gravity waves, scale asymptotics, and the pseudo-incompressible equations ulrich achatz...

Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations

Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations

Ulrich AchatzGoethe-Universität Frankfurt am Main

Rupert Klein (FU Berlin) and Fabian Senf (IAP Kühlungsborn)

13.9.10

Gravity waves in the

middle atmosphere

Gravity Waves in the Middle Atmosphere

Becker und Schmitz (2003)

Gravity Waves in the Middle Atmosphere

Becker und Schmitz (2003)

Without GW parameterization

linear GWs in the Euler equations

• no heat sources, friction, …• no rotation

pcR

p

pT

RTpDt

DDt

D

gpDt

D

/

0

z

0

0

1

v

ev


• no heat sources, friction, …• no rotation

pcR

p

pT

RTpDt

DDt

D

gpDt

D

/

0

z

0

0

1

v

ev

T

p

p

RTppcR

,/

0

are equivalent to

0

0

z

v

ev

v

p

c

R

Dt

DDt

D

gcDt

D


• linearization about atmosphere at rest:

0

')(

')(

'

gdz

dc

z

z

p

vv


• linearization about atmosphere at rest:

dz

dgN

gb

wvu

wNt

b

cc

Rgw

tc

bcDt

D

pv

p

p

2

2

z

''

)',','('

0''

0'''

'''

v

v

ev

0

')(

')(

'

gdz

dc

z

z

p

vv


• conservation of wave energy: linear equations satisfy

TRc

ccc

cN

bE

pt

E

v

psp

s

2222

2

2

22

,'2

''

2'

0'''

v

v


• conservation of wave energy: linear equations satisfy

TRc

ccc

cN

bE

pt

E

v

psp

s

2222

2

2

22

,'2

''

2'

0'''

v

v

• conservation suggests:

/'

/'

/1'

sc

Nb

v


• isothermal atmosphere: const. const., 2 Ncs

ti

tibb

mlkti

xk

xk

kxkvv

exp~'

exp~

'

),,( exp~'

00

0

0


• isothermal atmosphere:

22222

2222

222

2

4

14

1

Hmlkc

Hmlk

lkN

s

const. const., 2 Ncs

gravity waves

sound waves


• isothermal atmosphere:

22222

2222

222

2

4

14

1

Hmlkc

Hmlk

lkN

s

const. const., 2 Ncs

bkTc

m

Ni

bN

iw

bNk

miu

p

~~

~~

~~

22

2

2

2

gravity waves

sound waves

GW polarization relations:

GW breaking in the atmosphere

• Conservation

• Instabilität in großen Höhen• Impulsdeposition …

0

20 12 vv

Rapp et al. (priv. comm.)


• Conservation

• Instability at large altitudes• Impulsdeposition …

0

20 12 vv



• Conservation

• Instability at large altitudes• Turbulence• Momentum deposition…

0

20 12 vv


GW Parameterizations

• multitude of parameterization approaches (Lindzen 1981, Medvedev und Klaassen 1995, Hines 1997, Alexander and Dunkerton 1999, Warner and McIntyre 2001)

• too many free parameters

• reasons :– …– insufficent knowledge: conditions of wave breaking

• basic paradigm: breaking of a single wave– Stability analyses (NMs, SVs)– Direct numerical simulations (DNS)

GW breaking

GW stability in the Boussinesq theory

dzdg

Nz

gb

bwNbt

bpft

0

2

0

22

2

0

,

v

v

vkvkvv

Basic GW types

• Dispersionsrelation:

• Trägheitsschwerewellen (TSW):

• Hochfrequente Schwerewellen (HSW):

2/12222 cossin Nf

2

2

2

cot2

190f

Nf

cos90 N

Basic GW types

• Dispersion relation:

• Inertia-gravity waves (IGW):

• High-frequent GWs (HGW):

