# Weather analysis by hcj

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```									Chapter 12: Mountain waves &
downslope wind storms

trapped lee waves
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
12.1.1 linear theory:
sinusoidal mountains,
no shear, constant stability, 2D

vertically-propagating waves:

N2 > U2k2 or a > 2pU/N   (large-scale terrain)
cold
u’>0         u’<0
L   warm     H
a

evanescent waves:

N2 < U2k2 or a < 2pU/N    (small-scale terrain)
cold
u’>0
H
warm
u’<0
L

Fig. 12.3
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

a=1 km                                            a=100 km
witch of Agnesi
mountain

evanescent – vertically trapped                     vertically propagating
Fig. 12.4

linear theory – analytical solution               numerical solution

trapped lee
waves

Fig. 12.5

Fig. 12.6
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)
U
– in other words, Fr >>1                    Fr      Froude number
Nh
1 Nh
       non-dimensional mountain height
Fr U
• 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
– large u’ (wave steepening and breaking)
– transience
non-linear flow over an isolated 2D mountain:

Froude #
transient effects
•   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
Dz=14 km

– wave breaking over crest

2D numerical simulations over Agnesi mountain          Dx=256 km
The lines are isentropes                                           (Lin and Wang 1996)
12.2: flow over isolated peaks (3D): covered
in chapter 13 (blocked flow)

wind
12.3                                     Fig. 12. 9 & 10

downslope windstorms                                    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?
Boulder windstorm                                             Grand Junction CO

11 Jan 1972                                                00Z 19720112

tropopause
2D simulations by Ming Xue, OU
mountain halfwidth = 10 km
horizontal grid spacing = 1 km                                            stable
strong wind shear
input stability and wind profile                               less stable
stable

animations of zonal wind u, and potential temperature q:
(H=mountain height)
1. H= 1 km
2. H= 2 km
3. H= 3 km

This case study has been simulated by Doyle et al. (2000 in Mon. Wea. Rev.)
12.3.1 downslope windstorms: hydraulic jump analogy
12.3.1 downslope windstorms: shallow water theory

Fig. 12.12
12.3.2 downslope windstorms:
dividing streamline theory (Smith 1985)

dividing streamline

Smith 1985
Fig. 12.13
downslope windstorms
hmountain =200 m                          hmountain =300 m
•   Hydraulic theory requires
that N decreases with
height.
trapped lee waves
•   Durran (1986) examines the
effect of mountain height.

top of stable layer at 3 km in each case
hmountain =500 m
hmountain =800 m

trapped lee waves
severe
downslope
winds

Fig. 12.14
(numerical simulations by Durran 1986)
downslope windstorms
d stable layer = 1000 m
d stable layer = 2500 m

•   Durran (1986) also
examines the effect of
the depth of the low-level                                                                      severe
stable layer.                                                                                  downslope
winds

mountain height fixed at         d stable layer = 3500 m                   d stable layer = 4000 m
500 m in each case

trapped lee waves

Fig. 12.15
(numerical simulations by Durran 1986)
severe downslope winds: resonant amplification theory

Scinocca and Peltier (1993)

1. wave steepening & breaking
produces a well-mixed layer
aloft, above the lee slope
2. KH instability develops on top
of the surface stable layer,
squeezing that layer &
increasing the wind speed
(Bernouilli)
3. strong wind region expands
downstream

severe downslope winds:
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)
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 below and above
• reverse shear such that the wind aloft is weaker, possibly
even in reverse direction

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.
A downslope wind storm in the lee of the Sierra
Nevada picks dust in the arid Owens Valley.         12.4
rotor cloud
lee
rotors
wind

h (s-1)                                 blue line applies to
26 Jan                   the 26 Jan 2006
case, shown below

Haimov et al. 2008 (IGARRS)

reverse
flow

Fig. 12.17

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