Journal of Oceanography, Vol. 62, pp. 811 to 823, 2006
Reflection and Diffraction of Internal Solitary Waves by
a Circular Island
S HENN-YU C HAO1*, P ING-TUNG SHAW2, M ING-KUANG H SU3 and Y ING-JANG Y ANG4
1
Horn Point Laboratory, University of Maryland Center for Environmental Science,
Cambridge, MD 21613-0775, U.S.A.
2
Department of Marine, Earth, and Atmospheric Sciences, North Carolina State University,
Raleigh, NC 27695-8208, U.S.A.
3
Center for General Education, Northern Taiwan Institute of Science and Technology,
Taipei 112, Taiwan
4
Department of Marine Science, Chinese Naval Academy, Kaohsiung 813, Taiwan
(Received 9 November 2005; in revised form 26 June 2006; accepted 27 June 2006)
We have investigated the reflection and diffraction of first-mode and second-mode
solitary waves by an island, using a three-dimensional nonhydrostatic numerical
model. The model domain consists of a circular island 15 km in diameter in an ocean
300 m deep. We use prescribed density anomalies in an initially motionless ocean to
produce highly energetic internal solitary waves; their subsequent propagation is
subject to island perturbations with and without the effect of earth’s rotation. In
addition to reflected waves, two wave branches pass around the island and reconnect
behind it. Island perturbations to the first-mode and second-mode waves are qualitatively similar, but the latter is more profound because of the longer contact time and,
in the presence of earth’s rotation, the scale compatibility between Rossby radius of
the second baroclinic mode and the island diameter. Without earth’s rotation, reflected and diffracted waves are symmetrical relative to the longitudinal axis passing
through the island center. With earth’s rotation, the current following the wave front
veers to the right due to Coriolis deflection. For a westward propagating incoming
wave, the deflection favors northward wave propagation in the region between the
crossover point and the island, shifting the wave reconnection point behind the island
northward. It also displaces the most visible part of the reflected waves to the southeast. In the presence of earth’s rotation, a second-mode incoming wave produces island-trapped internal Kelvin waves, which are visible after the passage of the wave
front.
Keywords:
⋅ Internal solitary
waves,
⋅ wave impingement
on island,
⋅ nonhydrostatic
model.
(2002) have expanded the skill level beyond two dimensions by developing a two-layer model of weakly
nonlinear solitary waves over variable bottom topography. Although a two-layer model vertically compresses
first-mode waves to their rudimentary form, the ability
to cope with bottom depth variations and lateral wave
variations is a step forward. In this light, the present work
continues to expand modeling efforts in the three-dimensional realm.
During the pilot study of Asian Seas International
Acoustic Experiment (ASIAEX), Acoustic Doppler Current Profiler (ADCP) and thermistor chain moorings over
the shelf and upper continental slope south of China captured westward propagating, energetic, mode-1 and mode2 solitary waves in the springs of 1999 and 2000 (Yang et
al., 2004; Duda et al., 2004). Although the observations
1. Introduction
Internal solitary waves are ubiquitous features in the
ocean. In terms of numerical and laboratory modeling,
the state of the art has mostly focused on the two-dimensional realm, ignoring lateral variations of these waves.
In this limit, solitary waves of the first baroclinic mode
have been extensively studied, including their propagation (Vlasenko et al., 2000), breaking over the bottom
slope (Kao and Saffarinia, 1991; Helfrich, 1992), and
perturbation by sills (Vlasenko and Hutter, 2001;
Vlasenko and Alpers, 2005). Recently, Lynett and Liu
* Corresponding author. E-mail: [email protected]
Copyright©The Oceanographic Society of Japan/TERRAPUB/Springer
811
did not cover the Dongsha Island and its nearly circular
surrounding reefs to its immediate south (also known as
Pratas Islands), satellite images clearly showed diffraction of solitary waves by the reef (Fett and Rabe, 1977;
Hsu and Liu, 2000; Lynett and Liu, 2002). This work examines the reflection and diffraction of these highly
nonlinear internal solitary waves by a circular island in
an otherwise flat-bottomed ocean. A three-dimensional
nonhydrostatic general circulation model is used to facilitate the study. The size of the island is comparable to
that of the Dongsha Island reefs. The incoming wave can
be of the first or second mode. Numerical solutions with
and without the effect of earth’s rotation are compared.
2. Initialization Scheme
A brief preview of how we initialize first-mode and
second-mode waves helps to put the forthcoming discussion in perspective. Figure 1 illustrates initial density
anomalies and subsequent wave propagation with and
without the effect of earth’s rotation. Figure 1(a) illustrates a first-mode density anomaly in an initially motionless ocean and the immediately following upwelling
motion that splits the anomaly. To simplify the discussion, we use one density surface to depict responses in a
continuously stratified ocean. The vertical profile of the
density anomaly is of the first baroclinic mode. Horizontally, the steep-sided shape helps accelerate the development of first-mode solitary waves, because these waves
have steep fronts. With steep-sided initialization, the firstmode solitary wave in this model reaches full nonlinearity
in less than 20 km away from the source. If one initializes with a Gaussian profile instead of the top-hat profile, for example, it will take about 30 km of traveling
distance for the solitary wave to reach full nonlinearity.
The narrow width of the density anomaly regulates the
duration of currents following the passage of the waves.
With or without earth’s rotational effect, two waves soon
propagate away from the source region in opposite directions. The rear currents following the wave front are in
the same direction as the wave-induced currents if there
is no rotation (Fig. 1(b)). With rotation, the rear currents
veer to the right due to the presence of the Coriolis force
(Fig. 1(c)). Since first mode waves move fast, the Coriolis
deflection of rear current has little time to develop and is
therefore weak.
The initialization of second mode waves follows the
same idea, except with the added precaution to eliminate
generation of first mode waves. For an upper ocean
pycnocline, we start with a steep-sided density anomaly
having a vertical profile of the second baroclinic mode in
an initially motionless ocean (Fig. 1(d)). The steep edge
helps the wave to reach full nonlinearity less than 15 km
away from the source. Without the steep edge, the wave
will have to propagate over a longer distance away from
812
S.-Y. Chao et al.
Fig. 1. Initialization and subsequent propagation of internal
solitary waves with or without the effect of earth’s rotation. For the first baroclinic mode, panel (a) illustrates initial motionless density anomaly and immediately triggered
motion due to pressure imbalance. Panels (b) and (c) illustrate subsequent propagation of first mode solitary waves
without and with earth’s rotation, respectively. Corresponding illustrations for the second baroclinic mode wave initialization and propagation are given in panels (d), (e) and
(f).
the source to reach maturity. During the initial adjustment, the pressure imbalance triggers motions to split the
anomaly. The split is a highly nonlinear process, whereas
the second-mode vertical profile owes its existence to
linearization. The nonlinear effect will cause an asynchronous split of the upper dome and the lower trough. Consequently, all vertical modes of disturbances will be triggered and propagate away. The second-mode waves produced in this way are unstable and decay fast. To synchronize the split of upper dome and lower trough requires uniform enhancement of the upper dome density
or reduction of lower trough density by an empirical factor. The empirical adjustment will lead to the propagation of pure second mode waves, followed by brief rear
currents (Figs. 1(e) and (f)). Since second mode waves
propagate slowly, the Coriolis deflection of the rear currents is generally more profound than for first mode
waves.
The Coriolis deflection in Figs. 1(c) and (f), although
often weak, is a fundamental property of internal solitary
waves in the presence of earth’s rotation. In this light,
the deflection is not an artifact of initialization with localized density anomalies. One could, for example, initialize internal solitary waves with oscillating tidal currents over sills (e.g., Shaw and Chao, 2006). In that physical setting, Coriolis deflection also occurs as long as one
includes the effect of earth’s rotation.
