Download Chapter 4. Atmospheric Oscillations Linear Perturbation Theory

Chapter 7. Atmospheric Oscillations
Linear 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
1
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 u'
u
u '
u '
u '
u
 u u'
u
u'
u
x
x
x
x
x


Reinisch_ASD85.515_Chap#7
2
7.2 Properties of Waves
Wave motions = oscillations in field variables
that propagate in space
Familiar example of oscillation: the pendulum
Equation of motion
m
m
dV
 mg sin  , restoring force (Fig.7.1)
dt
   mg  d    g  , or
d l
dt
2
dt 2
l
d 2
g
2
2




0
where


(the book uses  instead of  )
2
dt
l
Reinisch_ASD85.515_Chap#7
3
Sinusoidal oscillations and waves
d 2
Solution for 2   2  0 :
dt
   c cos  t   s sin  t or   0 cos  t    ,or
 = 0 Re eit     Re  0ei t     Re  0e i eit 
  Re Cei t 
A sinusoidal wave propagating in the x direction is given by
  x, t   Re Cei kx  t    Cei kx  t  where Re 
 is kept in mind.
C= complex amplitude  C eic
 

  kx   t  phase, or   k  x  t 
k 

k  wave number,   angular frequency  2 f (here f = frequency)
Reinisch_ASD85.515_Chap#7
4
Phase velocity
How does a point of constant phase move through space?
  kx   t    const
D
Dx
0k
  0
Dt
Dt
Dx 
Phase speed c p 

Dt k
Reinisch_ASD85.515_Chap#7
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7.21 Fourier Series
The 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.
At time t = t 0  0 :

2 x
s 1
1
f  x    As sin s  Bs coss ; s k s x   s t0  k s x 
1

0

1
s
1
 2 x   2 x 
f  x  sin s ' dx   As  sin s sin s ' dx   As  sin 
s  sin 
s '  dx
s 1
s 1
 1   1

0
0


1
1
 2 x   2 x 
As ' since  sin 
s  sin 
s '  dx  0 for s  s '
2
 1   1

0
 As 
2
1
1 0
f  x  sin s dx and Bs 
2
1
1 0
f  x  coss dx
Reinisch_ASD85.515_Chap#7
6
7.22 Dispersion and Group Velocity
Usually 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 velocity cg 

.
k
Derivation: Assume the sum of two waves of equal amplitude, and
k1  k   k , k2  k   k , 1     ,  2    
  x, t   exp i  k   k  x      t   exp i  k   k  x      t 
 exp i  kx   t  exp i  k x   t   exp  i  k x   t  
  x, t   exp i  kx   t  cos  k x   t  or
  x, t   cos  kx   t  cos  k x   t 
The envelope has a wavelength of g 
2
. The envelope has a phase
k
g = k x   t.  Group velocity cg 



k
k
Reinisch_ASD85.515_Chap#7
7
7.3.1 Acoustic (Sound) Waves
Sound waves are longitudinal waves as illustrated in Fig.7.5. For simplicity
we assume U   u,0,0  and u  u  x, t  . The governing equations are:
Momentum
Continuity
Thermodynamic Energy
Du 1 p

0
 7.6 
Dt  x
1  u

0
 7.7 
 t x
Dln
 0 from  2.43 for adiabatic process
Dt
From poisson equation  2.44 
p 
R cp
p p 
R cp
p
R cp
s
1 R c p
s
 T s  
 ln   ln
 ln p
 ln 


R  p 
R
 p
cv 1
Dln
D ln  1 D ln p


 0 where1  R c p 
  7.9 
Dt
Dt
 Dt
cp 
Reinisch_ASD85.515_Chap#7
8
Combining  7.7  and  7.9  gives
1 D ln  u

0
 Dt
x
Perturbation method:
 7.10 
u  x, t   u  u '  x.t 
p  x, t   p  p '  x.t 
  x, t      '  x.t 
Neglecting products of primed quatities and considering
that u etc are constant:
Reinisch_ASD85.515_Chap#7
9
 
1 p '

0
  u  u '
x 
 x
 t
 
u '


u
p
'


p
0


x 
x
 t
Eliminate u '
 


u


x 
 t
2
 7.12 
 7.13
 

by applying   u  on  7.13 , and inserting  7.12  :
x 
 t
 

   u u'
2


 p 2 p '

t

x




p '  p
   u  p '
0
2
x
x 
 x
 t
Reinisch_ASD85.515_Chap#7
10
 
 p 2 p '

0
  u  p '
2
x 
 x
 t
Try the following solution:
2
 7.14 
p '  A exp ik  x  ct  
2
2  
 2
 
