Multiple-scale asymptotics
of the interaction between
gravity waves and synoptic-scale flow
and its implications for
soundproof modelling
Ulrich Achatz
Goethe-Universität Frankfurt am Main
Gravity waves
in the atmosphere
GW radiation: Spontaneous Imbalance
Source: ECMWF
Gravity Waves in the Middle Atmosphere
Becker und Schmitz (2003)
Gravity Waves in the Middle Atmosphere
Becker und Schmitz (2003)
Gravity Waves in the Middle Atmosphere
Without GW parameterization
Becker und Schmitz (2003)
GWs and Clear-Air Turbulence
Koch et al (2008)
Impact GWs on Weather Forecasts
Without GWs
Analysis
Palmer et al (1986)
Impact GWs on Weather Forecasts
without GWs
with GWs
Analysis
Palmer et al (1986)
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
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
• Approach:
DNS (everything resolved)
LES (turbulence parameterized, GWs resolved)
GW parameterization
DNS and LES of breaking IGW (Boussinesq)
Perturbation IGW by leading transverse SV: snapshot
1152 x 1152 direct numerical simulation
288 x 288 simulation with ALDM (INCA)
Fruman, Achatz (GU Frankfurt), Remmler and Hickel (TU München)
DNS and LES of breaking IGW (Boussinesq)
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)
GW breaking in the middle atmosphere
• Conservation

v
2
0
2

  v 
1
0
• Instability at large altitudes
• Momentum deposition…
Rapp et al. (priv. comm.)
GW breaking in the middle atmosphere
• Conservation

v
2
0
2

  v 
1
0
• Instability at large altitudes
• Momentum deposition…
• Competition between wave
growth and dissipation:
– not in Boussinesq theory
Rapp et al. (priv. comm.)
GW breaking in the middle atmosphere
• Conservation

v
2
0
2

  v 
1
0
• 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)
Rapp et al. (priv. comm.)
GW breaking in the middle atmosphere
• Conservation

v
2
0
Which soundproof
Rapp et al. (priv. comm.)
2

  v 
1
0
• Instability at large altitudes
• Momentum deposition…
• Competition
between
wave
model
should be
used?
growth and dissipation:
– not in Boussinesq theory
– Soundproof candidates:
– Anelastic
(Ogura and Philips 1962,
Lipps and Hemler 1982)
– Pseudo-incompressible
(Durran 1989)
Interaction between LargeAmplitude GW and Mean Flow:
Strong Stratification
No Rotation
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
x  xˆ / K
z  zˆ / K
K  characteristic wave number
• time scale set by GW frequency
t  tˆ / 
dispersion relation for
2 
K  1 / 2H
N 2k 2
k 2  m2 
  characteristic frequency
1
4H 2
Achatz, Klein, and Senf (2010)
N 2k 2
 2
k  m2


N
2
Scales: time and space
• for simplicity from now on: 2D
• non-hydrostatic GWs: same spatial scale in horizontal and vertical
x  Lxˆ
z  Lzˆ
L  1 / K  characteristic length scale
• time scale set by GW frequency
t  tˆ / 
dispersion relation for
2 
K  1 / 2H
N 2k 2
k 2  m2 
  characteristic frequency
1
4H 2
Achatz, Klein, and Senf (2010)
N 2k 2
 2
k  m2


N
2
Scales: time and space
• for simplicity from now on: 2D
• non-hydrostatic GWs: same spatial scale in horizontal and vertical
x  Lxˆ
z  Lzˆ
L  1 / K  characteristic length scale
• time scale set by GW frequency
t  Ttˆ
dispersion relation for
2 
characteristic time scale
K  1 / 2H
N 2k 2
k 2  m2 
T  1/  
1
4H 2
Achatz, Klein, and Senf (2010)
N 2k 2
 2
k  m2


N
2
Scales: time and space
• for simplicity from now on: 2D
• non-hydrostatic GWs: same spatial scale in horizontal and vertical
x  Lxˆ
z  Lzˆ
L  1 / K  characteristic length scale
• time scale set by GW frequency
t  Ttˆ
linear dispersion relation for
2 
N 2k 2
k 2  m2 
1
4H 2
Achatz, Klein, and Senf (2010)
T  1/  
characteristic time scale
K  1 / 2H
N 2k 2
 2
k  m2

N 
g
c pT00
Scales: velocities
• winds determined by linear polarization relations:
u~
m  ~
b
2
k N
 ~
i 2b
N
 i
~ 
w
what is
~
b?
• most interesting dynamics when GWs are close to breaking, i.e. locally
 ' d
b'

0 
 N 2w  0
z dz
z

~ N2
b 
m
Achatz, Klein, and Senf (2010)

v  Uvˆ

U
K
Scales: velocities
• winds determined by linear polarization relations:
u~
m  ~
b
2
k N
 ~
i 2b
N
 i
~ 
w
what is
~
b?
• most interesting dynamics when GWs are close to breaking, i.e. locally
 ' d
b'

0 
 N 2w  0
z dz
z

~ N2
b 
m
Achatz, Klein, and Senf (2010)

v  Uvˆ
 L
U 
K T
Non-dimensional Euler equations
• using:
x  Lxˆ
t  Ttˆ
v  Uvˆ
  ˆ
  T ˆ
0
yields
 2 Dvˆ
ˆ ˆ  e
 ˆ
z
2 Dtˆ
Dˆ


ˆˆ  vˆ  0
Dtˆ 1  
Dˆ
0
Dtˆ
Achatz, Klein, and Senf (2010)

1
 1
KH
1
1 d N 2


H   dz
g
Non-dimensional Euler equations
• using:
x  Lxˆ
t  Ttˆ
v  Uvˆ
  ˆ
  T ˆ
0
yields
Dvˆ
ˆ ˆ  e

 ˆ
z
Dtˆ
Dˆ


ˆˆ  vˆ  0
Dtˆ 1  
Dˆ
0
ˆ
Dt
2
Achatz, Klein, and Senf (2010)
L

 1
H
H 
cp
R
H
c pT00
g
isothermal potentialtemperature scale height
Scales: thermodynamic wave fields
~ N2
b 
m

• potential temperature:
'
T0
~

 b
T0 g
 O 
Achatz, Klein, and Senf (2010)
Scales: thermodynamic wave fields
~ N2
b 
m

• potential temperature:
'
T0
~

 b
T0 g
 O 
• Exner pressure:
~  i
2
 
m ~
2
b


'

O

N 2 cp Tk 2
Achatz, Klein, and Senf (2010)
Multi-Scale Asymptotics
Additional vertical scale needed:
•
 and  have vertical scale H   L

• scale of wave growth is
H
H

  
2
2
2K
Therefore: Multi-scale-asymptotic ansatz
(i )
 vˆ    v 
 (i ) 
 
i
 ˆ       xˆ , tˆ,  
 ˆ  i 0   (i ) 
 


also assumed:
ˆ  ( 0)  0

Achatz, Klein, and Senf (2010)
  zˆ
Multi-Scale Asymptotics
Additional vertical scale needed:
 and  have vertical scale H   L

H 
 L
• scale of wave growth is
 H 
2 2
2
•
Therefore: Multi-scale-asymptotic ansatz
(i )
 vˆ    v 
 (i ) 
 
i
 ˆ       xˆ , tˆ,  
 ˆ  i 0   (i ) 
 


also assumed:
ˆ  ( 0)  0

Achatz, Klein, and Senf (2010)
  zˆ
Multi-Scale Asymptotics
Additional vertical scale needed:
 and  have vertical scale H   L

H 
 L
• scale of wave growth is
 H 
2 2
2
•
• here assumed:
  O(1)
Achatz, Klein, and Senf (2010)
Large-amplitude WKBb
vˆ  ~
v (0)
~
ˆ  ˆ ( 0)   (1)
ˆ  ˆ ( 0)   2~ ( 2)
~
ˆ ( 0 )  V
ˆ (1)  o 
v (0)  V
(e.g.)
 ˆ (0)
i

(0)
(0)
ˆ
ˆ
ˆ
ˆ
V  V0 (t , x, zˆ )  V1 ( ,  ,  ) exp    ,  ,   



  
 ˆ (1)
i

(1)
(1)
ˆ
ˆ
V  V0 ( ,  ,  )   V ( ,  ,  ) exp    ,  ,   


 1 

Achatz, Klein, and Senf (2010)
Large-amplitude WKB
vˆ  ~
v (0)
~
ˆ  ˆ ( 0)   (1)
ˆ  ˆ ( 0)   2~ ( 2)
~
ˆ ( 0 )  V
ˆ (1)  o 
v (0)  V
(e.g.)
 ˆ (0)
i

(0)
(0)
ˆ
ˆ
ˆ
ˆ
V  V0 (t , x, zˆ )  V1 ( ,  ,  ) exp    ,  ,   



  
 ˆ (1)
i

(1)
(1)
ˆ
ˆ
V  V0 ( ,  ,  )   V ( ,  ,  ) exp    ,  ,   


 1 

Achatz, Klein, and Senf (2010)
Large-amplitude WKB
vˆ  ~
v (0)
~
ˆ  ˆ ( 0)   (1)
ˆ  ˆ ( 0)   2~ ( 2)
~
ˆ ( 0 )  V
ˆ (1)  o 
v (0)  V
(e.g.)
 ˆ (0)
i

(0)
(0)
ˆ
ˆ
ˆ
ˆ
V  V0 (t , x, zˆ )  V1 ( ,  ,  ) exp    ,  ,   



  
 ˆ (1)
i

(1)
(1)
ˆ
ˆ
V  V0 ( ,  ,  )   V ( ,  ,  ) exp    ,  ,   


 1 

Achatz, Klein, and Senf (2010)
Large-amplitude WKB
vˆ  ~
v (0)
~
ˆ  ˆ ( 0)   (1)
ˆ  ˆ ( 0)   2~ ( 2)
~
ˆ ( 0 )  V
ˆ (1)  o 
v (0)  V
(e.g.)
 ˆ (0)
i

(0)
(0)
ˆ
ˆ
ˆ
ˆ
V  V0 (t , x, zˆ )  V1 ( ,  ,  ) exp    ,  ,   



  
 ˆ (1)
i

(1)
(1)
ˆ
ˆ
V  V0 ( ,  ,  )   V ( ,  ,  ) exp    ,  ,   


 1 

Mean flow with only large-scale dependence
Achatz, Klein, and Senf (2010)
Large-amplitude WKB
vˆ  ~
v (0)
~
ˆ  ˆ ( 0)   (1)
ˆ  ˆ ( 0)   2~ ( 2)
~
ˆ ( 0 )  V
ˆ (1)  o 
v (0)  V
(e.g.)
 ˆ (0)
i

(0)
(0)
ˆ
ˆ
ˆ
ˆ
V  V0 (t , x, zˆ )  V1 ( ,  ,  ) exp    ,  ,   



  
 ˆ (1)
i

(1)
(1)
ˆ
ˆ
V  V0 ( ,  ,  )   V ( ,  ,  ) exp    ,  ,   


 1 

Wavepacket with
• large-scale amplitude
• wavenumber and frequency with large-scale dependence
Achatz, Klein, and Senf (2010)
k   (  , )



Large-amplitude WKB
vˆ  ~
v (0)
~
ˆ  ˆ ( 0)   (1)
ˆ  ˆ ( 0)   2~ ( 2)
~
ˆ ( 0 )  V
ˆ (1)  o 
v (0)  V
(e.g.)
 ˆ (0)
i

(0)
(0)
ˆ
ˆ
ˆ
ˆ
V  V0 (t , x, zˆ )  V1 ( ,  ,  ) exp    ,  ,   



  
 ˆ (1)
i

(1)
(1)
ˆ
ˆ
V  V0 ( ,  ,  )   V ( ,  ,  ) exp    ,  ,   


 1 

Achatz, Klein, and Senf (2010)
Large-amplitude WKB
vˆ  ~
v (0)
~
ˆ  ˆ ( 0)   (1)
ˆ  ˆ ( 0)   2~ ( 2)
~
ˆ ( 0 )  V
ˆ (1)  o 
v (0)  V
(e.g.)
 ˆ (0)
i

(0)
(0)
ˆ
ˆ
ˆ
ˆ
V  V0 (t , x, zˆ )  V1 ( ,  ,  ) exp    ,  ,   



  
 ˆ (1)
i

(1)
(1)
ˆ
ˆ
V  V0 ( ,  ,  )   V ( ,  ,  ) exp    ,  ,   


 1 

Next-order mean flow
Achatz, Klein, and Senf (2010)
Large-amplitude WKB
vˆ  ~
v (0)
~
ˆ  ˆ ( 0)   (1)
ˆ  ˆ ( 0)   2~ ( 2)
~
ˆ ( 0 )  V
ˆ (1)  o 
v (0)  V
(e.g.)
 ˆ (0)
i

(0)
(0)
ˆ
ˆ
ˆ
ˆ
V  V0 (t , x, zˆ )  V1 ( ,  ,  ) exp    ,  ,   



  
 ˆ (1)
i

(1)
(1)
ˆ
ˆ
V  V0 ( ,  ,  )   V ( ,  ,  ) exp    ,  ,   


 1 

Harmonics of the wavepacket due to nonlinear interactions
Achatz, Klein, and Senf (2010)
Large-amplitude WKB
vˆ  ~
v (0)
~
ˆ  ˆ ( 0)   (1)
ˆ  ˆ ( 0)   2~ ( 2)
~
ˆ ( 0 )  V
ˆ (1)  o 
v (0)  V
(e.g.)
 ˆ (0)
i

(0)
(0)
ˆ
ˆ
ˆ
ˆ
V  V0 (t , x, zˆ )  V1 ( ,  ,  ) exp    ,  ,   



  
 ˆ (1)
i

(1)
(1)
ˆ
ˆ
V  V0 ( ,  ,  )   V ( ,  ,  ) exp    ,  ,   


 1 

• collect equal powers in 
• collect equal powers in exp i /  
• no linearization!
Achatz, Klein, and Senf (2010)
Large-amplitude WKB: leading order
ˆ (0)  0
ik  V
1
ˆ (0) 

U
0
0
ik  1
  iˆ

(
0
)

 Wˆ

1
 iˆ  N im 
 0
 1 
ˆ (1)   0
 0

1 
N
 iˆ 0 

 N ˆ ( 0 ) 
 ik

im
0
0
ˆ ( 0 ) 
ˆ ( 2) 
1 

M ˆ , k 
ˆ    kUˆ 0( 0)
intrinsic frequency
Achatz, Klein, and Senf (2010)
Large-amplitude WKB: leading order
ˆ (0)  0
ik  V
1
ˆ (0) 

U
0
0
ik  1
  iˆ

(
0
)

 Wˆ

1
 iˆ  N im 
 0
 1 
ˆ (1)   0
 0

1 
N
 iˆ 0 

 N ˆ ( 0 ) 
 ik

im
0
0
ˆ ( 0 ) 
ˆ ( 2) 
1 

M ˆ , k 
ˆ    kUˆ 0( 0)
intrinsic frequency
dispersion relation and structure as
from Boussinesq
Achatz, Klein, and Senf (2010)
det M   0 
k2
ˆ  N 2
k  m2
 Uˆ 1( 0 ) 


(
0
)
 Wˆ1

 1 
ˆ (1)   Nullvector of M
1 

 N ˆ ( 0) 
ˆ ( 0 ) 
ˆ 1( 2) 


2
2
Large-amplitude WKB: 1st order
 Uˆ 1(1)  


 
(
1
)

 Wˆ1
 
ˆ ( 2)   
M ˆ , k  1 

1 


(
0
)
(
0
)
ˆ
ˆ
(0)

U

W
1 
ˆ
N
1
1
   


ˆ ( 0 ) 
ˆ (3)   


1 

Achatz, Klein, and Senf (2010)
Wˆ1( 0 )
ˆ ( 0)





ˆ ( 0 ) 
 
Large-amplitude WKB: 1st order
 Uˆ 1(1)  


 
(
1
)

 Wˆ1
 
ˆ ( 2)   
M ˆ , k  1 

1 


(
0
)
(
0
)
ˆ
ˆ
(0)

U

W
1 
ˆ
N
1
1
   


ˆ ( 0 ) 
ˆ (3)   


1 

Solvability condition
leads to wave-action
conservation
(Bretherton 1966,
Grimshaw 1975,
Müller 1976)
Wˆ1( 0 )
ˆ ( 0)





ˆ ( 0 ) 
 
  E' 
 E' 



 
0
  ,   c g
  ˆ 
 ˆ 
2
2
(0)

ˆ
(0)
(0) 
V
ˆ
ˆ  1
1 1 
E' 


 wave energy
2 ˆ (0)
2  2
2N 



 ˆ ( 0 ) ˆ ˆ 
c g  U 0 
,
group velocity

k m 

Achatz, Klein, and Senf (2010)
Large-amplitude WKB: 1st order
 Uˆ 1(1)  


 
(
1
)

 Wˆ1
 
ˆ ( 2)   
M ˆ , k  1 

1 


(
0
)
(
0
)
ˆ
ˆ
(0)

U

W
1 
ˆ
N
1
1
   


ˆ ( 0 ) 
ˆ (3)   


1 

Wˆ1( 0 )
ˆ ( 0)





ˆ ( 0 ) 
 
Same result
from pseudo-incompressible
and from
Solvability
condition
  E' 
 E ' anelastic theory (Klein 2011)
      ,    c g   0
leads to wave-action
  ˆ 
 ˆ 
conservation
(Bretherton 1966,
Grimshaw 1975,
Müller 1976)
2
(0)

ˆ
ˆ ( 0)  V1
1
E' 