3. Model Formulation
The three-dimensional nonhydrostatic numerical
model is described in detail in Shaw and Chao (2006). A
brief description of the equations and the parameters is
given below. Earlier versions of this model have been used
in several other studies (e.g., Chao and Shaw, 2002; Shaw
and Chao, 2003).
The model uses the three-dimensional momentum,
continuity, and density equations with the Boussinesq and
rigid-lid approximations. The equations are
1
ρ
Dv
∂2 v
+ 2Ωk ′ × v = −
∇P −
gk + A∇ 2 v + ν 2
ρ0
ρ0
Dt
∂z
(1)
∇⋅v = 0
(2 )
∂2 ρ
Dρ
= K∇ 2 ρ + k 2
Dt
∂z
(3)
where v is the three-dimensional velocity vector (u, v, w)
in the (x, y, z) directions, k′ is a unit vector pointing upward from the North Pole, k is the local upward unit vector, and ρ is the perturbation density about a reference
seawater density (ρ0 = 1022 kg m–3), Ω is the earth’s rotation rate, and P is the pressure. The rotation of the earth
is either zero or that at 20°N. The eddy viscosity and diffusivity are constant, so that A = K = 4 × 104 cm2s–1, ν =
1 cm 2s –1, and k = 0.1 cm 2s –1.
The Arakawa-C grid system is used for the spatial
discretization. Vertical discretization follows the z-coordinate formulation. Spatial derivatives are center
differenced to second-order accuracy. Integration in time
follows the Adams-Bashforth scheme. The model first
calculates the static pressure by vertically integrating the
hydrostatic equation. The depth-averaged velocity is then
found as in a conventional hydrostatic model. Deviation
from the hydrostatic pressure (the nonhydrostatic pressure) is solved by forming a three-dimensional Poisson
equation to enforce zero velocity divergence as required
by the continuity equation (Eq. (2)). The three-dimensional Poisson equation is solved by the preconditioned
conjugate gradient method (e.g., Pozrikidis, 1997). See
Shaw and Chao (2006) for details.
The initial perturbation density field consists of two
parts:
Fig. 2. (a) Stratification profile used in this study and (b) normalized modal structures for the first (solid curve) and second (dotted curve) baroclinic modes.
ρ = ρ b ( z ) + ρ n′ ( x, z ),
( 4)
where ρ b(z) is the ambient vertical stratification, and
ρ n ′(x, z) is a localized density anomaly of the n-th
baroclinic mode. Let x point eastward, y point northward,
and z point upward. The background stratification is given
by
ρb ( z) =
∆ρ 
 z − z0  ,
1 + tanh
 d  
2 
(5)
where ∆ρ = 6 kg m–3, z 0 = –80 m and d = 120 m. The
ocean depth is fixed at 300 m. Figure 2(a) shows the stratification profile employed in this study. The corresponding normalized modal structures for the first mode (φ 1(z))
and second mode (φ2(z)) for either density or vertical velocity are given in Fig. 2(b). The upper ocean pycnocline
in Fig. 2(a) resembles typical stratification in the northern South China Sea (Yang et al., 2004). The localized
density anomaly is given by
ρ n′ ( x, z ) = − R0 F( x )φ n ( z ),
(6 )
where R0 is the amplitude of the density anomaly and the
baroclinic mode number n can be either 1 or 2. The horizontal distribution of density forcing, F(x), is chosen to
be steep-sided to accelerate the development of solitary
waves. The optimum choice is piecewise unity between
x = –L and x = L,
F( x ) = H ( L − x ),
(7)
where H is the Heaviside unit function. The half-width
Island Perturbation of Solitary Waves
813
of density forcing (L) is 2 km. For the second mode initialization, the upper dome density anomaly (z > –120 m)
is uniformly enhanced by an empirical factor of 2.3 to
prevent generation of first mode waves.