2

u
p '  ikc  iku
  u  p '   2  u2
2 
x 
xt
x 
 t
 t
2


2
p'
 p 2 p '  p
2

ik
p'


2

x


For p '  A exp ik  x  ct   to be a solution requires therefore that

ikc  iku
cu

2

p
2
 ik   0 Dispersion relation

p
 u   RT

Reinisch_ASD85.515_Chap#7
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7.3.2 Shallow-Water Gravity Wave
Consider 2-fluid system pictured in Fi. 7.7. Stably stratified if 1   2 .
Waves may propagate along the interface. Assume incompressibility of
the 2 fluids (this avaoids sound waves). In hydrostatic equilibrium:
p  -  g z. Differentiate re x:
 p

 g
 0, if we assume no horizontal gradient in each fluid.
x z
x
This means also that the horizontal pressure gradient does not vary
with height:
 p  p
0

x z z x
Reinisch_ASD85.515_Chap#7
12
=0
Du u u u
u
1 p1
x-momentum equation
 u v w 
Dt =0
t
x y
z
1 x
=0
continuity equation
1  u v w
   0
1 t x y z
1 p1 g p h
But

from Fig. 7.7. Then:
1 x
1 x
u
u
u
g p h
u w 
. RHS is independent of z  u is indep. of z
t
x
z
1 x
u w
u

 0  w( h )  w(0)   h
x z
x
Dh h
h
But w  h  

 u , w 0  0
Dt t
x
Reinisch_ASD85.515_Chap#7

13
Internal Gravity (or Buoyancy) Waves
In 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 wave front means horizontal propagation. 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  :
D2
 z   N 2 z ,
2
Dt
N2  g
d ln  0
dz
Assume the wave front is tilted as in Fig. 7.8, then the restoring force is
  N 2 z  cos     N 2 s cos   cos     N cos    s
2
 7.24 
The momentum equation becomes now:
D2
2
 s    N cos    s   s  A exp  i  N cos   t  ; sinusoidal oscillation.
2
Dt
Reinisch_ASD85.515_Chap#7
14
Two-dimensional Internal Gravity Waves
Momentum equation neglecting Coriolis and friction:
DU
1
  p  g
Dt

u
u
w 1 p
u  w

0
 7.25
t
x
z  x
w
u
w 1 p
u  w

g 0
 7.26 
t
x
z  z
Continuity equation (using Boussinesq approximation):
u w

0
x z
Energy equation:
 7.27 
Dln 1 D




0
u
w
0
Dt
 Dt
t
x
z
 7.28
Reinisch_ASD85.515_Chap#7
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Perturbation - Linearization
R cp
R cp
p 
p  ps 
Poisson equation:   T  s 

 
p
R

 
 p
   0   ' (small density perturbance in momentum eq.)
_
_
_
u  u  u ', w  w ', p  p  z   p ',     z    '
_
 7.29 
 7.30 
_
Notice p and  are functions of height, 0 is not.


_
d p z
p  z   ps 
   0 g and  
dz
0  p_ z 
  
 7.29  becomes:
_
_
R cp
_
or ln   
1
_
ln p  z   ln  0  const
_ 

 _   ' 
 
p
'
 ' 
1






ln  1  _    ln p 1  _  ln   0 1     const
 

0  
    


p

 
Reinisch_ASD85.515_Chap#7
 7.34 
16
   ' 
  '  
'
ln   1  _    ln   ln 1  _    ln   _ , etc
    
   

'
'
1 p '
Using this in  7.34: _   _ 

p 0
'
p'
'
p'
  '   _ 0  _
  _ 0  2
 7.35 
cs

 p 

0
_
with c   p 0   RT . But in atmosphere
2
s
'
'
_

'
_

0
p'
, so
2
cs
0 . Substitue into linearized  7.25-28  :
Reinisch_ASD85.515_Chap#7
17
1 p '
 _  
0
  u  u '
x 
0 x
 t
1 p '  '
 _  
 _ g 0
  u  w '
x 
0 z 
 t
 7.37 
u ' w '

0
x x
 7.39 
 7.38
_
 
d

0
 7.40 
  u  ' w '
x 
dz
 t
Eliminate p' from first two by cross differentiation and subtraction:
_
Reinisch_ASD85.515_Chap#7
18
2
  _   u ' 1  p '

0
 u 
x  z 0 xz
 t
2
_



w
'
1

p ' g  '



 _
0
 u 
x  x 0 xz  x
 t
  _   w ' u '  g  '

