where u d is the mean wind velocity along the slope, h is the height of the cold air
layer, a the slope angle, C D is the joint drag coefficient caused by slope surface and
by the lower velocity airflow layer and Dh is the potential temperature difference
between the ambient air and air in the katabatic flow. The term I represents
acceleration, II and III, the slope and longitudinal advection transport, term IV the
buoyancy, term V the Coriolis force (at mid-latitude about 10
−4 s
−1 ), and VI the
turbulence drag.
Initially, the katabatic wind is mainly influenced by buoyancy (IV) and advection (II) (Stull 2000). At this stage, the mean velocity u d is given by
u d ¼ g
j j
Dh
h
xsin a
ð Þ
1=2
ð
Þ
ð5:16Þ
and after a 2nd stage, an equilibrium velocity, u eq , is attained where the gravity
(term IV) is compensated by frictional drag (term VI)
u eq ¼ g
Dh
h
h
C D
sin a
ð Þ
1=2
ð
Þ
ð5:17Þ
5.7 Descending Winds Under Inversion
Fohen and Bora-type winds are phenomena that can occur in the atmosphere under
conditions of thermal stability (cold air under warm air layer). These winds typically occur during the winter in temperate zones, under conditions when a synoptic
prevailing wind impacts on isolated or arrayed hills (Stull 2000).
If the height of the hill is higher than the height of the upwind cold air, then the
cold air is retained upwind and does not move uphill. Warm wind above the cold
layer, can jump the hilltop and descend adiabatically, without heat exchange with
the exterior ambient, along the downstream slope. This results in mild winds called
Foehn, typically found in several places in Europe (Fig. 5.15a). If the height z i of
the cold ascending air layer is greater than the height of the hilltop z hill , then a Bora
phenomenon occurs wherein very fast cold winds can move downwards along the
lee side of the hill (Fig. 5.15b). Under this phenomenom, the wind firstly accelerate
in the contraction between the hill and the overlying air layer with a pressure drop,
to P 1 (Fig. 5.16) according with the Bernoulli principle. Within the contraction
height, a modified Froude number F r
* becomes (Stull 1994)
F
Ã
r ¼
u
N BV z i À z hill
ð
Þ
ð5:18Þ
Accordingly, there can be two different types of flow depending on the wind
velocity. For low wind velocities, (F r
* < < 1) the acceleration over the hill can
move the inversion downward. When the winds are weak, flow separation occurs
154
5 Flow Over Modified Surfaces
layer, a the slope angle, C D is the joint drag coefficient caused by slope surface and
by the lower velocity airflow layer and Dh is the potential temperature difference
between the ambient air and air in the katabatic flow. The term I represents
acceleration, II and III, the slope and longitudinal advection transport, term IV the
buoyancy, term V the Coriolis force (at mid-latitude about 10
−4 s
−1 ), and VI the
turbulence drag.
Initially, the katabatic wind is mainly influenced by buoyancy (IV) and advection (II) (Stull 2000). At this stage, the mean velocity u d is given by
u d ¼ g
j j
Dh
h
xsin a
ð Þ
1=2
ð
Þ
ð5:16Þ
and after a 2nd stage, an equilibrium velocity, u eq , is attained where the gravity
(term IV) is compensated by frictional drag (term VI)
u eq ¼ g
Dh
h
h
C D
sin a
ð Þ
1=2
ð
Þ
ð5:17Þ
5.7 Descending Winds Under Inversion
Fohen and Bora-type winds are phenomena that can occur in the atmosphere under
conditions of thermal stability (cold air under warm air layer). These winds typically occur during the winter in temperate zones, under conditions when a synoptic
prevailing wind impacts on isolated or arrayed hills (Stull 2000).
If the height of the hill is higher than the height of the upwind cold air, then the
cold air is retained upwind and does not move uphill. Warm wind above the cold
layer, can jump the hilltop and descend adiabatically, without heat exchange with
the exterior ambient, along the downstream slope. This results in mild winds called
Foehn, typically found in several places in Europe (Fig. 5.15a). If the height z i of
the cold ascending air layer is greater than the height of the hilltop z hill , then a Bora
phenomenon occurs wherein very fast cold winds can move downwards along the
lee side of the hill (Fig. 5.15b). Under this phenomenom, the wind firstly accelerate
in the contraction between the hill and the overlying air layer with a pressure drop,
to P 1 (Fig. 5.16) according with the Bernoulli principle. Within the contraction
height, a modified Froude number F r
* becomes (Stull 1994)
F
Ã
r ¼
u
N BV z i À z hill
ð
Þ
ð5:18Þ
Accordingly, there can be two different types of flow depending on the wind
velocity. For low wind velocities, (F r
* < < 1) the acceleration over the hill can
move the inversion downward. When the winds are weak, flow separation occurs
154
5 Flow Over Modified Surfaces