Download White et al 2008 Flow separation and vertical

Available online at www.sciencedirect.com
Estuarine, Coastal and Shelf Science 77 (2008) 457e466
www.elsevier.com/locate/ecss
Flow separation and vertical motions in a tidal flow
interacting with a shallow-water island
Laurent White a,*, Eric Wolanski b,c
a
Université catholique de Louvain, Center for Systems Engineering and Applied Mechanics (CESAME),
4, Avenue G. Lemaı̂tre, B-1348 Louvain-la-Neuve, Belgium
b
Australian Institute of Marine Science (AIMS), PMB No 3, Townsville MC, Queensland 4810, Australia
c
James Cook University, Townsville, Queensland, Australia
Received 1 August 2007; accepted 5 October 2007
Available online 30 October 2007
Abstract
This paper reports on the case study of Rattray Island (Great Barrier Reef, northeast Australia), lying perpendicular to tidal flow in shallow
waters. At ebb and flood, attached (stable) eddies develop in the wake where swirls of turbidity suggest that sediment-laden waters are brought to
the surface as a result of vertical transport. Both eddy and tip upwellings are encountered in the tidal flow around Rattray Island but there is
currently no clear-cut answer as to which secondary flow generates upwelling with the largest intensity. This paper addresses this specific issue
through idealized and realistic high-resolution numerical experiments. The analysis is supported by physical arguments based on the theory of
flow separation. Given Rattray’s geometry and surrounding bathymetry, the mechanism of flow separation in shallow waters helps explain the
asymmetry in size of the eddies and their intensity. The results of idealized numerical experiments also suggest that eddy and tip upwellings may
be of similar intensity at Rattray Island.
Ó 2007 Elsevier Ltd. All rights reserved.
Keywords: Rattray Island; island wake; flow separation; upwelling; finite elements
1. Introduction
Topographically-induced secondary circulations in the
presence of islands, headlands and narrow passages can have
strong effects on marine ecosystems and geology (Hamner
and Hauri, 1981; Wolanski and Hamner, 1988; Coutis and
Middleton, 2002; Hoitink, 2004; Suthers et al., 2004). The
two-dimensional structure of these secondary circulations is
often quite complex and characterized by zones of converging,
diverging and curved flows. These features, combined with
dominant bottom friction, lead to potential mechanisms for
vertical motions (Hamner and Hauri, 1981; Wolanski et al.,
* Corresponding author. Princeton University/GFDL, 201 Forrestal Road,
Princeton, NJ 08540, USA.
E-mail addresses: [email protected], [email protected]
(L. White).
0272-7714/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ecss.2007.10.003
1984; Geyer, 1993; Alaee et al., 2004). Over the last two decades or so, particular interest has been devoted to the understanding of the two- and three-dimensional flow in the vicinity
of islands and headlands (Wolanski et al., 1984, 1996; Pattiaratchi et al., 1986; Ingram and Chu, 1987; Tomczak, 1988;
Deleersnijder et al., 1992; Geyer, 1993; Alaee et al., 2004;
Suthers et al., 2004; White and Deleersnijder, 2007).
This paper reports on the case study of Rattray Island in
northeast Australia (Fig. 1a). Since the tidal flow is almost perpendicular to the island’s main axis, flow separates and eddies
develop in the wake during rising and falling tides. These
eddies have been observed in aerial photographs (Wolanski
et al., 1984), measured with current meters (Wolanski et al.,
1984) and predicted with two- and three-dimensional numerical models (Falconer et al., 1986; Deleersnijder et al., 1992).
The swirls of turbidity encountered in the wake suggest that
sediment-laden waters are brought to the surface as a result
L. White, E. Wolanski / Estuarine, Coastal and Shelf Science 77 (2008) 457e466
458
(a)
24’
36’
48’ 149°E
N
12°S
(b)
12’
Rattray
N
48’
20°S
12’
1
15°S
24’
3
2
36’
18°S
0
0.8
21°S
0.6
0.4
0.2
Queensland
270
90
1 km
24°S
180
142°E
144°E
146°E
148°E
150°E
152°E
(c)
(d)
N
25
20
30
20
30
35
40
15
20
5
110
40
40
20
35
2
30
1
25
25
y
1km
30
35
25
25
x
Fig. 1. (a) Rattray Island is located in the Great Barrier Reef, northeast Australia. (b) Close-up view of Rattray with locations of (1) the 2-day mooring site (4e6
December), (2) overnight ADCP (4e5 December) and CTD casts on 5 December from 3:00 to 10:30, and (3) overnight ADCP (5e6 December). ADCP transects
shown in light gray. The island edge determined by GPS is represented by the red dots. The compass shows the depth-averaged horizontal velocity at site 1. (c)
Mesh used for the finite element computations. The resolution varies from 800 m to 60 m in the vicinity of Rattray. The three-dimensional mesh is obtained by
extruding the horizontal mesh downwards. (d) Computational domain (8 km by 11.8 km) used for all numerical simulations. The island and bathymetry (depth in
meters) are rotated by an angle (52.5 degrees clockwise) such that the far-field velocity may be considered parallel to the y-axis. The top and bottom boundaries are
open. Sites 1 and 2 identified by squares. (For interpretation of the references to color in the text, the reader is referred to the web version of this article.)
of vertical transport (by advection and mixing). The central
question is whether this transport of sediments is mainly
caused by eddy upwelling, upwelling along the island’s flanks
where the current is the swiftest (hereafter referred to as tip
upwelling) or a combination of both. Up to very recently,
eddy upwelling had been considered to be the only cause for
the presence of sediments in the surface layer (Wolanski and
Hamner, 1988; Deleersnijder et al., 1992; Wolanski et al.,
2003) whereas secondary circulations off the island’s tips
had been widely overlooked.
L. White, E. Wolanski / Estuarine, Coastal and Shelf Science 77 (2008) 457e466
Vigorous interest in Rattray Island has been recently renewed within the framework of a relevant benchmark for modern numerical models based on unstructured meshes (White,
2007). With the advent of these up-to-date techniques (e.g.,
Pietrzak et al., 2005, 2006) comes the prospect of improving
numerical predictions and the understanding of physical
processes. Rather than confirming previous results on eddy upwelling, new studies employing a finite element, unstructured
mesh model have suggested the occurrence of intense upwelling off the island’s tips while predicting upwelling of modest
intensity within the eddies (White and Deleersnijder, 2007).
Both eddy and tip upwellings are encountered in the tidal
flow around Rattray Island but there is currently no clear-cut
answer as to which secondary flow generates upwelling with
the largest intensity.
This paper addresses this specific issue through idealized
and realistic high-resolution numerical experiments. The analysis is supported by physical arguments based on the theory of
flow separation. Given Rattray’s geometry and the surrounding
bathymetry, the mechanism of flow separation in shallow
waters helps explain the asymmetry in size of the eddies and
their intensity. The results of idealized numerical experiments
also suggest that eddy and tip upwellings are of similar intensity at Rattray Island.
2. Methods
2.1. Data collection
Rattray Island, located in the Great Barrier Reef (northeast
Australia, 20 000 S, 148 380 E), is elliptical with the main axis
oriented northenortheast. The island is 1500-m long, 300-m
wide and lies in well-mixed water approximately 25-m deep
(Fig. 1).
Data were collected during the period 4e6 December
2006 aboard the RV Lady Basten. The wind blew from the
eastesoutheast at a mean speed of about 20 m s1. The island
shape was determined aboard a dinghy using a GPS recording locations every 30 s (see the red dots in Fig. 1b). In
the morning of 4 December (8:15 hours), a 600-kHz RDI
Instruments ADCP (Acoustic Doppler Current Profiler) was
deployed at site 1 (Fig. 1b) and retrieved on 6 December
(14:00 hours), recording 5-min ensemble averages of the
velocity binned into 50 cm intervals over the vertical. During
the same time span, 1-min averages of the sea level were
recorded every 5 min by a tide gauge located 4 m above
the seabed, at site 1.
On the night of 4 December, another ADCP was deployed
(between 17:05 and 10:11 hours) on the east of Rattray (site
2), recording 1-min ensemble averages of the velocity binned
into 1 m intervals. Finally, on the night of 5 December (between 18:20 and 07:11 hours), an ADCP was deployed on
the west of Rattray (site 3), recording velocity based on the
same characteristics as the previous one. During the period
4e6 December, a total of 21 ADCP transects were completed,
which are shown in Fig. 1b. These transects were either completed with dinghies equipped with a hull-mounted RDI
459
Instruments ADCP or aboard the RV Lady Basten. In all cases,
the boat speed was 1e2 m s1.
In the morning of 5 December (3:00 to 10:30 hours), 16
CTD (conductivity, temperature, depth) probe casts were carried out at site 2 every half hour. Tide reversal occurred around
6:00 hours, followed by flood and the formation of eddies. The
location of the CTD casts was very close to the center of the
northern eddy. This was observable from the vessel. A CTD
cast was completed far away from the island’s wake, at site
1, on 4 December at the time of the ADCP deployment.
This cast provided the so-called ‘‘reference values away
from island’’, which were undisturbed by topography.
2.2. Numerical model configuration
The three-dimensional shallow-water model SLIM1 was
used to reproduce the tidal flow around Rattray Island and investigate secondary currents and vertical motions. The model
is based on the finite element method, allowing the use of unstructured meshes with higher resolution in the neighborhood
of the island (Fig. 1c). The three-dimensional mesh is made up
of prismatic elements, built on the downward extrusion of twodimensional triangular meshes. The prismatic elements are
chosen to yield a consistent, conservative and stable numerical
scheme (White et al., in press).
The traditional, three-dimensional shallow-water equations
are solved with the additional approximation of constant density, justified by the absence of salinity and temperature contrasts (measured but not shown). A quadratic law is used for
bottom friction with a roughness length of 5 103 m, in
agreement with previous studies in that area (Black and Gay,
1987; Deleersnijder et al., 1992). Turbulent vertical momentum diffusion is parameterized by using a simple depthvarying parabolic profile for the viscosity coefficient nt. The
latter, introduced by Fischer et al. (1979), is acceptable for
unstratified shallow seas:
hþz
nt ¼ ku ðh þ zÞ 1 0:6
;
ð1Þ
H
where k is the von Karman constant, u* is the bottom friction
velocity, h is the unperturbed water depth, z is the vertical coordinate (z ¼ 0 is the unperturbed sea level) and H is the total
depth (i.e., depth perturbed by sea-surface fluctuations). Horizontal momentum diffusion is parameterized via a Smagorinsky scheme (Smagorinsky, 1963), the horizontal viscosity
coefficient depending on mesh size and the horizontal velocity
strain-rate tensor.
The computational domain is rotated so that the far-field
horizontal velocity direction may be considered parallel to
the side boundaries coinciding with the y-axis. These boundaries are deemed impermeable and only the top and bottom
boundaries (parallel to the x-axis) remain open. This configuration is shown in Fig. 1d. The rotation angle is determined by
1
Second-generation Louvain-la-Neuve Iceeocean Model, http://www.
climate.be/SLIM.
460
L. White, E. Wolanski / Estuarine, Coastal and Shelf Science 77 (2008) 457e466
taking the mean direction of the far-field depth-averaged horizontal velocity, measured at site 1. This angle is such that the
norm of the x-component of the rotated velocity field is minimum. At both open boundaries, the free-surface elevation and
the y-component of the rotated velocity are imposed in the
form of the Flather boundary condition (Flather, 1976), which
is equivalent to enforcing the incoming characteristic variable
(Blayo and Debreu, 2005):
period corresponds to falling tide when no eddies were present
on the east side of the island.
Salinity and temperature measurements by the CTD probe at
site 2 were almost uniform in time and over depth. They are not
shown. Backscatterance is plotted versus time and depth in
Fig. 3b and ranges from 1.0 to 3.5. The CTD cast at site 1 provided
a backscatterance value of 1.5 throughout the water column.
rffiffiffi
rffiffiffi
g
g ext
ext
un h ¼ un h ;
h
h
3.2. Numerical experiments
ð2Þ
where un is the depth-averaged normal velocity, g is the gravitational acceleration and h is the free-surface elevation. Variables superscripted by ext refer to field data. The phase lag
between both boundaries is less than 20 min and is neglected
since the same field data are used to impose the flow at both
open boundaries.
3. Results
3.1. Field data
Everywhere, the horizontal velocity field was vertically
uniform except near the bottom where the magnitude decreased by about 20 percent, due to bottom friction. The tidal
ellipse was strongly polarized, as suggested by the compass
plot (Fig. 1b) showing the depth-averaged horizontal velocity
at site 1, where one out of three measurements was plotted.
The main current direction switched between about 300 and
120 degrees. The current speed was on the order of
0.5 m s1 and the direction was almost perpendicular to the
island’s main axis.
The measured horizontal velocity field showed the presence
of eddies at rising tide (Fig. 2b, d), which confirmed the results
from the 1982 data collection (Wolanski et al., 1984). Eddies
on the west side of Rattray appeared at falling tide (Fig. 2a, c).
During the 1982 field trip, these eddies were observed but the
velocity field was not measured on the west side of Rattray.
All 21 transects also provided depth samplings that were
used to refine the bathymetry around Rattray Island (Fig. 1d).
Given the strong wind and ensuing free-surface waves,
none of these transects provided reliable measurements of vertical velocity, which turned out to be too noisy for valid interpretations. This is unfortunate considering that one of the
objectives was to quantify the turbulent processes in the free
shear layer.
The only usable measurements of vertical velocity were
given by the ship-born ADCP moored at site 2 during the night
of 4 December (Fig. 3a). During two 2-h periods (at about
20:00 hours on 4 December and about 8:00 hours on 5 December), positive vertical velocity of about 15 mm s1 was measured throughout the water column. These periods are one
tidal cycle apart and coincide with times of peak flood velocity. During the 6-h period starting at 22:00 hours on 4 December, vertical velocity was negative at about 15 mm s1. This
A 2-day simulation was carried out on the mesh shown in
Fig. 1c and comprising six layers of prisms. The open boundary conditions are shown in Fig. 2e, together with locations in
time when comparisons are carried out between the numerically predicted and measured velocity fields. These agree
well, as was already the case with the 1982 data set. Note
that the predicted velocity field is shown at one instant while
the ADCP transects typically took 30 min to complete. But
since the time scale of the eddies is significantly longer than
30 min, these transects give a good rendition of the instantaneous flow pattern.
Vertical motions may be quite intense in the vicinity of the
island (Fig. 4). The upwelling velocity is defined as the component of the vertical velocity from which the topographicallyinduced component is subtracted (Deleersnijder, 1989). The
four panels of Fig. 4 cover a full tidal cycle with snapshots
taken at moments of (a, c) peak velocity and (b, d) close to
tidal reversal when the eddies are big and still intense while
the far-field velocity is weak. A key characteristic of the
eddies is their asymmetry, the northern (right) eddy being systematically bigger than the southern (left) one, during rising
and falling tides. This was also observed and measured by Wolanski et al. (1984). The cause for this will be explored below.
An idealized island was employed in place of Rattray to
shed light on the principal mechanisms generating vertical
motions in a tidal flow (Fig. 5). In addition to its length l facing the main current, the island is fully characterized by the
aspect ratio:
w
a¼ ;
l
ð3Þ
where w is the island width. Both island’s tips are circular with
a radius of curvature r such that 2r ¼ w. That is, for a given
length, the aspect ratio a determines the remaining geometrical characteristics of the island.
A comparison is proposed between the flow structure and
upwelling velocity obtained with Rattray and with the idealized island presenting an aspect ratio a ¼ 0.10 and a length
l ¼ 1500 m (Fig. 5). The constant depth is set to 25 m. All parameterizations are identical and mesh resolutions are similar
in both simulations. Four types of vertical motions are identified by labels AeD. These vertical motions owe their existence to bottom friction, whose effect is to decelerate the
flow near the seabed. This leads to a vertical velocity profile
with maximum magnitude near the surface and minimum
magnitude near the bottom.
L. White, E. Wolanski / Estuarine, Coastal and Shelf Science 77 (2008) 457e466
(a)
461
(b)
5 Dec. 8h45
4 Dec. 14h20
S
1
S
1
E
2
E
500 m
0.5 m/s
500 m
0.5 m/s
(d)
(c)
5 Dec. 14h20
6 Dec. 8h45
S
1
1
E
S
E
500 m
(e)
0.5 m/s
2
1.5
1
0.5
0
−0.5
−1
−1.5
−2
b
a
4.6
Velocity [m s−1]
Elevation [m]
c
4.8
5
5.2
0.5 m/s
500 m
5.4
5.6
5.8
6
6.2
d
6.4
Time [day]
Fig. 2. Comparison between the numerically predicted and measured depth-averaged horizontal velocity fields along ADCP transects (continuous lines) and fixed
ADCP measurements (blue squares), when available at time of comparison. The predicted velocity field was interpolated on a structured grid for clarity. The starting and ending points of each transect are indicated by circled S and E. Transects completed between: (a) 14:15 and 14:41 hours, (b) 8:37 and 8:48 hours, (c) 14:20
and 15:06 hours, and (d) 8:26 and 8:55 hours. (e) Open boundary conditions and position in time when the comparisons (a)e(d) are carried out. (For interpretation
of the references to color in the text, the reader is referred to the web version of this article.)
The four mechanisms for vertical motion are summarized
in Fig. 6. In case of diverging depth-averaged flow (A), the
deficit in volume is larger near the surface than near the bottom. By continuity, water must come up to replace the greater
depletion at the surface, generating upwelling. When the
depth-averaged flow converges (B), the mechanism is opposite. Water accumulating near the surface must come down
to replace the deficit near the bottom, which brings about
Vertical velocity at site 2 [mm s−1]
(a)
45
30
Depth [m]
5
15
10
0
-15
15
-30
-45
20
20
22
0
2
4
4 Dec. [hr]
8
5 Dec. [hr]
Backscatterance at site 2 (ref. value away from island 1.5)
(b)
Depth [m]
6
5
3.0
10
2.5
15
2.0
20
1.5
25
3
4
5
6
7
8
9
10
5 Dec. [hr]
(c)
4h00
(d)
6h00
2
(e)
8h00
0.5 m s−1
2
2
(f)
10h00
2
Fig. 3. Time series of the measured vertical profiles for (a) the vertical velocity (zero contour represented by the thick line and contour value is 15 mm s1) and (b)
the backscatterance (contour value is 0.5) at site 2. Note that the time axes are not the same. (cef) Numerically predicted depth-averaged horizontal velocity field at
four different times on 5 December. The predicted velocity field was interpolated on a structured grid for clarity.
L. White, E. Wolanski / Estuarine, Coastal and Shelf Science 77 (2008) 457e466
(a)
463
(b)
4 Dec. 14h20
4 Dec. 17h00
7
7
5
5
3
3
1
1
−1
−1
−3
−3
−5
−5
−7
−7
−9
−9
−11
−11
−13
−13
−15
−15
(c)
(d)
4 Dec. 21h15
4 Dec. 23h10
2
7
7
5
5
3
3
1
1
−1
−1
−3
−3
−5
−5
−7
2
−7
−9
−9
−11
−11
−13
−13
−15
−15
0.5 m s−1
Fig. 4. Upwelling velocity (at mid-depth) in mm s1 at four different times during a complete tidal cycle around Rattray Island. Positive and negative values indicate upwelling and downwelling, respectively. The background predicted depth-averaged horizontal velocity field was interpolated on a structured grid for clarity.
(a) Peak ebb velocity, (b) end of falling tide, (c) peak flood velocity, and (d) end of rising tide. Notice the asymmetry in the size of the eddies, the northern (i.e., the
right-hand) eddy being systematically bigger both at rising and falling tides.
downwelling. In some respect, onshore flow (C) has a behavior
similar to that of converging flow, leading to downwelling.
This was observed in the field by Hamner and Hauri (1981).
A more delicate mechanism occurs with curved flows (D),
which are encountered both off the island’s tips and within
eddies. In curved flows, a balance is struck between the centripetal acceleration based on the depth-averaged velocity
and the positive pressure gradient due to the tilting of the
sea surface. This equality breaks down near the bottom where
the centripetal acceleration decreases because of bottom friction whereas the pressure gradient remains constant throughout the water column. This leads to inward convergence of
bottom water, which then flows upwards within the water column (e.g., Wolanski and Hamner, 1988). Eddy and tip
upwelling may both be encountered in the vicinity of Rattray,
though with magnitudes that can widely vary depending on the
radius of curvature, the flow intensity and the surface pressure
gradient.
4. Discussion
The measured ascending and descending motions (Fig. 3a)
are most likely caused by the mooring being located where the
bathymetry presents a steep slope (Fig. 1d). At ebb and at this
location, water is flowing northwestward, parallel to the
bathymetry gradient, descending the slope. At flood, water is
flowing in the opposite direction, going up the slope, yielding
a positive vertical velocity.
L. White, E. Wolanski / Estuarine, Coastal and Shelf Science 77 (2008) 457e466
464
(a)
(b)
B
9
9
7
7
5
A
5
B
3
3
A
1
1
−1
−1
B
−3
B
C
−3
−5
−5
D
C
D
−7
D
−7
−9
−9
−11
−11
−13
−13
0.5 m s−1
(c)
r
w = 2r
l
Fig. 5. Identification of the main zones of vertical motions (upwelling velocity at mid-depth in mm s1) around (a) Rattray Island, and (b) a geometrically simplified island, subject to the same tidal flow on a flat bottom. Positive and negative values indicate upwelling and downwelling, respectively. Horizontal depthaveraged velocity field interpolated on a structured grid for clarity. Different labels (A, B, C and D) refer to different upwelling or downwelling mechanisms.
A: diverging flow, B: converging flow, C: onshore flow on both sides of the island, D: curved flow. (c) Parameters used to define the idealized island geometry.
The aspect ration a ¼ w/l is equal to 0.10 in (b).
Backscatterance is directly linked to water turbidity and increases for sediment-laden water. Far away from the island
where water is relatively clear, backscatterance is about 1.5.
In the neighborhood of Rattray, any positive deviation from
this reference value is an anomaly indicative of higher turbidity.
During rising tide on 5 December starting at 6:00 hours, water is
clear with backscatterance close to the reference value (Fig. 3b).
After about 2- to 3-h into flood, backscatterance increases to
2.5. There is no clear vertical structure in the profile, which
could mean that sediments rapidly mix throughout the water
column to reach the surface. Water turbidity significantly
increases after peak flood velocity is reached (at about 9:00
hours), which is the time when eddies become much more
intense (Fig. 3e, f). At that time, the velocity magnitude off
the island’s tips also significantly increases. Both situations
generate upwelling via the mechanism D illustrated in Fig. 6.
Based on numerical experiments, when the aspect ratio of
the idealized island decreases, the eddy size, recirculation
strength and eddy upwelling increase while tip upwelling decreases. In particular, for the smallest aspect ratio considered
(a ¼ 0.05), eddy and tip upwellings are of similar intensity.
In that case, the size of the eddy is about the length of the island and the recirculation speed is about that of the free
stream. When the aspect ratio increases, eddies shrink in
size and intensity, eddy upwelling decreases while tip upwelling becomes dominant (Table 1).
The competition outcome between both mechanisms for
generating upwelling depends on the intricate details of flow
separation at the tips. By solving the boundary layer equation,
Signell and Geyer (1991) obtained an expression for the alongshore pressure gradient as the contributions of local acceleration (due to tidal flow acceleration), advection and bottom
friction. Bottom friction always contributes to a favoring
(i.e., negative) pressure gradient, which tends to prevent flow
separation. Therefore, in shallow-water flows with enhanced
bottom friction (e.g., very shallow or rough seabed), flow separation tends to occur further downstream along the curved island’s tip: flow sticks to the edge much longer before
separating. The opposite scenario occurs when bottom friction
is negligible or when the island’s tip is so sharp that separation
occurs much further upstream along the edge.
For large aspect ratios, tip upwelling dominates over eddy
upwelling because eddies are almost inexistant. The vorticity
input into the wake is weakened by the larger radius of curvature and the fact that separation occurs further downstream. In
these flows, tip upwelling gains in intensity because intense
curved flow occurs only off the tip. This, in turn, generates
a larger sea level depression and more intense converging
flow near the bottom (mechanism D in Fig. 6). For sharp
islands, flow separates too far upstream to ever be in the
case of curved flow at the tip. The vorticity input into the
wake increases and eddies increase in size and intensity. In
L. White, E. Wolanski / Estuarine, Coastal and Shelf Science 77 (2008) 457e466
Top view
465
Side view
(a)
Diverging flow
Surface water depletion replaced by
upwelled water
(b)
Converging flow
Downwelling of accumulated water
near the surface
(c)
Onshore flow
Downwelling of accumulated water
near the surface + tilting of
sea surface (adverse pressure gradient)
(d)
Separation point
Curved flow
Balance breakdown near the bottom
between pressure gradient and
centrifugal acceleration followed
by inward flow and upwelling
Eddy or
island’s tip
Pressure Centrifugal
gradient acceleration
Fig. 6. Summary of mechanisms generating vertical motions in shallow-water flows interacting with topography. All vertical motions owe their existence to the
prevalence of bottom friction. Both eddy and tip upwelling arise in curved flows (see panel D).
