We summarize a study that compares experimental laboratory data for gravity-driven coastal surface currents with corresponding theoretical results obtained from a new geostrophic model describing such currents. It is found that experiment and theory are, generally, in good agreement

GREGORIO, Sandy

HAIDVOGEL D.B.

LEVIN J.C.

LINDEN P.F.

TASKINOGLU E.S.

THOMAS P.J.

I-Revues

18 ème Congrès Français de Mécanique Grenoble, 27-31 août 2007
Investigation of Gravity-Driven Coastal Currents
Grégorio S.O.1, Thomas P.J.1, Linden P.F.2, Levin J.C.3. Haidvogel D.B.3, Taskinoglu E.S.3
1University of Warwick, Fluid Dynamics Research Centre, Coventry CV4 7AL, United Kingdom
2University of California, Department of Mechanical and Aerospace Engineering, San Diego, USA
3Rutgers University, Institute of Marine and Coastal Sciences, 71 Dudley Road, New Brunswick, USA
Email address of the corresponding author: S.Gregorio@warwick.ac.uk
Abstract :
We summarize a study that compares experimental laboratory data for gravity-driven coastal surface currents with
corresponding theoretical results obtained from a new geostrophic model describing such currents. It is found that
experiment and theory are, generally, in good agreement.
Key-words :
Gravity-Driven Flows; Rotating Flows; Coastal Currents
1 Introduction
When estuarine river water is discharged into the coastal zone, gravity-driven surface flows can
be established (?). These flows develop as a consequence of the density difference between the
buoyant estuarine fresh water and the denser, salty ocean water. For sufficiently large discharge
rates, i.e. when the current exceeds length scales larger than the Rossby deformation scale, the
current dynamics are affected by the Coriolis force arising from the rotation of the earth. As a
result the discharged fresh water is confined to the coastal zone where it forms a current flowing
along the coast.
? and, more recently, ?, derived scaling relations for the current width, its depth and its
velocity. However, since these relations were based on scaling arguments they did not yield the
constants of proportionality missing from the expressions obtained. We derived a new analytical
model which readily yields values for the constants. The expressions obtained are similar to
those of ?. Here we summarize how the predictions of our model compare to results from our
laboratory experiments - for a detailed comparison, refer to ?.
2 Experimental set-up
Two complementing sets of experiments, using different experimental facilities, were carried
out. The two facilities had substantially different spatial scales enabling the investigation of a
wide range of the independent experimental parameters. Small-scale laboratory experiments
were performed on a rotating turntable supporting a fluid-filled tank with a radius of approxi-
mately 0.5m. Large-scale experiments were conducted at the Coriolis Facility (LEGI, Grenoble)
where the rotating basin has a radius of 6.5m.
The principle of the experiments and the arrangement of the small-scale experimental fa-
cility are illustrated in Fig. 1. The tank was filled with salt water of density ρ2 representing
the ocean water. Fresh water of density ρ1, with ρ1 < ρ2, was released from a small source
mounted at the wall of the tank to simulate estuarine discharges. The source was adjusted to
be level with the surface of the dense salt water. The fluid was discharged vertically upwards,
rather than horizontally, as would be the case in the natural environment, in order to minimize
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18 ème Congrès Français de Mécanique Grenoble, 27-31 août 2007
Reservoir
Pump
Video Camera 1
(rigidly mounted on table
superstructure)
Video
Camera 2
PC
Source
(not shown flush)
Tank
Rotating turntable
Current
Figure 1: Small-scale setup.
Current, 1
Ambient
water, 2
x
y
z
u 0
Wall of tank
Source
h0
w0
l(t)
Figure 2: Nomenclature for model.
momentum-flux effects and mixing of fresh water and salt water near the source. Fluid was
released continuously and with a constant volumetric discharge rate q0. The density difference
between the fresh water and the salt water is characterized in terms of the reduced gravity, g′,
defined by g′ = ((ρ2−ρ1)/ρ1) g with g = 981 cm s−2. The rotation rate, Ω, of the turntable was
constant for each experiment. The salt water inside the tank was in solid-body rotation prior
to the start of fluid being released from the source. For visualization purposes the fresh water
discharging at the source was dyed with food colouring while the ambient salt water was clear.
The large-scale experiments were designed to mirror the small-scale study.
3 Theoretical description of the problem
Our model assumes that the flow velocities normal to the wall and in the vertical direction (see
Fig. 2) are negligible in comparison to the along-wall flow velocity. We allow for motion along
the x−axis but neglect all derivatives ∂/∂x. We use the geostrophic approximation, assume
that the potential vorticity is zero and conserved, and we also employ mass conservation. The
final, main theoretical results of ? are
h0 =
(
4Ω q0
g′
) 1
2
, w0 =
(
g′ q0
Ω3
) 1
4
, l(t) =
3
4
(q0g
′Ω)
1
4 t, (1)
where the meaning of h0, w0 and l(t) is apparent from Fig. 2. The expressions in Eq. (1) can
be conveniently expressed in dimensionless form. We use the non-dimensional time T = Ωt
and non-dimensionalize lengths by w0 = (g′q0/Ω3)1/4. Using capital letters to denote non-
dimensional variables, one obtains
H0 =
h0
w0
= 2I
5
4 , W =
w0
w0
= 1, L =
l(t)
w0
=
3
4
T (2)
where I = Ωq1/50 /g′3/5 is a dimensionless parameter that describes the aspect ratio, h0/w0, of
the current. For a detailed description of our model consult ?.
4 Experimental results
4.1 Qualitative observations
Figure 3 shows a flow visualization illustrating a typical current observed in the small-scale
facility for I = 0.3264. The picture shows a current viewed from above the circular tank,
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18 ème Congrès Français de Mécanique Grenoble, 27-31 août 2007
Figure 3: Current (dark fluid) flowing along the wall of the circular tank in the small-scale facility.
Turntable rotates anti-clockwise, current also propagates anti-clockwise keeping wall to its right.
looking vertically downwards onto the fluid surface. The source outlet, where the buoyant
current fluid is ejected, is located at the wall of the tank, just to the right of the anti-cyclonically
(i.e. clockwise) spinning gyre which has developed in the vicinity of the source. As discussed
in ?, the large-scale currents observed in the Coriolis Facility look qualitatively very similar to
those developing in the small-scale facility, but currents can appear quite different for different
values of I .
4.2 Current length
Figures 4 and 5 display summaries of 66 small-scale experiments and the 34 large-scale ex-
periments respectively. The figures display the non-dimensional current length, L, vs. non-
dimensional time, T . The solid line superposed onto both figures represents the theoretical pre-
diction L = 3/4 T from Eq. (2). Comparison of the two figures shows that they are very similar.
In both cases experiment and theory are in good agreement. Inspection of Figs. 4 and 5 reveals
that a superposition of the two figures collapses both data sets onto each other. Hence, the
small-scale and large-scale experiments display corresponding dynamic behaviour. The dashed
10-1 2 5 100 2 5 101 2 5 102 2 5 103 2
T
5
10-1
2
5
100
2
5
101
2
5
102
2
5
103
L
: Avicola & Huq (2002)
: Theory, L = 3/4 T
: Small-scale data
Figure 4: L vs. T ; Small-scale data.
10-1 2 5 100 2 5 101 2 5 102 2 5 103 2
T
5
10-1
2
5
100
2
5
101
2
5
102
2
5
103
L
: Avicola & Huq (2002)
: Theory, L = 3/4 T
: Large-scale data
Figure 5: L vs. T ; Large-scale data.
line superposed onto Figs. 4 and 5 represents the theoretical result of Avicola & Huq (2002)
which is L = 2T in our nomenclature (see ?). Figure 4 shows that, for the small-scale exper-
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18 ème Congrès Français de Mécanique Grenoble, 27-31 août 2007
iments, our model predicts the propagation of the current head overall slightly better than the
model of Avicola & Huq. For the large-scale data the opposite seems to be the case. However,
a detailed discussion, which is contained in ?, concurrently needs to take the data for the depth
into account. If this is done then the data analysis suggests that it is probably justified to con-
cluded that, overall, the agreement between the experimental data in both the small-scale and
the large-scale experiments, is in slightly better agreement with our present model than with the
model of Avicola & Huq (2002).
4.3 Current height
According to our model, the current height, h0, at the wall of the tank (see Fig. 2 and Eq. (1))
is constant, i.e. it is independent of the x-coordinate. However, the videos of the experiments
reveal that the current height decreases between the source region and the current head. A
sketch schematically illustrating the observed height profile at the wall (y = 0), in the x-z plane
of Fig. 2, is shown in Fig. 6. The current has a maximum height, hm in Fig. 6, just downstream
Current fluid supplied
from reservoir
Surface of fluid
Source (outlet vertically upwards)
Current fluid, 1
Ambient
fluid, 2
RL1 RL2 RL3 RL4
hm
Head of current
Figure 6: A side-view (x− z plane in Fig. 2)
of a current as recorded by Video Camera 2
shown in Fig. 1.
0.0 0.5 1.0 1.5 2.0 2.5 3.0
I
0
1
2
3
4
5
6
7
H
0
: q0 = 28.12 cm
3
s
-1
: q0 = 20.0 cm
3
s
-1
: q0 = 10.0 cm
3
s
-1
: q0 = 3.18 cm
3
s
-1
Small-scale exps.
: q0 = 300-900 cm
3
s
-1
: q0 = 3972 cm
3
s
-1
: q0 = 3000 cm
3
s
-1
: q0 = 1500 cm
3
s
-1
: q0 = 200 cm
3
s
-1
Large-scale exps.
: Theory, H0 H0
t
= 2 I5/4
: Least-squares fit, H0
e
= 2.10 I0.998
Figure 7: Non-dimensional maximum current
height, H0 = hm/w0, near the source as a
function of I .
of the source. This height hm was observed to be constant throughout each experiment. It was
used for comparison with the theoretical prediction for h0 of Eq. (1) viz. H0 of Eq. (2).
Figure 7 summarizes the measured values for the non-dimensional height H0 = hm/w0 as a
