Atmospheric Waves Chapter 6 - University of California,...

Atmospheric Waves Chapter 6

Paul A. Ullrich

[email protected]

Part 3: Internal Gravity Waves

Internal Gravity Waves

Paul Ullrich Atmospheric Waves March 2014

(Internal) gravity waves are generated by flow over topography, thunderstorms, other mesoscale systems

(Internal) gravity wave effects are: cloud streets, lenticularis clouds, rapid local temperature and pressure changes, clear air turbulence

They carry energy and momentum into the middle atmosphere (stratosphere, mesosphere)

Atmospheric gravity waves can only exist when the atmosphere is stably stratified (N2 > 0)

Buoyancy force is the restoring force for gravity waves

In an incompressible fluid (e.g. ocean) gravity waves travel primarily in the horizontal plane, since reflection at lower and upper boundary occurs for vertically traveling waves.

In the atmosphere: Due to compressibility and stratification, gravity waves may propagate vertically and horizontally.



Figure: Air moving over a mountain on an island triggers waves on the downwind (lee) side of the mountain, indicated by the cloud pattern

Topographic Gravity Waves


Undular Bore

Examples


Mountain-generated waves (lee waves)

Examples


Consider pure internal gravity waves: No rotation, no friction

Can be studied with the basic equations for 2D (x-z plane) motions with Boussinesq approximation (almost incompressible)


@u

@t

+ u

@u

@x

+ w

@u

@z

+1

⇢

@p

@x

= 0

@w

@t

+ u

@w

@x

+ w

@w

@z

+1

⇢

@p

@z

+ g = 0

@u

@x

+@w

@z

= 0

@✓

@t

+ u

@✓

@x

+ w

@✓

@z

= 0


Four equations in five unknowns

Take logarithms of both sides:

✓ = T

✓p0p

◆Rdcp

=p

⇢Rd

✓p0p

◆Rdcp

= Rdpcvcp ⇢�1p

Rdcp

0

ln ✓ =

cvcp

ln p� ln ⇢+ constant



Equations must be closed via an equation of state, such as the potential temperature equation:

Linearize the basic equations by letting

⇢ = ⇢0 + ⇢0

p = p(z) + p0

✓ = ✓(z) + ✓0

u = u+ u0

w = w0

dp

dz= �g⇢0 ln ✓ =

cvcp

ln p� ln ⇢0 + constant

IGWs: Derivation (1)


Hydrostatic Balance

The basic state must satisfy:

Definition of Potential Temperature

@u

@t

+ u

@u

@x

+ w

@u

@z

=@

@t

(u+ u

0) + (u+ u

0)@

@x

(u+ u

0) + w

0 @

@z

(u+ u

0),

=@u

0

@t

+ u

@u

0

@x

+ u

0 @u0

@x

+ w

0 @u0

@z

,

=@u

0

@t

+ u

@u

0

@x

Linearize the material derivative terms:



Constant Constant Constant

Small Small

@w

@t

+ u

@w

@x

+ w

@w

@z

=@w

0

@t

+ (u+ u

0)@w

0

@x

+ w

0 @w0

@z

,

=@w

0

@t

+ u

@w

0

@x

Small Small

@✓

@t

+ u

@✓

@x

+ w

@✓

@z

=@

@t

(✓ + ✓

0) + (u+ u

0)@

@x

(✓ + ✓

0) + w

0 @

@z

(✓ + ✓

0),

=@✓

0

@t

+ u

@✓

0

@x

+ u

0 @✓0

@x

+ w

0 @✓

@z

+ w

0 @✓0

@z

,

=@✓

0

@t

+ u

@✓

0

@x

+ w

0 @✓

@z

Linearize the thermodynamic equation:



