188
6 Internal Waves
The elementary solution of Eq. (6.8) takes the form:
((t) = cos(N t).
(6.9)
Equation (6.9) represents the simple harmonic motion of element of the fluid
moving up and down. The quantity N (z) is known as the Brunt-ViiisiiHi.
frequency or stability frequency:
{
d }I/2
N(z) = -~~
P dz
(6.10)
Usually frequency N (z) has a maximum in the thermocline where the density
gradient is the greatest, and it decreases both above and below this level, as the
water becomes more homogeneous (see Fig. 6.2). Sometimes multiple maxima
in the vertical profile of N (z) are observed; they correspond to the seasonal
thermoclines.
For the stable static equilibrium in an incompressible fluid (case a in Fig. 6.4),
o P I 0 z < 0, and N 2 > 0. The period 27r I N varies in the ocean from a few
minutes in the thermocline, to several hours in the deep ocean. In some circumstances, such as in lakes and in the deep ocean where oploz is very small,
the extra term -g2/C; (where Cs is the sound speed in the ocean) should be
included for the Brunt-Viiisiilii frequency N 2 .
In case b (Fig. 6.4), when oploz > 0, and N 2 < 0, Eq. (6.8) has no solution
of the wave type. In fact, the only solution is:
(6.11 )
and as t increases, the displacement goes exponentially to infinity. When N 2 =
0, the water mass is uniform in terms of density.
As will be shown in the following sections, the frequency N 2 plays a fundamental role in determining internal wave patterns.
6.3 Internal Waves in Two-Layer Ocean
In order to shed some light on the nature of internal waves, we first consider
a simple case of a two-layer ocean, which is a good approximation when the
thermocline and pycnocline change very sharply with water depth. We consider
the situation with an upper layer of depth hI and density PI, and a lower layer
of depth h2 and density P2, with an internal wave of length Li at the interface
between the layers (Fig. 6.5). It should be noted that at the level z = -hIl
the gradient of water density becomes infinitely large; thus the Brunt-Viiisiilii
frequency, N(z), cannot be defined at that level. The period T; and phase
6 Internal Waves
The elementary solution of Eq. (6.8) takes the form:
((t) = cos(N t).
(6.9)
Equation (6.9) represents the simple harmonic motion of element of the fluid
moving up and down. The quantity N (z) is known as the Brunt-ViiisiiHi.
frequency or stability frequency:
{
d }I/2
N(z) = -~~
P dz
(6.10)
Usually frequency N (z) has a maximum in the thermocline where the density
gradient is the greatest, and it decreases both above and below this level, as the
water becomes more homogeneous (see Fig. 6.2). Sometimes multiple maxima
in the vertical profile of N (z) are observed; they correspond to the seasonal
thermoclines.
For the stable static equilibrium in an incompressible fluid (case a in Fig. 6.4),
o P I 0 z < 0, and N 2 > 0. The period 27r I N varies in the ocean from a few
minutes in the thermocline, to several hours in the deep ocean. In some circumstances, such as in lakes and in the deep ocean where oploz is very small,
the extra term -g2/C; (where Cs is the sound speed in the ocean) should be
included for the Brunt-Viiisiilii frequency N 2 .
In case b (Fig. 6.4), when oploz > 0, and N 2 < 0, Eq. (6.8) has no solution
of the wave type. In fact, the only solution is:
(6.11 )
and as t increases, the displacement goes exponentially to infinity. When N 2 =
0, the water mass is uniform in terms of density.
As will be shown in the following sections, the frequency N 2 plays a fundamental role in determining internal wave patterns.
6.3 Internal Waves in Two-Layer Ocean
In order to shed some light on the nature of internal waves, we first consider
a simple case of a two-layer ocean, which is a good approximation when the
thermocline and pycnocline change very sharply with water depth. We consider
the situation with an upper layer of depth hI and density PI, and a lower layer
of depth h2 and density P2, with an internal wave of length Li at the interface
between the layers (Fig. 6.5). It should be noted that at the level z = -hIl
the gradient of water density becomes infinitely large; thus the Brunt-Viiisiilii
frequency, N(z), cannot be defined at that level. The period T; and phase