2/12222 cossin Nf

2

2

2

cot2

190f

Nf

cos90 N

sin

h24

22f

Coriolis parameter

Stability

dz

dgN

0

2

Basic GW types



• High-frequent gravity waves (HGW):

2/12222 cossin Nf

2

2

2

cot2

190f

Nf

cos90 N

sin

h24

22f

Coriolis parameter

Stability

dz

dgN

0

2

Basic GW types



• High-frequent GWs (HGW):

2/12222 cossin Nf

2

2

2

cot2

190f

Nf

cos90 N

sin

h24

22f

Coriolis parameter

Stability

dz

dgN

0

2

Traditional Instability Concepts

02

z

bN

z

Btot

• static (convective) instability:

amplitude reference (a>1)

• dynamic instability

4

1Ri 22

2

zv

zu

zb

N

• limited applicability to HGW breaking

Traditional Instability Concepts

02

z

bN

z

Btot

• static (convective) instability:

amplitude reference (a>1)

• dynamic instability

4

1Ri 22

2

zv

zu

zb

N

• BUT: limited applicability to GW breaking

NM analysis, 2.5D-DNS

||SVNM,SW

exp)()()0,,,(

kx

yibb

tyb

vvv

NMs: Growth Rates = 70°)

Achatz (2005, 2007b)

GW amplitude after a perturbation by a NM (DNS):

Achatz (2005, 2007b)

Energetics b

by

, deviation -

, and over average -

v

v

dbd

kubCdd

kuP

N

bAK

S

2

21

2

22

vv

v

DCPE

dtEdAKE

2

/

not the gradients in z matter,but those in !

instantaneous growth rate:

Energetics of a Breaking HGW with a0 < 1

Strong growth perturbation energy:buoyancy instability

DCPE

dtEd

2

/

Achatz (2007b)

Singular Vectors:

• normal modes: Exponential energy growth

• singular vectors: optimal growth over a finite time (Farrell 1988, Trefethen et al. 1993)

characteristic perturbations in a linear initial value problem:

NMs and SVs of an IGW (Ri > ¼!): growth within 5min

Achatz (2005) Achatz and Schmitz (2006a,b)

Dissipation rate breaking IGW (a< 1, Ri > ¼):

Measured dissipation rates 1…1000 mW/kg (Lübken 1997, Müllemann et al. 2003)

Achatz (2007a)

Summary GW breaking

In comparison to assumptions in GW parameterizations:

• GW breaking sets in at lower amplitudes , i.e.

it sets in earlier (at lower altitudes)

Summary GW breaking

In comparison to assumptions in GW parameterizations:

• GW breaking sets in at lower amplitudes , i.e.

it sets in earlier (at lower altitudes)

• GW dissipation stronger than assumed, i.e.

more momentum deposition

Soundproof Modelling and

Multi-Scale Asymptotics

Achatz, Klein, and Senf (2010)

GW breaking in the middle atmosphere

• Conservation

• Instability at large altitudes• Momentum deposition…

0

20 12 vv



• Conservation

• Instability at large altitudes• Momentum deposition…• Competition between wave

growth and dissipation:– not in Boussinesq theory

0

20 12 vv



• Conservation

• Instability at large altitudes• Momentum deposition…• Competition between wave growth and

dissipation:– not in Boussinesq theory– Soundproof candidates:– Anelastic

(Ogura and Philips 1962, Lipps and Hemler 1982)

– Pseudo-incompressible (Durran 1989)

0

20 12 vv



• Conservation

• Instability at large altitudes• Momentum deposition…• Competition between wave growth and

dissipation:– not in Boussinesq theory– Soundproof candidates:– Anelastic

(Ogura and Philips 1962, Lipps and Hemler 1982)

– Pseudo-incompressible (Durran 1989)

0

20 12 vv


Which soundproof model should be used?

Scales: time and space

• for simplicity from now on: 2D• non-hydrostatic GWs: same spatial scale in horizontal and vertical

KKzz

Kxx

/ˆ

/ˆ

characteristic wave number

• time scale set by GW frequency

/t̂t characteristic frequency

dispersion relation for

2

41 22

22

222

222 N

mk

kN

Hmk

kN

HK 2/1



KLzLz

xLx

/1 ˆ

ˆ

characteristic length scale


/t̂t characteristic frequency


2

41 22

22

222

222 N

mk

kN

Hmk

kN

HK 2/1





/1 ˆ TtTt characteristic time scale


2

41 22

22

222

222 N

mk

kN

Hmk

kN

HK 2/1

KLzLz

xLx

/1 ˆ

ˆ





characteristic time scale


0022

22

222

222

41 Tc

gN

mk

kN

Hmk

kN

p

HK 2/1

KLzLz

xLx

/1 ˆ

ˆ

/1 ˆ TtTt

Scales: velocities

• winds determined by polarization relations:

bN

iw

bNk

miu

~~

~~

2

2

what is ?b~

• most interesting dynamics when GWs are close to breaking, i.e. locally

~

0'

0'

2

2

m

Nb

wNz

b

dz

d

z

KU

U

vv ˆ

Scales: velocities

• winds determined by polarization relations:

bN

iw

bNk

miu

~~

~~

2

2

what is ?b~

• most interesting dynamics when GWs are close to breaking, i.e. locally

~

0'

0'

2

2

m

Nb

wNz

b

dz

d

z

T

L

KU

U

vv ˆ

Non-dimensional Euler equations

0ˆ

ˆ

0ˆˆˆ1ˆ

ˆ

ˆˆˆˆˆ

2

2

tD

D

tD

DtD

D

v

ev

z

• using:

ˆ

ˆ

ˆ

ˆ

ˆ

0T

U

tTt

L

vv

xx

yields

g

N

dz

d

H

KH211

11

Non-dimensional Euler equations

0ˆ

ˆ

0ˆˆˆ1ˆ

ˆ

ˆˆˆˆˆ2

tD

D

tD

DtD

D

v

ev

z

• using:

yields

g

TcH

R

cH

H

L

pp 00

1

ˆ

ˆ

ˆ

ˆ

ˆ

0T

U

tTt

L

vv

xx

isothermal potential-temperature scale height

Scales: thermodynamic wave fields

• potential temperature:

O

g

b

TT

~'

00

m

Nb

2~

Scales: thermodynamic wave fields

• potential temperature:

• Exner pressure:

m

Nb

2~

2

22

2

' ~~ Ob

kTc

m

Ni

p

O

g

b

TT

~'

00


Additional vertical scale needed:

• and have vertical scale

• scale of wave growth is

zti i

i

i

i ˆ ,ˆ,ˆ

ˆ

ˆˆ

0 )(

)(

)(

x

vv

L

H

K

HH

222

Therefore: Multi-scale-asymptotic ansatz

0ˆ )0( also assumed:





zti i

i

i

i ˆ ,ˆ,ˆ

ˆ

ˆˆ

0 )(

)(

)(

x

vv

L

HH

222



L

H





zti i

i

i

i ˆ ,ˆ,ˆ

ˆ

ˆˆ

0 )(

)(

)(

x

vv



L

H

L

HH

222





zti i

i

i

i ˆ ,ˆ,ˆ

ˆ

ˆˆ

0 )(

)(

)(

x

vv


0

0ˆ

)1(

)0(

also assumed:

L

HH

222

L

H

Scale asymptotics Euler: results

)0(

)0(

)0()0(

)0(

1

0ˆˆ

0ˆ

tt

Hydrostatic, large-scale, background


)0(

)0(

)0()0(

)0(

1

0ˆˆ

0ˆ

tt

0ˆ

ˆˆ

0ˆˆ

)0()0(

)1(0

)0(

)1()2()0(

)0(0

)2()0(

)0(0

wtD

D

ztD

wD

xtD

uD


Momentum equations and entropy equations as in Boussinesq or anelastic theory


)0(

)0(

)0()0(

)0(

1

0ˆˆ

0ˆ

tt

0ˆ

ˆˆ

0ˆˆ

)0()0(

)1(0

)0(

)1()2()0(

)0(0

)2()0(

)0(0

wtD

D

ztD

wD

xtD

uD

0ˆ1

0ˆ

)0()1()0(

)0()0(

)0(

w

w v

v


Momentum equations and entropy equations as in Boussinesq or anelastic theory

Exner-pressure equation:• leading order incompressibility (Boussinesq)• next order yields density effect on amplitude