No ,me dependence

Small

No x dependence

Linearize Remember some calculus: Linearization yields

1

⇢

@p

@z+ g =

1

⇢0 + ⇢0

✓@p

@z+

@p0

@z

◆+ g

1

⇢0 + ⇢0=

1

⇢0

1⇣1 + ⇢0

⇢0

⌘ ⇡ 1

⇢0

✓1� ⇢0

⇢0

◆⇢0

⇢0⌧ 1


For


1

⇢

@p

@z+ g ⇡ 1

⇢0

@p

@z

✓1� ⇢0

⇢0

◆+

1

⇢0

@p0

@z+ g =

1

⇢0

@p0

@z+

⇢0

⇢0g

Use 1

⇢0

@p

@z= �g

Potential temperature equation:

with

ln

✓

✓1 +

✓0

✓

◆�=

1

�ln

p

✓1 +

p0

p

◆�� ln

⇢0

✓1 +

⇢0

⇢0

◆�+ constant

� =cpcv


ln ✓ =

cvcp

ln p� ln ⇢+ constant

Linearize via


✓ = ✓ + ✓0, p = p+ p0, ⇢ = ⇢0 + ⇢0

Again, remember some calculus: Apply these rules to potential temperature equation:

To obtain linearized potential temperature equation:

ln(ab) = ln a+ ln b

ln(1 + ✏) ⇡ ✏ ✏ ⌧ 1

ln

✓

✓1 +

✓0

✓

◆�=

1

�ln

p

✓1 +

p0

p

◆�� ln

⇢0

✓1 +

⇢0

⇢0

◆�+ constant

✓0

✓⇡ 1

�

p0

p� ⇢0

⇢0⇢0 ⇡ �⇢0

✓0

✓+

p0

c2s c2s =p�

⇢0


Speed of sound


In gravity waves, density fluctuations due to pressure changes are small compared to those due to temperature changes: Therefore, to a first approximation, the potential temperature equation becomes:

��⇢0✓0

✓

�� p0

c2s

��

✓0

✓⇡ � ⇢0

⇢0



The linearized equation set for internal gravity waves becomes

✓@

@t

+ u

@

@x

◆u

0 +1

⇢0

@p

0

@x

= 0 (1)

✓@

@t

+ u

@

@x

◆w

0 +1

⇢0

@p

0

@z

� ✓

0

✓

g = 0 (2)

@u

0

@x

+@w

0

@z

= 0 (3)✓

@

@t

+ u

@

@x

◆✓

0 + w

0 @✓

@z

= 0 (4)



Leads to Now eliminate u’ from (5) by computing Leads to the equation:

✓@

@t

+ u

@

@x

◆✓@w

0

@x

� @u

0

@z

◆� g

✓

@✓

0

@x

= 0 (5)

@(5)

@x

+

✓@

@t

+ u

@

@x

◆@(3)

@z

✓@

@t

+ u

@

@x

◆✓@

2w

0

@x

2+

@

2w

0

@z

2

◆� g

✓

@

2✓

0

@x

2= 0 (6)

IGWs: Derivation (5) @(2)

@x

� @(1)

@z

Eliminate p’ by computing


Yields single equation for

✓@

@t

+ u

@

@x

◆2 ✓@

2w

0

@x

2+

@

2w

0

@z

2

◆+

g

✓

@✓

@z

@

2w

0

@x

2= 0

✓@

@t

+ u

@

@x

◆(6) +

g

✓

@

2(4)

@x

2

Eliminate θ’ from (6) by computing


Square of Brunt-Väisälä frequency: assume to be constant N 2


w0

Consider wave-like solutions:

w = wr + iwi m = mr + imi � = kx+mz � ⌫t

✓@

@t

+ u

@

@x

◆2 ✓@

2w

0

@x

2+

@

2w

0

@z

2

◆+N 2 @

2w

0

@x

2= 0



w

0= Re [w exp(i�)]

= Re [(wr + iwi) exp(i(kx+mrz � ⌫t)) exp(�miz)]

= [wr cos(Re(�))� wi sin(Re(�))] exp(�miz)

Horizontal wavenumber k is real, solution is always sinusoidal

The vertical wavenumber m is complex –  Real part mr describes the sinusoidal variation in z –  Imaginary part mi describes the exponential decay or

growth of the wave amplitude in z, depending on the sign of mi (positive / negative).

