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7th Int. Symp. on Stratified Flows, Rome, Italy, August 22 - 26, 2011
1
Dense Gravity Currents in a Rotating, Up-sloping, Converging Channel
Alan Cuthbertson1, Peter Davies2, Janek Laanearu3 and Anna Wåhlin4
1 School of the Built Environment, Heriot Watt University, UK, a.cuthbertson@hw.ac.uk
2 Department of Civil Engineering, University of Dundee, UK. p.a.davies@dundee.ac.uk
3 Institute of Mechanics, Tallinn University of Technology, Estonia, janek.laanearu@ttu.ee
4 Department of Earth Sciences, University of Gothenburg, Sweden, anna.wahlin@gu.se
Abstract
Results are presented from laboratory modelling experiments investigating the behaviour of
dense bottom gravity currents in an inclined, submerged, rotating and convergent vee-shaped
channel. High-resolution density and velocity profile measurements demonstrate that these
bottom currents adjust to geostrophic balance in the converging channel for a wide range of
parametric conditions. Comparisons with theoretical model predictions based on inviscid
rotating hydraulics and Ekman dynamics modelling approaches are shown to demonstrate
many qualitative aspects of the hydraulic behaviour of dense bottom gravity currents in such
rotating systems. Quantitative discrepancies between measurements and analytical model
predictions are attributed primarily to the omission, in both theoretical models, of shear-
induced turbulent entrainment and mixing processes between outflowing bottom currents and
overlying ambient waters.
1. Introduction
The hydraulics of gravity currents in rotating systems have wide environmental and
geophysical relevance, particularly for dense oceanic and estuarine/fjord outflows affected by
the rotation of the Earth, where deep ocean straits and complex seafloor topography exerts
crucial controls on the velocity and density fields within the flow and water exchange
between interconnected basins (e.g. Borenäs and Lundberg, 2004). Fundamental knowledge
of these topographic effects is currently of significant interest, especially with regard to (i) the
maintenance of geostrophic balance within oceanic outflows confined by complex bottom
topography, (ii) the prediction of slope, distortion and elevation of the interface between the
intrusive dense outflow layer and the overlying, relatively-quiescent receiving waters, (iii) the
role of boundary layer dynamics (Davies et al, 2006) and (iv) the limiting and restricting
effects of hydraulic control and transport capacity (Wåhlin, 2002) on the outflow dynamics.
1.1 Physical System
A schematic representation of the channel configuration under investigation is shown in
Figure 1. It consists of a symmetric, converging, vee-shaped channel with variable side slope
angles
(y) and an along-channel bed slope S0, inclined upwards towards the channel exit.
The Cartesian coordinate system (x, y, z) is orientated with x and y axes in the cross- and
along-channel directions, respectively, and the z axis taken as anti-parallel to the gravitational
acceleration vector g = (0, 0, -g). The initial, undisturbed configuration is one in which the
channel topography is submerged within a homogeneous body of ambient water of density
0
and depth H, set in a state of solid body rotation with angular velocity = (0, 0, ). At time t
= 0, a dense water source flow, having typical horizontal and vertical inflow dimensions l0
and h0, density
1 = [
0 + (
)0], dynamic viscosity
and initial volume flux Q0 is introduced
through a source arrangement located upstream of the constant width section of the channel.
7th Int. Symp. on Stratified Flows, Rome, Italy, August 22 - 26, 2011
2
Q0, 1
S0
2-2
4-4
6-6
2
4
6
z
x
y
1
zs
zb,min
1
1
2
2
3
3
4
4
5
5
6
6
0.375
0.375
0.375
0.375
1.5
0.8
4 probes @
100 mm spacing
4 probes @
50 mm spacing
6 ADV profiles @
50 mm spacing
Source
0.5
1
2
3
4
5
6
7
8
D/S control
section
2.5
Converging
channel section
U/S basin
section
P
P
Figure 1: (a) Schematic representation of physical system, (b) plan view of channel topography with
main attributes shown.
3. Experimental Arrangement and Procedure
The experiments utilised a 5 m diameter by 0.5 m-deep circular tank, rotating with a constant
angular velocity , within which the channel topography was installed with a longitudinal
bed slope S0 (= 1.7, 3.6) inclined upwards towards the channel exit. Prior to each
experimental run, the circular tank was filled with freshwater (
0 = 998 999 kg.m-3) to a
total depth Hb = 0.44 m, submerging the channel topography. The system was then spun-up to
solid-body rotation at the prescribed angular velocity of either 0.21 s-1 or 0.14 s-1. At the
start of each experiment, brine solutions (
1 = 1005 1019 kg.m-3) were introduced at the
upstream end of the channel at an initial volume flux Q0 = 0.3 l s-1. The reduced gravitational
acceleration g0 [= g.(
1
0)/
1] associated with these brine discharge conditions ranged
from g0 = 0.055 0.192 m.s-2 at the channel inlet. During each run, Q0 values were increased
incrementally (0.3 0.6 0.9 l s-1) at prescribed normalised elapsed times 2t.
Vertical density profiles were measured throughout the experiment duration to monitor spatial
and temporal development of the density field
(x,y,z,t) at fixed locations within the channel
using micro-conductivity probes mounted on motorised rack systems. Two profiling rigs were
sited at sections 4 and 6, each with four probes at fixed cross-channel measurement locations
(P1–P4 and P5–P8, Figure 1(b)). Corresponding velocity profile data for each outflow
condition was obtained using an acoustic Doppler velocimeter (ADV) mounted on a
motorised traversing system (at section 6).
4. Results
Figure 2 shows typical time series data at the downstream locations (P5–P8), plotted non-
dimensionally as density excess
= (
0)/
colour maps within the normalised time-
(2t) space (z/H) domain. Dashed vertical white lines in Figure 2 indicate normalised times
2t at which the inflow flux Q0 was increased incrementally. For all S0 = 3.6 runs, the
interface between the dense water outflow (
1) and overlying ambient fluid (
= 0)
remains sharply defined throughout the experiment (Figure 2(a)). The temporal development
of the dense outflow layer also indicates well-defined increases in interface elevation
corresponding to the incremental increases in Q0. After each adjustment, fully developed
outflow conditions were established along the channel prior to subsequent adjustment in Q0.
For runs with S0 = 1.7, the plots show a more diffuse interface forming between the
outflowing dense bottom water and overlying ambient fluid layer (Figure 2(b)), due to
significant shear-induced interfacial mixing – a property associated with the higher outflow
(a)
(b)
7th Int. Symp. on Stratified Flows, Rome, Italy, August 22 - 26, 2011
3
velocity (and, hence, lower gradient Richardson number) in the shallower bed slope case. The
temporal variation in the dense water outflow layer thickness (defined by the
= 0.2
isopycnal elevation) is shown to be qualitatively similar to the S0 = 3.6 run (Figure 2(a)),
with adjustments in the outflow layer thickness occurring after each prescribed increase in Q0.
(a)
(b)
Figure 2: Time sequences of density profiles at section 6 for (a) S0 = 3.6; Bu = 0.23, 0.20, 0.19 (P5-
P8) and (b) S0 = 1.7; Bu = 0.27, 0.30, 0.39 (P6-P8). Here, Burger number Bu = g0h0/l02(2Ω)2.
Figure 3 shows typical non-dimensional plots of cross-channel isopycnal (
= 0.10.9)
elevations measured at sections 4 (probes P1–P4) and 6 (probes P5–P8), indicating a
significant increase in cross-channel interface slope due to topographical convergence of the
outflow as the channel exit is approached. These plots show that isopycnals at the lower bed
slope of S0 = 1.7 (Figure 3(a)) are generally steeper and more diffuse than the tightly-packed
isopycnals obtained for S0 = 3.6 runs (Figure 3(b)), primarily due to the higher outflow
velocities and increased shear induced mixing associated with the S0 = 1.7 runs.