Scale asymptotics: pseudo-incompressible equations

scale-asymptotic analysis of

0

0

v

ev

z

Dt

D

gcDt

Dp



0ˆ1

0ˆ

ˆ

ˆˆ

0ˆˆ

)0()1()0(

)0()0(

)0(

)0()0(

)1(0

)0(

)1()2()0(

)0(0

)2()0(

)0(0

ww

wtD

D

ztD

wD

xtD

uD

v

v

results:

0

0

v

ev

z

Dt

D

gcDt

Dp



0ˆ1

0ˆ

ˆ

ˆˆ

0ˆˆ

)0()1()0(

)0()0(

)0(

)0()0(

)1(0

)0(

)1()2()0(

)0(0

)2()0(

)0(0

ww

wtD

D

ztD

wD

xtD

uD

v

v

results:

0

0

v

ev

z

Dt

D

gcDt

Dp

• same scale asymptotics as Euler• such a result cannot be obtained from the anelastic equations

Scale asymptotics: anelastic equations

0

0

'

v

ev

z

Dt

D

bp

Dt

D



01

ˆ1

0ˆ

0ˆ

ˆˆ

0ˆˆ

)0(

)0(

)0()0(

)0()1()0(

)0()0(

)0(

)0()0(

)1(0

)0(

)1()2()0(

)0(0

)2()0(

)0(0

w

ww

wtD

D

ztD

wD

xtD

uD

v

v

results:

0

0

'

v

ev

z

Dt

D

bp

Dt

D



01

ˆ1

0ˆ

0ˆ

ˆˆ

0ˆˆ

)0(

)0(

)0()0(

)0()1()0(

)0()0(

)0(

)0()0(

)1(0

)0(

)1()2()0(

)0(0

)2()0(

)0(0

w

ww

wtD

D

ztD

wD

xtD

uD

v

v

results:

0

0

'

v

ev

z

Dt

D

bp

Dt

D


anelastic equations only consistent if:

)0(

)0(

)0(

)0(

11

1

i.e. potential-temperature scale >> Exner-pressure scale

Large-Amplitude WKB

Achatz, Klein, and Senf (2010)

Large-amplitude WKB

1

)1()1(0

)1(

)0(1

)0(0

)0(

)1()0()0(

)2(2)0(

)1()0(

)0(

,,exp),,(ˆ),,(ˆˆ

,,exp),,(ˆ)ˆ,ˆ,ˆ(ˆˆ

(e.g.) oˆˆ~

~ˆˆ

~ˆˆ

~ˆ

i

izxt

VVV

VVV

VVv

vv

Large-amplitude WKB

1

)1()1(0

)1(

)0(1

)0(0

)0(

)1()0()0(

)2(2)0(

)1()0(

)0(

,,exp),,(ˆ),,(ˆˆ


(e.g.) oˆˆ~

~ˆˆ

~ˆˆ

~ˆ

i

izxt

VVV

VVV

VVv

vv

Mean flow with only large-scale dependence

Large-amplitude WKB

1

)1()1(0

)1(

)0(1

)0(0

)0(

)1()0()0(

)2(2)0(

)1()0(

)0(

,,exp),,(ˆ),,(ˆˆ


(e.g.) oˆˆ~

~ˆˆ

~ˆˆ

~ˆ

i

izxt

VVV

VVV

VVv

vv

Wavepacket with • large-scale amplitude• wavenumber and frequency with large-scale dependence

),(k

Large-amplitude WKB

1

)1()1(0

)1(

)0(1

)0(0

)0(

)1()0()0(

)2(2)0(

)1()0(

)0(

,,exp),,(ˆ),,(ˆˆ


(e.g.) oˆˆ~

~ˆˆ

~ˆˆ

~ˆ

i

izxt

VVV

VVV

VVv

vv

Large-amplitude WKB

1

)1()1(0

)1(

)0(1

)0(0

)0(

)1()0()0(

)2(2)0(

)1()0(

)0(

,,exp),,(ˆ),,(ˆˆ


(e.g.) oˆˆ~

~ˆˆ

~ˆˆ

~ˆ

i

izxt

VVV

VVV

VVv

vv

Wave induced mean flow

Large-amplitude WKB

1

)1()1(0

)1(

)0(1

)0(0

)0(

)1()0()0(

)2(2)0(

)1()0(

)0(

,,exp),,(ˆ),,(ˆˆ


(e.g.) oˆˆ~

~ˆˆ

~ˆˆ

~ˆ

i

izxt

VVV

VVV

VVv

vv

Harmonics of the wavepacket due to nonlinear interactions

Large-amplitude WKB

1

)1()1(0

)1(

)0(1

)0(0

)0(

)1()0()0(

)2(2)0(

)1()0(

)0(

,,exp),,(ˆ),,(ˆˆ


(e.g.) oˆˆ~

~ˆˆ

~ˆˆ

~ˆ

i

izxt

VVV

VVV

VVv

vv

• collect equal powers in • collect equal powers in • no linearization!

/exp i

Large-amplitude WKB: leading order

frequency intrinsic ˆˆ

0

ˆˆˆ

ˆ1

ˆ

ˆ

,ˆ

00

0ˆ0

ˆ0

00ˆ

0ˆ

)0(0

)2(1

)0(

)0(

)1(1

)0(1

)0(1

)0(1

Uk

N

W

U

M

imik

iN

imNi

iki

i

k

Vk

Large-amplitude WKB: leading order

frequency intrinsic ˆˆ

0

ˆˆˆ

ˆ1

ˆ

ˆ

,ˆ

00

0ˆ0

ˆ0

00ˆ

0ˆ

)0(0

)2(1

)0(

)0(

)1(1

)0(1

)0(1

)0(1

Uk

N

W

U

M

imik

iN

imNi

iki

i

k

Vk

dispersion relation and structure as from Boussinesq

M

N

W

U

mk

kN

M

of Nullvector

ˆˆˆ

ˆ1

ˆ

ˆ

ˆ

0det

)2(1

)0(

)0(

)1(1

)0(1

)0(1

22

222

Large-amplitude WKB: 1st order

)0(

)0(

)0(1

)0(1

)0(1

)3(1

)0(

)0(

)2(1

)1(1

)1(1

ˆ

ˆ

ˆ1ˆˆ

ˆˆˆ

ˆ1

ˆ

ˆ

,ˆ

WWUN

W

U

M

k


)0(

)0(

)0(1

)0(1

)0(1

)3(1

)0(

)0(

)2(1

)1(1

)1(1

ˆ

ˆ

ˆ1ˆˆ

ˆˆˆ

ˆ1

ˆ

ˆ

,ˆ

WWUN

W

U

M

k

velocitygroup ˆ

,ˆˆ

energy waveˆ

ˆ

2

1

2

ˆ

2

ˆ'

0ˆ'

ˆ'

)0(0

2

)0(

)0(1

2

2)0(

1)0(

,

mkU

NE

EE

g

g

c

V

cSolvability condition leads to wave-action conservation

(Bretherton 1966, Grimshaw 1975, Müller 1976)


)0(

)0(

)0(1

)0(1

)0(1

)3(1

)0(

)0(

)2(1

)1(1

)1(1

ˆ

ˆ

ˆ1ˆˆ

ˆˆˆ

ˆ1

ˆ

ˆ

,ˆ

WWUN

W

U

M

k

Solvability condition leads to wave-action conservation

(Bretherton 1966, Grimshaw 1975, Müller 1976)

velocitygroup ˆ

,ˆˆ

energy waveˆ

ˆ

2

1

2

ˆ

2

ˆ'

0ˆ'

ˆ'

)0(0

2

)0(

)0(1

2

2)0(

1)0(

,

mkU

NE

EE

g

g

c

V

c

Pseudo-incompressible divergence needed!

Large-amplitude WKB: 1st order, 2nd harmonics

1

)1()1( ,,exp),,(ˆˆ

iVV


)3(2

)0(

)0(

)2(2

)1(2

)1(2

ˆˆˆ

ˆ1

ˆ

ˆ

2,ˆ2

N

W

U

M k

1

)1()1( ,,exp),,(ˆˆ

iVV

:2


)3(2

)0(

)0(

)2(2

)1(2

)1(2

ˆˆˆ

ˆ1

ˆ

ˆ

2,ˆ2

N

W

U

M k

k2,ˆ2

ˆˆˆ

ˆ1

ˆ

ˆ

1

)3(2

)0(

)0(

)2(2

)1(2

)1(2

M

N

W

U

1

)1()1( ,,exp),,(ˆˆ

iVV

:2

2nd harmonics are slaved

Large-amplitude WKB: 1st order, higher harmonics

1

)1()1( ,,exp),,(ˆˆ

iVV


0

ˆˆˆ

ˆ1

ˆ

ˆ

,ˆ :2

)3()0(

)0(

)2(

)1(

)1(

N

W

U

M k

1

)1()1( ,,exp),,(ˆˆ

iVV

:2


0

ˆˆˆ

ˆ1

ˆ

ˆ

,ˆ :2

)3()0(

)0(

)2(

)1(

)1(

N

W

U

M k 2 0

ˆˆˆ

ˆ1

ˆ

ˆ

)3()0(

)0(

)2(

)1(

)1(

N

W

U

1

)1()1( ,,exp),,(ˆˆ

iVV

:2

higher harmonics vanish(nonlinearity is weak)

Large-amplitude WKB: mean flow

),,(ˆˆ

),,(ˆˆ

(e.g.) oˆˆ~

)1(0

)1(

)0(0

)0(

)1()0()0(

VV

VV

VVv


Horizontal flow horizontally homogeneous(non-divergent)