If m is real, total wave number may be regarded as a vector directed perpendicular to line of constant phase, and in the direction of phase increase.

k =2⇡

Lx

m =2⇡

Lz = ki+mk

IGWs: Wavenumbers m = mr + imi

w0= [wr cos(Re(�))� wi sin(Re(�))] exp(�miz)

with


IGW Frequency

Relative to the mean wind: –  Positive (+) sign means eastward phase propagation –  Negative (-) sign means westward phase propagation

(⌫ � uk)2(k2 +m2)�N 2k2 = 0

⌫ = uk ± Nkpk2 +m2

= uk ± Nk

IGWs: Dispersion w0

= Re [w exp(i�)]✓

@

@t

+ u

@

@x

◆2 ✓@

2w

0

@x

2+

@

2w

0

@z

2

◆+N 2 @

2w

0

@x

2= 0

Insert

into

IGW Dispersion Relation


If we let k > 0 and m < 0, then lines of constant phase φ = (kx + mz) tilt eastward with increasing height

The choice of the positive root then corresponds to an eastward and downward phase propagation (relative to the mean flow).

The zonal and vertical phase speeds (positive root) are

+

⌫ = uk ± Nkpk2 +m2

= uk ± Nk

cx

=⌫

k= u+

Npk2 +m2

cz

=⌫

m=

uk

m+

Nk

mpk2 +m2

IGWs: Frequency


Solutions for the other perturbation fields u’, p’, θ’ can be obtained by substituting w’ back into original linearized equations (let k > 0 and m < 0)

@u

0

@x

= �@w

0

@z

=) u

0 = �m

k

w

0

✓@

@t

+ u

@

@x

◆u

0 = � 1

⇢0

@p

0

@x

=) p

0 = ⇢0(⌫ � ku)

k

u

0

✓@

@t

+ u

@

@x

◆✓

0 = �w

0 @✓

@z

=) ✓

0 =i

(ku� ⌫)

@✓

@z

w

0

IGWs: Phase Relationships

Observe u’, w’, p’ are exactly in phase, with

However θ’ (or T’) out of phase by 1/4 wavelength: Warm air leads pressure ridge eastward

Transverse wave:

⌫ � ku > 0

k · u ⌘ ku0 +mw0 = w0hk⇣�m

k

⌘+m

i= 0

Perpendicular!


If we let k > 0 and m < 0, then lines of constant phase φ = (kx + mz) tilt eastward with increasing height

Horizontal distance

Height

Phase velocity

Perturbation velocity field

W C

W

C

W C Temperature perturbation

H H L

H L Pressure perturbation

IGWs: Cross Section

⌫ � ku > 0Assume

L

Phase lines


The zonal and vertical phase velocities are

Upper and lower signs chosen as above.

IGWs: Phase / Group Velocity

The components of the group velocity are

cx

=⌫

k= u± Np

k2 +m2

cz

=⌫

m=

uk

m± Nk

mpk2 +m2


cg,x

=@⌫

@k= u± Nm2

(k2 +m2)3/2

cg,z

=@⌫

@m= ⌥ Nkm

(k2 +m2)3/2

The vertical component of group velocity has a sign opposite to that of the vertical phase speed.

