Transport and Mixing Processes in Lakes and Reservoirs - Hydro-Environmental Analysis: Freshwater Environments - page 351

Wavelength = L

(a) D > L /2 (b) D < L /2

L /2

NTS

FIGURE 13.3

Relationship of orbital motion wave with wave height and wavelength.

The ratio of the wave height to the wavelength is highly variable, in a range of 1:100-1:10 (Wetzel
2001). Above a ratio of about 1:10, the wave becomes unstable and the peak collapses, forming
whitecaps.
Observations of short waves would seem to suggest that both the wave and the associated water
mass are traveling at the wave speed, but this is misleading. If that were the case, then a loating
object would be expected to move at or close to the wave speed (with some reduction due to drag) in
the direction of the wave. However, a closer observation over a short period indicates that the loat-
ing object simply bobs up and down without moving with the wave. While, over a longer period of
observation, there would be some residual circulation or movement of the loating object, this may
not occur over a shorter period. The reason that causal observation indicates that water is moving
at the speed of the wave is the orbital motion caused by wave action. Water at the surface does not
simply move vertically up and down with the rising and falling water surface as the wave passes.
Instead, the motion is circular in a vertical plane, making a complete revolution as each successive
wave passes, as illustrated in Figure 13.3. Thus, an important distinction for mixing is that wave
momentum and wave mass move in very different ways that tend to mix surface layers or layers at
an interface.
The wave-induced orbit will be greatest at the surface, where the radius is one-half of the wave
height ( H ). Below the water surface, the orbital radius of particles decreases (Figure 13.3). Since
short waves result in orbital moment with no net advection of water, the overall effect is to induce
mixing and not horizontal mass transport.
There is no appreciable orbital motion below a depth of approximately one-half of the wave-
length in an unstratiied low. This depth is often referred to as the wind-mixed depth (Figure 13.3).
The wind-mixed depth increases with fetch since the wave height and the wavelength increase with
increasing fetch. For example, Ragotzkie (1978) developed a relationship for lakes in Wisconsin and
Central Canada in which the mean depth of the thermocline ( D th ) is proportional to four times the
square root of the fetch (in kilometers):

DF

th = 4

(13.2)

As the wavelength becomes longer in relation to the depth or as the water becomes shallower,
the wave orbits become increasingly latter or elliptical, as illustrated in Figure 13.3. As the orbits
latten, the motion of the water becomes essentially a horizontal oscillation (Smith 1975) so that the
water motion due to long wavelengths is more organized rather than dispersive. Motion of this type
is characteristic of long waves, which have wavelengths that are much greater than the water depth.