# Chapter 12: Mountain waves & downslope wind storms

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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
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

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)

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

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

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)

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)

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)

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 windstorms example: 18 Feb 2009

12.3 downslope windstorms example: 18 Feb 2009

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?

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.)

12.3.1 downslope windstorms: (a) hydraulic jump analogy

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

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)

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)

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)

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: 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

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)

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.

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