 0.
  u 
 _
x  x z   x
 t
 _  
   u  '
2
_
   w ' u '  g  t
x 


u


 0. Using  7.40  :

 
 _
x   x z  
x
 t
_
   w ' u '  g d  w '



u



0.
Apply
:

 
 _
x   x z   dz x
x
 t
_
2
_
    w '  u '  g d  2w '

 0.
 _
 u   2 
2
x   x
xz   dz x
 t
_
2
2
2
Reinisch_ASD85.515_Chap#7
19
 2u '
2w '
From  7.39  :
 2 .
xz
z
2
   2w ' 2w ' 

w'

2
0
 u   2  2  N
2
x   x
z 
x
 t
2
_
_
_
g d
d ln 
with N  _
g
 const
dz
 dz
2
is the buoyancy frequency squared. Solution of PDE:
~
 ~ i kx  mz t  
w '  Re  we
where w  wr  iwi .
 7.43



With   kx  mz   t , w '  wr cos   wi sin  if m is real.
m can be complex: m  mr  imi . Then w '  e
Reinisch_ASD85.515_Chap#7
 mi z
 ~ i kx  mr z t  
Re  we


20
Dispersion relation
Substitute into PDE:
2
2
2







w'


2
0
  u   2  2  w ' N
2
x   x z 
x
 t
2
_
2


2
2
2 2

i


u
ik

k

m

N
k 0


 


_
 Nk
   uk 

where κ  k 2  m 2
κ
k 2  m2
_
 Nk
(We assumed here m  mr ).   "intrinsic frequency".
If we define the wave vector κ   k , l , m  , then
κ  r  kx  ly  mz and   κ  r   t
k  2 Lx , l  2 Ly , m  2 Lz  Ly   in our 2-D derivation 
Reinisch_ASD85.515_Chap#7
21
  kx  mz   t0  const
z    k m  x  const
z
k
dz
k
   0 in Figure
dx
m

k
k
x
Lx
Assume k  0, then m  0.
The wave vector points downward and eastward. The vertical
wavelength is Lz  2 m , the horizontal wavelength L x  2 k .
Reinisch_ASD85.515_Chap#7
22
_
The vertical phase speed relative to mean flow is cz 
 uk

k

k
_
u
_
The horizontal phase speed relative to mean flow is cx 
 uk
m
_
The phase speed in the direction of κ   k , 0, m  is c 
 uk
κ
 cx2  cz2
κ  k  k 2  m2
Group velocities:
 
k  k kk
k  k k k _
 _
cgx 
u N
u N
2
k
k
k2
_
Nm 2
cgx  u  3
k
cgz 

 Nkm

m
k3
 7.45a 
 7.45b 
Reinisch_ASD85.515_Chap#7
23
The tilt angle  against the vertical is given by
Lz
1m
k
k
cos  



2
2
2
2
2
2
k
Lx  Lz
k m
1 k   1 m 
On slide 14, the heuristic gravity frequency was given  7.14  :
   N cos  , i.e. less than the buoyancy frequency N.
cos    / N , i.e. tilt is independent of wave length!
Reinisch_ASD85.515_Chap#7
24
7.4.2 Topographic Waves
Air in statically stable conditions  d 0 dz  0  flowing over quasisinusoidal mountain ridge is forced to vertical displacements. The
resulting pattern of streamlines is stationary. This means no time
~
dependence of soultion, i.e.,   0 in  7.43 : w '  w exp i  kx  mz   .
Substitute into PDE  7.42  yields the dispersion relation :
2

 2
2
2 2

u
k
k

m

N
k  0, or






_
m N
2
2
2
 
2
u

k
 
 
_
Reinisch_ASD85.515_Chap#7
25
2
 
Discussion: m 2  N 2  u   k 2 . Assume k  0.
 
_
N
2
m  0 if u  . Then m is real,
k
_
w'=w ,0 cos  kx  mz   w  .
_
If u is a westerly (eastward flow) then the intrinsic frequency  7.44 
_
   u k   Nk k  0 (lower sign in 7.44 and 7.45). We associate
a negative frequency with phase propagation in the -z direction, i.e.,
downward. This means m  0. The vertical group velocity (7.45b):
cgz   Nkm k 3  0
is upward!
Reinisch_ASD85.515_Chap#7
26
Topographic example
Topographic height profile: h  x   hM cos kx
Setting   t  = 0 in the PDE  7.42  :
2
N

 w '  _  w '  0,

u
Any solution must satisfy:
 2
2
 2  2
z
 x
_
2
m 
c 2
N
=  _  - k 2.
 
u
_
w '  x, 0    Dh Dt  z 0  u h x   u khM sin kx.
Reinisch_ASD85.515_Chap#7
27
i  kx  mc z  

Try: w '  x, z    u khM Re ie
where m c  m  i  (m, real).