Figure 3 shows the model domain, which is 75 km
long and 80 km wide. A circular island 15 km in diameter
is centered at x = –30 km and y = 0 km. The horizontal
and vertical resolutions are 250 m and 20 m, respectively.
The time step is 30 s. Since the subsequent response is
symmetrical with respect to the center location of the
density anomaly (x = 0), density forcing is purposely
placed near the eastern ocean boundary (the stippled region) so that we can focus attention on westward propagating waves. Eastward propagating waves will soon leave
the model domain and have no effect on the motion of
interest. The distance between the forcing region and the
island is chosen to be long enough to ensure adequate
growth of solitary waves before impinging on the island.
The choice of this distance was done in preliminary experiments that removed the island but retained the density forcing to determine the location where the internal
solitary wave first began to form and propagate without
visible distortion. The island was then placed west of the
location where the soliton first appeared. However,
whether a solitary wave reaches full nonlinearity before
impinging on the island appears to be unimportant as
numerical results are qualitatively similar. On the east and
west open boundaries, waves are allowed to propagate
through without reflection using appropriate phase speeds
in the radiation condition. Zero normal gradients for horizontal velocities and perturbation density are enforced at
the ocean surface. No-slip boundary conditions for horizontal velocities and a zero normal gradient for the perturbation density are applied at the ocean bottom. The
north and south open boundaries are cyclical.
Table 1 lists four experiments to be discussed below. Experiment 1 produces a first-mode solitary wave
impinging on the island in the absence of earth’s rotation. In experiment 2, the same experiment is repeated by
including the earth’s rotation at 20°N. Experiment 3 produces a mode-2 solitary wave impinging on the island
without the effect of rotation. Experiment 4 repeats experiment 3 by including the earth rotation at 20°N.
4. Numerical Results
Fig. 3. Model domain and island. Stippled strip indicates the
region of initial density forcing.
4.1 First mode wave impingement
For fast-moving first-mode waves, wave perturbation by the island is small because of the short contact
time span. In the presence of earth’s rotation, the Rossby
radius of the first baroclinic mode (~58 km) far exceeds
the island diameter (15 km). The scale incompatibility
also reduces the effect of the island.
Figure 4, from experiment 1, shows the vertical structure of a first-mode wave around the circular island. The
incoming wave propagates along the x-axis without the
influence of earth’s rotation. In this non-rotational environment, the horizontal distributions of flow and density
are symmetrical with respect to the center axis, and the
maximum amplitude of reflected and transmitted waves
occurs along the x-axis. Just before the wave impinges
on the island (Fig. 4(a)), the Froude number in the incoming wave (maximum westward speed divided by the
phase speed of the first baroclinic mode) is 0.68. The
Table 1. List of experiments.
814
Experiment
Wave mode
Phase speed
(cm/s)
Rossby radius
(km)
Latitude
(°N)
Forcing strength
R 0 (kg m−3)
Forcing half-width
L (km)
1
2
3
4
1
1
2
2
126.3
126.3
58.3
58.3
∞
50.8
∞
23.45
0
20
0
20
−2
−2
1
1
2
2
2
2
S.-Y. Chao et al.
maximum westward current is 86 cm/s at the sea surface.
Downward displacement of isopycnals ranges up to 60 m
and the wavelength is about 2 km. Wave impingement
triggers wave reflection and upwelling at the point of
contact (Fig. 4(b)). The Froude number in the reflected
wave is 0.45, decreasing continuously in time due to the
radial expansion of the wave (Fig. 4(c)). The wavelength
of the reflected wave remains to be 2 km, but the radial
expansion of the wave continuously decreases the wave
amplitude. The radial expansion will be illustrated in Fig.
5. By hour 9, the incoming wave has passed the island
and appeared on the west side of the island (Fig. 4(d)).
Moving sufficiently away from the island, the Froude
number in the transmitted wave recovers to about 0.45
(Fig. 4(e)), but is still much lower than that of the incoming wave. The wavelength of the transmitted wave remains about 2 km, but the wave amplitude (downward
isopycnic displacement) decreases to about 40 m. The
maximum westward current speed for the transmitted
wave directly behind the island is 57 cm/s at the sea surface.
Figure 5, also from experiment 1, shows a sequence
of horizontal features at 90 m below the sea surface. Since
there is no rotation, the flow and density fields are symmetrical with respect to the x-axis. The radial spread of
the reflection wave is evident. In addition, Fig. 5(c) shows
that two wave branches passing around the island reconnect behind it. The reflected and diffracted waves essentially retain the vertical structure of the first baroclinic
mode; we therefore omit illustration of features at other
depths for simplicity.
Figure 6, from experiment 2, shows the corresponding sequence of horizontal features at 90 m depth in the
presence of earth’s rotation. The rotation induces several
asymmetric responses with respect to the x-axis. These
changes are relatively small because of the short time span
of the first-mode wave in contact with the island. A northward rear current following the incoming wave is induced
by the Coriolis deflection. Figure 6(c) shows that, after
waves have reconnected behind the island, a rear current
develops behind the reconnection point. The rear current
is generally northward except in a localized region north
Fig. 4. Vertical (x-z) sections of flow and density along the x-axis (y = 0) at hours 4, 5, 6, 9 and 11 from experiment 1, in which
a mode-1 wave impinges on the island without rotation. Contour interval is 1 kg m–3 for density. Maximum zonal (u) and
vertical (w) current speeds are given at the top of each panel.
Island Perturbation of Solitary Waves
815
Fig. 5. Flow and density fields 90 m below sea surface at
(a) hour 6, (b) hour 8 and (c) hour 10 from experiment 1, in
which a mode-1 solitary wave impinges on the island without the earth’s rotation. Contour interval is 0.2 kg m–3 for
density. Maximum zonal (u) and meridional (v) current
speeds are given in each panel.
816
S.-Y. Chao et al.
Fig. 6. As in Fig. 5 except from experiment 2, in which the
earth’s rotation at 20°N is included.
Fig. 7. Close-up views of density structure 90 m below sea surface near the mode-1 wave reconnection zone behind the island
from (a) experiment 1 without rotation and (b) experiment 2 with rotation. Contour interval is 0.01 kg m–3.
Fig. 8. Vertical (x-z) sections of flow and density along the x-axis at hours 8, 10, 13, 19 and 24 from experiment 3, in which a
mode-2 wave impinges on the island without rotation. The contour interval is 1 kg m –3 for density. Maximum zonal (u) and
vertical (w) current speeds are given in each panel.
Island Perturbation of Solitary Waves
817
Fig. 9. Horizontal sections of flow and density fields at
(a) hour 13, (b) hour 19 and (c) hour 24 from experiment 3,
in which a mode-2 wave impinges on the island without the
earth’s rotation. The horizontal slice is at z = –50 m for
density and z = –110 m for currents. The contour interval is
0.2 kg m–3 for density. Maximum zonal (u) and meridional
(v) current speeds are given in each panel.
818
S.-Y. Chao et al.
Fig. 10. As in Fig. 9 except from experiment 4, in which the
earth’s rotation at 20°N is included.
of the island. Further, the most visible part of the reflection wave in Fig. 6 (more notably in Figs. 6(a) and (c)) is
no longer symmetrical with respect to the x-axis, but shifts
slightly to the southeast. The upper-ocean flow conver-
gence associated with the northward rear current and the
flow in the reflection wave occurs only on the southeast
side of the island. The effect of this flow convergence is
responsible for the southeast shift of the wave front.
Figure 7 shows close-up views of density distribution in the wave reconnection region with and without
the effect of earth’s rotation. Effect of earth’s rotation
shifts the reconnection point slightly to the north in Fig.
7(b). The northward shift is conceivable because the rear
current following the wave has a northward component.