0
0.2
0.4
0.6
0.8
1
-1 -0.5 0 0.5 1
X/H
Z/H
P2
P3
P4
P1
Section 4
(a)
0
0.2
0.4
0.6
0.8
1
-1 -0.5 0 0.5 1
X/H
Z/H
P1
P2
P3
P4
Section 4
(b)
0
0.2
0.4
0.6
0.8
1
-1 -0.5 0 0.5 1
X/H
Z/H
P1
P2
P3
P4
Section 4
(c)
0
0.2
0.4
0.6
0.8
1
-1 -0.5 0 0.5 1
X/H
Z/H
P5
P6
P7
P8
Section 6
(a)
0
0.2
0.4
0.6
0.8
1
-1 -0.5 0 0.5 1
X/H
Z/H
P5
P6
P7
P8
Section 6
(b)
0
0.2
0.4
0.6
0.8
1
-1 -0.5 0 0.5 1
X/H
Z/H
P5
P6
P7
P8
Section 6
(c)
Figure 3: Non-dimensional plots of cross-channel isopycnals = 0.1 (navy) – 0.9 (red) at sections 4
and 6 for runs with S0: Bu values of (a) 3.6: 0.190, (b) 1.7: 0.273 and (c) 1.7: 0.030.
Comparison of Figures 3(b) and 3(c) shows that cross-channel interface slope and outflow
layer thickness both increase as Bu decreases, suggesting that the relative influence of density
stratification (through g0) diminishes while rotation effects (through 2) becomes more
7th Int. Symp. on Stratified Flows, Rome, Italy, August 22 - 26, 2011
4
dominant. Indeed, values of the cross-channel interface slopes αi, derived from measured
isopycnal elevations (Figure 3) are shown to agree well with the assumption of geostrophy
within the outflow along the converging channel (Figure 4), with geostrophic slope αg written
as
g = V(2)/g, where V is the depth-averaged outflow velocity and g' = g(
)/
0 is the
local, modified gravitational acceleration based upon measured density profiles.
0
10
20
30
010 20 30
g
i
(b)
Figure 4: Comparison of measured cross-channel interface slopes
i (section 6) and calculated
geostrophic slopes
g (Equation 1). Error bars show average variance for S0 = 1.7 (triangles) and 3.6
(diamonds) data sets.
Figure 5 shows typical profiles of the along- v(x,z) channel velocities measured by the ADV
at six cross-channel locations at section 6 (see Figure 1(b)) for three separate runs. These
plots show clearly that the magnitude of the along-channel velocity v within the outflowing
dense water layer is relatively high compared to the passive conditions typically observed
within the overlying ambient water layer. The plots also indicate that the maximum along-
channel outflow velocities vmax occur at or close to the channel centreline (x/H 0) in all
cases, while diminishing in magnitude at more outlying measurement locations where the
outflow layer thickness also diminishes. Magnitudes of v values in the outflow layer were
notably higher for the lower bed slope runs (S0 = 1.7, Figure 5(b)(d)); a property reflected
in the increased levels of shear-induced mixing observed in these runs (see Figure 2(b)).
100
134
167
200
234
267
301
-150 -100 -50 050 100 150 200
z/H
x/H
0
-50
50
v veloci ty (mm s-1)
(a)
-0.45
-0.3
-0.15
0
0.15
0.3
0.45
0.6
0.3
0.4
0.5
0.6
0.7
0.8
0.9
33
67
100
134
167
200
234
-150 -100 -50 050 100 150 200
z/H
x/H
0
-50
50
v veloci ty (mm s-1)
(b)
-0.45
-0.3
-0.15
0
0.15
0.3
0.45
0.6
0.1
0.2
0.3
0.4
0.5
0.6
0.7
33
67
100
134
167
200
234
-150 -100 -50 050 100 150 200
z/H
x/H
0
-50
50
v veloci ty (mm s-1)
(a)
-0.45
-0.3
-0.15
0
0.15
0.3
0.45
0.6
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Figure 5: Typical plots of along-channel ADV velocity profiles v(x,z) measured at section 6 for runs
with S0: Bu values of (a) 3.6: 0.022, (b) 1.7: 0.030 and (c) 1.7: 0.120
Direct comparison of velocity profile plots in Figure 5(b) and (c) shows the effect on the
along-channel velocities v of increasing the relative influence of stratification over rotation
(i.e. an increase in Bu, for otherwise identical conditions). Specifically, while the maximum v
values in the outflow layer increase at all x/H locations, the corresponding layer thickness is
shown to decrease. This finding is consistent with the expectation that more mixing (and,
hence, more ambient fluid entrainment), will occur when the density difference between the
two fluid layers is reduced (i.e. at lower g0 values).
(c)
7th Int. Symp. on Stratified Flows, Rome, Italy, August 22 - 26, 2011
5
5. Theoretical Considerations
5.1 Inviscid Rotating Hydraulic Model (IRHM)
The model developed by Borenäs and Lundberg (1986) to describe the hydraulics of rotating,
inviscid, dense outflows may be applied. By conservation of potential vorticity,
0
h
f
dt
dz
(1)
where
z is the vertical component of relative vorticity of the disturbance flow, h is the
(variable) dense fluid depth and f (= 2Ω) is the Coriolis parameter. In a channel where along-
channel variations are much smaller than cross-channel variations [∂/∂y << ∂/∂x], the cross-
channel component of the momentum equation reduces to the geostrophic balance for the
along-channel velocity v(x), such that
x
gxvf
(2)
where
is the interface height. Furthermore, for an inviscid flow, the potential vorticity is
conserved along a streamline and equation (1) can be simplified to
h
f
h
xvf
(3)
where h is the potential depth [i.e. the water depth at which relative vorticity (v/x)
vanishes], which, for comparison with the laboratory model results, is fixed at h = H/2 for all
computations. A unique solution for the interface height η(x) = h(x) + zb(x) can be derived
from equations (2) and (3) by eliminating the velocity term v(x). Integration of the resulting
equation twice in respect of x and specifying boundary conditions at the edges of the outflow
where the deep layer thickness vanishes (i.e. at intersection between interface and bottom
boundary), yields the following equation for cross-sectional interface height in a triangular
channel:
xzh
ba
xa
a
xabx
ba
ah
xs
sinh
sinh
2sinhsinh
sinh
2
)(
(4)
where x = (-a, b) represents the lateral coordinates of the intersection with the channel bed of
the interface between the dense gravity current and the overlying, quiescent receiving fluid
(see figure 2) and λ = (g'h∞)1/2/f is the Rossby radius of deformation based on the potential
depth h. The geostrophic velocity v(x) can be found directly from equations (2) and (4), with
the geostrophic transport maximum determined by the critical flow variables x = (acrit, bcrit)
corresponding to the appropriate Froude number condition Fr* = 1 for the rotating triangular
channel topography (see Laanearu, 2001), such that:
2
tanhv
ˆ
&
2
coth1v
,
2
tanh
ˆ
v
ˆ
2
coth
ˆ
v
ˆ
v22
ba
k
h
a
f
g'baa
f
g'
ba
h
ba
hFr
(5)
where
v
and
v
ˆ
are half the sum and difference, respectively, of the outflow velocities at x =
a and b and
h
ˆ
= (h2g/fQ)1/2 is the dimensionless potential depth. The solutions obtained
from equations (2) – (5) are obtained assuming critical flow conditions (i.e. Fr* = 1), with the
maximum geostrophic flux matched to each different experimental flow rate considered.