2  2
2N 2

 ˆ ( 0 ) ˆ ˆ 
c g  U 0 
,


k

m


Achatz, Klein, and Senf (2010)
ˆ (0)

1
ˆ ( 0)
2


 wave energy


group velocity
Large-amplitude WKB: 1st order, 2nd harmonics
ˆ
V
 ˆ (1)
i

    V ( ,  ,  ) exp   ,  ,   


 1 

(1)
Achatz, Klein, and Senf (2010)
Large-amplitude WKB: 1st order, 2nd harmonics
ˆ
V
 ˆ (1)
i

    V ( ,  ,  ) exp   ,  ,   


 1 

(1)
  2:
 Uˆ 2(1) 

  
(
1
)
 Wˆ 2  
 
ˆ ( 2)     
M 2ˆ ,2k  1 
2 


(
0
)

 N ˆ   
ˆ ( 0 ) 
ˆ ( 3)   
2 

Achatz, Klein, and Senf (2010)
Large-amplitude WKB: 1st order, 2nd harmonics
ˆ
V
 ˆ (1)
i

    V ( ,  ,  ) exp   ,  ,   


 1 

(1)
  2:
 Uˆ 2(1) 

  
(
1
)
 Wˆ 2  
 
ˆ ( 2)     
M 2ˆ ,2k  1 
2 


(
0
)

 N ˆ   
ˆ ( 0 ) 
ˆ ( 3)   
2 

 Uˆ 2(1) 
 


(
1
)
 
 Wˆ 2 


1
 1 
(
2
)

ˆ
 M 2ˆ ,2k  
2



(
0
)


 N ˆ 
 
ˆ ( 0 ) 
 
ˆ ( 3) 
2 

2nd harmonics are slaved
Achatz, Klein, and Senf (2010)v
Large-amplitude WKB: 1st order, higher harmonics
ˆ
V
 ˆ (1)
i



    V
(

,

,

)
exp


,

,






 1 

(1)
Achatz, Klein, and Senf (2010)
Large-amplitude WKB: 1st order, higher harmonics
ˆ
V
 ˆ (1)
i



    V
(

,

,

)
exp


,

,






 1 

(1)
  2:
  2:
 Uˆ (1) 


(
1
)
 Wˆ 
ˆ ( 2)   0 
M ˆ , k  1 
 

 N ˆ ( 0 ) 
ˆ ( 0 ) 
ˆ ( 3) 
 

Achatz, Klein, and Senf (2010)
Large-amplitude WKB: 1st order, higher harmonics
ˆ
V
 ˆ (1)
i



    V
(

,

,

)
exp


,

,






 1 

(1)
  2:
  2:
 Uˆ (1) 


(
1
)
 Wˆ 
ˆ ( 2)   0 
M ˆ , k  1 
 

 N ˆ ( 0 ) 
ˆ ( 0 ) 
ˆ ( 3) 
 

 Uˆ (1) 


(
1
)
 Wˆ 
 1 
ˆ ( 2)   0
 

 N ˆ ( 0 ) 
ˆ ( 0 ) 
ˆ ( 3) 
 

  2
higher harmonics vanish
(nonlinearity is weak)
Achatz, Klein, and Senf (2010)
Large-amplitude WKB: mean flow
~
ˆ ( 0)  V
ˆ (1)  o 
v (0)  V
ˆ (0)  V
ˆ ( 0) ( ,  ,  )  
V
0
ˆ (1)  V
ˆ (1) ( ,  ,  )  
V
0
Achatz, Klein, and Senf (2010)
(e.g.)
Large-amplitude WKB: mean flow
~
ˆ ( 0)  V
ˆ (1)  o 
v (0)  V
ˆ (0)  V
ˆ ( 0) ( ,  ,  )  
V
0
Uˆ 0( 0 )
0

Wˆ ( 0 )  0
ˆ (1)  V
ˆ (1) ( ,  ,  )  
V
0
• acceleration by GWmomentum-flux
divergence
0
ˆ (1)  0

0
ˆ (00 )
Uˆ 0( 0 ) ˆ ( 0 ) 
U

   Fˆ GW


(1) 2
1
ˆ
ˆ


( 0 ) 
W
ˆ
ˆ


   FGW 
2
ˆ


2ˆ ( 0 )
ˆ (0)


(
1
)

Wˆ0
   Fˆ GW
0

(0)
0
(e.g.)
( 2)
0
(0)
Achatz, Klein, and Senf (2010)
Large-amplitude WKB: mean flow
Uˆ 0( 0 )
0

Wˆ ( 0 )  0
~
ˆ ( 0)  V
ˆ (1)  o 
v (0)  V
ˆ (0)  V
ˆ ( 0) ( ,  ,  )  
V
0
ˆ (1)  V
ˆ (1) ( ,  ,  )  
V
0
• acceleration by GWmomentum-flux
divergence
0
ˆ (1)  0

0
ˆ (00 )
Uˆ 0( 0 ) ˆ ( 0 ) 
U

   Fˆ GW


2
(1)
(0)
( 2)
ˆ
ˆ

 0

1
W
ˆ ( 0 )
 (00 )    FˆGW

2

ˆ
2ˆ ( 0 )
(0)
ˆ



Wˆ0(1)
   Fˆ GW
0

Achatz, Klein, and Senf (2010)
(e.g.)
• p.-i. wave-correction
of hydrostatic
balance not
appearing in
anelastic dynamics
Large-amplitude WKB: Validation
1D GW packet in
isothermal atmosphere at rest
Pseudo-Incompressible
WKB
Buoyancy Wave 1
z
Buoyancy Wave 2
Buoyancy Wave 3
t
Rieper, Achatz and Klein (submitted)
Large-amplitude WKB: Validation
2D GW packet in
isothermal atmosphere at rest
Pseudo-Incompressible
WKB
Mean horizontal wind
t
z
Horizontal gradient
mean buoyancy
Horizontal gradient
Exner pressure
x
Rieper, Achatz and Klein (submitted)
Large-amplitude WKB: Pseudo-Incompressible Effect
2D GW packet in
isothermal atmosphere at rest
(1) 2
1
ˆ
ˆ


( 0 ) 
W
ˆ
ˆ


    FGW 


 2ˆ ( 0) 2
ˆ





term2 
term0
term1
(0)
0
( 2)
0
(0)
Rieper, Achatz and Klein (submitted)
Large-amplitude WKB: Pseudo-Incompressible Effect
2D GW packet in
isothermal atmosphere at rest
(1) 2
1
ˆ
ˆ


