Ulrich Achatz Goethe- Universität Frankfurt am Main

Ulrich AchatzGoethe-Universität Frankfurt am Main

Multiple-scale asymptotics of the interaction between

gravity waves and synoptic-scale flowand its implications for

soundproof modelling

Gravity waves in the atmosphere

GW radiation: Spontaneous Imbalance

Source: ECMWF

Gravity Waves in the Middle Atmosphere

Becker und Schmitz (2003)

Becker und Schmitz (2003)

Gravity Waves in the Middle Atmosphere

Becker und Schmitz (2003)

Without GW parameterization

Gravity Waves in the Middle Atmosphere

GWs and Clear-Air Turbulence

Koch et al (2008)

Impact GWs on Weather Forecasts

Analysis

Without GWs

Palmer et al (1986)

Analysis

without GWs

with GWs

Palmer et al (1986)

Impact GWs on Weather Forecasts

Open Questions

Open Questions and Research Strategy

How well do present parameterizations describe the GW impact? (theory needed!)

(e.g. Linzen 1981, Palmer et al 1986, McFarlane 1987, Medvedev and Klaassen 1995, Hines 1997, Lott and Miller 1997, Alexander and Dunkerton 1999, Warner and McIntyre 2001)

How well do we understand GW breaking?

How well do we understand GW propagation?

Presently no parameterization of spontaneous imbalance

How well do present parameterizations describe the GW impact? (theory needed!)

(e.g. Linzen 1981, Palmer et al 1986, McFarlane 1987, Medvedev and Klaassen 1995, Hines 1997, Lott and Miller 1997, Alexander and Dunkerton 1999, Warner and McIntyre 2001)

How well do we understand GW breaking?

How well do we understand GW propagation?

Presently no parameterization of spontaneous imbalance

Approach:

DNS (everything resolved)LES (turbulence parameterized, GWs resolved)GW parameterization

Open Questions and Research Strategy

DNS and LES of breaking IGW (Boussinesq)

1152 x 1152 direct numerical simulation 288 x 288 simulation with ALDM (INCA)

Fruman, Achatz (GU Frankfurt), Remmler and Hickel (TU München)

Perturbation IGW by leading transverse SV: snapshot

Random perturbation IGW Random perturbation NH GW

Horizontally averaged vertical spectrum of total energy density vs. wavenumber

Fruman, Achatz (GU Frankfurt), Remmler and Hickel (TU München)

DNS and LES of breaking IGW (Boussinesq)

Conservation

Instability at large altitudesMomentum deposition…

0

20 12 vv

Rapp et al. (priv. comm.)

GW breaking in the middle atmosphere

GW breaking in the middle atmosphere

Conservation

Instability at large altitudesMomentum deposition…Competition between wave growth and dissipation:

not in Boussinesq theory

0

20 12 vv

Rapp et al. (priv. comm.)

Conservation

Instability at large altitudesMomentum deposition…Competition between wave growth and dissipation:

not in Boussinesq theorySoundproof candidates:Anelastic (Ogura and Philips 1962, Lipps and Hemler 1982)Pseudo-incompressible (Durran 1989)

0

20 12 vv

Rapp et al. (priv. comm.)

GW breaking in the middle atmosphere

Conservation

Instability at large altitudesMomentum deposition…Competition between wave growth and dissipation:

not in Boussinesq theorySoundproof candidates:Anelastic (Ogura and Philips 1962, Lipps and Hemler 1982)Pseudo-incompressible (Durran 1989)

0

20 12 vv

Rapp et al. (priv. comm.)

GW breaking in the middle atmosphere

Which soundproof model should be used?

Interaction between Large-Amplitude GW and Mean Flow:

Strong Stratification

No Rotation

Achatz, Klein, and Senf (2010)

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

KKzzKxx

/ˆ/ˆ

characteristic wave number

• time scale set by GW frequency

/t̂t characteristic frequency

dispersion relation for

2

41 22

22

222

222 N

mkkN

Hmk

kN

HK 2/1

Achatz, Klein, and Senf (2010)

Scales: time and space

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

KLzLzxLx

/1 ˆˆ

characteristic length scale

• time scale set by GW frequency

/t̂t characteristic frequency

dispersion relation for

2

41 22

22

222

222 N

mkkN

Hmk

kN

HK 2/1

Scales: time and space

Achatz, Klein, and Senf (2010)

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

characteristic length scale

• time scale set by GW frequency

/1 ˆ TtTt characteristic time scale

dispersion relation for

2

41 22

22

222

222 N

mkkN

Hmk

kN

HK 2/1

KLzLzxLx

/1 ˆˆ

Scales: time and space

Achatz, Klein, and Senf (2010)

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

characteristic length scale

• time scale set by GW frequency

characteristic time scale

linear dispersion relation for

0022

22

222

222

41 Tc

gNmkkN

Hmk

kN

p

HK 2/1

KLzLzxLx

/1 ˆˆ

/1 ˆ TtTt

Achatz, Klein, and Senf (2010)

Scales: time and space

Scales: velocities

• winds determined by linear polarization relations:

bNiw

bNk

miu

~~

~~

2

2

what is ?b~

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

~

0' 0'

2

2

mNb

wNzb

dzd

z

KU

U

vv ˆ

Achatz, Klein, and Senf (2010)

• winds determined by linear polarization relations:

bNiw

bNk

miu

~~

~~

2

2

what is ?b~

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

~

0' 0'

2

2

mNb

wNzb

dzd

z

TL

KU

U

vv ˆ

Scales: velocities

Achatz, Klein, and Senf (2010)

Non-dimensional Euler equations

0ˆˆ

0ˆˆˆ1ˆ

ˆ

ˆˆˆˆˆ

2

2

tDDtD

DtD

D

v

evz

• using:

ˆ

ˆˆ

ˆˆ

0T

UtTtL

vv

xx

yields

gN

dzd

H

KH211

11

Achatz, Klein, and Senf (2010)

0ˆˆ

0ˆˆˆ1ˆ

ˆ

ˆˆˆˆˆ2

tDDtD

DtD

D

v

evz

• using:

yields

gTc

HRc

H

HL

pp 00

1

ˆ

ˆˆ

ˆˆ

0T

UtTtL

vv

xx

isothermal potential-temperature scale height

Non-dimensional Euler equations

Achatz, Klein, and Senf (2010)

Scales: thermodynamic wave fields

• potential temperature:

Ogb

TT

~'

00

mNb

2~

Achatz, Klein, and Senf (2010)

• potential temperature:

• Exner pressure:

mNb

2~

222

2

' ~~ ObkTcm

Ni

p

Ogb

TT

~'

00

Scales: thermodynamic wave fields

Achatz, Klein, and Senf (2010)

Multi-Scale AsymptoticsAdditional vertical scale needed:

• and have vertical scale

• scale of wave growth is

zti i

i

i

i ˆ ,ˆ,ˆˆ

ˆˆ

0 )(

)(

)(

xvv

LH

KHH

222

Therefore: Multi-scale-asymptotic ansatz

0ˆ )0( also assumed:

Achatz, Klein, and Senf (2010)

Additional vertical scale needed:

• and have vertical scale

• scale of wave growth is

zti i

i

i

i ˆ ,ˆ,ˆˆ

ˆˆ

0 )(

)(

)(

xvv

LHH

222

Therefore: Multi-scale-asymptotic ansatz

0ˆ )0( also assumed:

LH

Multi-Scale Asymptotics

Achatz, Klein, and Senf (2010)

Additional vertical scale needed:

• and have vertical scale

• scale of wave growth is

• here assumed:

LHH

222

LH

)1(O

Multi-Scale Asymptotics

Achatz, Klein, and Senf (2010)

Large-amplitude WKBb

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

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

Achatz, Klein, and Senf (2010)

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

Achatz, Klein, and Senf (2010)

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

Mean flow with only large-scale dependence

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

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

),(k

Large-amplitude WKB

Achatz, Klein, and Senf (2010)

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

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

Next-order mean flow

Large-amplitude WKB

Achatz, Klein, and Senf (2010)

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

Harmonics of the wavepacket due to nonlinear interactions

Large-amplitude WKB

Achatz, Klein, and Senf (2010)

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

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

/exp i

Achatz, Klein, and Senf (2010)

Large-amplitude WKB

Large-amplitude WKB: leading order

frequency intrinsic ˆˆ

0

ˆˆˆˆ1

ˆˆ

,ˆ

000ˆ0

ˆ000ˆ

0ˆ

)0(0

)2(1

)0(

)0(

)1(1

)0(1

)0(1

)0(1

Uk

N

WU

M

imikiN

imNiiki

i

k

Vk

Achatz, Klein, and Senf (2010)

frequency intrinsic ˆˆ

0

ˆˆˆˆ1

ˆˆ

,ˆ

000ˆ0

ˆ000ˆ

0ˆ

)0(0

)2(1

)0(

)0(

)1(1

)0(1

)0(1

)0(1

Uk

N

WU

M

imikiN

imNiiki

i

k

Vk

dispersion relation and structure as from Boussinesq

M

N

WU

mkkN

M

of Nullvector

ˆˆˆˆ1

ˆˆ

ˆ

0det

)2(1

)0(

)0(

)1(1

)0(1

)0(1

22

222

Large-amplitude WKB: leading order

Achatz, Klein, and Senf (2010)

Large-amplitude WKB: 1st order

)0(

)0(

)0(1

)0(1

)0(1

)3(1

)0(

)0(

)2(1

)1(1

)1(1

ˆˆ

ˆ1ˆˆˆˆˆˆ1

ˆˆ

,ˆWWUN

WU

M

k

Achatz, Klein, and Senf (2010)

)0(

)0(

)0(1

)0(1

)0(1

)3(1

)0(

)0(

)2(1

)1(1

)1(1

ˆˆ

ˆ1ˆˆˆˆˆˆ1

ˆˆ

,ˆWWUN

WU

M

k

velocitygroup ˆ

,ˆˆ

energy waveˆˆ

21

2

ˆ

2ˆ

'

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)

Achatz, Klein, and Senf (2010)

Large-amplitude WKB: 1st order

)0(

)0(

)0(1

)0(1

)0(1

)3(1

)0(

)0(

)2(1

)1(1

)1(1

ˆˆ

ˆ1ˆˆˆˆˆˆ1

ˆˆ

,ˆWWUN

WU

M

k

velocitygroup ˆ

,ˆˆ

energy waveˆˆ

21

2

ˆ

2ˆ

'

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)

Achatz, Klein, and Senf (2010)

Large-amplitude WKB: 1st order

Same result from pseudo-incompressible and from anelastic theory (Klein 2011)

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

1

)1()1( ,,exp),,(ˆˆ

iVV

Achatz, Klein, and Senf (2010)

)3(2

)0(

)0(

)2(2

)1(2

)1(2

ˆˆˆˆ1

ˆˆ

2,ˆ2

N

WU

M k

1

)1()1( ,,exp),,(ˆˆ

iVV

:2

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

Achatz, Klein, and Senf (2010)

)3(2

)0(

)0(

)2(2

)1(2

)1(2

ˆˆˆˆ1

ˆˆ

2,ˆ2

N

WU

M k

k2,ˆ2

ˆˆˆˆ1

ˆˆ

1

)3(2

)0(

)0(

)2(2

)1(2

)1(2

M

N

WU

1

)1()1( ,,exp),,(ˆˆ

iVV

:2

2nd harmonics are slaved

Achatz, Klein, and Senf (2010)v

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

Large-amplitude WKB: 1st order, higher harmonics

1

)1()1( ,,exp),,(ˆˆ

iVV

Achatz, Klein, and Senf (2010)