That is, downward phase propagation implies upward energy propagation


cx

=⌫

k= u± Np

k2 +m2

cz

=⌫

m=

uk

m± Nk

mpk2 +m2


cg,x

=@⌫

@k= u± Nm2

(k2 +m2)3/2

cg,z

=@⌫

@m= ⌥ Nkm

(k2 +m2)3/2

Horizontal phase and group velocities point in the same direction (no sign change)

Remarkable property of internal gravity waves: Group velocity travels perpendicular to the direction of the phase propagation

That is, in internal gravity waves energy travels parallel to the crests and throughs (in contrast to shallow water gravity waves)



Height Phase velocity

Phase lines (only the crests are shown here)

Group velocity (energy travels in this direction)

α

Angle of phase lines to the local vertical:

α


Lx

LLz

Lx

cos↵ = ± L

Lx

= ± Lzp

L2x

+ L2z

= ±�2⇡m

�q�

2⇡k

�2+

�2⇡m

�2 = ± kpk2 +m2

= ±k


pL2x

+ L2z

Consequence: gravity wave intrinsic frequency (relative to mean flow) must be less than buoyancy frequency

cos↵ = ± L

Lx

= ± Lzp

L2x

+ L2z

= ±�2⇡m

�q�

2⇡k

�2+

�2⇡m

�2 = ± kpk2 +m2

= ±k

⌫ = ⌫ � uk = ± Nkpk2 +m2

= ±Nk

⌫ ⌘ ⌫ � uk = ±N cos↵

IGWs: Frequency Angle of phase lines to the local vertical:

Frequency of internal gravity waves:

Intrinsic Frequency


Propagation of an internal gravity wave packet

https://www.youtube.com/watch?v=cDsNmnpq9_o

http://www.youtube.com/watch?v=RpP62QSJM0g

Observed waves in the atmosphere

IGWs: Movies


For example, idealized mountain height profile:

⌫ = �uk < 0

Topographic Gravity Waves Topographic gravity waves are stationary internal waves driven by topography (a lower boundary condition on w).

For stationary waves (ν=0) this means that the intrinsic frequency is (westward relative to the mean flow)

Wavelength k is determined by the wavelength of the topography (flow follows topography near the ground)

h(x) = hM cos(kx)


For stationary waves, the dispersion relationship for IGWs ✓

@

@t

+ u

@

@x

◆2 ✓@

2w

0

@x

2+

@

2w

0

@z

2

◆+N 2 @

2w

0

@x

2= 0

✓@

2w

0

@x

2+

@

2w

0

@z

2

◆+

N 2

u

2 w

0 = 0

Topographic Gravity Waves Topographic gravity waves are stationary internal waves driven by topography (a lower boundary condition on w).

simplifies to


Seek stationary ( ) wave solutions: yields the dispersion relationship

If , m2 > 0 (m must be real). Solutions have the form of vertically propagating waves (with constant amplitude):

⌫ = 0, ⌫ = uk

w0= Re [w exp(i�)] = wr cos�� wi sin�

w = wr + iwi, m = mr + imi, � = kx+mz

m2 =N 2

u2 � k2

|u| < N/k

w

0= w exp(i(kx+mz))



Wide (broad) mountain small k m real Ridges and trough lines slope with height, opposite to u Upward flow of energy and momentum

Intrinsic phase speed vector

Wave number vector (k,m)

Intrinsic group velocity vector

Absolute group velocity vector u

u


Vertically Propagating Waves

If , m2 < 0 (m is purely imaginary). Solutions have the form of vertically trapped waves (that decay with height):

m2 =N 2

u2 � k2



|u| > N/k

w

0= w exp(ikx) exp(�miz)

decay

Vertically Trapped Waves


Narrow mountain large k m imaginary Ridges and trough lines aligned with height No tilt, no momentum transport

w

0= w exp(ikx) exp(�miz)



Ver$cal propaga$on of gravity waves is only possible when is less than the buoyancy frequency Stable stra,fica,on, wide ridges and rela,vely weak zonal flow provide favorable condi,ons for ver,cally propaga,ng gravity waves. If energy is transported upward, phase must be downward.

|uk| N

Summary

Figure: Circulation is more complex in real gravity waves: For example, rotors, breaking waves, wind shear, varying

More Realistic IGWs

N


Figure: Lenticularis cloud forming on the lee side of the mountain

Lenticularis Cloud

Movie: https://www.youtube.com/watch?v=YKAfKHSeWZc


Atmospheric Waves Chapter 6 - University of California,...

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