However, the shift is likely too small to be significant.
The northward rear current also affects the crossover wave
pattern behind the reconnection point. The northern
branch of the crossover waves is visibly stronger, apparently enhanced by the northward advection of the rear
current.
4.2 Second mode wave impingement
For slower moving second-mode waves, wave perturbation by an island is more profound because of longer
contact time with the island. If the effect of earth’s rotation is included, the Rossby radius for the second mode
(~23 km) is close to the island diameter (15 km). The
scale compatibility ensures larger responses for modetwo waves than mode-one waves.
Figure 8, from experiment 3 without earth’s rotation,
shows vertical features of a second-mode wave propagating along the x-axis. Shortly before impingement, the
Froude number in Fig. 8(a) is about 1.13. The vertical
displacement of isopycnals in the incoming wave is up to
60 m; the wavelength (longitudinal extent) is about 3~4
km. Maximum westward current speed for the incoming
wave is 66 cm/s at the core depth. Subsequent impingement at hour 10 (Fig. 8(b)) further raises isopycnals in
the upper pycnocline and depresses isopycnals in the
lower pycnocline at the contact point. Shortly after reflection, the Froude number of the reflected wave is about
0.6 (Fig. 8(c)), decreasing continuously thereafter because
of the radial expansion. The isopycnic displacement of
the reflected wave decreases continuously with distance
away from the island, but its wavelength remains about
3~4 km. Moving past the island, the Froude number of
the transmitted wave is about 0.65 on the west side of the
island (Fig. 8(e)). Thus, the Froude number decreases by
a factor of 0.57 after passing the island. The corresponding damping factor is about 0.66 for the first mode wave
in Fig. 4. Compared to the first mode wave impingement,
island damping is therefore more severe for the second
mode wave. As a result, upward and downward isopycnic
displacements of the transmitted wave reduce to about
30 m. Nevertheless, the wavelength of the transmitted
wave remains about 3~4 km. The maximum westward
current speed of the transmitted wave directly behind the
island is 38 cm/s at the core depth.
Fig. 11. Close-up views of currents at 90 m below sea surface
from experiment 4 with earth’s rotation at 20°N at (a) hour
21 and (b) hour 24. Solid and dashed lines delineate the
most visible parts of an island-trapped second-mode
Kelvin wave. Maximum zonal (u) and meridional (v) current speeds are given at the top of each panel.
Figure 9, also from experiment 3, shows a series of
horizontal features below the sea surface. The horizontal
slice for density is at 50 m below the sea surface to better
illustrate the upper ocean density distribution. For currents, the slice is at 110 m below the sea surface to capture the core of the second-mode waves. The absence of
earth’s rotation again ensures symmetric responses with
respect to the x-axis. The radial spread of the reflected
wave is much slower because of the slower phase speed
of the second baroclinic mode. The diffracted waves cross
Island Perturbation of Solitary Waves
819
Fig. 12. Close-up views of density structure 50 m below sea surface near the mode-2 wave reconnection zone behind the island,
from (a) experiment 3 without the earth rotation and (b) experiment 4 with earth rotation. Contour interval is 0.01 kg m –3.
Fig. 13. MODIS image around Dongsha Reef showing island
perturbation of an incident wave and reflection wave.
over and reconnect behind the island. In general, both
reflected and diffracted waves largely retain the vertical
structure of the second baroclinic mode.
Figure 10, from experiment 4, shows corresponding
horizontal features in the presence of earth’s rotation.
Rotation induces much more profound changes with the
incoming wave having the structure of the second
baroclinic mode because of longer contact period with
the island and scale compatibility between Rossby radius
and the island diameter. Even without the island, the effect of Coriolis deflection is more profound for the slow
820
S.-Y. Chao et al.
Fig. 14. RADARSAT ScanSAR image in the northern South
China Sea on April 26, 1998, showing wave reconnection
after passing around the Dongsha Reef.
moving second mode wave. In all panels of Fig. 10, the
rear current immediately behind the wave front veers to
the right and gains a northward component. As the wave
front moves sufficiently away, the rear northward current farther behind veers further to the right and gains an
eastward component. The continuous production of subsurface countercurrent far behind the wave is essentially
a “Coriolis drag” which dampens the second mode wave.
( a ) Without Eath Rotation
( b ) With Earth Rotation
Fig. 15. Illustration of wave reflection and diffraction by an island (a) without earth’s rotation and (b) with earth’s rotation.
Dashed lines in the wave crossover region behind the island indicate weaker waves. Arrows are not to scale.
Around the island, the rotation-induced asymmetry
with respect to the x-axis is much more visible for the
second-mode waves. First, the reflected wave is most visible to the southeast of the island because of the flow
convergence between the reflected wave and the northward rear current at the core depth. Second, the northward rear current shifts the reconnection point behind the
island to the north. Third, island-trapped second-mode
coastal Kelvin waves are evident after the wave passage.
These island-trapped waves propagate clockwise around
the island in the presence of ever-changing ambient currents that are not island-trapped; determination of their
amplitude, e-folding width scale, wavelength and period
is subject to great uncertainty. To select a depth where
ambient currents are not so prevalent, we display currents
at 90 m below the sea surface to highlight the propagation of Kelvin waves at hour 21 (Fig. 11(a)) and hour 24
(Fig. 11(b)). Solid and dashed lines delineate the most
visible parts of an island-trapped Kelvin wave. It becomes
evident that the second-mode internal Kelvin wave has a
wavelength of half of the island circumference; this makes
sense, as island trapping requires the circumference to be
a multiple of wavelength (Caldwell and Eide, 1976). Using the maximum current speed at the core depth of mode2 wave as a measure, the internal Kelvin wave amplitude
is about 20 cm/s. The theoretical propagation speed of
second-mode wave is about 58.3 cm/s (Table 1) if stratification does not change in time, but the estimated clockwise propagation speed from hour 21 to hour 24 in Fig.
11 is about 33 cm/s. Two reasons could account for the
discrepancy. First, the ever-changing ambient currents at
all depths and all times are capable of interfering with
the propagation speed. Second, the stratification around
the island weakens by a varying amount after the passage
of an internal solitary wave, decreasing the phase speed
of internal Kelvin waves. The theoretical e-folding width
of mode-2 Kelvin waves is 23.45 km (Table 1) if the stratification remains constant in time. After the passage of
the internal solitary wave, the presence of ever-changing
ambient currents makes corresponding estimate from
model results impossible, and the weakened stratification
is highly variable around the island, decreasing the e-folding width scale of the mode-2 Kelvin waves by a varying
amount.
The northward current between the reconnection
point and the island in Fig. 10(c) is quite strong. Closeup views of density distribution in the wave reconnection region (Fig. 12) show more rotation-induced asymmetry. Flow advection significantly enhances the northern branch but weakens the southern branch of the crossover wave pattern in the presence of earth’s rotation (Fig.
12(b)).
5. Satellite Images around Dongsha Reef
The dimension of Dongsha Reef is somewhat larger
than the model island diameter (15 km). In particular, its
north-south extent is smaller than the zonal extent; the
former is more relevant because it is more or less perpendicular to the incoming wave direction. Insofar as the
major numerical results are concerned, our preliminary
numerical experiments suggest that the model is not sensitive to modest changes in island diameter, enabling us
to make comparisons with satellite images around
Dongsha Reef to lowest order.
