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

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trapped lee waves

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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 Try a real-time animation of a cross section of isentropes/winds over the Snowy Range or Sierra Madre (source: NCARs WRF runs) stack of lenticular clouds

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linear theory: sinusoidal mountains, no shear, constant stability, 2D vertically-propagating waves: N 2 > U 2 k 2 or a > 2 U/N (large-scale terrain) evanescent waves: N 2 < U 2 k 2 or a < 2 U/N (small-scale terrain) a u<0 u>0 warm cold L H u>0 cold L u<0 warm H Fig. 12.3

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linear theory: isolated mountain (k=0), no shear, constant stability, 2D a=1 kma=100 km a > N a >> U/N or U/a >> N evanescent – vertically trappedvertically propagating Fig (Durran 1986) witch of Agnesi mountain z x ~0

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linear/steady vs non-linear/unsteady trapped lee waves both 2D Fig linear theory – analytical solutionnumerical solution Fig time-independent, i.e. steady

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non-linear flow over 2D mountains Linear wave theory assumes that –mountain height h << flow depth, and –that u<__>1 In reality Fr is often close to 1 –Fr E k –Fr >1 : flow over mountain, E k >E p Non-linear effects caused by –terrain amplitude –large u (wave steepening and breaking) –transience Froude number non-dimensional mountain height
__

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2D numerical simulations over Agnesi mountain The 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 # z=14 km x=256 km non-linear flow over an isolated 2D mountain: transient effects Fr

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2D numerical simulations over Agnesi mountain The 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 # z=14 km x=256 km non-linear flow over an isolated 2D mountain: transient effects Fr linear theory VP waves (a >> U/N)

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

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

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12.3 downslope windstorms example: 18 Feb 2009

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12.3 downslope windstorms Fig Fig & 10 (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

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Boulder windstorm 11 Jan D 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 : (H=mountain height) 1.H= 1 kmH= 1 km 2.H= 2 kmH= 2 km 3.H= 3 kmH= 3 km stable less stable stable tropopause strong wind shear Grand Junction CO 00Z 1972/01/12 This case study has been simulated by Doyle et al. (2000 in Mon. Wea. Rev.)

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downslope windstorms: (a) hydraulic jump analogy

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

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downslope windstorms: (a) hydraulic theory - dividing streamline (Smith 1985) Fig Smith 1985 dividing streamline assumptions: steady (Bernouilli) inviscid hydrostatic & Boussinesq Longs equation Scorer parameter (l)

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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 (N low >> N up ) constant wind U environment. Here the mountain height is varied. h mountain =200 m h mountain =800 m h mountain =500 m h mountain =300 m (numerical simulations by Durran 1986) trapped lee waves severe downslope winds trapped lee waves Fig top of stable layer at 3 km in each case ~ subcritical flow

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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 m d stable layer = 3500 m d stable layer = 2500 m Fig (numerical simulations by Durran 1986) ~ subcritical flow

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

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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 min t=66 min t=20 min

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

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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.

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

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