0ˆˆ

ˆ

ˆ2ˆ

ˆ

ˆˆˆ

ˆˆ

ˆˆ

0ˆ

0ˆ

0ˆ

)0()1(

0

2)0(

2)1(1

)0(

)2(0

)0(0)0(

)0(0)0(

)0(0

)1(0

)0(0

)0(0

GW

WGW

UGW

W

F

U

W

U

F

F

),,(ˆˆ

),,(ˆˆ

(e.g.) oˆˆ~

)1(0

)1(

)0(0

)0(

)1()0()0(

VV

VV

VVv


No zero-order vertical flow

0ˆˆ

ˆ

ˆ2ˆ

ˆ

ˆˆˆ

ˆˆ

ˆˆ

0ˆ

0ˆ

0ˆ

)0()1(

0

2)0(

2)1(1

)0(

)2(0

)0(0)0(

)0(0)0(

)0(0

)1(0

)0(0

)0(0

GW

WGW

UGW

W

F

U

W

U

F

F

),,(ˆˆ

),,(ˆˆ

(e.g.) oˆˆ~

)1(0

)1(

)0(0

)0(

)1()0()0(

VV

VV

VVv


No first-order potential temperature

0ˆˆ

ˆ

ˆ2ˆ

ˆ

ˆˆˆ

ˆˆ

ˆˆ

0ˆ

0ˆ

0ˆ

)0()1(

0

2)0(

2)1(1

)0(

)2(0

)0(0)0(

)0(0)0(

)0(0

)1(0

)0(0

)0(0

GW

WGW

UGW

W

F

U

W

U

F

F

),,(ˆˆ

),,(ˆˆ

(e.g.) oˆˆ~

)1(0

)1(

)0(0

)0(

)1()0()0(

VV

VV

VVv


0ˆˆ

ˆ

ˆ2ˆ

ˆ

ˆˆˆ

ˆˆ

ˆˆ

0ˆ

0ˆ

0ˆ

)0()1(

0

2)0(

2)1(1

)0(

)2(0

)0(0)0(

)0(0)0(

)0(0

)1(0

)0(0

)0(0

GW

WGW

UGW

W

F

U

W

U

F

F

),,(ˆˆ

),,(ˆˆ

(e.g.) oˆˆ~

)1(0

)1(

)0(0

)0(

)1()0()0(

VV

VV

VVv

• acceleration by GW-momentum-flux divergence


0ˆˆ

ˆ

ˆ2ˆ

ˆ

ˆˆˆ

ˆˆ

ˆˆ

0ˆ

0ˆ

0ˆ

)0()1(

0

2)0(

2)1(1

)0(

)2(0

)0(0)0(

)0(0)0(

)0(0

)1(0

)0(0

)0(0

GW

WGW

UGW

W

F

U

W

U

F

F

• acceleration by GW-

momentum-flux divergence

• GWs induce lower-order potential temperature variations

),,(ˆˆ

),,(ˆˆ

(e.g.) oˆˆ~

)1(0

)1(

)0(0

)0(

)1()0()0(

VV

VV

VVv

0ˆˆ

ˆ

ˆ2ˆ

ˆ

ˆˆˆ

ˆˆ

ˆˆ

0ˆ

0ˆ

0ˆ

)0()1(

0

2)0(

2)1(1

)0(

)2(0

)0(0)0(

)0(0)0(

)0(0

)1(0

)0(0

)0(0

GW

WGW

UGW

W

F

U

W

U

F

F


),,(ˆˆ

),,(ˆˆ

(e.g.) oˆˆ~

)1(0

)1(

)0(0

)0(

)1()0()0(

VV

VV

VVv

• acceleration by GW-momentum-flux divergence

• GWs induce lower-order potential temperature variations

• p.-i. wave-correction not appearing in anelastic dynamics

0ˆˆ

ˆ

ˆ2ˆ

ˆ

ˆˆˆ

ˆˆ

ˆˆ

0ˆ

0ˆ

0ˆ

)0()1(

0

2)0(

2)1(1

)0(

)2(0

)0(0)0(

)0(0)0(

)0(0

)1(0

)0(0

)0(0

GW

WGW

UGW

W

F

U

W

U

F

F


),,(ˆˆ

),,(ˆˆ

(e.g.) oˆˆ~

)1(0

)1(

)0(0

)0(

)1()0()0(

VV

VV

VVv

• Wave-induced lower-order vertical flow vanishes

Summary

• multi-scale asymptotics:

• nonlinear dynamics of GWs near breaking• linear theory used for obtaining scale estimates

• velocity• thermodynamic wave fields

• unique small parameter:

• result:• same asymptotics for Euler equations and pseudo-incompressible equations• no such result for anelastic theory

• conclusion:• use pseudo-incompressible equations for studies of GW dynamics with altitude-dependent amplitude

11

KH

Summary





• result:• same asymptotics for Euler equations and pseudo-incompressible equations• no such result for anelastic theory


11

H

L

KH

Summary





• result:• same asymptotics for Euler equations and p.-i. equations• no such result for anelastic theory


11

H

L

KH

Summary





• result:• same asymptotics for Euler equations and p.-i. equations• no such result for anelastic theory• large-amplitude WKB (consistent with p.i. equations)


11

H

L

KH

gravity waves, scale asymptotics, and the pseudo-incompressible equations ulrich achatz...

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