Geoscience Reference
In-Depth Information
Figure 9.3 Left: Steady, horizontally homogeneous, nonturbulent flow (shown for
the northern hemisphere) is parallel to the isobars. Right: With turbulence the mean
flow has a cross-isobaric angle
α
.
that
fz
i
/u
1(
Csanady
,
1974
)). This limit
(Problem 9.14)
is “channel flow” - i.e.,
flow down the mean pressure gradient.
The cross-isobaric angle enters ABL mean-flow energetics as follows. In steady, hori-
zontally homogeneous conditions the mean horizontal momentum equation
(8.65)
in the
ABL is
∗
∂u
i
u
j
∂x
j
1
ρ
0
∂P
∂x
i
−
0
=−
−
2
ij k
j
U
k
,
i
=
1
,
2
.
(9.3)
This expresses a mean balance among turbulent stress divergence, pressure-gradient, and
Coriolis forces in the horizontal plane. Multiplying this equation by
U
i
and rewriting the
first term yields the steady budget of
U
i
U
i
/
2, the mean-horizontal-flow kinetic energy
per unit mass (MKE):
∂
∂x
j
∂U
i
∂x
j
−
U
i
ρ
0
∂P
∂x
i
0
=−
U
i
u
i
u
j
+
u
i
u
j
.
(9.4)
The first term on the right, a transport term, integrates to zero over the ABL (
Problem
it represents the mean rate of exchange of kinetic energy between the mean horizontal
flow and the turbulence. The final term is the mean rate of production of MKE by mean
horizontal flow down the mean pressure gradient. Maintaining the mean ABL flow against
its rate of loss of kinetic energy to the turbulence requires this final term to be nonzero.
This in turn requires a nonzero cross-isobaric angle
α
(
Figure 9.3
)
.
In the stable ABL (i.e., one with
∂/∂z
positive,
Chapter 12
) both buoyant production
and viscous dissipation are loss rates of TKE,
Eq. (8.70)
, so the turbulence must extract
kinetic energy from the mean flow in order to survive. As a result its cross-isobaric angle
tends to be larger, typically on the order of 45 degrees near the surface; the adjustment to
geostrophic flow occurs over the depth of the stable ABL. In contrast, most of the TKE
