reinisch_asd85.515_chap#71 chapter 7. atmospheric oscillations linear perturbation theory

Reinisch_ASD85.515_Chap#7 1

Chapter 7. Atmospheric OscillationsLinear Perturbation Theory

To describe large-scale atmospheric motions with some accuracy requires numerical techniques. This makes it difficult to understand the fundamental processes and the balances of forces. Meteorological disturbances often have a wave-like character. Do discuss waves in the atmosphere or in fluids, we linearize the governing equations using the perturbation method.


7.1 Perturbation Method

1. Field variable = (static portion) + (pertubation portion)

Assume u, v, p, T, etc. are time and longitude-averaged

variables, then ( , ) ' ,u x t u u x t

2. Products of perturbation terms can be neglected.

Example:

' ' ' '' '

u uu u u uu u u u u u

x x x x x


7.2 Properties of WavesWave motions = oscillations in field variables

that propagate in space

Familiar example of oscillation: the pendulum

Equation of motion

sin , restoring force (Fig.7dV

m mgdt

2

2

22 2

2

.1)

, or

0 where (the book uses instead of )

d l d gm mg

dt dt l

d g

dt l


Sinusoidal oscillations and waves

22

2

0

0 0 0

Solution for 0 :

cos sin or cos ,or

= Re Re Re

Re

c s

i t i t i i t

i t

d

dtt t t

e e e e

Ce

A sinusoidal wave propagating in the x direction is given by

, Re where Re is kept in mind.

C= complex amplitude

phase, or

wave number, angular f

c

i kx t i kx t

i

x t Ce Ce

C e

kx t k x tk

k

requency 2 (here f = frequency)f


Phase velocity

How does a point of constant phase move through space?

kx t const

0 0

Phase speed p

D Dxk

Dt DtDx

cDt k


7.21 Fourier SeriesThe wave number k = 2 is defined by the characteristics of the medium

in which the wave propagates. Usually k depends also on the frequency.

Each wave package (disturbance) can be represented as a sum

of waves.

1 1 1

0

01 1

' '1 1 1 10 0 0

1'

1 1

At time t = t 0 :

2sin cos ;

2 2sin sin sin sin sin '

2 2 since sin sin ' 0 for

2

s s s s s s s ss

s s s s ss s

s

xf x A B k x t k x s

x xf x dx A dx A s s dx

x xA s s dx

1

1 1

0

1 10 0

'

2 2sin and cos s s s s

s s

A f x dx B f x dx


7.22 Dispersion and Group VelocityUsually every spectral component propagates with its own phase speed.

This leads to dispersion of a wave package. A wave package consisting of

components around k and propagates with the group velocit y .gc k

1 2 1 2

Derivation: Assume the sum of two waves of equal amplitude, and

, , ,

, exp exp

exp exp exp

, exp cos

k k k k k k

x t i k k x t i k k x t

i kx t i k x t i k x t

x t i kx t k x

g

g

or

, cos cos

2The envelope has a wavelength of . The envelope has a phase

= . Group velocity g

t

x t kx t k x t

k

k x t ck k


7.3.1 Acoustic (Sound) Waves

Sound waves are longitudinal waves as illustrated in Fig.7.5. For simplicity

we assume ,0,0 and , . The governing equations are:

1Momentum 0 7.6

1Continuity 0 7.7

Thermodynamic En

u u u x t

Du p

Dt x

u

t x

U

Dlnergy 0 from 2.43 for adiabatic process

Dt

1

From poisson equation 2.44

ln ln ln ln

Dln ln 1 ln 10 where1 7.9

Dt

p p p

p

R c R c R cR cs s s

vp

p

p p ppT p

p R p R

cD D pR c

Dt Dt c


Combining 7.7 and 7.9 gives

1 ln0 7.10

Perturbation method:

, ' .

, ' .

, ' .

