6.4 Internal Waves when the Density Varies Continuously with Depth
191
below, each of which is almost homogeneous. However, in most cases in the
open sea, water density varies continuously with depth in a more complicated
manner.
6.4 Internal Waves when the Density Varies Continuously with Depth
6.4.1 Modal Structure of Internal Waves
In general, ocean water is continuously stratified, i. e. the water density varies
continuously with the depth, as shown in Fig. 6.2. In such situations, internal
waves still occur, but in contrast to the two layer model, the internal waves
can also propagate in non-horizontal direction. Thus, wave number should be
treated as the vector k = k(kx, ky, kz) in three dimensional space. This fact is
of basic importance for the dynamics of ocean water and biological life in the
ocean, as internal waves propagating in the vertical direction transport energy
and matter from the surface to the bottom of the ocean and vice versa.
Let us briefly describe the conditions which are satisfied by internal wave
motion when the Brunt-Viiisiilii frequency, N, changes with depth. Consider
an ocean with a constant water depth, h. At the impermeable bottom, the
vertical component of the internal wave velocity should be zero, i. e. w = 0 at
z = -h. As we have mentioned above, the internal waves produce only small
disturbances of the free surface, with amplitudes 0 (b.p / p) times the amplitude
of internal waves. Because of this, at the free surface we can also assume that
w = 0 at z = 0, as far as the internal waves are concerned. In fact the surface is
essentially a rigid surface for the internal waves because stratification is weak.
Therefore, the top and bottom conditions, with w = 0 at z = 0 and z = -h,
act as a wave guide for internal waves.
For the prescribed Brunt-Vaisala frequency, N, the frequency of internal
waves, w, cannot be arbitrary. At a given depth, this frequency is bounded
above by the Brunt-Viiisalii frequency, N (z), and below by the inertial frequency f = 2WEsin
minimum frequency that the free internal wave motion can possess. Assuming
that N(z) may be as large as 0.01 rad S-l (TN ~ 10 min), we obtain the band
of possible internal wave frequencies as shown in Fig. 6.6. At a given depth z,
the Brunt-V iiisiilii frequency, N (z), determines the maximal frequency for the
waves which occur at that depth. Since N(z) is greatest in the thermocline
where the variation of density with depth is greatest, internal waves of the
largest frequency occur in the thermocline.
For a given profile of the Brunt-Viiisiilii frequency, N (z), the structure of
internal waves can be determined only numerically and will not be discussed
here. For more details of such solutions, the reader should consult books by
Roberts (1975), Phillips (1977), Le Blond and Mysak (1978), Miropolskiy and
191
below, each of which is almost homogeneous. However, in most cases in the
open sea, water density varies continuously with depth in a more complicated
manner.
6.4 Internal Waves when the Density Varies Continuously with Depth
6.4.1 Modal Structure of Internal Waves
In general, ocean water is continuously stratified, i. e. the water density varies
continuously with the depth, as shown in Fig. 6.2. In such situations, internal
waves still occur, but in contrast to the two layer model, the internal waves
can also propagate in non-horizontal direction. Thus, wave number should be
treated as the vector k = k(kx, ky, kz) in three dimensional space. This fact is
of basic importance for the dynamics of ocean water and biological life in the
ocean, as internal waves propagating in the vertical direction transport energy
and matter from the surface to the bottom of the ocean and vice versa.
Let us briefly describe the conditions which are satisfied by internal wave
motion when the Brunt-Viiisiilii frequency, N, changes with depth. Consider
an ocean with a constant water depth, h. At the impermeable bottom, the
vertical component of the internal wave velocity should be zero, i. e. w = 0 at
z = -h. As we have mentioned above, the internal waves produce only small
disturbances of the free surface, with amplitudes 0 (b.p / p) times the amplitude
of internal waves. Because of this, at the free surface we can also assume that
w = 0 at z = 0, as far as the internal waves are concerned. In fact the surface is
essentially a rigid surface for the internal waves because stratification is weak.
Therefore, the top and bottom conditions, with w = 0 at z = 0 and z = -h,
act as a wave guide for internal waves.
For the prescribed Brunt-Vaisala frequency, N, the frequency of internal
waves, w, cannot be arbitrary. At a given depth, this frequency is bounded
above by the Brunt-Viiisalii frequency, N (z), and below by the inertial frequency f = 2WEsin
minimum frequency that the free internal wave motion can possess. Assuming
that N(z) may be as large as 0.01 rad S-l (TN ~ 10 min), we obtain the band
of possible internal wave frequencies as shown in Fig. 6.6. At a given depth z,
the Brunt-V iiisiilii frequency, N (z), determines the maximal frequency for the
waves which occur at that depth. Since N(z) is greatest in the thermocline
where the variation of density with depth is greatest, internal waves of the
largest frequency occur in the thermocline.
For a given profile of the Brunt-Viiisiilii frequency, N (z), the structure of
internal waves can be determined only numerically and will not be discussed
here. For more details of such solutions, the reader should consult books by
Roberts (1975), Phillips (1977), Le Blond and Mysak (1978), Miropolskiy and