Geoscience Reference
In-Depth Information
along the interface in the positive x direction would have the same form as a surface
wave, i.e.:
z
¼
A
0
sin
ð
kx
ot
Þ:
ð
4
:
16
Þ
The corresponding velocity potential describing motion within the two layers may be
obtained by essentially the same methods as used for the surface waves (see Phillips,
1966
, p. 167) to give:
oA
0
k
cosh kz
sinh kh
1
cos
upper layer
:
f
¼
ð
kx
ot
Þ
0
>
z
>
h
1
ð
4
:
17
Þ
ð
þ
Þ
oA
0
k
cosh k
h
z
lower layer
:
f
¼
cos
ð
kx
ot
Þ
h
1
>
z
>
h
2
sinh kh
2
from which we can find expressions for the horizontal and vertical particle velocities
as in the surface wave case. The phase speed c of the waves is given by:
o
2
k
2
¼
g
0
k
ð
coth kh
2
Þ
1
c
2
¼
coth kh
1
þ
ð
4
:
18
Þ
where g
0
¼
is the reduced gravity.
In many situations in shelf seas, we are concerned with waves of long period
(up to the semi-diurnal (
g
ð=
0
Þ
24 hours) periods). For such
waves, the wavelength is long compared with the depth (
∼
12 hours) and diurnal (
∼
l ≫
h) so that kh
≪
1and
kh
2
Þ
1
etc. so
Equation (4.18)
then simplifies to:
we can approximate coth kh
2
ð
h
1
h
2
h
1
þ
h
1
h
2
h
c
2
g
0
g
0
¼
h
2
¼
ð
4
:
19
Þ
where h
¼
h
1
þ
h
2
is the water column depth. If, in addition, the upper layer thickness
g
0
h
p
. This is of the
same form as the expression for the speed of long surface waves, but the speeds
involved are much lower, mainly because g
0
/g
h
1
≪
h, the phase speed can be further approximated by c
¼
10
3
which makes the
¼ D
r/r
0
∼
internal waves travel
∼
30 times slower than their surface counterparts.
4.2.2
Particle motions in internal waves
The motion described by the velocity potential of
Equation (4.17)
, termed the lowest
mode or the first vertical mode, is an important reference solution which we will now
explore in more detail. The particle velocities u and v are readily obtained by
differentiation of the velocity potentials in the two layers. For example, in the bottom
layer (
h
2
<
z
<
h
1
) at a particular location (x
¼
0) we have:
¼
@
f
oA
0
cosh k
ð
h
þ
z
Þ
x
¼
u
sin ot
@
sinh kh
2
ð
4
:
20
Þ
¼
@
f
@
oA
0
sinh k
ð
h
þ
z
Þ
w
z
¼
cos ot
sinh kh
2
Search WWH ::
Custom Search