3Quasi-stationary lenticular clouds result from trapped lee waves stack of lenticular cloudsTry a real-time animation of a cross section of isentropes/winds over the Snowy Range or Sierra Madre (source: NCAR’s WRF runs)MODISAmsterdam Island,Indian Ocean
412.1.1 linear theory: sinusoidal mountains, no shear, constant stability, 2D 𝑑𝑈 𝑑𝑧 = 𝑑 2 𝜃 𝑑 𝑧 2 =0vertically-propagating waves:N2 > U2k2 or a > 2pU/N (large-scale terrain)coldu’>0u’<0LHwarmaevanescent waves:N2 < U2k2 or a < 2pU/N (small-scale terrain)coldu’>0Hwarmu’<0LFig. 12.3
512.1.2 linear theory: isolated mountain (k=0), no shear, constant stability, 2D a << U/N or U/a >> Na >> U/N or U/a >> Nlzlx ~0a=1 kma=100 kmwitch of Agnesimountainevanescent – vertically trappedvertically propagatingFig (Durran 1986)
612.1.3 linear/steady vs non-linear/unsteady both 2Dlinear theory – analytical solutionnumerical solutiontime-independent, i.e. steadytrapped lee wavesFig. 12.5witch of Agnesi mtn, N constant,U increases linearly with height ( 𝑑 2 𝑈 𝑑𝑧 2 =0)from Durran (2003)Fig. 12.6
7non-linear flow over 2D mountains Linear wave theory assumes thatmountain height h << flow depth, andthat u’<<U (wave pert. wind << mean wind)in other words, Fr >>1In reality Fr is often close to 1Fr <1 : blocked flow, Ep>EkFr >1 : flow over mountain, Ek>EpNon-linear effects caused byterrain amplitudelarge u’ (wave steepening and breaking)transienceFroude numbernon-dimensional mountain height
8non-linear flow over an isolated 2D mountain: transient effects Froude #Fr ~1.3:no mtn wave breaking, no upstream blockingresemble linear vert. prop. mtn wavesFr ~1:A weakly non-linear, stationary internal jump forms at the downstream edge of the breaking wave.strong downslope winds near the surfaceFr ~0.7:jump propagates a bit downstream, and becomes ~stationaryupstream blockingFr ~0.4:upstream flow firmly blockedwave breaking over crestFrDz=14 km2D numerical simulations over Agnesi mountainThe lines are isentropes. The ND time Ut/a = 50.4Dx=256 km(Lin and Wang 1996)
9non-linear flow over an isolated 2D mountain: transient effects Froude #Fr ~1.3:no mtn wave breaking, no upstream blockingresemble linear vert. prop. mtn wavesFr ~1:A weakly non-linear, stationary internal jump forms at the downstream edge of the breaking wave.strong downslope winds near the surfaceFr ~0.7:jump propagates a bit downstream, and becomes ~stationaryupstream blockingFr ~0.4:upstream flow firmly blockedwave breaking over crestFrlinear theoryVP waves (a >> U/N)Dz=14 km2D numerical simulations over Agnesi mountainThe lines are isentropes. The ND time Ut/a = 50.4Dx=256 km(Lin and Wang 1996)
10isentropes wind anomalies 3.5 hrs 14 hrs 3.5 hrs 14 hrs early late no wave breaking aloftno upstream blockingheight (km)wave breaking aloftno upstream blockingheight (km)wave breaking aloftupstream blockingwave breaking is firstheight (km)wave breaking aloftupstream blockingblocking is firstheight (km)Froudenumber3.5 hrs14 hrstime(for U=10 m/sa = 10 km)3.5 hrs14 hrs(Lin and Wang 1996)
1112.2: flow over isolated peaks (3D): covered in chapter 13 (blocked flow) wind
1612.3 downslope windstorms q (K) u (m/s) Fig & 10q (K)less stablestableu (m/s)Fig. 12.8Is 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?
17Boulder windstorm 11 Jan 1972 2D simulations by Ming Xue, OU Grand Junction CO00Z 1972/01/12tropopause2D simulations by Ming Xue, OUmountain halfwidth = 10 kmhorizontal grid spacing = 1 kminput stability and wind profile animations of zonal wind u, and potential temperature q:(H=mountain height)H= 1 kmH= 2 kmH= 3 kmstablestrong wind shearless stablestableThis case study has been simulated by Doyle et al. (2000 in Mon. Wea. Rev.)
21downslope windstorms – hydraulic theory hmountain =200 mShallow 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) discontinuityDurran (1986) does this, using a two layer (Nlow>> Nup) constant wind U environment. Here the mountain height is varied.hmountain =300 mtrapped lee waves~ subcritical flowhmountain =500 mhmountain =800 mtrapped lee wavessevere downslope windstop of stable layer at 3 km in each caseFig(numerical simulations by Durran 1986)
22downslope windstorms – hydraulic theory d stable layer = 1000 md stable layer = 2500 mDurran (1986) also examines the effect of the depth of the low-level stable layer.severe downslope windsmountain height fixed at 500 m in each cased stable layer = 3500 md stable layer = 4000 m~ subcritical flowtrapped lee wavesFig(numerical simulations by Durran 1986)
23plunging flow in Laramie, east of the Laramie Range plunging flow + hydraulic jump?barrier jet ?
24downslope 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 slopeThis 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 downstreamt=0 mint=20 mint=66 mint=96 mint=160 mint=166 minshading shows isentropic layers
25downslope windstorms: resonant amplification theory transient flow (4 different times), non-linearlinearFr=20, Ri=0.1non-linearFr=20, Ri=0.1shaded regions: Ri <0.25Wang and Lin (1999)
26downslope windstorms: forecast clues asymmetric mountain, with gentle upstream slope and steep lee slopestrong cross-mountain wind (>15 m s-1) at mtn top levelcross mountain flow is close to normal to the ridge linestable 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.
27A downslope wind storm in the lee of the Sierra Nevada picks dust in the arid Owens Valley. 12.4 lee rotorsrotor cloudwindh (s-1)blue line applies to the 26 Jan 2006 case, shown below26 JanHaimov et al (IGARRS)vorticity sheet(no-slip BC)downslopewindstormreverseflowFig