Neglecting products of primed quatities and considering

that u etc are constant:

D u

Dt x

u x t u u x t

p x t p p x t

x t x t


2 2 2

2

1 '' 0 7.12

t

'' 0 7.13

t

Eliminate ' by applying on 7.13 , and inserting 7.12 :t

''t

' ' 0t t

pu u

x x

uu p p

x x

u ux

u up px

u p p u px x x x


2 2

2

2 2 2 2 22

2 2

22

2

'' 0 7.14

t

Try the following solution:

' exp

' 2 ' 't t t

''

For ' exp to be a solution requ

p pu p

x x

p A ik x ct

u p u u p ikc iku px x x

p p pik p

x

p A ik x ct

2 2

ires therefore that

0 Dispersion relationp

ikc iku ik

pc u u RT


7.3.2 Shallow-Water Gravity Wave1 2Consider 2-fluid system pictured in Fi. 7.7. Stably stratified if .

Waves may propagate along the interface. Assume incompressibility of

the 2 fluids (this avaoids sound waves). In hydrostatic equi

librium:

- . Differentiate re x:

0, if we assume no horizontal gradient in each fluid.

This means also that the horizontal pressure gradient does not vary

with height:

0

p g z

pg

x z x

p p

x z z

x


1

1 1

1

1 from Fig. 7.7. Then:

. RHS is independent of z u is indep. of z

0 ( ) (0)

But , 0 0

p g p h

x x

u u u g p hu w

t x z x

u w uw h w h

x z xDh h h

w h u wDt t x

But

1

1

1

1x-momentum equation

1continuity equation 0

pDu u u u uu v w

Dt t x y z x

u v w

t x y z

=0

=0=0


Internal Gravity (or Buoyancy) WavesIn a stably stratified atmosphere gravity waves can propagate. The wave normal

can have horizontal and vertical components, i.e., the wave front (plane of

constant phase) is tilted (Fig.7.8). Vertical

2

22

wave front means horizontal pro-

pagation. In that case the buoancy drives an air parcel vertically up and down as

discussed in section 2.7.3, and the momentum equation is given by 2.52 :

,D

z N z NDt

2 0lndg

dz

22 2

22

2

Assume the wave front is tilted as in Fig. 7.8, then the restoring force is

cos cos cos cos 7.24

The momentum equation becomes now:

cos exp cos ; sinusoidal osci

N z N s N s

Ds N s s A i N t

Dt

llation.


Two-dimensional Internal Gravity Waves

Momentum equation neglecting Coriolis and friction:

D 1

10 7.25

10 7.26

Continuity equation (using Boussinesq approximation):

0 7.27

Energy e

pDt

u u w pu w

t x z x

w u w pu w g

t x z z

u w

x z

Ug

quation:

Dln 1 D0 0 7.28

Dt Dtu w

t x z


Perturbation - Linearization

0

_ _ _

_ _

0

_

0

Poisson equation: 7.29

' (small density perturbance in momentum eq.)

', ', ', ' 7.30

Notice and are functions of height, is not.

p pR c R c

s sp ppT

p R p

u u u w w p p z p z

p

d p z

dz

__ _ _

10_

0

_ _1

0_ _0

and or ln ln ln

7.29 becomes:

' ' 'ln 1 ln 1 ln 1 7.34

pR c

sp z p

g p z constp z

pp const

p


_ _ _

1_ _

0

0 0_ _ _ 2

0

_2

0 0_ 2

' ' 'ln 1 ln ln 1 ln ,

' ' 'Using this in 7.34:

' ' ' '' 7.35

' 'with . But in atmosphere , so

'

s

ss

etc

p

p

p p

cp

pc p RT

c

0_

'. Substitue into linearized 7.25-28 :


_

0

_

_0

__

1 '' 0 7.37

1 ' '' 0 7.38

' '0 7.39

' ' 0 7.40

Eliminate p' from first two by cross differentiation and subtraction:

pu u

t x x

pu w g

t x z

u w

x x

du w

t x dz


2_

0

2_

_0

_

_

' 1 '0

' 1 ' '0

' ' '0.

u pu

t x z x z

w p gu

t x x x z x

w u gu

t x x z x

_

2_

_

_2_

_

_2 2 2 2_

_2 2

'' '

0. Using 7.40 :

' ' '0. Apply :

x

' ' '0.

uw u g t x

ut x x z x

w u g d wu

t x x z dz x

w u g d wu

t x x x z dz x


2 2

2

2 2 2 2_2

2 2 2

_ _

2_

' 'From 7.39 : .

