1. Chapter 12: Mountain waves & downslope wind storms
see also: COMET Mountain Wave Primer

2. trapped lee waves

3. Quasi-stationary lenticular clouds result from trapped lee waves
stack of lenticular clouds
Try a real-time animation of a cross section of isentropes/winds over the Snowy Range or Sierra Madre (source: NCAR’s WRF runs)
MODIS
Amsterdam Island,
Indian Ocean

4. 12.1.1 linear theory: sinusoidal mountains, no shear, constant stability, 2D
𝑑𝑈 𝑑𝑧 = 𝑑 2 𝜃 𝑑 𝑧 2 =0
vertically-propagating waves:
N2 > U2k2 or a > 2pU/N (large-scale terrain)
cold
u’>0
u’<0 L H
warm a
evanescent waves:
N2 < U2k2 or a < 2pU/N (small-scale terrain)
cold
u’>0 H
warm
u’<0 L
Fig. 12.3

5. 12.1.2 linear theory: isolated mountain (k=0), no shear, constant stability, 2D
a << U/N or U/a >> N
a >> U/N or U/a >> N lz
lx ~0
a=1 km
a=100 km
witch of Agnesi
mountain
evanescent – vertically trapped
vertically propagating
Fig (Durran 1986)

6. 12.1.3 linear/steady vs non-linear/unsteady
both 2D
linear theory – analytical solution
numerical solution
time-independent, i.e. steady
trapped lee waves
Fig. 12.5
witch of Agnesi mtn, N constant,
U increases linearly with height ( 𝑑 2 𝑈 𝑑𝑧 2 =0)
from Durran (2003)
Fig. 12.6

7. non-linear flow over 2D mountains
Linear wave theory assumes that
mountain height h << flow depth, and
that u’<<U (wave pert. wind << mean wind)
in other words, Fr >>1
In reality Fr is often close to 1
Fr <1 : blocked flow, Ep>Ek
Fr >1 : flow over mountain, Ek>Ep
Non-linear effects caused by
terrain amplitude
large u’ (wave steepening and breaking)
transience
Froude number
non-dimensional mountain height

8. non-linear flow over an isolated 2D mountain: transient effects
Froude #
Fr ~1.3:
no mtn wave breaking, no upstream blocking
resemble linear vert. prop. mtn waves
Fr ~1:
A weakly non-linear, stationary internal jump forms at the downstream edge of the breaking wave.
strong downslope winds near the surface
Fr ~0.7:
jump propagates a bit downstream, and becomes ~stationary
upstream blocking
Fr ~0.4:
upstream flow firmly blocked
wave breaking over crest Fr
Dz=14 km
2D numerical simulations over Agnesi mountain
The lines are isentropes. The ND time Ut/a = 50.4
Dx=256 km
(Lin and Wang 1996)

9. non-linear flow over an isolated 2D mountain: transient effects
Froude #
Fr ~1.3:
no mtn wave breaking, no upstream blocking
resemble linear vert. prop. mtn waves
Fr ~1:
A weakly non-linear, stationary internal jump forms at the downstream edge of the breaking wave.
strong downslope winds near the surface
Fr ~0.7:
jump propagates a bit downstream, and becomes ~stationary
upstream blocking
Fr ~0.4:
upstream flow firmly blocked
wave breaking over crest Fr
linear theory
VP waves (a >> U/N)
Dz=14 km
2D numerical simulations over Agnesi mountain
The lines are isentropes. The ND time Ut/a = 50.4
Dx=256 km
(Lin and Wang 1996)

10. isentropes wind anomalies 3.5 hrs 14 hrs 3.5 hrs 14 hrs early late
no wave breaking aloft
no upstream blocking
height (km)
wave breaking aloft
no upstream blocking
height (km)
wave breaking aloft
upstream blocking
wave breaking is first
height (km)
wave breaking aloft
upstream blocking
blocking is first
height (km)
Froude
number
3.5 hrs
14 hrs
time
(for U=10 m/s
a = 10 km)
3.5 hrs
14 hrs
(Lin and Wang 1996)

11. 12.2: flow over isolated peaks (3D): covered in chapter 13 (blocked flow)
wind

12. 12.3 downslope windstorms example: 18 Feb 2009

13. 12.3 downslope windstorms example: 18 Feb 2009

14. 12.3 downslope windstorms example: 18 Feb 2009

15. 12.3 downslope windstorms example: 18 Feb 2009

16. 12.3 downslope windstorms q (K) u (m/s)
Fig & 10
q (K)
less stable
stable
u (m/s)
Fig. 12.8
Is this downslope acceleration & lee ascent dynamically the same as a hydraulic jump in water?
Or is it due to wave energy reflection on a self-induced critical level & local resonance?

