(iii) if F r > 1, the flow is supercritical. In this case, the current velocity is higher
than the velocity of the waves, so that these are dragged in the same
direction.
To apply the Froude number to airflow around or over topography, it is assumed
that under thermal stability, layers of disturbed air oscillate vertically in the Brunt–
Väisälä, N BV , frequency
N BV ¼ À
g
h 0
@ h
@z
1=2
ð5:7Þ
where h is the mean air temperature potential and h 0 the temperature potential at
height z 0 . The frequency defined in Eq. (5.7) is the natural frequency of the internal
gravitational waves, or downwind waves.
The oscillation of an air mass with this frequency, in an air mass with a mean
velocity u, induces a wave in this flow with a natural wavelength k BV :
k BV ¼
2p u
N BV
ð5:8Þ
The Froude number for flow over topography with a characteristic length C, can
be expressed as (Oke 1992)
F r ¼
p u
N BV C
ð5:9Þ
The Froude number defined by Eq. (5.9) represents the ratio between inertia and
buoyancy forces. It is about the same order of magnitude as the inverse square of
the Richardson mass number (Arya 1988)
F r % R
À2
im
ð5:10Þ
Thermal stratification influences flow over topography in a significant way. This
influence in flow patterns is qualitatively like that under neutrality conditions,
although with different intensity. Other effects, such as the generation of gravity
waves, vortices formation, hydraulic jumps, the inability of low-level fluid to go
over hills, and upstream blocking can only occur in stable stratified flows over or
around topography (Arya 1988).
If F r << 1, the stratification is strong, while for F r >> 1, conditions are near
neutral. For strictly neutral conditions, F r ¼ 1 corresponding to a case wherein the
Froude number no longer represents the flow dynamics. A variety of airflow
conditions can be envisaged for an isolated hill, depending on the Froude number
(Stull 1994). For Froude numbers about 0.1, corresponding to thermal stability, air
5.5 Flow Over Gently Sloping Hills
147
than the velocity of the waves, so that these are dragged in the same
direction.
To apply the Froude number to airflow around or over topography, it is assumed
that under thermal stability, layers of disturbed air oscillate vertically in the Brunt–
Väisälä, N BV , frequency
N BV ¼ À
g
h 0
@ h
@z
1=2
ð5:7Þ
where h is the mean air temperature potential and h 0 the temperature potential at
height z 0 . The frequency defined in Eq. (5.7) is the natural frequency of the internal
gravitational waves, or downwind waves.
The oscillation of an air mass with this frequency, in an air mass with a mean
velocity u, induces a wave in this flow with a natural wavelength k BV :
k BV ¼
2p u
N BV
ð5:8Þ
The Froude number for flow over topography with a characteristic length C, can
be expressed as (Oke 1992)
F r ¼
p u
N BV C
ð5:9Þ
The Froude number defined by Eq. (5.9) represents the ratio between inertia and
buoyancy forces. It is about the same order of magnitude as the inverse square of
the Richardson mass number (Arya 1988)
F r % R
À2
im
ð5:10Þ
Thermal stratification influences flow over topography in a significant way. This
influence in flow patterns is qualitatively like that under neutrality conditions,
although with different intensity. Other effects, such as the generation of gravity
waves, vortices formation, hydraulic jumps, the inability of low-level fluid to go
over hills, and upstream blocking can only occur in stable stratified flows over or
around topography (Arya 1988).
If F r << 1, the stratification is strong, while for F r >> 1, conditions are near
neutral. For strictly neutral conditions, F r ¼ 1 corresponding to a case wherein the
Froude number no longer represents the flow dynamics. A variety of airflow
conditions can be envisaged for an isolated hill, depending on the Froude number
(Stull 1994). For Froude numbers about 0.1, corresponding to thermal stability, air
5.5 Flow Over Gently Sloping Hills
147