Yakun
Guo and Peter
A Davies
Department of Civil Engineering, University of Dundee, Dundee DD1 4HN, UK
E-mail: y.guo@dundee.ac.uk, E-mail: p.a.davies@dundee.ac.uk
Abstract:
A series of laboratory experiments has been carried out to investigate the
interaction of an internal solitary wave of depression with a bottom ridge in a
two-layer stratified fluid system. Density, velocity and vorticity fields
induced by the wave propagation have been measured upstream, downstream and
above the ridge, over a wide range of model parameters. The results show for
cases in which the upper layer is stably-stratified that wave breaking may occur
for large wave amplitude and strong blockage effect of the ridge. Consequently,
mixing and overturning events take place, as measured with the use of an array
of micro-conductivity probes. Some calculations, such as the buoyancy anomaly,
the mixedness parameter, are carried out based on the measured density profiles.
Keywords:
internal solitary wave, mixing and overturning, blockage, breaking
As the interest
in the exploitation of the ocean resources has increased, more and more offshore
structures, such as offshore floating platforms, have been being built around
the world. The stability of such offshore structures is a key problem to be
considered in their design. Loads and vibrations of the offshore structures
induced by the possible impact of large internal (solitary) waves are a
continuing concern. Internal solitary waves (ISWs), observed at many oceanic
locations (e.g. in the Andaman Sea (Osborne and Burch, 1980), in the Sulu Sea (Apel,
et al. 1985)), can be generated when
tidal currents encounter bottom topography in stratified oceans. The properties
and dynamics of the motion of the ISWs in a two-layer system of constant water
depth have been carried out extensively in the past decades (Koop and Butler,
1981; Kao et al., 1985; Michallet and
Barthélemy, 1998; Grue et al., 1999
among others). The shoaling processes (breaking, mixing and estimation of energy
loss) of ISWs over a sloping bottom have been examined respectively by Kao et
al (1985), Helfrich (1992) and Michallet and Ivey (1999). However, few
investigations have been done on the propagation of ISWs of depression over a
bottom ridge. The most closely related previous studies have been made by Guo et
al (2000) and Grue et al (2000).
The former modelled the motion of ISWs over a bottom ridge in a two-layer fluid
system and the latter studied the breaking and broadening of ISWs in a
stratified fluid. In their experiments, Grue et
al. (2000) were able to observe the wave breaking taking place in the upper
layer for a moderate wave amplitude. The present study considers the motion of
ISWs over a bottom ridge in a stratified (upper layer) fluid, a counterpart
investigation to that of Guo et al (2000)
and Grue et al (2000).
Fig.1 is a
sketch of the experimental channel with dimensions 6.4 m long ´
0.4 m wide ´
0.6 m deep. The channel was filled with saltier water of density ρ2
to a depth h2. Then an upper layer of linearly-stratified fluid was
filled to a depth h1 through the use of floating sponges and the two
reservoir technique. The associated buoyancy frequency N0 of the
upper layer is defined as N0=(g/r0)(Dr)/h1,
with g being the gravitational acceleration, r0
and Dr
being respectively the density at mid-depth and the density difference between
the top and bottom of the upper layer fluid. A watertight movable gate was
installed at one end of the channel to allow internal solitary waves with a
prescribed amplitude (realized by adjusting the location of the gate and the
volume of lighter salt water behind the gate) to be generated by using the
step-like pool techniques (Kao et al., 1985; Grue et al.,
1999; Guo et al., 2000).
Fig. 1 Schematic representation of physical system
The experiment was initiated by moving the gate. An initial disturbance then evolved into a solitary wave propagating along the wave channel. The density, velocity and vorticity fields induced by the propagation of the solitary wave were measured with the use of computer-controlled microconductivity probe arrays (Head, 1983) and the Digimage particle tracking technique (Dalziel, 1992). Measurements were taken at three camera locations, namely 3.2 m (C1), 4.05 m (C2) and 4.9 m (C3) downstream of the gate (see Fig 1). Reference runs without the presence of the ridge were performed for comparison purposes. Three different ridges (placed at C2 position) were used in the experiments. Table 1 lists the model parameters investigated in this study.
Table 1 Experimental parameters (see Figure 1 for symbols)
|
N0 (s-1) |
h2/h1 |
H(cm) |
a (0) |
hr (cm) |
a/h2 |
|
0.65 ® 1.41 |
3.1 ® 4.1 |
30 ® 39 |
11.3 ® 18.4 |
10 ® 15 |
0.1® 0.27 |
The observations of the experiments show that there exist two distinct wave motion patterns, namely:
Type I: for small and moderate wave amplitude, no mixing occurs in the upper layer as the wave propagates over the ridge. However, the vertical density structures measured through the wave trough are stretched as a result of the passage of the wave, thereby weakening the upper layer stratification.
Type II: for large wave amplitude, vortices are initially formed at the back and middle part of the wave. The vortices roll and move forward, causing mixing and overturning. This also happens for the cases in which no ridge is present, though the excursion of mixing and overturning is smaller than those with the presence of ridge. The mixing and overturning can extend below the initially undisturbed level of the interface due to the blockage effect (see density measurements below). In the extreme cases involving the strong blockage effect due to the presence of the ridge, not only mixing and overturning occur at the upper layer of the wave region, but also the wave profile is distorted and broken as it moves over the ridge. Vortices are observed to be generated around the ridge as wave passes.
The experiments have been carried out for both cases with and without the presence of the ridge for the purpose of comparison. Figs 2a, b illustrate the typical Type I velocity profiles measured through the wave trough at C1 and C3 for experiments with and without the ridge for almost identical initial input conditions. The velocity data shown in Fig 2 are normalized by the long wave speed c0, which can be calculated from (within the Boussinesq approximation) (Grue, et al, 2000):
X / cot (X) + h1 /h2 = 0 (1)
Where X=(N0h1)/c0 and c0 is solved for X in the interval (p/2, p). The results shown in Figure 2 indicate that (i) the experiments are very repeatable and (ii) the ridge has little effect on the upstream and downstream velocity fields induced by the propagation of Type I waves except the region over the ridge. We also note that there is a slight reduction of the velocity values measured at C3, which is ascribed to viscous effects. The gradual change of the velocity in the upper layer, which is different from that in the two-layer cases, is associated with the linear density stratification.
Fig. 2 Type I wave velocity profiles at (a) C1 and (b) C3
For the Type II waves, the formation of vortices in the upper layer causes local mixing and overturning, thus modifying the local density and velocity fields. Figs 3a, b show typical Type II wave velocity profiles measured through the wave trough at C1 and C3 with and without the ridge in position. The mixing and overturning occuring in the upper layer make the local region be more or less homogeneous. This feature is illustrated in Figs. 3a,b (and the density measurements, see below) where the values of the velocity close to the free surface are almost uniform, which is the case in the two-layer fluid system. The data shown in Fig 3a indicate that two experiments (with and without ridge) have almost the same incoming wave as measured at C1. Fig 3b reveals that the velocity values of both the upper and lower layers measured at C3 for the ridge case are appreciably smaller than those without the ridge, indicating that the ridge applies a significant blocking effect on the propagation of the wave, resulting in a reduction of the values of the velocity field induced by the propagation of the wave. The fluid velocity close to the free surface shown in Fig. 3a is as large as 1.7 times of the linear long wave speed for this large wave amplitude case. Referred to Fig. 3 in Grue et al. (2000), the fluid velocity in Fig. 3a is approximately the same value as the phase speed of the wave.
Fig. 3 Typical Type II velocity profiles measured at (a) C1 and (b) C2
For the Type I waves, no mixing and overturning occurs in the upper layer. However, the upper layer density structure within the wave is stretched, as shown in Fig. 4a. Solid lines in Fig. 4a, b are the initially undisturbed density profiles. The data shown in Fig. 4a indicate that the ridge has no significant influence on the density field induced by the propagation of Type I waves.
Fig. 4 Typical density profiles for (a) Type I and (b) Type II wave
In contrast, the ridge applies a significant effect on the propagation of Type II waves, thus modifying the velocity (see above) and density fields. Fig. 4b is a typical Type II wave density profile measured at C2, to show that the mixing and overturning occur at the upper layer of the wave region. It can be seen in Fig. 4b that mixing and overturning take place above y=0 for the case without the ridge, while the mixing region extends below y=0 when the ridge is present, indicating that the presence of the ridge enhances the mixing process.
As seen, vortices, mixing and overturning may take place in the upper layer for Type II waves. These events can be quantified conveniently by calculating the buoyancy anomaly profile and the mixedness parameter g within the mixing region. The buoyancy anomaly is defined as (De Silva et al., 1997)
g’(yi)=g{ρ(yi) – ρT (yi)}/ρm (2)
where g is the gravitational acceleration, ρ(yi) the measured instantaneous density profile at C2, ρT (yi) the corresponding Thorpe-ordered density profile and ρm the mean density. The mixedness g is defined as (De Silva and Fernando, 1992; Folkard, Davies and Fernando,1997)
g=1 - (N/N0)2 (3)
where N is the buoyancy frequency of the Thorpe-ordered density profile of the mixed region (see Fig. 4b for reference). Fig. 5 is the buoyancy anomaly profile based on the density profiles of Fig. 4b, in which the buoyancy anomaly with no ridge is offset by 0.4 cm/s2. The non-zero value of the buoyancy anomaly at a particular vertical position indicates that mixing and overturning processes occur there. The greater the magnitude of the buoyancy anomaly, the stronger the mixing and overturning processes. Thus, the data shown in Fig. 5 demonstrate that the degree and excursion of the mixing are increased by the presence of the ridge.
(a) (b)
Fig. 5 Type II buoyancy anomaly profiles.
Fig. 6 Mixedness parameters measured for all experiments Symbols: □ with ridge, ■ no ridge and Δ no mixing.
Fig. 6 shows that the mixedness increases with increasing values of a/h2. The square symbols (hollow and solid) represent the experiments in which mixing and overturning occur in the upper layer, while the triangle symbol denotes the experiment without mixing and overturning taking place. It is seen that there is a jump in the values of the mixedness parameter when the mixing and overturning take place.
The experimental results are presented to show the propagating properties of an internal solitary wave in a stratified fluid system and its interaction with a bottom ridge. Two distinct wave motion patterns are observed in the present study. For small wave amplitude, the ridge has little effect on the propagation of the ISWs. Vortices, mixing and overturning may take place in the upper layer of the wave for large wave amplitude. These events also occur for cases without the ridge, though the mixing excursion and associated attenuation of the velocity field are enhanced by the blockage of the ridge.
Acknowledgements
The authors are grateful for the support of this work by the UK Engineering and Physical Science Research Council (EPSRC).
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