( 0 ) 
W
ˆ
ˆ


    FGW 


 2ˆ ( 0) 2
ˆ





term2 
term0
term1
(0)
0
( 2)
0
(0)
Rieper, Achatz and Klein (submitted)
p.i. effect 30% ~ O()
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:
𝐻𝑠 = 𝐻𝜌 ≈
𝑅𝑇00
𝑔
𝐿2
𝑂 1 =
Internal Rossby deformation radius
thus
𝜿=
𝑹
𝒄𝒑
= 𝑶 𝑹𝒐 ≪ 𝟏
Geostrophic scaling → … →
𝑑
𝐿2𝑠
𝐿2𝑑𝑖
𝑠
=
0
𝐿2𝑠
𝑁2 𝐻𝑠2 /𝑓02
𝑼𝒔
=
𝜿𝟑/𝟐 𝑹𝑻𝟎𝟎
𝑳𝒔
=
𝜿𝟏/𝟐 𝑹𝑻𝟎𝟎 /𝒇𝟎
𝑯𝒔
=
𝜿𝟗/𝟐 𝑹𝑻𝟎𝟎 /𝒇𝟎
𝑾𝒔
=
𝜿𝟏𝟏/𝟐 𝑹𝑻𝟎𝟎 /𝒇𝟎
𝒈
=
𝜿−𝟗/𝟐 𝒇𝟎 𝑹𝑻𝟎𝟎
𝒑′
𝒑
𝜽′
𝜽
Achatz, Senf, and Klein (in prep.)
𝐿2
𝑂 𝑅𝑜 = 𝐿2𝑠 = 𝑔𝐻 𝑠/𝑓2
External Rossby deformation radius
𝟐
=
𝑶 𝜿
=
𝑶 𝜿𝟐
𝝆′
𝝆
𝝅′
𝝅
≈
𝑅
𝐿2𝑠
𝑐𝑝 𝑔𝐻𝑠 /𝑓02
=
𝑶 𝜿𝟐
=
𝑶 𝜿𝟑
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 𝑹𝑻𝟎𝟎 /𝒇𝟎
𝜿
𝟏𝟏/𝟐
𝑯
𝑻
=
=
𝜿𝑯𝒔
𝜿𝑻𝒔
=
=
𝑹𝑻𝟎𝟎 /𝒇𝟎
𝟏/𝒇𝟎
𝑼
=
𝑼𝒔
=
𝜿𝟑/𝟐 𝑹𝑻𝟎𝟎
𝑾
=
𝑾𝒔
=
𝜿𝟏𝟏/𝟐 𝑹𝑻𝟎𝟎
′
𝜽
𝜽
𝝅′
𝝅
=
⇒
𝜿𝟒
𝑫𝒖
+ 𝒇𝒆𝒛 × 𝒖
𝑫𝒕
𝑫𝒘
𝜿𝟏𝟐
𝑫𝒕
𝑫𝜽
𝑫𝒕
𝑫𝝆
+ 𝝆𝛁 ⋅ 𝒗
𝑫𝒕
−𝜽𝛁𝒉 𝝅
=
−𝜽
=
𝟎
=
𝟎
𝑶 𝜿𝟐
𝝆
=
=
𝑶 𝜿𝟒
Achatz, Senf, and Klein (in prep.)
=
𝝏𝝅
− 𝜿𝟐
𝝏𝒛
𝟏−𝜿
𝝅 𝜿
𝜽
Large-Amplitude WKB in Near-Isothermal Stratification
Multiple-Scale Ansatz, WKB:
near-isothermal stratification ⇒ reference atmosphere
𝜽, 𝝅 = 𝜽, 𝝅 (𝜻 = 𝜿𝟐 𝒛)
Slow and long scales for synoptic scales and wave amplitude
𝑋, 𝑻 = 𝜅𝑥, 𝜅𝑡
thus WKB multi-scale ansatz
field = reference + synoptic-scale part + wave
∞
∞
𝒋
𝜿𝒋 𝑽𝟎 𝑿, 𝑻 + 𝕽
𝒗=
𝒋=𝟎
∞
𝟎
𝜽 = 𝚯𝟎
𝒋
𝜿𝒋 𝚯𝟎 𝑿, 𝑻 + 𝕽
𝜻 +
𝟎
𝑿, 𝑻 𝒆𝒊 𝝓(𝑿,𝑻)/𝝐
𝒋
𝑿, 𝑻 𝒆𝒊 𝝓(𝑿,𝑻)/𝝐
𝒋
𝑿, 𝑻 𝒆𝒊 𝝓(𝑿,𝑻)/𝝐
𝜿𝒋 𝚯𝒋
𝒋=𝟐
∞
𝒋
𝜿𝒋 𝚷𝟎 𝑿, 𝑻 + 𝕽
𝜻 +
𝒋
𝒋=𝟎
∞
𝒋=𝟐
∞
𝝅 = 𝚷𝟎
𝜿𝒋 𝑽𝒋
𝒋=𝟑
Achatz, Senf, and Klein (in prep.)
𝜿𝒋 𝚷𝒋
𝒋=𝟒
WKB in Near-Isothermal Stratification: Balance Equations
• Hydrostatic equilibrium reference atmosphere
𝟎
𝒅𝚷𝟎
𝟏
= − (𝟎)
𝒅𝜻
𝚯
𝟎
• Hydrostatic and geostrophic equilibrium synoptic-scale flow
𝟑
𝟎
𝟎 = −𝚯𝟎
(𝟎)
𝟎
𝟐
𝝏𝚷𝟎
𝚯𝟎
+ 𝟎
