Advanced Dynamical Meteorology
Roger K. Smith
CH03
Waves on moving stratified flows
Small-amplitude internal gravity waves in a stratified shear
flow U = (U(z),0,0), including the special case of uniform flow
U(z) = constant.
Linearized anelastic equations:
u t  Uu x  wUz   Px
w t  Uw x  Pz  b
b t  Ub x  N 2 w  0
u x  w z  w / Hs  0
Travelling wave solutions
Substitute
ˆ
ˆ
(u, w, P, b)  (u(z),
w(z),
P(z), b(z)) exp[i(kx  t)]
a set of ODEs or algebraic relationships between u
 (z) etc.

  wU
 z  ikP
i(  Uk)u
ˆ
ˆ b
ˆ  P
i( Uk)w
z
ˆ  N2w
ˆ 0
i(  Uk)b

w
z
iku  w
0
Hs
ˆ bˆ to
Relate quantities û, P,
:
w
*
2
i 






ˆ
ˆ
w
i

N
w


*
ˆ  w
ˆ b)
ˆ P,
ˆ z   kU z 
ˆ ,
(u,
 ˆz
 , 2  w
w
* 
Hs  k 
Hs  
 k 


 i 
*    Uk
is the intrinsic frequency of the wave.
The frequency measured by an observer
moving with the local flow speed U(z).
Then the second equation gives an ODE for

w
L
M
M
N
O
F
I
F
I
N
 0
 G  1Jk P
w
G
J
H K H K P
Q
1
k
k Uz
1


w zz 
w z  * U zz  *

Hs

 Hs
Hs
Put
2
2
*2
z
~ (z)
 (z)  exp(z / 2Hs )w
w
c/k
*  k(c  U)
Then
Scorer
parameter
2
2 ~
~
wzz  (l (z)  k )w  0
2
N
l 2 (z) 
Uc
a f
2

Scorer’s
equation
U zz  U z / Hs
1

Uc
4 Hs2
Free waves
 When w satisfies homogeneous boundary conditions (e.g. w = 0)
at two particular levels, we have an eigenvalue problem.
w = 0 at z = 0, H
~  0 at z  0, H
w
 For a given horizontal wavenumber k, free waves are possible
only for certain phase speeds c.
 Alternatively, when c is fixed, there is a constraint on the
possible wavenumbers.
 The possible values of c(k), or given c, the possible values of k,
and the corresponding vertical wave structure are obtained as
solutions of the eigenvalue problem.
 Often the eigenvalue problem is analytically intractable.
Example 1: Stratified shear flow between rigid horizontal
boundaries
w=0
z=H
N(z)
z
U(z)
z=0
x
w=0
Example 2: Wave-guide for waves on a surface-based stable
layer underlying a deep neutrally-stratified layer
no vertical wave propagation!
neutrally-stable layer
N=0
z
stable layer
N(z)
U(z)
x
A model for the “Morning-Glory” wave-guide
The Morning-Glory
The Gulf of Carpentaria Region
Morning-Glory
cloud lines
A Morning Glory over Bavaria
Forced waves
 If the waves are being generated at a particular source
level, the boundary conditions on Scorer’s equation are
inhomogeneous and a different type of mathematical
problem arises: -
Example 1: Waves produced by the motion of a (sinusoidal)
corrugated boundary underlying a stably-stratified
fluid at rest.
z
x
c
corrugated boundary
Assume:
1. no basic flow (U = 0)
2. the Boussinesq approximation is valid (1/Hs = 0)
3. N and c are given constants
the vertical wavenumber m is determined by the
dispersion relation
m 2  N 2 / c2  k 2
k is the horizontal wavenumber
c is the speed of the boundary
the wavenumber of the corrugations
The group velocity of the waves is
c3
c g  2 ( m 2 , mk )
N
The vertical flux of mean wave energy is
F  pw  Ewg
the mean wave
energy density
Here c < 0
the vertical component
of the group velocity
sgn(wg) = sgn(mk)
k
these waves propagate
energy vertically
phase
lines
c
The solution of
2
2 ~
~
wzz  (l (z)  k )w  0 is determined by
~
1. A condition at z = 0 relating w to the amplitude
of the corrugations, and the speed of boundary motion, c.
2. A radiation condition as z  
 the condition at z = 0 is an inhomogeneous condition
 the condition as z   ensures that the direction of
energy flux at large heights is away from the wave source,
i.e. vertically upwards.
This condition is applied by ensuring that only wave
components (in this case there is only one) with a positive
group velocity are contained in the solution.
Conditions 1 and 2 determine the forced wave solution uniquely
Example 2: Waves produced by the flow of a stably-stratified
fluid over a corrugated boundary.
U
z
x
This flow configuration is a Galilean transformation of the first:
the boundary is at rest and a uniform flow U (> 0) moves over it.
Mountain waves
When a stably-stratified airstream crosses an isolated ridge or, more
generally, a range of mountains, stationary wave oscillations are frequently
observed over and to the lee of the ridge or range. These so-called mountain
waves, or lee waves, may be marked by smooth lens-shaped clouds, known
as lenticular clouds.
Linear Theory
Assume: 1. Uniform flow U
2. Boussinesq fluid (1/Hs = 0)
3. Small amplitude sinusoidal topography:
z  h(x)  hm cos kx
Surface boundary condition: streamline slope equals the
terrain slope
(U + u, w) = (dx, dh)
linearize
w
dh