Table 1
Characterization of secondary circulations depending on the idealized island’s
aspect ratio a, as defined by Eq. (3). The far-field speed is noted U. The length
of the island facing the current is noted l. As the island becomes sharper, the
eddy size, recirculation strength and eddy upwelling increase. For the largest
aspect ratio, flow separation occurs further downstream along the curved edge
and eddies barely develop. Tip upwelling is at its maximum. Eddy and tip
upwellings are of similar intensities for the smallest aspect ratio
Aspect
ratio
Eddy size
Eddy intensity
Tip upwelling
[mm/s]
Eddy upwelling
[mm/s]
0.05
0.10
0.15
0.25
l
0.85l
0.75l
0.50l
U
0.8U
0.6U
0.2U
3e5
4e6
6e8
8e10
2e3
1e2
1
< 0.5
these cases, eddy and tip upwellings may be of similar strength
(Table 1).
In the case of Rattray, those considerations suggest that the
bathymetry and topography in the vicinity of the island should
be known as accurately as possible. The flow separation mechanisms are most likely the most important aspects of the
dynamics controlling the size, position and intensity of the
eddies as well as the intensity of eddy and tip upwelling.
The northern tip lies in deeper water than the southern tip.
Hence, bottom friction is enhanced close to the southern tip.
The geometry of the northern tip is also sharper. These topographical features help explain why the northern eddy is systematically bigger than the southern one, both at ebb and
flood (Fig. 4). The northern tip does not have the same geometry facing rising and falling tide. For a northwestward flow,
the northern tip has a larger radius of curvature while for
466
L. White, E. Wolanski / Estuarine, Coastal and Shelf Science 77 (2008) 457e466
a southeastward flow, the tip is sharper. This could explain
why tip upwelling is more significant during falling tide
than during rising tide (Fig. 4). Given Rattray’s topography,
it falls into the category of smaller aspect ratios, where eddy
and tip upwellings are of similar intensity at the time of
peak tidal speed. The topography also helps explain why the
eddy size is on the order of the island’s length and the recirculation speed is on the order of the free stream magnitude.
5. Conclusion
This paper reported on the case study of Rattray Island
(Great Barrier Reef, northeast Australia), subject to tidal
flow in shallow waters. At ebb and flood, attached (stable)
eddies develop in the wake where swirls of turbidity suggest
that sediment-laden waters are brought to the surface as a result
of vertical transport. Both eddy and tip upwellings are encountered in the tidal flow around Rattray Island. Given Rattray’s
geometry and surrounding bathymetry, the mechanism of
flow separation in shallow waters allows to explain the asymmetry in size of the eddies and their intensity. The results of
idealized numerical experiments also suggest that eddy and
tip upwellings may be of similar intensity at Rattray Island.
Acknowledgements
The authors are grateful to the crew and captain of the RV
Lady Basten. LW is a Research fellow with the Belgian National Fund for Scientific Research (FNRS). LW carried out
this study within the scope of the project ‘‘A second-generation
model of the ocean system’’, which is funded by the Communauté Française de Belgique, as Actions de Recherche Concertées, under contract ARC 04/09-316. This work is a contribution
to the construction of SLIM, the Second-generation Louvainla-Neuve Iceeocean Model (http://www.climate.be/SLIM).
References
Alaee, M.J., Ivey, G., Pattiaratchi, C., 2004. Secondary circulation induced by
flow curvature and Coriolis effects around headlands and islands. Ocean
Dynamics 54, 27e38.
Black, K.P., Gay, S.L., 1987. Eddy formation in unsteday flows. Journal of
Geophysical Research 92 (C9), 9514e9522.
Blayo, E., Debreu, L., 2005. Revisiting open boundary conditions from the
point of view of characteristic variables. Ocean Model 9, 231e252.
Coutis, P.F., Middleton, J.H., 2002. The physical and biological impact of
a small island wake in the deep ocean. Deep-Sea Research, Part I: Oceanographic Research Papers 49, 1341e1361.
Deleersnijder, E., 1989. Upwelling and upsloping in three-dimensional marine
models. Applied Mathematical Modelling 13, 462e467.
Deleersnijder, E., Norro, A., Wolanski, E., 1992. A three-dimensional model
of the water circulation around an island in shallow water. Continental
Shelf Research 12 (7/8), 891e906.
Falconer, R.A., Wolanski, E., Mardapitta-Hadjipandeli, L., 1986. Modeling
tidal circulation in an island’s wake. Journal of Waterway, Port, Coastal
and Ocean Engineering 112, 234e254. ASCE.
Fischer, H.B., List, E.Y., Koh, R.C.Y., Imberger, J., Brooks, N.H., 1979.
Mixing in Inland and Coastal Waters. Academic Press, New York.
Flather, R.A., 1976. A tidal model of the northwest European continental
shelf. Mémoires de la Société Royale des Sciences de Liège 6 (10),
141e164.
Geyer, W.R., 1993. Three-dimensional tidal flow around headlands. Journal of
Geophysical Research 98 (C1), 955e966.
Hamner, W.M., Hauri, I.R., 1981. Effects of island mass: water flow and
plankton pattern around a reef in the Great Barrier Reef lagoon, Australia.
Limnology & Oceanography 26 (6), 1084e1102.
Hoitink, A.J.F., 2004. Tidally-induced clouds of suspended sediment
connected to shallow-water coral reefs. Marine Geology 208, 13e31.
Ingram, R.G., Chu, V.H., 1987. Flow around islands in Ruppert Bay: an investigation of the bottom friction effect. Journal of Geophysical Research 92
(C13), 14521e14533.
Pattiaratchi, C., James, A., Collins, M., 1986. Island wakes and headland
eddies: a comparison between remotely sensed data and laboratory experiments. Journal of Geophysical Research 92 (C1), 783e794.
Pietrzak, J., Deleersnijder, E., Schröter, J. (Eds.), 2005. The Second International Workshop on Unstructured Mesh Numerical Modelling of Coastal,
Shelf and Ocean Flows. Ocean Model, vol. 10. Elsevier.
Pietrzak, J., Iskandarani, M., Schröter, J., Lyard, F. (Eds.), 2006. The Third
International Workshop on Unstructured Mesh Numerical Modelling of
Coastal, Shelf and Ocean Flows. Ocean Model, vol. 15. Elsevier.
Signell, R.P., Geyer, W.R., 1991. Transient eddy formation around headlands.
Journal of Geophysical Research 96 (C2), 2561e2575.
Smagorinsky, J., 1963. General circulation experiments with the primitive
equations. 1. The basic experiment. Monthly Weather Review 91 (5),
99e165.
Suthers, I.M., Taggart, C.T., Kelley, D., Rissik, D., Middleton, J.H., 2004.
Entrainment and advection in an island’s tidal wake, as revealed by light
attenuance, zooplankton, and ichthyoplankton. Limnology & Oceanography
49, 283e296.
Tomczak, M., 1988. Island wakes in deep and shallow water. Journal of Geophysical Research 93 (C5), 5153e5154.
White, L., 2007. Accuracy and consistency in finite element ocean modeling.
PhD thesis, Université catholique de Louvain. Electronic copy available
from: http://edoc.bib.ucl.ac.be:81/ETD-db/collection/available/BelnUcetd03192007-143839.
White, L., Deleersnijder, E., 2007. Diagnoses of vertical transport in a threedimensional finite-element model of the tidal circulation around an island.
Estuarine, Coastal and Shelf Science 74, 655e669.
White, L., Legat, V., Deleersnijder, E. Tracer conservation for threedimensional, finite element, free-surface, ocean modeling on moving prismatic meshes. Monthly Weather Review, in press.
Wolanski, E., Asaeda, T., Tanaka, A., Deleersnijder, E., 1996. Threedimensional island wakes in the field, laboratory experiments and numerical models. Continental Shelf Research 16 (11), 1437e1452.
Wolanski, E., Brinkman, R., Spagnol, S., McAllister, F., Steinberg, C.,
Skirving, W., Deleersnijder, E., 2003. Merging scales in models of water
circulation: perspectives from the Great Barrier Reef. In: Lakhan, V.C.
(Ed.), Advances in Coastal Modeling. Elsevier Oceanography Series,
vol. 67. Elsevier Science, pp. 411e429 (Chapter 15).
Wolanski, E., Hamner, W.M., 1988. Topographically controlled fronts
in the ocean and their biological influence. Science 241 (4862),
177e181.
Wolanski, E., Imberger, J., Heron, M.L., 1984. Island wakes in shallow
coastal waters. Journal of Geophysical Research 89 (C6), 10553e
10569.