Weather analysis by hcj


									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

                                                                   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)


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)
    u’>0         u’<0
     L   warm     H

                                             evanescent waves:

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

                                 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


            a=1 km                                            a=100 km
                     witch of Agnesi

evanescent – vertically trapped                     vertically propagating
                                       Fig. 12.4
12.1.3 linear/steady vs non-linear/unsteady                                       both 2D

       linear theory – analytical solution               numerical solution

                                                                              trapped lee

                                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)
   – in other words, Fr >>1                    Fr      Froude number
                                              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)

        12.3                                     Fig. 12. 9 & 10

downslope windstorms                                    q (K)

                                   less 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

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

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
                                                                                                     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

                                                                                                             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

      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
  3. strong wind region expands

 shading shows isentropic layers
   severe downslope winds:
resonant amplification theory
   transient flow (4 different times), non-linear

                                                       Fr=20, Ri=0.1


                                                      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

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

                                                                Haimov et al. 2008 (IGARRS)


                                      Fig. 12.17

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