on z  h( x)
U  u dx
dh
wU
at z  0
dx
Let (x, z, t ) be the vertical displacement of a fluid parcel.
w
(x, z, t )
D
Dt

wU
x
linearize

U = constant
F
G
H
satisfies the same equation as w
IJ
K
d 2
N2
2 


k
0
2
2
dz
U
The general solution is
  ikU
w
m2  N2 / U2  k 2
 (z)  Aeimz  Be  imz
constants
 (z)  Aeimz  Be  imz
The boundary condition at the ground is ( x,0)  h m e ikx
 (0)  h m
The upper boundary condition and hence the solution depends
on the sign of m 2  N 2 / U 2  k 2
call
There are two cases:
l2
0 < |k| < l
l < |k|
l
N
0
U
m real
m imaginary
Two cases:
I. 0 < |k| < l
m real
if m, k > 0, we must choose B = 0 to satisfy
the radiation condition.
The phase lines slope upstream with height
Then the complete wave solution is
 (x, z)  h m e i( kx  mz )
The real part is implied
sgn(mk) > 0
II.
l < |k|
Then
m imaginary
(= im0 say) where m0   k 2  l2
 (z)  Ae  m0z  Be m0z
The upper boundary condition requires that  (z)
remains bounded as z  
B=0
Then the complete wave solution is
 (x, z)  h me(
k 2 l 2 )z ikx
Here, the phase lines are vertical
Streamline patterns for uniform flow over sinusoidal
topography
vertical propagation: U|k| < N
0 < |k| < l
U
vertical decay: N < U|k|
 (x, z)  h m e i( kx  mz )
l < |k|
U
 (x, z)  h me(
k 2 l 2 )z ikx
Exercise
Calculate the mean rate of working of pressure forces at the
boundary z = 0 for the two solutions
Show that for  (x, z)  h m e
drag on the airstream
whereas for  (x, z)  h me(
i ( kx  mz )
the boundary exerts a
k 2 l 2 )z ikx
it does not.
Show also that, in the former case, there exists a mean
downward flux of horizontal momentum and that this is
independent of height and equal to the drag exerted at
the boundary.
Flow over isolated topography
U
hm
b
Consider flow over an isolated ridge with
h(x) = hm/[1 + (x/b)2]
maximum height of the ridge = hm
characteristic half width = b
The ridge is symmetrical about x = 0
it may be expressed as a Fourier cosine integral
h( x ) 
z
z

h( k )cos( kx)dk
0
 Re

h( k )e ikx dk
0
where
2
h( k ) 

