aguidedtourthroughthetheoryfor...

A  guided  tour  through  the  theory  for      Downslope  Windstorm  Flows  (DWFs)  

Rich  Rotunno  

Large  Amplitude  Mountain  Waves  and  Downslope  Windstorms  

DWFs  in  the  Meteor  Crater  

Theory  in  the  beginning…  

Long  (1954,  Tellus)  

momentum  

xz

Orographic Flow / Inversion Density Stratification  

�

a

�

uU

= f ha,g'd∞

U, hd∞

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

�

h

�

θ∞(z) =θ∞(0)θ∞(0) + Δθ∞

⎧ ⎨ ⎩

�

θ∞(0)�

θ∞(z0) + Δθ∞

�

d∞

�

z < d∞z > d∞

�

g'≡ g Δθ∞

θ∞(0)

�

u∞ =U

In  the  beginning…  

Long  (1954,  Tellus)  

SupercriJcal  

SubcriJcal  

TranscriJcal  

�

u(x)g'D(x)

>1

�

u(x)g'D(x)

<1

�

u(0)g'D(0)

=1

�

D(x) ≡ d(x) − h(x)

SupercriJcal  

SubcriJcal  

TranscriJcal  

Durran  (1986,  JAS)    Durran  (2003,  Encyclopedia  of  Atmos.  Sci.  )  

�

u(x)g'D(x)

>1

�

u(x)g'D(x)

<1

�

u(0)g'D(0)

=1

�

D(x) ≡ d(x) − h(x)

Shallow  Water  Theory  

Houghton&Kasahara  (1968,  Comm.  Pure  &  Appl.  Math)  

 Regime  Diagram  for  Hydraulic  Flow  over  Terrain  

A  

B  

C  

D  

(Houghton  &  Kasahara,  1968)  

SupercriJcal  everywhere  

PropagaJng  lee-­‐slope  jump  

StaJonary  lee-­‐slope  jump  

SubcriJcal  everywhere  

�

F0 =Ug'h0

, Mc =Hc

h0

Three  layers  to  represent  more  complex  straJficaJon  

Long  (1953,  Tellus)  

momentum  

xz

Orographic Flow / Density Stratification / Lid  

�

a

�

uU

= fha,NhU,NHU

⎛ ⎝ ⎜

⎞ ⎠ ⎟

�

h

�

θ∞(z) = θ∞(0) 1+N 2

gz

⎛

⎝ ⎜

⎞

⎠ ⎟

�

θ∞(z1)�

θ∞(z2)

�

θ∞(z3)

�

H�

u∞ =U

Long  (1955,  Tellus)  

Orographic Flow / Density Stratification / Lid  

momentum  

xz

Orographic Flow / Density Stratification /No Lid  

�

a

�

uU

= f ha,NhU

⎛ ⎝ ⎜

⎞ ⎠ ⎟

�

h

�

θ∞(z) = θ∞(0) 1+N 2

gz

⎛

⎝ ⎜

⎞

⎠ ⎟

�

θ∞(z1)�

θ∞(z2)�

θ∞(z3)

�

u∞ =UUpward Energy Propagation  

For  a  single  Fourier  component:  

2Nfor0 12211

22

311

π=

−=⎟⎟

⎠

⎞⎜⎜⎝

⎛=

U)zz(N

Uz

NNNHN),x(umax

(Klemp  &  Lilly,  1975)  

Orographic Flow / Density Stratification / Tropopause Linear Theory  

Klemp&Lilly  (1975,  JAS)  

�

N1 = 0N2 = N3 = NU 2

g'd∞

=1

�

U

�

N varying  

Orographic Flow / Density Stratification / Tropopause Linear Theory  

(Nonlinear Model, No Tropopause)

�

N = 0�

N = 0.01s−1

Orographic Flow / Density Stratification Nonlinear Theory  

�

U

�

N =  constant  

PelJer&Clark  (1979,  JAS)  

Smith  (1985,  JAS)  Durran  and  Klemp  (1987,  JAS)  

Orographic Flow / Density Stratification Hybrid Nonlinear Theory  

SoluJon  Curves  for  DeflecJon  vs  Terrain  Height  

(Smith,  1985)  

)hH(h

cc −+=

δδ

0cos

�

l ≡ NU

Durran  (1986,  JAS)  

Orographic Flow / Density Stratification / Tropopause Linear and Nonlinear Theory  

�

NL

�

NU

�

U

Durran  (1986,  JAS)  

Orographic Flow / Density Stratification / No Lid  

SupercriJcal  

SubcriJcal  

TranscriJcal  

Durran  (1986,  JAS)    Durran  (2003,  Encyclopedia  of  Atmos.  Sci.  )  

�

u(x)g'D(x)

>1

�

u(x)g'D(x)

<1

�

u(0)g'D(0)

=1

�

D(x) ≡ d(x) − h(x)

Durran  (2003,  Encyclopedia  of  Atmos.  Sci.)  

Orographic Flow / Density Stratification / No Lid  

1.  When  a  standing  mountain  wave  in  a  deep  cross-­‐mountain  flow  achieves  sufficient  amplitude  to  overturn  and  breakdown  at  some  level  in  the  troposphere    2.    When  standing  mountain  waves  break  and  dissipate  at  a  criJcal  level  in  a  shallow  cross-­‐mountain  flow    3.    When  there  is  sufficient  staJc  stability  near  mountain-­‐top  level  in  the  cross-­‐mountain  flow  to  create  high  downslope  winds  even  without  wave  breaking.  

Intense  downslope  winds  occur  can  occur  in  three  broad  categories: