7. WKB Theory For internal Gravity Waves – Non-rotating Case
Slowly Varying Medium
So far we have considered N = constant, Now we shall assume that N is a slowly varying
function with respect to the phase of the wave (N represents the medium).
N is a much stronger function of z than of (x,y) so let us consider N = N(z) only. Then
from the ray equations, that describe the motion of a wave package in a slowly varying
medium, we have
∂K
+ cg • ∇K = −∇Ω
∂t
K
where Ω = N H
K
that is
∂k
+ c g • ∇k = 0 ⇒ k = k o
∂t
initial value is conserved!
∂l
+ cg • ∇l = 0 ⇒ l = l o
∂t
initial value is conserved
∂m
∂Ω
+ c g • ∇m = −
∂t
∂z
only m is changed when the wave goes
through a region of variable N
Also
∂ω
∂Ω
+ c g • ∇ω =
=0
∂t
∂t
ω = ωo initial value
(k,l,ω) are constant for an observer moving with the wave packet (but vary at a fixed
position). The governing equation is the same:
∂2
∂t
2
∇ 2w + N 2(z)∇ 2H w = 0
(1)
Look for a solution of the form:
W = A(z)e i(kx+ly−ωt+θ(z))
1
with
dA
dθ
>>
dz
dz
d2A
and
dz 2
<< A
⎧
d
1
⎪ dz ~ o( λ )
⎪
⎪ dA
1
⎪
~ o(
)
⎨ dz
Lm
⎪
⎪ 2
⎪ d A =~ θ( 1 )
⎪ dz 2
L m2
⎩
As N is varying slowly, locally the solution will look like a plane wave. Then we define
m(z) =
dθ
dz
(For rigorously plane wave in a homogeneous medium θ = mz and m =
dθ
= cons tan t )
dz
Inserting the solution into the governing equation (1) we have
−ω 2 [−(k 2 + l2 )A + Azz − Aθ2z ] − N 2 (k 2 + l2 )A
−iω 2 (2θz Az + θzz A) = 0
real part
imaginary part
or
A zz + A[
(N 2 − ω 2 )K 2H
ω2
− θ2z ] − 2iω 2θ1z / 2
∂
(Aθ1z / 2 ) = 0
∂z
Real and imaginary parts must both be zero. We assumed θz~0(1) and Azz/A<<1, then
the dominant term in the real part is
(N 2 − ω 2 ) 2
K H − θ2z ~0
2
ω
m2 = θ2z =
m(z) =
N2 − ω2
ω
2
which gives
K 2H
∂θ N 2 − ω 2 1 / 2
=(
) KH
∂z
ω2
w = A(z)e i(kx+ly−ωt+θ(z))
and
2
−ω 2 [−(k 2 + l2 )A +
∂2
∂z
2
w] − N 2 (k 2 + l2 )A = 0
dA i (...)
dθ
∂
w=
e
+ iA ei (...) and the second differentiation gives
dz
dz
∂z
∂ 2w
dθ
d 2 A i(kx+ly−ωt+θ(z)) dA dθ i(...)
d 2θ i(...)
−A( ) 2 e i(...)
=
e
+
i
e
+iA
e
2
2
2
dz)
dz dz
∂z
dz
dz
or
−ω 2 [−(k 2 + l2 )A + A zz − Aθ2z ] − ω 2 (k 2 + l2 )A −iω 2[2Azθz + Aθzz ] = 0
Then the vertical phase factor is
z
N 2 (z) − ω 2
zo
ω2
θ(z) = ∫ K H
dz
as N(z) is varying
zo = initial position
We must equate to zero also the imaginary part of the equation for A:
∂
∂
(Aθ1z / 2 ) = (Am1 / 2 ) = 0 => Am1/2 = constant = A(zo)mo1/2
∂z
∂z
or
A(z) =
A(z o )
(m/mo )1/ 2
As m gets larger, i.e. in a region of larger N, A decreases
If the wave propagates to a zT where ω>N, m becomes im, the solution becomes
exponentially decaying beyond zT. In other words, N(z) acts as an index of refraction for
w, and ω=N is the point of total reflection. Then we can find the point of total reflection,
or the turning point for the wave packet (the ray) by using the ray equations. For two
dimensions:
3
dz
km
= c gz = −N 3
dt
K
m2
dx
= c gx = +N 3
dt
K
dz c gz
k
=
=−
dx c gx
m
or
the ray path
Rewriting m = (
N2 − ω2 1/ 2
) k
ω2
as
2
ω =
k
ω
=
m
N2 − ω2
N 2 (z)k 2
and
m2 + k 2
ω
dz
=−
dx
N2 − ω2
Consider the region near zT where N(zT) ~ ω. Expand N2 around zT, the turning point
N 2 (z) = N 2 (z T ) − (z T − z)
dN 2
|z + …
dz T
So
(N 2 − ω 2 ) z T = −(z T − z)
and
dz
=
dx
−ω
(−
or
dN 2
|z
dz T
dN 2
| )(z T − z)
dz z T
(z T − z)1 / 2 dz =
−ω dx
−(
dN 2
)z
dz T
4
But
ω
z
2
∫ (z T − z)1 / 2 = (z T − z) 3 / 2 = −
3
zT
and
zT − z = [
3
2
−(
dN
)z
dz T
−(
2
+ω
xT
2
∫ dx =
2
ω
x
−(
(x T − x)
dN
)z
dz T
]2 / 3 (x T − x) 2 / 3
dN
)z
dz T
The ray path has a cusp at the turning point (xT,zT).
5
MIT OpenCourseWare
http://ocw.mit.edu
12.802 Wave Motion in the Ocean and the Atmosphere
Spring 2008
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.