Downslope W

ind Storm

s

Equation of M

otion for frictionless flow:

∂V ∂t

=l−∇k−α∇p+g

If w

e assu

me a horizo

ntally homogen

eous, hydrostatic referen

ce state w

here su

bsrript "o

":

α

o∂p

o/∂z=−g

and so, su

btrac

ting:

∂V ∂t=l−∇k−αo∇p+α'/αog

where primes are dev

iations from base state. Integrate equations of motion for frictionless flow

along the trajec

tory of the flow (⋅ds):

∂V ∂t

s∫⋅ds=l

s∫⋅ds−

∇k⋅ds

s∫−

αo∇p'⋅ds

s∫−

α'/αo

()g⋅ds

s∫Allow only for flow in x/z plane, assu

me Lam

b vec

tor is small l:0

(),

and flow is stea

dy state ∂V ∂t

=0

, an

d then

we get B

ernouli's equation

1

/2V

2

+α'/αo

()gz+

αodsp

s∫

How does

acc

eleration over the

wing affec

t pressure field?

Flow M

ountain R

idge

•In

finitely long m

ountain, no flow aro

und ridge

•Consider first an airplane wing:

α'/αog=αo∂p

'/∂s

Small Ridge

•Sim

ilar to airplane wing:

α'/αog=αo∂p

'/∂s

Mes

o-B

eta Sca

le R

idge

•Res

onan

t gravity res

ponse

is involved

, low pressure shifts

increa

singly down strea

m as the sc

ale of the ridge

bec

omes

larger:

α'/αog:αo∂p

'/∂s

Witch

of Agnes

i Ridge

•Lets co

nsider a “W

itch

of Agnes

i”, bell sh

aped

mountain

(norm

ally use

d for an

alytica

l mountain w

ave studies)

hav

ing the fo

rmula:

–a is the half-width, h is the max

imum height, and d is

the distance

fro

m the ridge top, an

d z_s is the

topography height.

2

22

s

az

ha

d=

+

More about W

itch

of Agnes

i

•Eas

y for an

alytica

l so

lutions

•NOT a sine wav

e, is a bell sh

aped

ridge that

contains a sp

ectrum of wav

e co

mponen

ts

represe

nting m

any w

avelen

gths

•Some parts of the ridge may

be at super

Rossby rad

ius sc

ale an

d some may

be at

sub-R

ossby rad

ius sc

ale fo

r instan

ce

Fro

udeNumber

•Im

portan

t influen

ces on atm

osp

here

resp

onse

to flow over an object:

–(a) Len

gth sca

le of the object

–(N

) Bru

nt-Vas

allaifreq

uen

cy, the vertica

l stab

ility pro

viding a res

toring force fo

r gravity

wav

es:

–(U

) velocity of flow norm

al to the ridge

1 2g

Nzθ

θ∂

=

∂

Fro

udeNumber

•Define Fro

udeNumber:

inertial frequen

cy/

Bru

nt-Vas

allai Frequen

cyr

Ua

FN

==

Inertial C

utoff, ie

Rossby

Number

•The co

riolis param

eter is an

other importan

t

param

eter. If the mountain is big enough,

we get lee

cyclogen

esis, not gravity w

aves

!

So w

e must consider the Rossby N

umber,

ie:

o

RaR

L=

Flow O

ver a R

idge

•W

e co

nsider flow over shallow (h << dep

th of

troposp

here) ridges

of se

veral half-widths an

d look at the

resu

lts of a linea

r an

alytica

l so

lution for the W

itch

of

Agnes

imountain.

•The so

lution to the linea

r pro

blem yields a wav

e eq

uation

of the fo

rm:

w-vertica

l velocity

z-height ab

ove su

rfac

e

k –

vertica

l wav

e number

l –Sco

rer Param

eter(

)2

20

ttzz

wl

kw

+−

=

Vertica

l W

ave Number

•L_x is the horizo

ntal wav

elen

gth of the

gravity w

ave. This param

eter in the vertica

l

wav

e eq

uation is purely nonhydrostatic!

2

x

kLπ

=

Sco

rer Param

eter

•This param

eter is related to the

tran

smissivityof the atmosp

here to gravity

wav

es considering only hydro

static

pro

cesses

22

2

22

1N

dU

lU

Udz

=−

When

Gravity W

aves

?

•Gravity w

ave so

lutions only exist when

•Therefore, there is a “sh

ort w

ave cu

toff”sc

ale, below

which gravity w

aves

can

not ex

ist:

–L_z is the vertica

l wav

elen

gth of the gravity w

ave

–L_x is the horizo

ntal wav

elen

gth of the gravity w

ave

22

0l

k−

>

22

22

zx

LL

ll

kππ

=>

−

Narro

w R

idge:

Evan

esce

nt wav

es

Med

ium R

idge:

Mountain (gravity)

wav

es

Typical M

ountain W

ave (L

enticu

lar) C

loud

Mountain W

ave –Len

ticu

larCloud

Double W

ave (L

enticu

lar) C

loud

Flying Sau

cer W

ave Cloud

Len

ticu

larCloud

Bro

ad R

idge:

Lee

Cyclogen

esis

for larg

er m

odes

,

GW

for sm

aller

modes

Med

ium-N

arro

w ridge, but with

Sco

rer Param

eter (l) varying w

ith

height. This “trap

s”sh

orter w

aves

of the “W

itch

of Agnes

i”

mountain, but tran

smitts

vertica

lly

the longer ones

, lead

ing to lee

wav

es.

-This is mostly a

nonhydro

static

effect –

why?

-The sh

orter w

aves

hav

e

solutions in low lev

els

where l is large, but do not

above, so they

reflect off

Lee

Wav

es

Lee

Wav

es

Mountain (Gravity) W

aves

•; i.e. static stab

ility dominates

over inertia

•or ; i.e. effec

t of stab

ility

dominates

over C

oriolis

•, i.e. sca

le is larg

er than

short-

wav

e cu

toff for gravity w

aves

xR

LL

<

2/

xL

lπ

>

1oR

>

1rF<<

Vertica

lly Pro

pag

ating G

ravity

Wav

es

Gravity w

ave ab

sorb

ed at critical lev

el w

here

phas

e sp

eed equals wind spee

d and air

statically stable above

Effec

t of moisture on M

ountain

Wav

es

•Effec

t is to les

sen the Bru

nt Vas

allai

freq

uen

cy bec

ause

laten

t hea

t reduce

s lapse

rate:

2

2

2

1ln

1

vls

vls

lmoist

vls

p

pLr

Lr

rRT

Ng

Lr

zcTz

z

cRT

θε

+

∂

∂∂

=+

−

∂

∂∂

+

•In

crea

ses dep

th of mountain w

ave

•In

crea

ses horizo

ntal wav

elen

gth

•M

ay cau

se some trap

ping of sh

orter

wav

elen

gths

Theo

ry of DownslopeW

ind

Storm

s

•They

go by a number of nam

es:

–Chinook w

inds (R

ock

ies, Indian nam

e that

mea

ns “s

now eater”

–Foeh

nwind, nam

e use

d in E

uro

pe

–San

ta A

na wind, nam

e use

d in Southern

Californ

ia

•Downslopewind storm

s are related to

mountain w

aves

•M

ountain w

aves

will loca

lly incrreas

ethe

winds on the lee side of the mountain, but

typically not to sev

ere levels

•But in downslopewind cas

es they

get very

stro

ng rea

ching sev

ere levels ro

utinely (>

55 kts)

•Lets look at a famous docu

men

ted

windstorm

hitting B

oulder C

olorado on 11

January, 1972

Klemp and L

illy T

heo

ry

•Bas

ed on hydro

static sim

ulations

•Partial reflection of gro

up velocity off of

tropopau

secrea

ting res

onan

ce

•Nee

d tro

popau

seheight to be integer

number of half wav

elen

gths ab

ove su

rfac

e

•Res

onan

ce increa

ses am

plitude of mountain

wav

e…no w

ave break

ing in their

hydro

static theo

ry

Clark

and Pelteir

(1977)

•Sam

e effect but upper w

ave break

s

•The break

ing upper w

ave des

tabilizes

upper

troposp

here an

d lower stratosp

here ducting

the underlying m

ountain w

ave more

•Strong amplifica

tion of lower tro

posp

here

wav

e

•Critica

l level at ¾

optimal

zL

Influen

ce of M

id-L

evel Inversion

•Created

by a cold pool to the wes

t an

d to

the ea

st, su

ch as a Great B

asin H

igh to w

est

of Rock

ies an

d A

rctic High to eas

t

•In

version nea

r or just above ridge top

•In

version traps wav

e en

ergy below, lead

ing

to large am

plifica

tion down low and

form

ation of a hydraulic jump

Hydraulic Ju

mp A

nalogy

•Curren

t thinking among m

ountain m

eteo

rologists

•Im

agine flow along a rock

y strea

m bed

:

–W

ater under air is an

alogous to the layer of co

ld stable

air at the su

rfac

e under les

s stab

le air above! Notice

the

water w

aves

are trapped

fro

m m

oving upward into the

air as

the wav

es in the stab

le lay

er of air are trap

ped

from m

oving upward into the less stable air.

–W

hen

water is much

dee

per than

rock

s, turb

ulence

, water flows ac

ross the ro

cks with little turb

ulence

. Y

ou

could tak

e a boring raft trip down such

a lam

inar

stream

.

•Now imag

ine that the water lowers to be just

dee

per than

the ro

cks. Now you hav

e whitew

ater!

The water plunges

down the lee side of the ro

cks

and even

digs a little hole, dep

ressing the su

rfac

e

and blowing out ro

cks etc.

•The sa

me is tru

e fo

r the downslopewind. T

rapped

ben

eath the inversion, the wav

e am

plifies

and

break

s, clowingout Boulder!

Wind Spee

dPotential T

emperature

Class C

ase Study: Feb

ruary 3, 1999 w

est of Boulder

1400 U

TC