2h m

Re
0 h ( x ) cos kx dx 

z
za f
z
2h m b

Re


0
e ikx dx
1 x / b

0
e ikbu du
1  u2
2
The solution for flow over the ridge is just the Fourier synthesis
of the two solutions for 0 < |k| < l and l < |k|.
z
L
(x, z)  Re Mh( k)e
N
l
i[ kx ( l 2  k 2 )1/ 2 z ]
0
dk 
z

h( k ) e
ikx ( k 2  l 2 )1/ 2 z
dk
l
O
P
Q
Put kb = u
(x, z)  h m
z
z
L
ix I
L
F
Re Mexp M
u 1 
 id
l b u i
N NH b K
ix I
L
F

exp M
u 1 
d
u l b i
NH b K
lb
2 2
0

lb
= I1 + I2 , say
2
O
P
Q
z OO
du P
P
b QQ
2 1/ 2
2 2 1/ 2
z
du
b
Two limiting cases:
narrow ridge, lb << 1
I1 << I2 and
z
ix z IO
L
F
( x, z)  h Re
exp M
u 1   P
du
N H b b KQ
b I
h
F

Hb  z K[1  x / (b  z) ] .

m
0
m
2
2
 each streamline has the same general shape as the ridge
 the width of the disturbed portion of a streamline increases
linearly with z
 its maximum displacement decreases in proportion to
1/(1 + z/b) , becoming relatively small for heights a few times
greater than the barrier width.
narrow ridge, lb << 1  Nb/U << 1
Steady flow of a homogeneous fluid over an isolated
two-dimensional ridge, given by
b I
F
(x, z) 
Hb  z K[1  x
hm
2
2 .
/ ( b  z) ]
The highest wind speed and the lowest pressure occurs at
the top of the ridge.

broad ridge, lb >> 1
Nb/U >> 1
I1 >> I2 and
( x, z)  h m Re{e ilz
z

0
 hm
a
f
exp  u 1  ix / b du}  h m
cos lz  ( x / b)sin lz O
L
.
M
P
N 1  (x / b) Q
L
be O
Re M P
Nb  ix Q
ilz
2
 In this limit, essentially all Fourier components propagate
vertically.
Buoyancy-dominated
hydrostatic flow over
an isolated twodimensional ridge,
given by
(x,z)  h m 
 cos lz  (x / b)sin lz 


2
1

(x
/
b)


 The pressure difference across the mountain results in a net
drag on it.
 The drag can be computed either as the horizontal pressure
force on the mountain
z

dh
D
p(x,0) dx

dx
or as the vertical flux of horizontal momentum in the wave
motion
D  0 (z)
Boussinesq approximation
z

uwdx

0 (z)   *
moderate ridge, lb  1
Nb / U  1
The integrals I1 and I2 are too difficult to evaluate analytically,
but their asymptotic expansions at large distances from the
mountain (compared with 1/l) are revealing.
0/2
Let u  lb cos where
x  r cos  ,
Then
I1  h m lb Re
z  r sin 
z
z

2
where
0
(sin  e  lb cos  ) e irl cos( ) d
0
I1  Re

2
h(  ) e
irl g (  )
d
0
Asymptotic expansion for large r by stationary phase method
I1  h m lb Re
z

2
(sin  e  lb cos  ) e irl cos( ) d
0
Far field behaviour of I1
  ix  
I2  h m max exp  u 1   
lb  u 
b 
 
 h m b exp (lb) / z.


e  (u lb)z / b du
lb
I2 is at most O(b/r)
Since lb  (1) , I1 + I2 is always the same order as I1
not surprising since I1 contains all the vertically propagating
wave components.
Then for x > 0 and  not to close to 0 or 
(x, z)  h m lb sin e
 lb cos 
F I cos Flr   I
H K H 4K
2
lr
1/ 2
Lee waves in the case where lb  1
Scorer’s Equation
2
2 ~
~
wzz  (l (z)  k )w  0
2
N
l 2 (z) 
Uc
a f
2
U zz  U z / Hs
1


Uc
4 Hs2
Trapped lee waves
 It was pointed out by Scorer (1949) that, on occasion, a long
train of lee waves may exist downstream of a mountain
range.
 Scorer showed that a long wave train is theoretically possible
if the parameter l2(z) decreases sufficiently rapidly with
height.
 Eigensolutions of
~  ( l 2 ( z )  k 2 )w
~ 0
w
zz
are difficult to obtain when l2 varies with z.
A two-layer model
perturbation streamfunction
U
l12  N12 / U 2
,
1
layer 1
U
l 22  N 22 / U 2
,
2
layer 2
~
~
In the Boussinesq approximation  satisfies the same ODE as w
~
d2
2
2 ~
n

(
l

k
) n  0 , ( n  1,2)
n
2
dz
u x  wz  0

w   x

~
~  ik
w
~
d2
2
2 ~
n

(
l

k
) n  0 , ( n  1,2)
n
2
dz
Assume stationary wave solutions so that c = 0
then l2 = N2/U2 as before
Solutions:
~  exp(imz)

where m2 = l2  k2
Assume that m22 > 0, so that vertical wave propagation is
possible in the lower layer.
Case (a):
upper layer more stable (l1 > l2)
k1
layer 1
m1
k
transmitted wave
layer
interface
z=0
k
layer 2
k2
k2
- m2
reflected wave
m2
k
incident wave
Streamfunctions:
transmitted wave
1   e
i ( kx  m1z )
incident plus reflected wave
m1 , m2 > 0
 2  a ei( kxm2z)  ei( kxm2z)
a given
Wave solutions are coupled by conditions expressing the
continuity of interface displacement and pressure at z = 0.
Continuity of interface displacement at z = 0
 1   2
at z  0
1 w
2
w
at z  0
1 
2

provided, as here,
that U1 = U2
at z  0
When the density is continuous across the interface,
continuity of pressure implies that
 1z  
 2z

at z  0
F
G
H
see Ex. 3.6
IJ
K
2 m 2a
m 2  m1

and  
a
m1  m 2
m1  m 2
always a transmitted wave
no trapped "resonant"
solutions in the lower layer
Case (a):
upper layer less stable (l1 < l2)
Assume that
m12 < 0
exponential decay
layer 1
k
layer
interface
z=0
k
layer 2
k
reflected wave
incident wave
The appropriate solution in the upper layer is the one which
decays exponentially with height:
1  e
ikx  m1 z
 2  a ei( kxm2z)  ei( kxm2z)
 = a
 2  Aeikx sin m 2 z
w 2  ik 2  ikAeikx sin m 2 z
w 2  0 at
z  h if
sin m2 h  0
m2h
tan m 2 h  
| m1| h
put x = m2h
y

 m2h  
2
3
 m2 h  2
2


2

2

3
2
x
y
y  tan x
x
| m1| h

 m2h  
2
2
2
2
2

m

l

k
2
2
4h 2
Recall that
m12 < 0
l12  k 2
2

l12  k 2  l 22 
4h2
To illustrate the principles involved in obtaining solutions for
trapped lee waves, we assume that the upper layer behaves as
a rigid lid for some appropriate range of wavelengths
w 0  0
z=H
U
l2  N2 / U2
,

z=0
Solution
( x, z)  Ae  ikx sin m( H  z)
constant
m2  l2  k 2
The general solution may be expressed as a Fourier integral of
( x, z)  Ae  ikx sin m( H  z)
 (x, z) 
1
2
Uh m

2 
1  ( x / b)
z
z

A( k )e  ikx sin m(H  z)dk

1
2

A( k )e  ikx sin mH dk

A solution showing trapped lee waves.
Summary
Lee waves for lb  1  Nb/U  1
narrow ridge, lb << 1  Nb/U << 1
Steady flow of a homogeneous fluid over an isolated
two-dimensional ridge, given by
b I
F
(x, z) 
Hb  z K[1  x
hm
2
2 .
/ ( b  z) ]
The highest wind speed and the lowest pressure occurs at
the top of the ridge.
Broad ridge 1 << lb
 Nb/U << 1
(x,z)  h m 
 cos lz  (x / b)sin lz 


2
1

(x
/
b)


Cross-section of potential temperature field (K) during a severe downslope
wind situation in the Colorado Rockies on 11 January 1972. From Klemp
and Lilly, 1975.
The End