function of I . A least squares fit interpolating all data points gives He0 = 2.10I
0.998 and is shown
as the solid line in Fig. 7. The dashed line in Fig. 7 is the theoretical prediction H0 = 2I5/4.
It can be seen that experiment and theory are overall in good agreement. A similarly good
agreement is also found for the current width as is discussed in ?.
5 Froude number
We calculate experimental values for the Froude number, Fr(Ti) = ue0/
√
g′hm, based on the
measured current mean velocity ue0 = li/ti and the measured maximum depth hm near the
source. The results obtained in this way are displayed in Figs. 8 and 9. The two figures reveal
that, for both our sets of experiments, the Froude number is in the range of approximately
0.1 ≤ Fr ≤ 1.51 and the figures illustrate how Fr decreases with time, T .
In Figs. 8 and 9, the data points for the different runs have been grouped into three different
sets according to their values of the parameter I . Both figures reveal that there is a general
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18 ème Congrès Français de Mécanique Grenoble, 27-31 août 2007
0 100 200 300 400 500 600 700
T
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Fr
: I > 1.0
: 0.5 < I 1.0
: 0 I 0.5
Small-scale data
: Fr = 1.2 T-0.22
Large-scale data
: Fr = 0.85 T-0.09
Figure 8: Fr vs. T ; Small-scale data.
0 50 100 150 200 250 300 350
T
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Fr
: I > 0.2
: 0.05 < I 0.2
: 0 I 0.05
Large-scale data
: Fr = 0.85 T-0.09
Small-scale data
: Fr = 1.2 T-0.22
Figure 9: Fr vs. T ; Large-scale data.
trend for the Froude number to adopt lower values for smaller values of I . Low values of I
correspond to high values of the Ekman number (see ?) and, hence, to the parameter regime
where viscous effects become increasingly important and where our inviscid model is no longer
valid.
For the vast majority of all data points for highest values of I (lowest Ek), the Froude
number lies within the interval 0.35 ≤ Fr ≤ 0.78. The solid lines, given by Fr = 0.85T−0.09
and Fr = 1.2T−0.22, interpolating the data points in Figs. 8 and 9 were determined by plotting
all data in double-logarithmic representation and then fitting lines through the regions displaying
highest concentrations of data points by eye. These lines are rough guidelines summarizing the
data behaviour for the higher values of the parameter I; they do not represent least-squares fits.
To facilitate easy intercomparison, the solid line of Fig. 8 corresponds to the dotted line in Fig. 9
and vice versa. The two figures show that most data lie within 0.35 ≤ Fr ≤ 0.78. This is in
overall good agreement with our model which predicts Fr = 0.5303. For a comparison with
corresponding Froude number values found by other authors, refer to ?.
6 Conclusions
We compared theoretical predictions of a new geostrophic model for gravity-driven coastal
surface currents to experimental data. Experiment and model are, generally, in good agreement.
A detailed summary and discussion of our results is available in ?.
Our present model only describes currents flowing along vertical boundaries. We have re-
cently begun to investigate the more realistic scenario where the currents flow along sloping
coastlines. To this end, we are currently preparing an experimental study funded through the
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18 ème Congrès Français de Mécanique Grenoble, 27-31 août 2007
EU’s HYDRALAB III initiative. This study will be conducted in May 2007 employing the 5m-
diameter Coriolis tank of the Marine Systems Research Infrastructure (www.ntnu.no/trondheim-
marine-RI).
Figure 10: Visualization of a current from our computational simulations.
In collaboration with our colleagues from the Institute of Marine and Coastal Sciences
(IMCS) at Rutgers University, we are currently also initiating numerical simulations of the flow.
While we do not yet have data available that lend themselves to a quantitative comparison
with our experiments, we already obtained some first preliminary computational results. One
typical current simulated with Rutgers’ ROMS code (www.myroms.org) is shown in Fig. 10
(Ω = 0.5 rads−1, g′ = 86.3 cms−2, q0 = 20.0 cm3s−1, I = 0.063 at t = 30 s). A comparison of
Fig. 10 with the photograph in Fig. 3 already reveals promising qualitative agreement.
References
Avicola, G. & Huq, P. 2002 Scaling analysis for the interaction between a buoyant coastal
current and the continental shelf: Experiments and observations. J. Phys. Oceanography 32,
3233-3248.
Davies, P. A., Jacobs, P. T. & Mofor, L. A. 1993 A laboratory study of buoyant fresh water
boundary currents in tidal crossflows. Oceanologica Acta 16, 489-503.
Griffiths, R. W. 1986 Gravity currents in rotating systems. Ann. Rev. Fluid Mech. 18, 59-89.
Lentz, S. J. & Helfrich, K. R. 2002 Buoyant gravity currents along a sloping bottom in a rotating
fluid. J. Fluid Mech. 464, 251-278.
Thomas, P.J., Linden, P.F. 2007 Rotating Gravity Currents: Small-scale and large-scale experi-
ments and a geostrophic model. In print: J. Fluid Mech..
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