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![Page 1: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/1.jpg)
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
![Page 2: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/2.jpg)
Gravity waves in the
middle atmosphere
![Page 3: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/3.jpg)
Gravity Waves in the Middle Atmosphere
Becker und Schmitz (2003)
![Page 4: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/4.jpg)
Gravity Waves in the Middle Atmosphere
Becker und Schmitz (2003)
![Page 5: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/5.jpg)
Gravity Waves in the Middle Atmosphere
Becker und Schmitz (2003)
Without GW parameterization
![Page 6: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/6.jpg)
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
![Page 7: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/7.jpg)
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
T
p
p
RTppcR
,/
0
are equivalent to
0
0
z
v
ev
v
p
c
R
Dt
DDt
D
gcDt
D
![Page 8: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/8.jpg)
linear GWs in the Euler equations
• linearization about atmosphere at rest:
0
')(
')(
'
gdz
dc
z
z
p
vv
![Page 9: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/9.jpg)
linear GWs in the Euler equations
• 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
![Page 10: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/10.jpg)
linear GWs in the Euler equations
• 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
![Page 11: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/11.jpg)
linear GWs in the Euler equations
• 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
![Page 12: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/12.jpg)
linear GWs in the Euler equations
• isothermal atmosphere: const. const., 2 Ncs
ti
tibb
mlkti
xk
xk
kxkvv
exp~'
exp~
'
),,( exp~'
00
0
0
![Page 13: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/13.jpg)
linear GWs in the Euler equations
• isothermal atmosphere:
22222
2222
222
2
4
14
1
Hmlkc
Hmlk
lkN
s
const. const., 2 Ncs
gravity waves
sound waves
![Page 14: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/14.jpg)
linear GWs in the Euler equations
• 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:
![Page 15: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/15.jpg)
GW breaking in the atmosphere
• Conservation
• Instabilität in großen Höhen• Impulsdeposition …
0
20 12 vv
Rapp et al. (priv. comm.)
![Page 16: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/16.jpg)
GW breaking in the atmosphere
• Conservation
• Instability at large altitudes• Impulsdeposition …
0
20 12 vv
Rapp et al. (priv. comm.)
![Page 17: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/17.jpg)
GW breaking in the atmosphere
• Conservation
• Instability at large altitudes• Turbulence• Momentum deposition…
0
20 12 vv
Rapp et al. (priv. comm.)
![Page 18: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/18.jpg)
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)
![Page 19: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/19.jpg)
GW breaking
![Page 20: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/20.jpg)
GW stability in the Boussinesq theory
dzdg
Nz
gb
bwNbt
bpft
0
2
0
22
2
0
,
v
v
vkvkvv
![Page 21: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/21.jpg)
Basic GW types
• Dispersionsrelation:
• Trägheitsschwerewellen (TSW):
• Hochfrequente Schwerewellen (HSW):
2/12222 cossin Nf
2
2
2
cot2
190f
Nf
cos90 N
![Page 22: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/22.jpg)
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
![Page 23: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/23.jpg)
Basic GW types
• Dispersion relation:
• Inertia-gravity waves (IGW):
• 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
![Page 24: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/24.jpg)
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
![Page 25: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/25.jpg)
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
![Page 26: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/26.jpg)
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
![Page 27: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/27.jpg)
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
![Page 28: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/28.jpg)
NM analysis, 2.5D-DNS
||SVNM,SW
exp)()()0,,,(
kx
yibb
tyb
vvv
![Page 29: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/29.jpg)
NMs: Growth Rates = 70°)
Achatz (2005, 2007b)
![Page 30: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/30.jpg)
GW amplitude after a perturbation by a NM (DNS):
Achatz (2005, 2007b)
![Page 31: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/31.jpg)
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:
![Page 32: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/32.jpg)
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:
![Page 33: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/33.jpg)
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:
![Page 34: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/34.jpg)
Energetics of a Breaking HGW with a0 < 1
Strong growth perturbation energy:buoyancy instability
DCPE
dtEd
2
/
Achatz (2007b)
![Page 35: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/35.jpg)
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:
![Page 36: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/36.jpg)
NMs and SVs of an IGW (Ri > ¼!): growth within 5min
Achatz (2005) Achatz and Schmitz (2006a,b)
![Page 37: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/37.jpg)
Dissipation rate breaking IGW (a< 1, Ri > ¼):
Measured dissipation rates 1…1000 mW/kg (Lübken 1997, Müllemann et al. 2003)
Achatz (2007a)
![Page 38: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/38.jpg)
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)
![Page 39: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/39.jpg)
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
![Page 40: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/40.jpg)
Soundproof Modelling and
Multi-Scale Asymptotics
Achatz, Klein, and Senf (2010)
![Page 41: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/41.jpg)
GW breaking in the middle atmosphere
• Conservation
• Instability at large altitudes• Momentum deposition…
0
20 12 vv
Rapp et al. (priv. comm.)
![Page 42: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/42.jpg)
GW breaking in the middle atmosphere
• Conservation
• Instability at large altitudes• Momentum deposition…• Competition between wave
growth and dissipation:– not in Boussinesq theory
0
20 12 vv
Rapp et al. (priv. comm.)
![Page 43: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/43.jpg)
GW breaking in the middle atmosphere
• 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
Rapp et al. (priv. comm.)
![Page 44: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/44.jpg)
GW breaking in the middle atmosphere
• 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
Rapp et al. (priv. comm.)
Which soundproof model should be used?
![Page 45: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/45.jpg)
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
![Page 46: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/46.jpg)
Scales: time and space
• for simplicity from now on: 2D• non-hydrostatic GWs: same spatial scale in horizontal and vertical
KLzLz
xLx
/1 ˆ
ˆ
characteristic length scale
• 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
![Page 47: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/47.jpg)
Scales: time and space
• 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
mk
kN
Hmk
kN
HK 2/1
KLzLz
xLx
/1 ˆ
ˆ
![Page 48: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/48.jpg)
Scales: time and space
• 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
dispersion relation for
0022
22
222
222
41 Tc
gN
mk
kN
Hmk
kN
p
HK 2/1
KLzLz
xLx
/1 ˆ
ˆ
/1 ˆ TtTt
![Page 49: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/49.jpg)
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
![Page 50: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/50.jpg)
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
![Page 51: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/51.jpg)
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
![Page 52: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/52.jpg)
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
![Page 53: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/53.jpg)
Scales: thermodynamic wave fields
• potential temperature:
O
g
b
TT
~'
00
m
Nb
2~
![Page 54: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/54.jpg)
Scales: thermodynamic wave fields
• potential temperature:
• Exner pressure:
m
Nb
2~
2
22
2
' ~~ Ob
kTc
m
Ni
p
O
g
b
TT
~'
00
![Page 55: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/55.jpg)
Multi-Scale Asymptotics
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:
![Page 56: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/56.jpg)
Multi-Scale Asymptotics
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
HH
222
Therefore: Multi-scale-asymptotic ansatz
0ˆ )0( also assumed:
L
H
![Page 57: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/57.jpg)
Multi-Scale Asymptotics
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
Therefore: Multi-scale-asymptotic ansatz
0ˆ )0( also assumed:
L
H
L
HH
222
![Page 58: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/58.jpg)
Multi-Scale Asymptotics
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
Therefore: Multi-scale-asymptotic ansatz
0
0ˆ
)1(
)0(
also assumed:
L
HH
222
L
H
![Page 59: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/59.jpg)
Scale asymptotics Euler: results
)0(
)0(
)0()0(
)0(
1
0ˆˆ
0ˆ
tt
Hydrostatic, large-scale, background
![Page 60: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/60.jpg)
Scale asymptotics Euler: results
)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
Hydrostatic, large-scale, background
Momentum equations and entropy equations as in Boussinesq or anelastic theory
![Page 61: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/61.jpg)
Scale asymptotics Euler: results
)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
Hydrostatic, large-scale, background
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
![Page 62: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/62.jpg)
Scale asymptotics: pseudo-incompressible equations
scale-asymptotic analysis of
0
0
v
ev
z
Dt
D
gcDt
Dp
![Page 63: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/63.jpg)
Scale asymptotics: pseudo-incompressible equations
scale-asymptotic analysis of
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
![Page 64: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/64.jpg)
Scale asymptotics: pseudo-incompressible equations
scale-asymptotic analysis of
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
![Page 65: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/65.jpg)
Scale asymptotics: anelastic equations
0
0
'
v
ev
z
Dt
D
bp
Dt
D
scale-asymptotic analysis of
![Page 66: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/66.jpg)
Scale asymptotics: anelastic equations
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
scale-asymptotic analysis of
![Page 67: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/67.jpg)
Scale asymptotics: anelastic equations
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
scale-asymptotic analysis of
![Page 68: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/68.jpg)
Scale asymptotics: anelastic equations
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
scale-asymptotic analysis of
anelastic equations only consistent if:
)0(
)0(
)0(
)0(
11
1
i.e. potential-temperature scale >> Exner-pressure scale
![Page 69: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/69.jpg)
Large-Amplitude WKB
Achatz, Klein, and Senf (2010)
![Page 70: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/70.jpg)
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
![Page 71: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/71.jpg)
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
![Page 72: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/72.jpg)
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
![Page 73: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/73.jpg)
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
Mean flow with only large-scale dependence
![Page 74: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/74.jpg)
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
![Page 75: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/75.jpg)
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
![Page 76: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/76.jpg)
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
Wave induced mean flow
![Page 77: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/77.jpg)
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
Harmonics of the wavepacket due to nonlinear interactions
![Page 78: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/78.jpg)
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
• collect equal powers in • collect equal powers in • no linearization!
/exp i
![Page 79: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/79.jpg)
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
![Page 80: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/80.jpg)
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
![Page 81: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/81.jpg)
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
![Page 82: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/82.jpg)
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
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)
![Page 83: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/83.jpg)
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
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!
![Page 84: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/84.jpg)
Large-amplitude WKB: 1st order, 2nd harmonics
1
)1()1( ,,exp),,(ˆˆ
iVV
![Page 85: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/85.jpg)
Large-amplitude WKB: 1st order, 2nd harmonics
)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
![Page 86: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/86.jpg)
Large-amplitude WKB: 1st order, 2nd harmonics
)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
![Page 87: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/87.jpg)
Large-amplitude WKB: 1st order, higher harmonics
1
)1()1( ,,exp),,(ˆˆ
iVV
![Page 88: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/88.jpg)
Large-amplitude WKB: 1st order, higher harmonics
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
![Page 89: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/89.jpg)
Large-amplitude WKB: 1st order, higher harmonics
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)
![Page 90: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/90.jpg)
Large-amplitude WKB: mean flow
),,(ˆˆ
),,(ˆˆ
(e.g.) oˆˆ~
)1(0
)1(
)0(0
)0(
)1()0()0(
VV
VV
VVv
![Page 91: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/91.jpg)
Large-amplitude WKB: mean flow
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
![Page 92: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/92.jpg)
Large-amplitude WKB: mean flow
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
![Page 93: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/93.jpg)
Large-amplitude WKB: mean flow
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
![Page 94: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/94.jpg)
Large-amplitude WKB: mean 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
• acceleration by GW-momentum-flux divergence
![Page 95: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/95.jpg)
Large-amplitude WKB: mean 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
• 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
![Page 96: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/96.jpg)
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
Large-amplitude WKB: mean flow
),,(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
![Page 97: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/97.jpg)
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
Large-amplitude WKB: mean flow
),,(ˆˆ
),,(ˆˆ
(e.g.) oˆˆ~
)1(0
)1(
)0(0
)0(
)1()0()0(
VV
VV
VVv
• Wave-induced lower-order vertical flow vanishes
![Page 98: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/98.jpg)
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
![Page 99: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/99.jpg)
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
![Page 100: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/100.jpg)
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
![Page 101: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/101.jpg)
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
H
L
KH
![Page 102: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/102.jpg)
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 p.-i. equations• no such result for anelastic theory
• conclusion:• use pseudo-incompressible equations for studies of GW dynamics with altitude-dependent amplitude
11
H
L
KH
![Page 103: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/103.jpg)
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 p.-i. equations• no such result for anelastic theory
• conclusion:• use pseudo-incompressible equations for studies of GW dynamics with altitude-dependent amplitude
11
H
L
KH
![Page 104: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/104.jpg)
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 p.-i. equations• no such result for anelastic theory• large-amplitude WKB (consistent with p.i. equations)
• conclusion:• use pseudo-incompressible equations for studies of GW dynamics with altitude-dependent amplitude
11
H
L
KH
![Page 105: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/105.jpg)
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 p.-i. equations• no such result for anelastic theory• large-amplitude WKB (consistent with p.i. equations)
• conclusion:• use pseudo-incompressible equations for studies of GW dynamics with altitude-dependent amplitude
11
H
L
KH
![Page 106: Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e725503460f94b72004/html5/thumbnails/106.jpg)