Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz...

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

Ulrich AchatzGoethe-Universität Frankfurt am Main

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

13.9.10

Gravity waves in the

middle atmosphere

Gravity Waves in the Middle Atmosphere

Becker und Schmitz (2003)

Without GW parameterization

linear GWs in the Euler equations

• no heat sources, friction, …• no rotation

RTppcR

are equivalent to

• linearization about atmosphere at rest:

)',','('

• conservation of wave energy: linear equations satisfy

• conservation suggests:

• isothermal atmosphere: const. const., 2 Ncs

),,( exp~'

• isothermal atmosphere:

const. const., 2 Ncs

gravity waves

sound waves

• isothermal atmosphere:

const. const., 2 Ncs

gravity waves

sound waves

GW polarization relations:

GW breaking in the atmosphere

• Conservation

• Instabilität in großen Höhen• Impulsdeposition …

20 12 vv

Rapp et al. (priv. comm.)

• Conservation

• Instability at large altitudes• Impulsdeposition …

20 12 vv

• Conservation

• Instability at large altitudes• Turbulence• Momentum deposition…

20 12 vv

GW Parameterizations

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

• too many free parameters

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

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

GW breaking

GW stability in the Boussinesq theory

vkvkvv

Basic GW types

• Dispersionsrelation:

• Trägheitsschwerewellen (TSW):

• Hochfrequente Schwerewellen (HSW):

2/12222 cossin Nf

cos90 N

Basic GW types

• Dispersion relation:

• Inertia-gravity waves (IGW):

• High-frequent GWs (HGW):

2/12222 cossin Nf

cos90 N

Coriolis parameter

Stability

Basic GW types

• High-frequent gravity waves (HGW):

2/12222 cossin Nf

cos90 N

Coriolis parameter

Stability

Basic GW types

• High-frequent GWs (HGW):

2/12222 cossin Nf

cos90 N

Coriolis parameter

Stability

Traditional Instability Concepts

• static (convective) instability:

amplitude reference (a>1)

• dynamic instability

1Ri 22

• limited applicability to HGW breaking

1Ri 22

• limited applicability to HGW breaking

1Ri 22

• BUT: limited applicability to GW breaking

NM analysis, 2.5D-DNS

||SVNM,SW

exp)()()0,,,(

NMs: Growth Rates = 70°)

Achatz (2005, 2007b)

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

Achatz (2005, 2007b)

Energetics b

, deviation -

, and over average -

kubCdd

dtEdAKE

not the gradients in z matter,but those in !

instantaneous growth rate:

Energetics b

, deviation -

kubCdd

dtEdAKE

Energetics b

, deviation -

kubCdd

dtEdAKE

Energetics of a Breaking HGW with a0 < 1

Strong growth perturbation energy:buoyancy instability

Achatz (2007b)

Singular Vectors:

• normal modes: Exponential energy growth

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

characteristic perturbations in a linear initial value problem:

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

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

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

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

Achatz (2007a)

Summary GW breaking

In comparison to assumptions in GW parameterizations:

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

it sets in earlier (at lower altitudes)

Summary GW breaking

In comparison to assumptions in GW parameterizations:

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

it sets in earlier (at lower altitudes)

• GW dissipation stronger than assumed, i.e.

more momentum deposition

Soundproof Modelling and

Multi-Scale Asymptotics

Achatz, Klein, and Senf (2010)

GW breaking in the middle atmosphere

• Conservation

• Instability at large altitudes• Momentum deposition…

20 12 vv

• Conservation

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

growth and dissipation:– not in Boussinesq theory

20 12 vv

• Conservation

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

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

(Ogura and Philips 1962, Lipps and Hemler 1982)

– Pseudo-incompressible (Durran 1989)

20 12 vv

• Conservation

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

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

(Ogura and Philips 1962, Lipps and Hemler 1982)

– Pseudo-incompressible (Durran 1989)

20 12 vv

Which soundproof model should be used?

Scales: time and space

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

characteristic wave number

• time scale set by GW frequency

/t̂t characteristic frequency

dispersion relation for

HK 2/1

characteristic length scale

/t̂t characteristic frequency

HK 2/1

/1 ˆ TtTt characteristic time scale

HK 2/1

characteristic time scale

HK 2/1

/1 ˆ TtTt

Scales: velocities

• winds determined by polarization relations:

what is ?b~

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

Scales: velocities

• winds determined by polarization relations:

what is ?b~

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

Non-dimensional Euler equations

0ˆˆˆ1ˆ

ˆˆˆˆˆ

• using:

yields

Non-dimensional Euler equations

0ˆˆˆ1ˆ

ˆˆˆˆˆ2

• using:

yields

isothermal potential-temperature scale height

Scales: thermodynamic wave fields

• potential temperature:

Scales: thermodynamic wave fields

• potential temperature:

• Exner pressure:

' ~~ Ob

Additional vertical scale needed:

• and have vertical scale

• scale of wave growth is

i ˆ ,ˆ,ˆ

Therefore: Multi-scale-asymptotic ansatz

0ˆ )0( also assumed:

i ˆ ,ˆ,ˆ

also assumed:

Scale asymptotics Euler: results

)0()0(

Hydrostatic, large-scale, background

)0()0(

)1()2()0(

)2()0(

Momentum equations and entropy equations as in Boussinesq or anelastic theory

)0()0(

)1()2()0(

)2()0(

)0()1()0(

)0()0(

Momentum equations and entropy equations as in Boussinesq or anelastic theory

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

Scale asymptotics: pseudo-incompressible equations

scale-asymptotic analysis of

)0()1()0(

)0()0(

)1()2()0(

)2()0(

results:

)0()1()0(

)0()0(

)1()2()0(

)2()0(

results:

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

Scale asymptotics: anelastic equations

)0()0(

)0()1()0(

)0()0(

)1()2()0(

)2()0(

results:

)0()0(

)0()1()0(

)0()0(

)1()2()0(

)2()0(

results:

)0()0(

)0()1()0(

)0()0(

)1()2()0(

)2()0(

results:

anelastic equations only consistent if:

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

Large-Amplitude WKB

Achatz, Klein, and Senf (2010)

Large-amplitude WKB

)1()1(0

)1()0()0(

)2(2)0(

)1()0(

,,exp),,(ˆ),,(ˆˆ

,,exp),,(ˆ)ˆ,ˆ,ˆ(ˆˆ

(e.g.) oˆˆ~

Large-amplitude WKB

)1()1(0

)1()0()0(

)2(2)0(

)1()0(

,,exp),,(ˆ),,(ˆˆ

(e.g.) oˆˆ~

Large-amplitude WKB

)1()1(0

)1()0()0(

)2(2)0(

)1()0(

,,exp),,(ˆ),,(ˆˆ

(e.g.) oˆˆ~

Large-amplitude WKB

)1()1(0

)1()0()0(

)2(2)0(

)1()0(

,,exp),,(ˆ),,(ˆˆ

(e.g.) oˆˆ~

Mean flow with only large-scale dependence

Large-amplitude WKB

)1()1(0

)1()0()0(

)2(2)0(

)1()0(

,,exp),,(ˆ),,(ˆˆ

(e.g.) oˆˆ~

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

Large-amplitude WKB

)1()1(0

)1()0()0(

)2(2)0(

)1()0(

,,exp),,(ˆ),,(ˆˆ

(e.g.) oˆˆ~

Large-amplitude WKB

)1()1(0

)1()0()0(

)2(2)0(

)1()0(

,,exp),,(ˆ),,(ˆˆ

(e.g.) oˆˆ~

Wave induced mean flow

Large-amplitude WKB

)1()1(0

)1()0()0(

)2(2)0(

)1()0(

,,exp),,(ˆ),,(ˆˆ

(e.g.) oˆˆ~

Harmonics of the wavepacket due to nonlinear interactions

Large-amplitude WKB

)1()1(0

)1()0()0(

)2(2)0(

)1()0(

,,exp),,(ˆ),,(ˆˆ

(e.g.) oˆˆ~

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

/exp i

Large-amplitude WKB: leading order

frequency intrinsic ˆˆ

ˆˆˆ

Large-amplitude WKB: leading order

frequency intrinsic ˆˆ

ˆˆˆ

dispersion relation and structure as from Boussinesq

of Nullvector

ˆˆˆ

Large-amplitude WKB: 1st order

ˆ1ˆˆ

ˆˆˆ

ˆ1ˆˆ

ˆˆˆ

velocitygroup ˆ

energy waveˆ

cSolvability condition leads to wave-action conservation

(Bretherton 1966, Grimshaw 1975, Müller 1976)

ˆ1ˆˆ

ˆˆˆ

Solvability condition leads to wave-action conservation

(Bretherton 1966, Grimshaw 1975, Müller 1976)

velocitygroup ˆ

energy waveˆ

Pseudo-incompressible divergence needed!

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

)1()1( ,,exp),,(ˆˆ

ˆˆˆ

)1()1( ,,exp),,(ˆˆ

ˆˆˆ

k2,ˆ2

ˆˆˆ

)1()1( ,,exp),,(ˆˆ

2nd harmonics are slaved

Large-amplitude WKB: 1st order, higher harmonics

)1()1( ,,exp),,(ˆˆ

ˆˆˆ

,ˆ :2

)3()0(

)1()1( ,,exp),,(ˆˆ

ˆˆˆ

,ˆ :2

)3()0(

M k 2 0

ˆˆˆ

)3()0(

)1()1( ,,exp),,(ˆˆ

higher harmonics vanish(nonlinearity is weak)

Large-amplitude WKB: mean flow

),,(ˆˆ

(e.g.) oˆˆ~

)1()0()0(

Horizontal flow horizontally homogeneous(non-divergent)

ˆˆˆ

)0()1(

)0(0)0(

),,(ˆˆ

(e.g.) oˆˆ~

)1()0()0(

No zero-order vertical flow

ˆˆˆ

)0()1(

)0(0)0(

),,(ˆˆ

(e.g.) oˆˆ~

)1()0()0(

No first-order potential temperature

ˆˆˆ

)0()1(

)0(0)0(

),,(ˆˆ

(e.g.) oˆˆ~

)1()0()0(

ˆˆˆ

)0()1(

)0(0)0(

),,(ˆˆ

(e.g.) oˆˆ~

)1()0()0(

• acceleration by GW-momentum-flux divergence

ˆˆˆ

)0()1(

)0(0)0(

• acceleration by GW-

momentum-flux divergence

• GWs induce lower-order potential temperature variations

),,(ˆˆ

(e.g.) oˆˆ~

)1()0()0(

ˆˆˆ

)0()1(

)0(0)0(

),,(ˆˆ

(e.g.) oˆˆ~

)1()0()0(

• acceleration by GW-momentum-flux divergence

• GWs induce lower-order potential temperature variations

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

ˆˆˆ

)0()1(

)0(0)0(

),,(ˆˆ

(e.g.) oˆˆ~

)1()0()0(

• Wave-induced lower-order vertical flow vanishes

Summary

• multi-scale asymptotics:

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

• velocity• thermodynamic wave fields

• unique small parameter:

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

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

Summary

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

Summary

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

Summary

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

Summary

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

Gravity Waves, Scale Asymptotics, and the Pseudo-Incompressible Equations Ulrich Achatz...

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