17. Boulder windstorm 11 Jan 1972 2D simulations by Ming Xue, OU
Grand Junction CO
00Z 1972/01/12
tropopause
2D simulations by Ming Xue, OU
mountain halfwidth = 10 km
horizontal grid spacing = 1 km
input stability and wind profile 
animations of zonal wind u, and potential temperature q:
(H=mountain height)
H= 1 km
H= 2 km
H= 3 km
stable
strong wind shear
less stable
stable
This case study has been simulated by Doyle et al. (2000 in Mon. Wea. Rev.)

18. 12.3.1 downslope windstorms: (a) hydraulic jump analogy

19. downslope windstorms: (a) hydraulic theory: shallow water theory
Fig

20. downslope windstorms: (a) hydraulic theory - dividing streamline (Smith 1985)
Fig
assumptions:
steady (Bernouilli)
inviscid
hydrostatic & Boussinesq
𝜕 2 𝛿 𝜕𝑧 2 + 𝑙 2 𝛿=0
Long’s equation
𝑙 2 = 𝑁 2 𝑈 2 − 1 𝑈 𝑑 2 𝑈 𝑑𝑧 2
Scorer parameter (l)

21. downslope windstorms – hydraulic theory
hmountain =200 m
Shallow water eqns assume a density discontinuity (free water surface). Results qualitatively similar to a hydraulic jump can be produced in a numerical model with a stability (N) discontinuity
Durran (1986) does this, using a two layer (Nlow>> Nup) constant wind U environment. Here the mountain height is varied.
hmountain =300 m
trapped lee waves
~ subcritical flow
hmountain =500 m
hmountain =800 m
trapped lee waves
severe downslope winds
top of stable layer at 3 km in each case
Fig
(numerical simulations by Durran 1986)

22. downslope windstorms – hydraulic theory
d stable layer = 1000 m
d stable layer = 2500 m
Durran (1986) also examines the effect of the depth of the low-level stable layer.
severe downslope winds
mountain height fixed at 500 m in each case
d stable layer = 3500 m
d stable layer = 4000 m
~ subcritical flow
trapped lee waves
Fig
(numerical simulations by Durran 1986)

23. plunging flow in Laramie, east of the Laramie Range
plunging flow + hydraulic jump?
barrier jet ?

24. downslope windstorms: (b) resonant amplification theory
Clark and Peltier (1984)
Scinocca and Peltier (1993)
Resonant amplification due to wave energy reflection at the level of wave breaking. The storm is transient, with this evolution according to 2D inviscid simulations:
wave steepening & breaking produces a well-mixed layer aloft, above the lee slope
This results in a (self-induced) critical level (U=0)
Ri<0.25  KH instability develops on top of the surface stable layer, squeezing that layer & increasing the wind speed (Bernouilli)
strong wind region expands downstream
t=0 min
t=20 min
t=66 min
t=96 min
t=160 min
t=166 min
shading shows isentropic layers

25. downslope windstorms: resonant amplification theory
transient flow (4 different times), non-linear
linear
Fr=20, Ri=0.1
non-linear
Fr=20, Ri=0.1
shaded regions: Ri <0.25
Wang and Lin (1999)

26. downslope windstorms: forecast clues
asymmetric mountain, with gentle upstream slope and steep lee slope
strong cross-mountain wind (>15 m s-1) at mtn top level
cross mountain flow is close to normal to the ridge line
stable layer near mountain top (possibly a frontal surface), less stable air above
(not always) reverse shear such that the wind aloft is weaker, possibly even in reverse direction ( pre-existing critical level)
Note: The Front Range area sees less downslope winds than the Laramie valley in winter in part because of strong lee stratification, due to low-level cold air advected from the Plains states. Thus the strong winds often do not make it down to ground level.

27. A downslope wind storm in the lee of the Sierra Nevada picks dust in the arid Owens Valley.
12.4 lee rotors
rotor cloud
wind
h (s-1)
blue line applies to the 26 Jan 2006 case, shown below
26 Jan
Haimov et al (IGARRS)
vorticity sheet
(no-slip BC)
downslope
windstorm
reverse
flow
Fig