Laboratory modelling of the motion of an internal solitary wave over a
ridge in a stratified fluid
Yakun Guo, Peter A Davies, J Kirstian Sveen, John Grue & Ping Dong
Department of Civil Engineering, University of Dundee, Dundee DD1 4HN, UK
Division of Mechanics, Department of Mathematics, University of Oslo,
Oslo N0316, Norway
Laboratory measurements are presented for the case of an internal solitary wave of depression
propagating in a gravitationally-stable two layer fluid system in which the upper and lower layer
are stably stratified and homogeneous respectively. The effect of the presence of a bottom ridge
upon the propagation characteristics of the wave are investigated. Results confirm that that the
strongest encounters with the ridge are observed when large amplitude waves and tall ridges are
involved. Such strong encounters are manifested by (i) wave breaking, (ii) vortex generation in
both layers and (iii) enhanced mixing in the upper layer. The data indicate that the strength of
the mixing can be quantified conveniently in terms of (i) the buoyancy anomaly profiles within
the layers and (ii) the thickness of the upper mixed layer. Non-dimensionalised plots are
presented to show the dependence of both of these parameters upon the normalised wave
amplitude and ridge height, respectively.
1. Introduction
Internal solitary waves (ISWs), observed at many oceanic locations (Osborne and Burch, 1980;
Apel, et al. 1985), can be generated when tidal currents encounter bottom topography in
stratified oceans (Farmer and Armi, 1999). Such waves are known to produce significant
hydrodynamical loading on offshore structures in deep water oceanic conditions. The properties
of the ISWs in a two-homogeneous-layer system of constant water depth have been extensively
investigated in the past decades (Koop and Butler, 1981; Kao et al., 1985; Michallet and
Barth lemy, 1998; Grue et al., 1999 among others). The shoaling processes of ISWs over a
sloping bottom have been examined respectively by Kao et al (1985), Helfrich (1992) and
Michallet and Ivey (1999). However, comparatively few investigations have been undertaken 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-homogeneous-layer fluid system and the latter
studied the breaking and broadening of ISWs in a two layer fluid with an upper stratified layer.
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, as a complementary study to the work
reported by Guo et al (2000) and Grue et al (2000).
2. Experiments
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 salt water of density _2 to a depth h2. Then an upper layer of
linearly-stratified saline fluid was filled to a depth h1 through the use of floating sponges and the
two reservoir technique. The corresponding buoyancy frequency N0 of the upper layer is defined
as N 0=(g/ρ0)(∆ρ)/h1, with g being the gravitational acceleration, ρ0 and ∆ρ 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).
gate
_1
C1
C2
y
C3
_(y)
h1
x
a
H
h2
_(y)
_2
_
hr
Fig. 1 Schematic representation of physical system.
The density, velocity and vorticity fields induced by the propagation of the generated 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. Parameter values investigated in the present study are: 0.65 † N0 † 1.41 s -1; 3.1 †
h2/h1† 4.1; 30 † H† 39 cm; 11.3 † α † 18.4 0; 10 † hr † 15 cm; 0.1 † a/h2 † 0.27.
3. Results
3.1 Observations
Observations show that the wave patterns characterizing the resulting encounter with the ridge
depend primarily upon (i) the amplitude of the wave, (ii) the height of the ridge and (iii) the
buoyancy frequency of the upper layer.
For sufficiently small wave amplitude, the ridge has little influence on the propagation of wave.
(Note that the wave amplitude is defined here as the distance between the initially-undisturbed
interface between the two constituent layers and the final sharp boundary through the wave trough
between the upper stratified layer and the lower homogeneous layer, as determined from the density
measurements, see below). For such weak encounter, no mixing occurs in the upper layer as the
wave propagates over the ridge though 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.
For moderate and 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 provided that the wave amplitude is
sufficiently large (as observed by Grue et al. 2000), 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. In the
extreme cases involving strong blockage and large wave amplitude, 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 propagates over the ridge. Vortices are observed to be generated around the ridge as
wave passes.
3.2 Velocity profiles
Figs 2a, b illustrate the typical velocity profiles for weak encounter cases, measured through the
wave trough at 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 defined
as (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 (π/2, π). The solid line in Fig 2 are the
predictions of the fully nonlinear theory (Grue et al., 2000) for the constant depth (no ridge) case.
The results shown in Figure 2 indicate that (i) the ridge has little effect on the velocity fields for
such weak encounter and (ii) the experimental results agree well with the nonlinear theory except
close to the free surface where the theoretical predictions exceed slightly the experimental
results. 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.
1
1
0
0
y/h1
y/h1
-1
-1
-2
-2
-3
ridge
-3
-0.4
-0.2
0
0.2
u /Co
no ridge
0.4
theory
0.6
0.8
-4
-0.8
-0.4
0
0.4
0.8
1.2
1.6
2
u/Co
Fig 2. Velocity profiles for weak encounter Fig 3. Velocity profiles for strong encounter
As noted above, for large wave amplitude the formation of vortices in the upper layer causes
local mixing and overturning, thus modifying the local density and velocity fields. Fig 3 shows
such an example in which the velocity profiles were measured through the wave trough at C3
with ( _) and without ( _) the ridge in position (the incoming wave at C1 is nominally identical
for two cases), where the solid (for ridge case) and circle-solid (for no ridge case) lines are the
nonlinear theory results for amplitude a/h1 being 0.8 and 1, respectively. The values of the
velocity in the region close to the free surface are almost uniform since mixing and overturning
occuring there make the local density structure more or less homogeneous. It is seen that the
velocities in both the upper and lower layers 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 local reduction of the values of the velocity. The
discrepancy between the experimental results and those predicted by the fully nonlinear theory
close to the free surface is expected because the theory does not take into account the mixing
effect.