Figure 13 shows a MODIS image collected on March
28, 2003 around the Dongsha Reef. At the time of the
snapshot, the two branches of the first-mode incoming
wave have nearly completed their propagation around the
Island Perturbation of Solitary Waves
821
island. The southern branch appears to move faster, mostly
because the water is deeper on the south side of the island. The effect of earth’s rotation alone is not sufficient
to enhance the propagation of the southern branch to such
a degree (Fig. 6). Because the two branches propagate at
different speeds around the island, there is some uncertainty regarding the incident direction of the incoming
wave. Despite the uncertainty, the most visible segment
of the reflected wave appears to shift to the southeast of
the island. This southeast shift appears to be common in
many of the images we have seen. Conceivably, the effect of earth’s rotation is capable of producing the shift
(Fig. 6). Our model findings further suggest an even more
profound southeast shift of the reflected wave for a second-mode incident wave. Unfortunately, the diffraction
and reflection of second mode waves have not been well
documented around the Dongsha Reef.
Figure 14, adapted from Hsu and Liu (2000), show a
RADARSAT ScanSAR wide image collected on April 26,
1998. Dongsha Island appears as a white dot located on
the northwest perimeter of the coral reefs. At least four
packets of internal solitary waves are seen to propagate
westward in Fig. 14. Their wave propagation speed of
about 1.9 m/s is consistent with the phase speed of the
first baroclinic mode based on stratification profiles observed in the vicinity (Bole et al., 1994; Yang et al., 2004).
The distance between two adjacent waves decreases from
about 100 km in the east to about 50 km in the west. Apparently, the shoaling bottom toward the west decelerates wave propagation. For the wave packet just past the
Dongsha Reef, the two disconnected branches reconnect
behind the reef. The crossover wave pattern following
the reconnection point is similar to the model-simulated
pattern. Other model findings, such as the northward shift
of the wave reconnection point induced by earth’s rotation, are relatively minor and cannot be conclusively identified from a single image.
6. Conclusions
With the Dongsha Reef in the northern South China
Sea in mind, we have investigated the reflection and diffraction of first-mode and second-mode solitary waves
by a circular island, using a three-dimensional
nonhydrostatic numerical model. Since the subject rarely
appears in the literature, we have neglected site-specific
features such as bottom topography variations, tidal characteristics and ambient currents in exchange for generality, in the hope that the reference solutions will be useful
to other similar settings as well. Some of our findings,
such as reflected and diffracted wave patterns, are readily verifiable from satellite images around the Dongsha
Reef. Several other findings, most notably the island
perturbations of second-mode incident waves, still await
future observational confirmation. Whether the incom-
822
S.-Y. Chao et al.
ing wave is of first or second baroclinic mode, the results
are qualitatively similar. Quantitatively, island
perturbations are more profound for second-mode incoming waves because of longer contact hours and scale compatibility between the relevant Rossby radius and the island diameter.
Figure 15 schematically illustrates dominant responses with and without the effect of earth’s rotation. In
the absence of rotation, reflected and diffracted waves
are symmetrical with respect to the center latitude of the
island. With earth’s rotation, the Coriolis deflection induces northward rear current trailing behind the wave
front. The northward rear current in turn produces three
asymmetric responses relative to the center latitude of the
island. First, the northward rear current shifts the wave
reconnection point behind the island slightly to the north.
Second, the northern branch of the crossover waves behind the island becomes visibly stronger than the southern branch. Third, the reflected wave becomes more visible to the southeast of the island. All three changes mentioned above become more profound if the incoming wave
is of the second mode. In addition, island-trapped internal Kelvin waves become visible after the passage of wave
front for the second mode wave impingement.
Acknowledgements
Author SYC was supported by the Physical Oceanography Program of Office of Naval Research under
contracts N00014-04-1-0419 and N00014-05-1-0279 as
part of the Nonlinear Internal Wave Initiative. Author PTS
was supported by the same program under contracts
N00014-04-1-0430 and N00014-05-1-0280. Author MKH
was supported by the National Science Council of Taiwan under grant NSC93-2611–M-149-001. We are indebted to Antony K. Liu of NASA Goddard Space Flight
Center for sharing knowledge with us. The Canadian
Space Agency provided the RADARSAT image. This is
UMCES contribution no. 3999.
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