Geoscience Reference
In-Depth Information
The first term on the right side can amplify and tilt the gradient of a conserved scalar
in both two- and three-dimensional flow. Satellite photos reveal large-scale, quasi-
two-dimensional turbulent water vapor fields that because of this stretching and
tilting process can look verymuch like the familiar three-dimensional ones. Because
in two-dimensional flow the single component of vorticity
ω
3
is a conserved scalar,
˜
its gradient
ω
3
,i
satisfies
Eq. (3.22)
.
The absence of vortex stretching in two-dimensional turbulence means there is
no downscale energy cascade, but there is a cascade of scalar variance. Vertical
vorticity, being a scalar, participates in this cascade. The distortion of the vorticity
field generates spatial structure in vorticity that extends to the dissipative scales,
with an intervening inertial range having a cascade of vorticity variance rather than
kinetic energy.
˜
7.5.2 Interscale transfer of enstrophy
If we represent vorticity in two-dimensional turbulence, a scalar, as the sum of
ensemble-mean and fluctuating parts,
ω
3
˜
=
+
ω
, the equation for fluctuating
vorticity
ω
is
∂ω
∂t
+
,j
u
j
+
ω
,j
U
j
+
ω
,j
u
j
−
ω
,j
u
j
=
νω
,jj
.
(7.55)
Multiplying by
ω
, ensemble averaging, rewriting the molecular term, and neglect-
ing viscous diffusion gives the evolution equation for one-half the mean-square
fluctuating vorticity, or
enstrophy
:
†
ω
2
2
u
j
ω
2
2
∂ω
2
∂t
+
U
j
1
2
,j
+
u
j
ω
,j
+
=−
νω
,j
ω
,j
.
(7.56)
,j
This is identical to
Eq. (5.7)
for scalar variance. A displacement in the presence
of a
mean vorticity gradient
,j
generates a vorticity fluctuation, so we recognize
u
j
ω
,j
as a mean-gradient production term. The mean advection and turbulent
transport terms, being divergences, simply move vorticity variance around on the
plane. We conclude that the steady, global, physical-space balance of enstrophy
implied by
Eq. (7.56)
is
rate of production of enstrophy through mean vorticity gradient
=
rate of destruction of enstrophy through viscosity
.
(7.57)
†
According to
Frisch
(
1995
), this term is due to C. Leith.