3.3 Density fields
For low amplitude waves, no overturning occurs in the upper layer, though the density profiles
taken through the trough of the wave show that isopycnals in the upper layer are stretched vertically
above the wave. In contrast, for high wave amplitude, mixing and overturning take place in the
upper layer and these events are enhanced by the presence of a bottom ridge. Fig 4 is a typical
wave density profiles measured at C2 for cases with (_) and without (_) the ridge and nominally
identical input conditions in which the solid line represents the density profile taken before
experiment. It is seen that the partially mixed region extends further into the lower layer when
the ridge is present.
1
1
0
y/h1
y/h1
0
-1
-1
ridge
no ridge
ridge
-2
-0.4
-2
1046
no ridge
0
0.4
0.8
g’(cm/s^2)
1048
1050
density (kg/m^3)
Fig 4. Typical density profiles for strong
encounter
Fig 5. Buoyancy anomaly deduced from Fig 4.
3.4 Mixing and overturning
In order (i) to quantify the vigor of the overturning events described above and (ii) to determine
their vertical distribution, it is convenient to compute the buoyancy anomaly profiles g′(yi) from the
measured, discretised density profiles _(yi). The buoyancy anomaly is defined as (De Silva et al.,
1997)
(2)
g_(yi)=g{_(yi) — _T (yi)}/_m
where g is the gravitational acceleration, _(yi) the measured instantaneous density profile at C2,
_T (yi) the corresponding Thorpe-ordered density profile and _m the mean density. Fig 5 shows
the buoyancy anomaly profile deduced from the density profiles of Fig 4, in which the buoyancy
anomaly with no ridge case (_) is offset by 0.4 cm/s2. The non-zero value of the buoyancy
anomaly at a particular vertical position indicates that overturning occurs there. The greater the
magnitude of the buoyancy anomaly, the stronger the overturning processes. Thus, the data
shown in Fig 5 demonstrate that the degree and excursion of the mixing are enhanced by the
presence of the ridge.
The bulk mixing in the upper layer can be quantified by introducing the r.m.s buoyancy anomaly
g′rms, derived from the discretised buoyancy anomaly profiles g′(y′i) as:
(g′rms)2 = Σ(g′(y′i))2/n
i = 1,2,.... n
(3)
where n is the total number of points in the section of the profile in which overturning events are
found. A suitable normalising scale for g′rms is N02h1, the value of g′ associated with the undisturbed
0.08
0.3
ridge I
ridge II
ridge III
no ridge
(a1-a3)/a1
g’rms/g’o
0.2
0.04
ridge I
ridge II
0.1
ridge III
no ridge
no mixing
0
0
0
0.1
0.2
a/h2
Fig 6. g′rms/N02h1 versus a1/h2
0.3
0.4
0
0.1
0.2
0.3
0.4
a1/h2
Fig 7. Plot of wave amplitude attenuation (a1-a3)/a1
versus a1/h2
density stratification in the upper layer. The summary plot in Fig 6 illustrates the dependence of
(g′rms)/N02h1 upon a1/h2 and hr/h2, for all experimental data. The plot confirms the findings of the
velocity and density measurements described above, namely that the vigor of the mixing in the
upper layer increases with increasing wave amplitude, even with no ridge in position. Where there
is topographic blockage to the flow, the dependence of (g′rms)/N02h1 on ridge height is relatively
weak for low (< 0.24, say) values of a1/h2, though for the highest values of a1/h2, the vigor of the
mixing increases significantly with increasing values of hr/h2 for a given wave amplitude.
3.5 Wave amplitude attenuation
As seen, the attenuation of the wave amplitude due to the combination effects of the viscous
damping and the local loss due to the bottom ridge blockage can be conveniently evaluated from
the sequence of the measured density profiles. The summary results are plotted in Fig 7 in which
the dimensionless integrated wave amplitude attenuation is calculated as (a1-a3)/a1 with a3 being
the wave amplitude measured at C3. It is seen that there is a jump in the wave amplitude
damping for high wave amplitude where the mixing occurs and such damping can be
significantly enhanced by the presence of the bottom ridge. However, for low wave amplitude,
the ridge has little influence on the wave amplitude attenuation. Compared with the previous
results (Guo et al, 2000), the wave amplitude attenuation in present study is found to be smaller
than that in two-layer system.
4. Conclusions
The experimental results are presented to show the propagating properties of an internal solitary
wave in a stratified fluid system and its encounter with a bottom ridge. For significantly small
wave amplitude, the ridge is observed to apply negligible effect on the propagation of the ISWs.
Vortices, mixing and overturning may take place in the upper layer of the wave for moderate and
large wave amplitude. These events also occur for cases without the ridge for sufficiently high
wave amplitude, though the mixing excursion and associated attenuation of the velocity field and
wave amplitude are enhanced when the bottom ridge is in position.
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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