𝝏𝒛
𝚯𝟎
𝟑
𝒇𝒆𝐳 × 𝑼𝟎 = −𝚯𝟎 𝛁𝒙,𝒉 𝚷𝟎 ,
𝟎
𝑾𝟎 = 𝟎
WKB in Near-Isothermal Stratification: Wave Dynamics
• Dispersion relation and polarization relations from
(𝟎)
−𝒊𝝎 −𝒇
𝟎
𝒇
−𝒊𝝎
𝟎
𝟎
𝟎
−𝒊𝝎𝜿𝟖
𝟎
𝟎
𝑵𝟎
𝒊𝒌
𝒊𝒍
𝒊𝒎
𝟎
𝟎
−𝑵𝟎
−𝒊𝝎
𝟎
𝒊𝒌
𝒊𝒍
𝒊𝒎
𝟎
𝟎
𝑼𝟏
𝟎
𝑽𝟏
(𝟎)
𝑾𝟏
=𝟎
𝟐
𝑩𝟏 /𝑵𝟎
(𝟎)
(𝟎)
𝚯𝟎 𝚷𝟏
⇒
𝟐 𝟐
𝟐 𝒎𝟐
𝑵
𝒌
+
𝒇
𝟎
𝒉
𝝎𝟐 = 𝟖 𝟐
𝜿 𝒌𝒉 + 𝒎𝟐
…
• Wave-action conservation as before
𝜕𝐴 𝜕𝑡 + 𝛻 ⋅ 𝑐𝑔 𝐴 = 0
WKB in Near-Isothermal Stratification: Wave Dynamics
• Dispersion relation and polarization relations from
(𝟎)
−𝒊𝝎 −𝒇
𝟎
𝒇
−𝒊𝝎
𝟎
𝟎
𝟎
−𝒊𝝎𝜿𝟖
𝟎
𝟎
𝑵𝟎
𝒊𝒌
𝒊𝒍
𝒊𝒎
𝟎
𝟎
−𝑵𝟎
−𝒊𝝎
𝟎
𝒊𝒌
𝒊𝒍
𝒊𝒎
𝟎
𝟎
𝑼𝟏
𝟎
𝑽𝟏
(𝟎)
𝑾𝟏
=𝟎
𝟐
𝑩𝟏 /𝑵𝟎
No Linearization!
(𝟎)
(𝟎)
𝚯𝟎 𝚷𝟏
⇒
𝟐 𝟐
𝟐 𝒎𝟐
𝑵
𝒌
+
𝒇
𝟎
𝒉
𝝎𝟐 = 𝟖 𝟐
𝜿 𝒌𝒉 + 𝒎𝟐
…
• Wave-action conservation as before
𝜕𝐴 𝜕𝑡 + 𝛻 ⋅ 𝑐𝑔 𝐴 = 0
WKB in Near-Isothermal Stratification: Mean Flow (1)
• Mean flow:
𝝏
(𝟎)
(𝟎)
(𝟎)
(𝟏)
+ 𝑼𝟎 ⋅ 𝛁𝐗 𝑼𝟎 + 𝜷𝒀𝒆𝒛 × 𝑼𝟎 + 𝒇(𝟎) 𝒆𝒛 × 𝑼𝟎
𝝏𝑻
𝟏
𝟏
(𝟎) 𝟎
𝟎
𝟒
𝟎 ∗
= −𝚯𝟎 𝛁𝑿,𝒉 𝚷𝟎 − ℜ 𝟎 𝛁𝑿 ⋅ 𝝆𝟎 𝑽𝟏 𝑼𝟏
𝟐 𝝆
𝝏
(𝟎)
(𝟐)
+ 𝑼𝟎 ⋅ 𝛁𝐗 𝚯𝟎 +
𝝏𝑻
𝟎
𝟎
𝒅𝚯
(𝟏)
𝟎
𝑾𝟎
𝒅𝜻
𝟎
𝟏
𝟏
(𝟎) 𝟎
𝟐 ∗
= − ℜ 𝟎 𝛁𝑿 ⋅ 𝝆𝟎 𝑽𝟏 𝚯𝟏
𝟐 𝝆
𝟎
𝟏
𝛁𝑿 ⋅ 𝝆𝟎 𝑽𝟎
=𝟎
• Same result in both pseudo-incompressible and anelastic theory!
• There is a higher-order correction to hydrostatic equilibrium
4
3
𝜕Π
Θ
𝟏
𝟏
(𝟎) 𝟎
0
𝟎 ∗
0
0
−Θ0
+ 0 = − ℜ 𝟎 𝛁𝑿 ⋅ 𝝆𝟎 𝑽𝟏 𝑾𝟏
𝜕𝑍
𝟐 𝝆
Θ
0
𝟎
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
𝝏
𝝏 𝟏
(𝟎)
+ 𝑼𝟎 ⋅ 𝛁𝐗 𝑷 = −
𝛁 ⋅ 𝒄𝒈 𝒍𝑨
𝝏𝑻
𝝏𝑿 𝝆 𝟎 𝑿
𝟎
(𝟎)
(𝟑)
𝟐
𝑷 = 𝛁𝑿,𝒉
𝚯𝟎 𝚷𝟎
Balanced
(EP flux)
flow
forced
+𝒇
𝟎
by
+ 𝜷𝒀 +
𝒇
𝟎
𝟎
𝝆𝟎
𝝏 𝟏
+
𝛁 ⋅ 𝒄𝒈 𝒌𝑨
𝝏𝒀 𝝆 𝟎 𝑿
𝟎
𝟎
𝝏 𝝆𝟎 𝝏
(𝟎) (𝟑)
𝚯
𝚷
𝝏𝒁 𝑵𝟐𝟎 𝝏𝒁 𝟎 𝟎
pseudo-momentum-flux
convergencies
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
𝑹
𝜿=
→𝟎
𝒄𝒑
•
consistent with Smolarkiewicz and Dörnbrack (2008),
Klein (2009), Smolarkiewicz and Szmelter (2011)