Reinisch_ASD85.515_Chap#7 1 Chapter 7. Atmospheric Oscillations Linear Perturbation Theory To describe large-scale atm ospheric m otions with som e accuracy requires numerical techniques. This makes it difficultto understand the fundam entalprocesses 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 equationsusing the perturbation m ethod.
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Reinisch_ASD85.515_Chap#71 Chapter 7. Atmospheric Oscillations Linear Perturbation Theory.
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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.
Reinisch_ASD85.515_Chap#7 2
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
Reinisch_ASD85.515_Chap#7 3
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
Reinisch_ASD85.515_Chap#7 4
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
Reinisch_ASD85.515_Chap#7 5
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
Reinisch_ASD85.515_Chap#7 6
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
Reinisch_ASD85.515_Chap#7 7
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
Reinisch_ASD85.515_Chap#7 8
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
Reinisch_ASD85.515_Chap#7 9
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
Reinisch_ASD85.515_Chap#7 10
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
Reinisch_ASD85.515_Chap#7 11
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
Reinisch_ASD85.515_Chap#7 12
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
Reinisch_ASD85.515_Chap#7 13
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
Reinisch_ASD85.515_Chap#7 14
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.
Reinisch_ASD85.515_Chap#7 15
Two-dimensional Internal Gravity Waves
Momentum equation neglecting Coriolis and friction: