aguidedtourthroughthetheoryfor...
TRANSCRIPT
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: