ulrich achatz goethe- universität frankfurt am main
DESCRIPTION
Ulrich Achatz Goethe- Universität Frankfurt am Main. Multiple-scale asymptotics of the interaction between gravity waves and synoptic-scale flow and its implications for soundproof modelling. Gravity waves in the atmosphere. GW radiation : Spontaneous Imbalance. Source: ECMWF. - PowerPoint PPT PresentationTRANSCRIPT
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)