' ' '0

lnwith N

is the buoyancy frequency squared.

u w

x z z

w w wu N

t x x z x

g d dg const

dz dz

~ ~

~

Solution of PDE:

' Re where . 7.43

With , ' cos sin if m is real.

m can be complex: . Then ' Re ri

i kx mz tr i

r i

i kx m z tm zr i

w we w w iw

kx mz t w w w

m m im w e we


Dispersion relation

2 2 2 2_2

2 2 2

2_2 2 2 2

_2 2

2 2

Substitute into PDE:

'' 0

0

where

(We assumed here ). "intrinsic frequency".r

wu w N

t x x z x

i u ik k m N k

Nk Nku k k m

k m

m m

κκ

If we define the wave vector , , , then

and

2 , 2 , 2 in our 2-D derivationx y z y

k l m

kx ly mz t

k L l L m L L

κ

κ r κ r


0

0 in Figure

kx mz t const

z k m x const

dz k

dx m

z x

Assume 0, then 0.

The wave vector points downward and eastward. The vertical

wavelength is L 2 , the horizontal wavelength L 2 .

k m

m k

Lz

=kx+mz=const

z

x

Lx


__

_

_

2 2

2 2

The vertical phase speed relative to mean flow is

The horizontal phase speed relative to mean flow is

The phase speed in the direction of ,0, is

Gr

z

x

x z

u kc u

k k

u kc

m

u kk m c c c

k m

κκ

κ

_ _

2 2

2_

3

3

oup velocities:

7.45

7.45

gx

gx

gz

kkk kc u N u Nk

Nmc u a

Nkmc b

m


2 2 2 2 2 2

The tilt angle against the vertical is given by

1cos

1 1

On slide 14, the heuristic gravity frequency was given 7.14 :

cos , i.e. less than the buoyancy frequency N.

cos /

z

x z

L m k k

L L k mk m

N

, i.e. tilt is independent of wave length!N


7.4.2 Topographic Waves

0Air in statically stable conditions d dz 0 flowing over quasi-

sinusoidal mountain ridge is forced to vertical displacements. The

resulting pattern of streamlines is stationary. This means no time

dep

~

2_2 2 2 2

2_2 2 2

endence of soultion, i.e., 0 in 7.43 : ' exp .

Substitute into PDE 7.42 yields the dispersion relation :

0, or

w w i kx mz

u k k m N k

m N u k


2_2 2 2

_2

0

_

_

Discussion: . Assume 0.

0 if . Then m is real,

w'=w cos .

If is a westerly (eastward flow) then the intrinsic frequency 7.44

k 0 (lower sign in 7.44 and

w

m N u k k

Nm u

k

kx mz

u

Nku

,

3

7.45). We associate

a negative frequency with phase propagation in the -z direction, i.e.,

downward. This means 0. The vertical group velocity (7.45b):

0

is upward!

gz

m

c Nkm


Topographic example

2 22 2

2 22 2 _ _

_ _

0

Topographic height profile: cos

Setting = 0 in the PDE 7.42 :

' ' 0, = - . x

Any solution must satisfy:

' ,0 sin .

M

c

Mz

h x h kx

t

N Nw w m k

z u u

w x Dh Dt u h x u kh kx


_

2 __

2_

_ _

2_22

_

Try: ' , Re where (m, real).

