Chapter 12: Mountain waves & downslope wind storms

see also: COMET Mountain Wave Primer

trapped lee waves

Quasi-stationary lenticular clouds result from trapped lee waves

Amsterdam Island,Indian Ocean

MODIS

Try a real-time animation of a cross section of isentropes/winds over the Snowy Range or Sierra Madre (source: NCAR’s WRF runs)

stack of lenticular clouds

12.1.1 linear theory: sinusoidal mountains,

no shear, constant stability, 2D

vertically-propagating waves:

N2 > U2k2 or a > 2pU/N (large-scale terrain)

evanescent waves:

N2 < U2k2 or a < 2pU/N (small-scale terrain)

a

u’<0

u’>0

warm

cold

L

H

u’>0cold

Lu’<0

warm H

Fig. 12.3

𝑑𝑈𝑑𝑧

=0𝑑2𝜃𝑑 𝑧 2=0

12.1.2 linear theory: isolated mountain (k=0), no shear, constant stability,

2D

a=1 km a=100 km

a << U/N or U/a >> N a >> U/N or U/a >> N

evanescent – vertically trapped vertically propagatingFig. 12.4 (Durran 1986)

witch of Agnesimountain

lzlx ~0

12.1.3 linear/steady vs non-linear/unsteady

trapped lee waves

both 2D

Fig. 12.5

linear theory – analytical solution numerical solution

Fig. 12.6

time-independent, i.e. steady

witch of Agnesi mtn, N constant, U increases linearly with height (from Durran (2003)

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

Nh

UFr Froude number

U

Nh

Fr

1non-dimensional mountain height

2D numerical simulations over Agnesi mountainThe lines are isentropes. The ND time Ut/a = 50.4

• 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

(Lin and Wang 1996)

Froude #

Dz=

14

km

Dx=256 km

non-linear flow over an isolated 2D mountain:

transient effects Fr

2D numerical simulations over Agnesi mountainThe lines are isentropes. The ND time Ut/a = 50.4

• 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

(Lin and Wang 1996)

Froude #

Dz=

14

km

Dx=256 km

non-linear flow over an isolated 2D mountain:

transient effects Fr

linear theoryVP waves (a >> U/N)

heig

ht

(km

)heig

ht

(km

)heig

ht

(km

)heig

ht

(km

)

Froude

num

ber

3.5 hrs 14 hrs 3.5 hrs 14 hrstime(for U=10 m/s

a = 10 km)

no wave breaking aloftno upstream blocking

wave breaking aloftno upstream blocking

wave breaking aloftupstream blocking

wave breaking is first

wave breaking aloftupstream blocking

blocking is first

isentropesearly late early late

wind anomalies

(Lin and Wang 1996)

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

wind

12.3 downslope windstorms

example: 18 Feb 2009

12.3 downslope windstorms

example: 18 Feb 2009

12.3 downslope windstormsexample: 18 Feb 2009

12.3 downslope windstormsexample: 18 Feb 2009

12.3downslope windstorms

Fig. 12.8

Fig. 12. 9 & 10

q (K)

u (m/s)

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?

stable

less stable

Boulder windstorm 11 Jan 1972

2D simulations by Ming Xue, OUmountain halfwidth = 10 km horizontal grid spacing = 1 kminput stability and wind profile

animations of zonal wind u, and potential temperature q:(H=mountain height)1. H= 1 km2. H= 2 km3. H= 3 km

stable

less stable

stable

tropopause

strong wind shear

Grand Junction CO00Z 1972/01/12

This case study has been simulated by Doyle et al. (2000 in Mon. Wea. Rev.)

12.3.1 downslope windstorms: (a) hydraulic jump analogy

downslope windstorms: (a) hydraulic theory: shallow water theory

Fig. 12.12

downslope windstorms: (a) hydraulic theory - dividing streamline (Smith

1985)

Fig. 12.13

Smith 1985

dividing streamline

assumptions: steady (Bernouilli)inviscidhydrostatic & Boussinesq

𝜕2𝛿𝜕 𝑧2 +𝑙

2 𝛿=0

𝑙2=𝑁2

𝑈 2 −1𝑈

𝑑2𝑈𝑑𝑧2

Long’s equation

Scorer parameter (l)

downslope windstorms – hydraulic theory

• 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 =200 m

hmountain =800 mhmountain =500 m

hmountain =300 m

(numerical simulations by Durran 1986)

trapped lee waves

severe downslope

winds

trapped lee waves

Fig. 12.14

top of stable layer at 3 km in each case

~ subcritical flow

downslope windstorms – hydraulic theory

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

d stable layer = 1000 m

mountain height fixed at 500 m in each case

severe downslope

winds

trapped lee waves

d stable layer = 4000 md stable layer = 3500 m

d stable layer = 2500 m

Fig. 12.15 (numerical simulations by Durran 1986)

~ subcritical flow

plunging flow in Laramie, east of the Laramie

Range

plunging flow + hydraulic jump?

barrier jet ?

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:

1. wave steepening & breaking produces a well-mixed layer aloft, above the lee slope

2. This results in a (self-induced) critical level (U=0)

3. Ri<0.25 KH instability develops on top of the surface stable layer, squeezing that layer & increasing the wind speed (Bernouilli)

4. strong wind region expands downstream

shading shows isentropic layers

t=0 min

t=166 mint=160 min

t=96 mint=66 min

t=20 min

downslope windstorms:resonant amplification theory

shaded regions: Ri <0.25

linear

non-linear

Fr=20, Ri=0.1

Fr=20, Ri=0.1

Wang and Lin (1999)

transient flow (4 different times), non-linear

• 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)

downslope windstorms: forecast clues

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.

12.4lee

rotorswind

Haimov et al. 2008 (IGARRS)

blue line applies to the 26 Jan 2006 case, shown below

26 Jan

Fig. 12.17

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

rotor cloud

h (s-1)

vorticity sheet

(no-slip BC)

reverseflow

downslopewindstorm