intuitive introduction to acoustic-gravity waves

Intuitive Introduction To Acoustic-gravity Waves

EP711 Supplementary MaterialThursday, January 26th, 2012

Jonathan B. Snively Embry-Riddle Aeronautical University

1

Contents

• Observational Basis• Theoretical Basis• Propagation Characteristics


2

Atmospheric gravity waves...

[Images from: Wikipedia Commons]

• Comprise the range of atmospheric waves governed primarily by buoyancy, rather than rotation or compression.

• Exhibit spatial scales ranging from kilometers to hundreds of kilometers, and time scales from a few minutes to a hours.

• Propagate to high (middle and upper atmospheric) altitudes, where they may interact, break, trigger instabilities, and ultimately dissipate.

• Provide coupling by transporting energy and momentum between atmospheric regions.

• Are readily observable via perturbations to clouds, ice layers, and airglow.

3

This thing.

Solitary Wave Roll Cloud? ... Extends as far as visible from ERAU Lehman Building!

4

Gravity Waves in Noctilucent Clouds (NLCs)... aka Polar Mesospheric Clouds, or Mesospheric Ice Clouds

[Over Kiruna, Sweden, from Dalin et al., 2004]

In comparison to tropospheric clouds < ~10 km altitude,NLCs form near mesopause at ~85 km. Wave scale sizes are

typically much larger at higher altitude!

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Gravity Waves in Upper-Mesospheric Airglow... Green emission due to O, red (and near-infrared) emissions due to OH and O2, yellow emission due to Na.

[Mesospheric bore (nonlinear ducted wave) over Texas from Smith et al., 2003]

6

Unstable “Ripples” in Clouds

Taken w/ iPhone Camera at ERAU Physics Dept.

In-situ generated unstable wave-like features, evolving from coherent patterns into fine small-scale turbulent structure.

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... iPhone pics of wave clouds from an airplane

Gravity Wave?

Ripple Instability?

8

Narrow-Field

InGaAs Cam

era

BLO All-Sky Imager

[Courtesy of M. J. Taylor and P-D. Pautet]

Observable Scales:OH airglow data at small and large fields of view, high and low resolutions.

Unw

arpe

d D

ata

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Contents



10

Time

The Brunt-Väisälä Frequency N:

If a parcel is displaced vertically in a gravity-stratified atmosphere,

... it will oscillate at a characteristic frequency N (~0.01-0.02 rad/sec)

The Earth’s atmosphere is stratified, such that gravity and buoyancy maintain equilibrium. Density and pressure decrease exponentially with altitude at a characteristic scale height H~ 6-8 km.

N2 =g

�o

d�o

dz� g2

cs(� � 1)

This is the highest frequency at which gravity waves can propagate!

11

�vx

�t= � 1

�o

�p

�x(1)

�vz

�t= � 1

�o

�p

�z� �g

�o(2)

��

�t= ��o

�vx

�x� �o

�vz

�z� vz

��o

�z(3)

�p

�t= �vz

�po

�z� �po

��vx

�x+

�vz

�z

�(4)

Linearized Equations of Motion:

Substituting vz = (�o/�s)�1/2wz, a wave equation for the normalized verticalperturbation velocity wz is obtained:

�2wz

�z2+

��2 � �2

o

c2s

� �2 �N2

v2�x

�wz = 0.

which is governed by the linear dispersion relation

k2z =

�2 � �2o

c2s

� �2 �N2

v2�x

,

where �o = g�2cs

is the acoustic cut-o� frequency and N = gcs

�� 1 is the

Brunt-Vaisala resonance frequency.12

0 =!2 � !2

o

c2s

� !2 �N2

v2�x

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050

100

200

300

400

500

600Ve

loci

ty (m/s

)

Frequency (rad/s)

Example Phase and Group Velocity:Horizontal Phase Progression kz=0

PhaseGroup

Acoustic Waves (Infrasonic)

Gravity Waves(Approaching Evanescence)

Zo

N

13

Gravity waves are transverse wave motions below the Brunt-Vaisala fre-quency N and the acoustic cut-o� frequency �o.

Neglecting the acoustic terms, a convenient form for the incompressible dis-persion relation relates kx and kz via the wave vector angle � to the horizontal(see Figure) [e.g., Nappo, 2002, p. 32]:

� =kxN

(k2x + k2

z)1/2= N cos �.

k2z =

�2 � �2o

c2s

� �2 �N2

v2�x

As stated, the compressible dispersion relation is:

Interesting Conclusion: Frequency ω of gravity wave (relative to N) determines λx, λz, and thus direction of propagation!

λx

λz

β

vφ

vg Group and Phase Velocity are Perpendicular!

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Dynamical Model:

• Finite volume method (FVM) solution for nonlinear, compressible, stratified, Euler equations [e.g., LeVeque, 1997].

• Viscosity, thermal conduction, and gravity wave sources included via a time-split approach [Snively and Pasko, 2003, 2008].

• Coupled solutions for minor species densities (via FVM) for airglow studies [Snively et al., 2005, 2010].

Allow simulation of linear and nonlinear gravity wave dynamics, and perturbations to airglow photochemistry.

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Model Equations of Motion:We model the atmosphere as a non-rotating, fully-nonlinear, compressible

gas:

⇧⌅

⇧t+⇧ · (⌅�v) = 0 (1)

⇧

⇧t(⌅�v) +⇧ · (⌅�v�v) = �⇧p� ⌅�g (2)

⇧E

⇧t+⇧ ·{ (E + p)�v} = �⌅gvz (3)

where the energy equation and the equation of state for an ideal gas are definedas:

E =p

(� � 1)+

12⌅(�v · �v) (4)

where ⌅ is density, p is pressure, �v is the fluid velocity, along with energy densityE. The Euler equations are coupled with equations for viscosity and thermalconduction:

⇧�v

⇧t= ⇤⇧2�v (5)

⇧T

⇧t= ⇥⇧2T (6)

Mass Density:

Momentum:

Energy:

State:

Viscosity:

Conduction:

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Contents



17

HORIZONTALPERTURBATION

VELOCITY

VERTICALPERTURBATIONVELOCITY

Acoustic Wave Propagation:Above N, gravity waves are not generated

T=40 sec. period z=+300 km

Wave Steepening

Altit

ude

0 km x=+300 kmy=-300 km18

Gravity Wave Propagation:Effects of Wind and Temperature

Gravity wave vertical wavelength �z varies with wind U as it depends onthe Doppler-shifted intrinsic frequency � = � � kxU , while ground-relative fre-quency � and horizontal wavenumber kx (and wavelength �x) remain constant.

Therefore: When wind varies only with altitude, Doppler shifts modify only the intrinsic frequency relative to the flow. Frequency relative to the ground (extrinsic) remains constant!

Shift to higher frequency: Shift to lower frequency:Wind -U Wind U

Zero Wind

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Gravity Wave Propagation:Isothermal, Windless

Altit

ude

(km

)

Scaledw (m/s) t=3300 s

10

20

30

40

50

60

70

80

90

100

Horizontal Distance (km)0 5 10 15 20

t=3600 s

0 5 10 15 20

t=3900 s

0 5 10 15 20

t=4200 s

0 5 10 15 20

t=4500 s

0 5 10 15 20

�x

= 20km

! =Np2

Wind:Zero!

Wave:

Result:The gravity wave propagates upwards through the horizontally-periodic domain. This simple case never occurs in reality!

20

�x

= 20km

! =Np2

Wave:

Wind:+20 m/s Peak at 50 km.Result:The gravity wave is Doppler-shifted to lower frequency at the wind peak, where vertical wavelength (and group velocity) is reduced.

Gravity Wave Propagation:Isothermal, with Positive Wind

Altit

ude

(km

)


10

20

30

40

50

60

70

80

90

100


t=3600 s

0 5 10 15 20

t=3900 s

0 5 10 15 20

t=4200 s

0 5 10 15 20

t=4500 s

0 5 10 15 20

+20 m/s

21

�x

= 20km

! =Np2

Wave:

Wind:+60 m/s Peak at 50 km.

Result:The gravity wave is Doppler-shifted towards zero frequency, leading to dissipation and blockage of the wave’s passage.

Gravity Wave Propagation:Isothermal, Critical Level

Altit

ude

(km

)


10

20

30

40

50

60

70

80

90

100


t=3600 s

0 5 10 15 20

t=3900 s

0 5 10 15 20

t=4200 s

0 5 10 15 20

t=4500 s

0 5 10 15 20

+60 m/s

22

�x

= 20km

! =Np2

Wave:

Gravity Wave Propagation:Isothermal, with Negative Wind

Altit

ude

(km

)


10

20

30

40

50

60

70

80

90

100


t=3600 s

0 5 10 15 20

t=3900 s

0 5 10 15 20

t=4200 s

0 5 10 15 20

t=4500 s

0 5 10 15 20

Wind:-20 m/s Peak at 50 km.

Result:The gravity wave is Doppler-shifted to higher frequency. Vertical wavelength increases, and it is partially reflected. This is the basis for ducting!

-20 m/s

23

Gravity Wave Propagation:Realistic Thermal Variation

�x

= 20km

! =Np2

Wave:

Wind:Zero!

Result:The gravity wave characteristics are modified relative to variations in the Brunt frequency, leading to refraction and partial reflection.

Temperature:MSISE90 Profile

Altit

ude

(km

)


10

20

30

40

50

60

70

80

90

100


t=3600 s

0 5 10 15 20

t=3900 s

0 5 10 15 20

t=4200 s

0 5 10 15 20

t=4500 s

0 5 10 15 20

24

Wave perturbation amplitudes increase with altitude, eventually leading to breaking.

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I – Summary• The gravity wave spectrum is the range of waves for which buoyant

effects dominate over compression or rotation, the maximum frequency being the Brunt-Väisälä frequency N.

• Waves of high frequency propagate with nearly vertical trajectories, and are thus most likely to reach high altitudes.

• Gravity wave propagation is strongly influenced by ambient wind and temperature profiles of the middle atmosphere, which may lead to strong refraction, or reflection.

• Upward propagating waves experience growth in amplitude due to decreasing density, and therefore may be subject to breaking.


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intuitive introduction to acoustic-gravity waves

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