Richard RotunnoRichard Rotunno

National Center for Atmospheric Research , USA

Dynamical Mesoscale Mountain Meteorology

• Dynamic Of or pertaining to force producing motion

• Meso-(intermediate)scale Length ~ 1-100km Time ~1h – 1day

• Mountain Meteorology Science of atmospheric phenomena caused by mountains

Topics

Lecture 1 : Introduction, Concepts, Equations

Lecture 2: Thermally Driven Circulations

Lecture 3: Mountain Waves

Lecture 4: Mountain Lee Vortices

Lecture 5: Orographic Precipitation

Thermally Driven Circulations

Whiteman (2000)Whiteman (2000)

Jane

Eng

lish

Mt.

Sha

sta

Mountain Waves

Haw

aii

Mountain Lee Vortices East

Spa

ce S

hutt

le

Orographic Precipitation

What do all these phenomena have in common?

Buoyancy

par

parenvgBρ

ρρ −=

δenvρ

parρg

Displacement

= density“env” = environment“par” = parcel

ρ

Buoyancy is acceleration

gz

ppar

parpar ρδρ −

∂∂

−=&&

gz

p

z

penv

envpar ρ−=∂

∂≈

∂∂

Bgpar

parenv ≡−

=ρ

ρρδ&&

forcesofsumonacceleratimass

=×

To a goodapproximation...

= pressure = vertical coordinatezp

δenvρ

parρg

δρ

ρρ 2NgBpar

parenv −≡−

=

( ) ( ) δρρρδρρρpar

envparenv

envenv zz ∂

∂+≅

∂

∂+≅ 0;0

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

−∂∂

−=parenv zz

gN

ρρ

ρ2

Stability

= “static stability” = “Brunt-Väisälä Frequency”NN 2

= 0 for incompressibledensity-stratifed fluid

(unstable)(stable) 0,0 22 <> NN

(unstable)(stable) 0,0 >×<× BB δδ

Air is a compressible fluid…

0≠∂∂

parzρ

RT

p=ρGas Law

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

−∂∂

=parenv z

T

z

T

T

gN 2

1st Law of Thermo (adiabatic)

18.91 −−≅

∂∂

⇒−≈∂

∂=

∂∂ kmC

zTg

zp

zTc

par

par

parparp

o

ρ

in terms of Temperature2,NB

= specific heat at constant pressure , R = gas constant for dry airpc

env

envpar

T

TTgB

−=

Air Parcel Behavior in a Stable Atmosphere

)(zTenv

)(δparT

00 2 <−==⇒> δδδ NB&&

δ

Temperature

z

02 >N

Air Parcel Behavior in an Unstable Atmosphere

)(zTenv

)(δparT

δ

Temperature

z

02 <N

00 2 >−==⇒> δδδ NB&&

env

envpargBθ

θθ −=

pcR

mb

pT⎟⎠⎞

⎜⎝⎛≡≡1000

; ππ

θ

in terms of potential temperature BN ,2 θ

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

−∂∂

=parenv zz

gN

θθ

θ2

Air Parcel Behavior in Stable or Unstable Atmosphere

δ

Potential Temperature

)()(

stablezenvθ

)()(

unstablezenvθ

.ctpar =θ

z

envz

gN

∂∂

=θ

θ2

Dynamic Mesoscale Mountain Meteorology

Governing Equations

In terms of and …..

1st Law of Thermodynamics

Dt

Dp

Dt

DQ

Dt

DTcp ρ

1+=

Dt

DQ

cDt

D

pπθ 1

=π

Dt

DQ

Dt

cDT p =

)ln( θWith previous definitions

Common form…

θ

Newtons 2nd Law

FkgcDt

uDp

rr+−∇−= ˆπθ

xx

y

z

),,( wvuu =r

zyx kji ∂+∂+∂=∇ ˆˆˆ

zyxt wvuDt

D∂+∂+∂+∂=

pcp ∇=∇ρ

πθ 1

With previous definitions

= frictional force/unit massFr

01

=⋅∇+ uDtD rρ

ρ

),( θπρρ =

Mass Conservation

uDt

D r⋅∇=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ −+ π

κθ ln

11ln

With previous definitions

pcR≡κ

Summary of Governing Equations

FkgcDt

uDp

rr+−∇−= ˆπθ

Dt

DQ

cDt

D

pπθ 1

=

uDt

D r⋅∇=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ −+ π

κθ ln

11ln

Conservation of

momentum

energy

mass

Simplify Governing Equations I

kgcDt

uDp

ˆ−∇−= πθr

0=DtDθ

uDt

D r⋅∇=⎟

⎠⎞

⎜⎝⎛ −

πκ

ln11

Conservation of

momentum

energy

mass

Neglect molecular diffusion 0,0 == FQr

Simplify Governing Equations II

0000 1)( θππθ

pzp c

gzzgc −=⇒−=∂

Conservation of momentum

Boussinesq Approximation

1~

,1

~~,

~

0000 <<<<+=+= π

πθ

θπππθθθ

kBkg

cDt

uDp

ˆˆ~

)~(0

0 +−∇=+−∇≅ ϕθθπθ

r

Simplify Governing Equations III

0=DtDB

Conservation of energy

θθθ ~0 +=With

Simplify Governing Equations IVConservation of mass

uDt

D r⋅∇=⎟

⎠⎞

⎜⎝⎛ −

πκ

ln11

)1000/ln(ln mbpκπ =By definition

uDt

Dp

c

r⋅∇=− 2

1ρ

0≅⋅∇ ur

1,1,122

22

2

2

<<<<<<cgL

cL

cU ω3 conditions for effective

incompressibility(Batchelor 1967 pp. 167-169)

= speed of soundc

= velocity, length, frequency scales

ω,,LU

Summary of Simplified Governing Equations

kBDt

uD ˆ+−∇= ϕr

0=DtDB

0=⋅∇ ur

Conservation of

momentum

energy

mass

Still nonlinear (advection)

Filtering equations for mean / turbulent fluxes ofB and u (see Sullivan lectures)

Summary

- Buoyancy is a fundamental concept for dynamical mountain meteorology

- Boussinesq approximation simplifies momentum equation

- For most mountain meteorological applications, velocity field approximately

solenoidal ( )0=⋅∇ ur