorticity
ω(r)
of eddies of size
r
is related to the
three-dimensional vorticity spectrum
ψ(κ)
by
10
−
2
ms
−
1
and
η
Kolmogorov scales are
υ
=
=
2
[
ω(r)
]
∼
κψ(κ).
(2.70)
κ
2
E(κ)
, so that the vorticity spectrum
Since
ω(r)
∼
u(r)/r
, it follows that
ψ(κ)
∼
in the inertial subrange behaves as
κ
2
E(κ)
2
/
3
κ
1
/
3
,
ψ(κ)
∼
∼
(2.71)
which increases with increasing wavenumber. The increase is ultimately damped
by viscosity at
κ
∼
1
/η
.
2.7 Turbulent pressure
Taking the divergence of theNavier-Stokes
equation (1.26)
gives a Poisson equation
for the pressure field:
1
ρ
∇
∂u
j
∂x
i
∂u
i
∂x
j
.
2
p
−
=
(2.72)
We can express a velocity gradient as a sum of strain-rate and rotation-rate tensors
s
ij
and
r
ij
:
∂u
i
∂x
j
+
∂u
i
∂x
j
−
∂u
i
∂x
j
=
1
2
∂u
j
∂x
i
1
2
∂u
j
∂x
i
+
=
s
ij
+
r
ij
.
(2.73)
s
ij
and
r
ij
are symmetric and antisymmetric tensors, respectively.
Bradshaw
and Koh
(
1981
) pointed out that with
Eq. (2.73)
one can rewrite the Poisson
equation (2.72)
in terms of
s
ij
and
r
ij
:
1
ρ
∇
2
p
−
=
(s
ij
+
r
ij
)(s
ij
−
r
ij
)
=
s
ij
s
ij
−
r
ij
r
ij
.
(2.74)
They offered a physical interpretation of
Eq. (2.74)
:
The rate-of-strain contribution comes from near saddle points in the streamline pattern
(streamlines approaching a given point fromnorth and south, and leaving it fromeast towest,