2If - 0 ( ), wide ridge:

' , Re sin

If - 0 0,

ci kx m z cM

cx

i kx mzM M

c c

w x z u kh ie m m i

N uk m m real u k N L

Nu

w x z u kh ie u kh kx mz

Nk m i m u k

u

_

_

2 narrow:

' , sin

x

zM

uN L

N

w x z u kh e kx


7.5.1 Inertial Oscillation

Foe lrager scale gravity waves (few hundred km), Coriolis can no

longer be neglectd, the parcel oscillations are elliptical rather than

straight lines. For simplicity we assume that the parcel motion

2

2

does

not perturb the pressure field. Then from (2.24-25):

17.49

17.50

Leads to ,

Inertial stability for 0

g

g

Du p Dyfv fv f

Dt x Dt

Dv pfu f u u

Dt y

D y Mf y M fy u

Dt y

M


7.5.2 Inertio-gravity Waves

2

We use again the heuristic approach, examining Fig.7.11. Consider an

air parcel oscillating along a slantwise path in the y,z plane. The verical

buoyancy force is cos . The meridional Coriolis foN z

2

22 2

2

2 2 2 2

22 2

2

rce

component is from 7.53 ysin ysin . The

oscillator equation along s becomes

cos ysin

cos sin

cos sin i t

f M y f

D sN z f

Dt

N s f s

D sN f s s e

Dt


2 22

2 8 2 2 20

4

Dispersion relation: cos sin 7.56

Since 10 usually recall ln

, for 0 vertical propagation

110

for 90 horizontal propagation

3

N f

f N f N g d dz

f N N

sff

h


7.7 Rossby Waves

The important waves for large-scale meteorological processes are the

planetary -- or Rossby -- waves. They are a result of the -effect, .

We limit the discussion to free barotropic Rossby wave

df

dy

s. It was shown in

section 4.5 that for a fluid in which =0, the absolute vorticity is

conserved, 0 4.27

0 7.88

since

h

h

D f Dt

u v vt x y

f f f fD f Dt u v v v

t x y y

U


_ _ _

_ _ _

_

Perturbation:

', ', '

' ' '' 0. If we assume 0 :

' '' 0

' 'To solve this PDE, we recall: '

'Introduce function ' , , , such that

u u u v v v

u v v vt x y

u vt x

v u v u

x y x y

x y t

_

2

'', '. Then

' , and the PDE becomes:

'' 0 7.89

v and ux y

ut x x


2 2 2

_ _2 2 2

_ _2 2

2 2

Try ' exp exp ,

' '

' '

''. This gives the dispersion relation

0 7.91

'' Re ex

i kx ly t i kx ly t

k l

u k l i u ikt x

ikx

kk l u k k u k

k l

u ily

p sin

'' Re exp sin

i kx ly t l kx ly t

v ik i kx ly t k kx ly tx


Phase speed of Rossby waves

_2 2 2

2

_

2

Phase speed in the x direction is

where , or

0 7.92

The Rossby zonal phase propagation relative to the mean zonal

flow is always (Fig.7.14).

Also, the zonal speed is

x

x

c u K k lk K

c uK

westward2 1 , i.e., increases with increasing

wavelength.xc K


2 2_

x 2 2 2

2 5 12

2 6

For a typical midlatitude synoptic disturbance with a wavelength of

6000 km and k=l, the Rossby wave speed is

c 2 sin 2 sin2 8 8

7 10 36 10sin ~ c

4 4 6.4 10 10

x x

x

L Ld du

k dy Rd

L d

R d

1

1

_

x

os 45 7

The mean zonal wind speed is generally to the east and larger than

7 . The Rossby wave pattern moves therefore to the east at a

lower speed. If c , the Rossby pattern is stationary re

ms

ms

u

ground.

reinisch_asd85.515_chap#71 chapter 7. atmospheric oscillations linear perturbation theory

Documents

topographic waves slide

buoyancy waves slide

properties of waves

phase velocity slide

group velocity slide

dispersion relation

inertial oscillation

inertiogravity waves