190
6 Internal Waves
Let us assume that densities PI and P2 are the same as above, and additionally
let hI = h2 = h. Thus, Eq. (6.16) gives:
P2 - PI gh =
P2 + PI
P2 - PI G = 0.038G.
P2 + PI
(6.17)
Again the phase speed of the internal wave is much less than that of a long
surface wave of this length.
In coastal regions, near river estuaries, the upper layer is usually very thin,
say hI < Li/20, over a deep lower layer, where usually h2 > Li/2. Using these
water depth estimates in Eq. (6.13), we obtain:
P2 - PI h
Gi = - - - g 1·
PI
(6.18)
These waves are non-dispersive (not dependent on wave frequency) and travel
slowly because of small density difference. For example, when hI = 5 m,
PI = 1 000 kg/m 3 (freshwater), and P2 = 1 023 kg/m 3 , the phase velocity, Gi ,
becomes 0.15v' ghl = 1.05 m/s.
By analogy with surface waves, the orbits of internal waves are circular (short
waves) or elliptical (long waves), and orbit dimensions decay away from the
interface. We note that internal waves in the ocean produce only very small
vertical displacement of the free surface; usually they are smaller by a factor
of the order (P2 - PI)/PI (say rv 10- 3 ) than the internal wave displacement.
Suppose now that some object is located at water depth z = -hi (see
Fig. 6.5). This object can be stationary when the balance between its weight
and the buoyancy force is maintained. However, due to motion of the internal
waves along the pycnocline, the object's buoyancy varies depending on the density of the surrounding fluid. In 1963, the nuclear submarine U.S.S. Thresher,
with all crew members, was lost in the West Atlantic. There had been no indication of equipment malfunction, or unusual storm weather. As submerged
submarines attain neutral buoyancy by flooding or jettisoning sea water from
ballast tanks, there are speculations that U.S.S. Thresher was probably cruising along a pycnocline when it encountered a large internal wave and suddenly
dropped to a greater depth because of the lower density of the ambient water.
Evidently the incident occurred too rapidly and submarine crew was not able
to arrest the ship's fall (Pinet, 1992).
The model of internal waves in which a layer of water of uniform lower density
is located over a layer of uniform higher density, with a sharp interface is quite
realistic for coastal regions. River runoff occupies the upper layer of low density,
over the deep layer of much higher salinity, and a sharp gradient of salinity (i. e.
density) occurs between them. Two-layer model is also applicable when one
tries to describe an upper oceanic, well-mixed layer located over deeper water.
The thermocline is then fairly abrupt and separates water masses above and
6 Internal Waves
Let us assume that densities PI and P2 are the same as above, and additionally
let hI = h2 = h. Thus, Eq. (6.16) gives:
P2 - PI gh =
P2 + PI
P2 - PI G = 0.038G.
P2 + PI
(6.17)
Again the phase speed of the internal wave is much less than that of a long
surface wave of this length.
In coastal regions, near river estuaries, the upper layer is usually very thin,
say hI < Li/20, over a deep lower layer, where usually h2 > Li/2. Using these
water depth estimates in Eq. (6.13), we obtain:
P2 - PI h
Gi = - - - g 1·
PI
(6.18)
These waves are non-dispersive (not dependent on wave frequency) and travel
slowly because of small density difference. For example, when hI = 5 m,
PI = 1 000 kg/m 3 (freshwater), and P2 = 1 023 kg/m 3 , the phase velocity, Gi ,
becomes 0.15v' ghl = 1.05 m/s.
By analogy with surface waves, the orbits of internal waves are circular (short
waves) or elliptical (long waves), and orbit dimensions decay away from the
interface. We note that internal waves in the ocean produce only very small
vertical displacement of the free surface; usually they are smaller by a factor
of the order (P2 - PI)/PI (say rv 10- 3 ) than the internal wave displacement.
Suppose now that some object is located at water depth z = -hi (see
Fig. 6.5). This object can be stationary when the balance between its weight
and the buoyancy force is maintained. However, due to motion of the internal
waves along the pycnocline, the object's buoyancy varies depending on the density of the surrounding fluid. In 1963, the nuclear submarine U.S.S. Thresher,
with all crew members, was lost in the West Atlantic. There had been no indication of equipment malfunction, or unusual storm weather. As submerged
submarines attain neutral buoyancy by flooding or jettisoning sea water from
ballast tanks, there are speculations that U.S.S. Thresher was probably cruising along a pycnocline when it encountered a large internal wave and suddenly
dropped to a greater depth because of the lower density of the ambient water.
Evidently the incident occurred too rapidly and submarine crew was not able
to arrest the ship's fall (Pinet, 1992).
The model of internal waves in which a layer of water of uniform lower density
is located over a layer of uniform higher density, with a sharp interface is quite
realistic for coastal regions. River runoff occupies the upper layer of low density,
over the deep layer of much higher salinity, and a sharp gradient of salinity (i. e.
density) occurs between them. Two-layer model is also applicable when one
tries to describe an upper oceanic, well-mixed layer located over deeper water.
The thermocline is then fairly abrupt and separates water masses above and