_
2
_
N
_
2 u
2
c
If  _  - k  0  m  m (real ), u k  N  Lx 
wide ridge:
 
N
u
_
w '  x, z    u khM Re  ie
i  kx  mz 
_
   u khM sin  kx  mz 

2
_
N
_
2 u
2
c
c 2
If  _  - k  0  m  i    m   0, u k  N  Lx 
narrow:
 
N
u
_
w '  x, z    u khM e   z sin kx
Reinisch_ASD85.515_Chap#7
28
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 does
not perturb the pressure field. Then from (2.24-25):
Du
1 p
Dy
 fv 
 fv  f
Dt
 x
Dt
Dv
1 p
  fu 
  f u  ug 
Dt
 y
 7.49 
 7.50 
D 2 y
M
Leads to
f
 y, M  fy  u g
2
Dt
y
Inertial stability for M  0
Reinisch_ASD85.515_Chap#7
29
7.5.2 Inertio-gravity Waves
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  N 2 z cos  . The meridional Coriolis force
component is from  7.53  f  M y   y sin   f 2 y sin . The
oscillator equation along  s becomes
D 2 s
2
2


N

z
cos



f
 y sin 



2
Dt
   N 2 s cos 2      f 2 s sin 2 
D 2 s
2
2

   N cos     f sin    s   s
2


Dt
Reinisch_ASD85.515_Chap#7
ei t
30
Dispersion relation:  2   N cos     f sin 
2
Since f 2
2
 7.56 
108  usually N 2  f 2  recall N 2  g d ln  0 dz 
 f    N,
  N for   0 vertical propagation
1
4 

10
s

  f for   90 horizontal propagation  f

  3h 


Reinisch_ASD85.515_Chap#7
31
7.7 Rossby Waves
The important waves for large-scale meteorological processes are the
df
.
dy
We limit the discussion to free barotropic Rossby waves. It was shown in
planetary -- or Rossby -- waves. They are a result of the  -effect,  
section 4.5 that for a fluid in which   U =0, the absolute vorticity is
conserved, Dh   f  Dt  0
 4.27 



  u  v    v  0
x
y 
 t
f
f
f
f
since Dh f Dt   u  v  v   v
t
x
y
y
 7.88
Reinisch_ASD85.515_Chap#7
32
Perturbation:
_
_
_
u  u  u ', v  v  v ',      '
_
 ' _  ' _  '
u
v
  v '  0. If we assume v  0 :
t
x
y
 ' _  '
u
 v'  0
t
x
v u
v ' u '
To solve this PDE, we recall:      ' 

x y
x y
 '
 '
Introduce function  '  x, y, t  , such that
 v ', and
 u '. Then
x
y
 '  , and the PDE becomes:
 '
 _  2
0
  u    ' 
x 
x
 t
Reinisch_ASD85.515_Chap#7
 7.89 
33
Try  '   exp i  kx  ly   t    exp i ,    kx  ly   t 
 2 '    k 2  l 2  '
_
 _   2


2
2
  u    '    k  l   i  u ik  '
x 


 t
 '

  ik '. This gives the dispersion relation
x
_
_
k


2
2
  k  l     u k    k  0    u k  2 2
 7.91
k l


 '
u' 
  Re  il exp i  kx  ly   t    l sin  kx  ly   t 
y
 '
v' 
 Re  ik exp i  kx  ly   t    k sin  kx  ly   t 
x
Reinisch_ASD85.515_Chap#7
34
Phase speed of Rossby waves
Phase speed in the x direction is
cx 

k
_
_
 u
cx  u 

2

K2
where K 2  k 2  l 2 , or
 7.92 
0
K
The Rossby zonal phase propagation relative to the mean zonal
flow is always westward (Fig.7.14).
Also, the zonal speed is cx  1 K 2 , i.e., increases with increasing
wavelength.
Reinisch_ASD85.515_Chap#7
35
For a typical midlatitude synoptic disturbance with a wavelength of
6000 km and k=l, the Rossby wave speed is

L2x d
L2x d
cx  u   2  2
2 sin    2
2 sin 
2k
8 dy
8 Rd
_
L2x d
7  105  36  1012
1

sin

~
c
os
45


7
m
s
4 R 2 d
4  6.4 106 10
The mean zonal wind speed is generally to the east and larger than
7 m s 1. The Rossby wave pattern moves therefore to the east at a
_
lower speed. If c x  u , the Rossby pattern is stationary re ground.
Reinisch_ASD85.515_Chap#7
36