0

ˆˆˆˆ1

ˆˆ

,ˆ :2

)3()0(

)0(

)2(

)1(

)1(

N

WU

M k

1

)1()1( ,,exp),,(ˆˆ

iVV

:2

Achatz, Klein, and Senf (2010)

Large-amplitude WKB: 1st order, higher harmonics

0

ˆˆˆˆ1

ˆˆ

,ˆ :2

)3()0(

)0(

)2(

)1(

)1(

N

WU

M k 2 0

ˆˆˆˆ1

ˆˆ

)3()0(

)0(

)2(

)1(

)1(

N

WU

1

)1()1( ,,exp),,(ˆˆ

iVV

:2

higher harmonics vanish(nonlinearity is weak)

Large-amplitude WKB: 1st order, higher harmonics

Achatz, Klein, and Senf (2010)

Large-amplitude WKB: mean flow

),,(ˆˆ),,(ˆˆ

(e.g.) oˆˆ~

)1(0

)1(

)0(0

)0(

)1()0()0(

VV

VV

VVv

Achatz, Klein, and Senf (2010)

0ˆˆˆ

ˆ2ˆ

ˆˆˆˆ

ˆˆˆˆ0ˆ0ˆ

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

Large-amplitude WKB: mean flow

Achatz, Klein, and Senf (2010)

0ˆˆˆ

ˆ2ˆ

ˆˆˆˆ

ˆˆˆˆ0ˆ0ˆ

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

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

Achatz, Klein, and Senf (2010)

Large-amplitude WKB: mean flow

Rieper, Achatz and Klein (submitted)

Pseudo-Incompressible WKB

Buoyancy Wave 1

Buoyancy Wave 2

Buoyancy Wave 3

z

t

1D GW packet in isothermal atmosphere at rest

Large-amplitude WKB: Validation

Pseudo-Incompressible WKB

Mean horizontal wind

Horizontal gradient mean buoyancy

Horizontal gradientExner pressure

z

x

2D GW packet in isothermal atmosphere at rest

t

Rieper, Achatz and Klein (submitted)

Large-amplitude WKB: Validation

Large-amplitude WKB: Pseudo-Incompressible Effect

2D GW packet in isothermal atmosphere at rest

1

ˆ22

ˆ

0

ˆˆˆˆ

2)0(

2)1(1

)0(

)2(0

)0(0)0(

termterm

F

term

WGW

Rieper, Achatz and Klein (submitted)

2D GW packet in isothermal atmosphere at rest

1

ˆ22

ˆ

0

ˆˆˆˆ

2)0(

2)1(1

)0(

)2(0

)0(0)0(

termterm

F

term

WGW

p.i. effect 30% ~ O()

Large-amplitude WKB: Pseudo-Incompressible Effect

Rieper, Achatz and Klein (submitted)

Interaction between Large-Amplitude GW and Synoptic-Scale Flow:

Near-Isothermal Stratification

With Rotation

Achatz, Senf, and Klein (in prep.)

WKB in Near-Isothermal Stratification: QG Scaling

QG theory for synoptic-scale flow: Basic assumptions and consequences

Vertical scale: External Rossby deformation radius Internal Rossby deformation radius

thus

Geostrophic scaling 𝑼 𝒔 ¿ 𝜿𝟑 /𝟐√𝑹𝑻 𝟎𝟎

𝑳𝒔 ¿ 𝜿𝟏/𝟐√𝑹𝑻 𝟎𝟎 / 𝒇 𝟎

𝑯 𝒔 ¿ 𝜿𝟗/𝟐√𝑹𝑻 𝟎𝟎 / 𝒇 𝟎

𝑾 𝒔 ¿ 𝜿𝟏𝟏/𝟐√𝑹𝑻 𝟎𝟎 / 𝒇 𝟎

𝒈 ¿ 𝜿−𝟗/𝟐 𝒇 𝟎√𝑹𝑻 𝟎𝟎

𝒑 ′

𝒑 ¿ 𝑶 (𝜿𝟐 ) ¿𝝆 ′

𝝆 ¿ 𝑶 (𝜿𝟐 )

𝜽 ′

𝜽 ¿ 𝑶 (𝜿𝟐 ) ¿ 𝝅′

𝝅 ¿ 𝑶 (𝜿𝟑 )

Achatz, Senf, and Klein (in prep.)

WKB in Near-Isothermal Stratification: IGW Scaling

Scaling and non-dimensionalization for IGW close to breaking:

• IGW shorter and faster than synoptic-scale flow

• large amplitude close to static instability

• linear polarization relations

Then 𝑳 ¿ 𝜿𝑳𝒔 ¿ 𝜿𝟑/𝟐√𝑹𝑻 𝟎𝟎 / 𝒇 𝟎

𝑯 ¿ 𝜿𝑯 𝒔 ¿ 𝜿𝟏𝟏/𝟐√𝑹𝑻𝟎𝟎 / 𝒇 𝟎𝑻 ¿ 𝜿𝑻 𝒔 ¿ 𝟏/ 𝒇 𝟎

𝑼 ¿ 𝑼 𝒔 ¿ 𝜿𝟑 /𝟐√𝑹𝑻 𝟎𝟎

𝑾 ¿ 𝑾 𝒔 ¿ 𝜿𝟏𝟏/𝟐√𝑹𝑻 𝟎𝟎

𝜽 ′

𝜽 ¿ 𝑶 (𝜿𝟐 )

𝝅 ′

𝝅 ¿ 𝑶 (𝜿𝟒 )

𝜿𝟒(𝑫�⃗�𝑫𝒕 + 𝒇 �⃗�𝒛×�⃗�) ¿ −𝜽𝛁𝒉𝝅

𝜿𝟏𝟐 𝑫𝒘𝑫𝒕 ¿ −𝜽 𝝏 𝝅

𝝏 𝒛 −𝜿𝟐

𝑫𝜽𝑫𝒕 ¿ 𝟎

𝑫𝝆𝑫𝒕 +𝝆𝛁 ⋅ �⃗� ¿ 𝟎

𝝆 ¿ 𝝅𝟏−𝜿𝜿

𝜽

Achatz, Senf, and Klein (in prep.)

Large-Amplitude WKB in Near-Isothermal Stratification

Multiple-Scale Ansatz, WKB:

near-isothermal stratification reference atmosphere

Slow and long scales for synoptic scales and wave amplitude

thus WKB multi-scale ansatz

field = reference + synoptic-scale part + wave

Achatz, Senf, and Klein (in prep.)

WKB in Near-Isothermal Stratification: Balance Equations

• Hydrostatic equilibrium reference atmosphere

• Hydrostatic and geostrophic equilibrium synoptic-scale flow

WKB in Near-Isothermal Stratification: Wave Dynamics

• Dispersion relation and polarization relations from

…

• Wave-action conservation as before

WKB in Near-Isothermal Stratification: Wave Dynamics

• Dispersion relation and polarization relations from

…

• Wave-action conservation as before

No Linearization!

WKB in Near-Isothermal Stratification: Mean Flow (1)

• Mean flow:

• Same result in both pseudo-incompressible and anelastic theory!• There is a higher-order correction to hydrostatic equilibrium but this has no non-anelastic correction, and it does not matter anyway!

WKB in Near-Isothermal Stratification: Mean Flow (2)

Final Result:PV conservation in QG approximation

Balanced flow forced by pseudo-momentum-flux convergencies (EP flux)

Summary

• multi-scale asymptotics of dynamics of GWs in interaction with

synoptic-scale flow• large-amplitude WKB theory through a distinguished limit• Pseudo-incompressible theory yields same results as

general equations• Differences anelastic theory to pseudo-incompressible theory

become negligible to the same precision as QG theory holds,

i.e. as

• consistent with Smolarkiewicz and Dörnbrack (2008),

Klein (2009), Smolarkiewicz and Szmelter (2011)