7th Int. Symp. on Stratified Flows, Rome, Italy, August 22 - 26, 2011
6
5.2 Ekman Dynamics Frictional Model (EDFM)
Previous treatments of frictional controls of dense outflows in rotating channel geometries
have utilised an Ekman layer model in which the dynamic balance is between the effects of
along- and cross-pressure gradients, Coriolis acceleration and bottom friction (Davies et al,
2006). As demonstrated by Wåhlin & Walin (2001), bottom friction can be represented either
as a bulk drag force acting on the main flow or by resolving the Ekman spiral. Both
approaches give dynamically similar results if the magnitude of the drag coefficient is chosen
such that the net frictional force is equal in the different representations. The Ekman layer
solution may be written conveniently as follows:
2
sin
2
cos1;
2
sin
2
cos1 2222 z
eu
z
evv
z
ev
z
euu
z
g
z
g
z
g
z
g
(6)
where δ = (υ/2f)1/2 is the thickness of the Ekman layer, υ is the kinematic viscosity and (ug, vg)
are the geostrophic velocities in the (x, y) directions (Davies et al, 2006). For the case of a
moving lower dense layer and an overlying quiescent, upper ambient layer, with the
assumption of v/x << f and /y << /x, a frictionally-controlled system can be derived and
solved with respect to the dense layer thickness h(x,y). Solutions for different channel
topographies have been obtained previously (Wåhlin, 2002; Wåhlin 2004; Davies et al, 2006;
Borenäs et al, 2007). For example, for a dense gravity current flowing down an inclined vee-
shaped channel (Davies et al., 2006):
01;012
xAe
S
xhxeAxh ii S
x
i
S
x
(7)
where h is the thickness of the outflowing dense layer, Si is the along-channel interface slope
and A is an integration constant determined from boundary conditions. Equation (7) is the
solution to a first-degree differential equation and hence requires one boundary condition to
determine the solution. Horizontal integration of the quantity vh across the channel gives the
along-channel transport flux Q according to
0
2
22
0a
F
i
bb
a
AQ
S
f
g
dxvhdxvhdxvhQ
(8)
where QF(A) = [2A 2 + ln(1 2/A)] and integral limits x = (a, b) are the interface/side-
slope intersections. For the up-sloping, vee-channel considered here, both A and Si are
determined using two prescribed conditions: (i) the right-hand intersection between the
outflow layer and channel bed boundary (looking downstream) is set horizontal, and (ii) the
along-channel transport Q is equal to the source flux Q0. The first condition can be written as
follows
ub hbzbh
(9)
where h(b) and zb(b) are respectively the outflow layer thickness and bed elevation at x = b
and hu is the upstream outflow layer thickness. By using equation (9) in (7), Si can be
expressed as a function of A, i.e.
0
xz
Sb
i
eA
(10)
Substitution of equations (10) into (8) provides two roots of Si for each Q0 value, the smaller
of the two being more physically appealing since both the along- and cross-channel interface
slopes then increases as the downstream end section of the channel is approached.
7th Int. Symp. on Stratified Flows, Rome, Italy, August 22 - 26, 2011
7
6. Comparisons
Figure 6 shows comparisons between measured and predicted cross-channel variation in
interface elevation at sections 4 and 6. Comparisons of outflow layer thickness hi and cross-
channel interface slope
i are presented in Figure 7. The predictions of interface elevation by
the Ekman dynamics friction model (EDFM) at section 4 are consistently close to (but lower
than) the measured interface elevations (with correspondingly lower hi values, see Figure
7(a)). By contrast, EDFM predictions at the channel exit (section 6) are significantly higher
than observed in the experiments. This discrepancy relates to the required specification of the
upstream elevation hu of the interface and a fixed value for the along-channel slope Si
[equations (13) and (11)] as input parameters. By contrast, the IRHM predictions of interface
elevation and outflow layer thickness hi are significantly lower than the experimental
measurements at section 6. This effect is attributed to (i) the frictionless bottom condition
adopted in IRHM, which reduces the flow area (and hence hi) required for the critical and
maximal discharge conditions to develop at the downstream channel control, or (ii) the
intrinsic assumption of critical outflow conditions (Fr*2 = 1.0) at the channel exit.
0
0.2
0.4
0.6
0.8
1
-1.5 -1 -0.5 0 0.5 1 1.5
X/H
Z/H
0.171: 5.60e-4
0.126: 11.2e-4
0.120: 16.8e-4
SECTION 4
0
0.2
0.4
0.6
0.8
1
-1 -0.5 0 0.5 1
X/H
Z/H
SECTION 6
(a)
0
0.2
0.4
0.6
0.8
1
-1.5 -1 -0.5 0 0.5 1 1.5
X/H
Z/H
0.104: 20.8e-4
0.083: 41.7e-4
0.069: 62.5e-4
0
0.2
0.4
0.6
0.8
1
-1 -0.5 0 0.5 1
X/H
Z/H
(b)
Figure 6: Non-dimensional cross-channel interface profiles for runs with S0 = 1.7 and Bu values as
shown. Key: measured
= 0.2 isopycnal elevations (diamonds sections 4 & 6); EDFM predictions
(solid lines sections 4 & 6); and IRHM predictions (dotted lines section 6 only).
Comparisons between the measured and predicted values of cross-channel interface slope
i
(Figure 7(b)) show both EDFM and IRHM underestimating
i at both sections (IRHM at
section 6 only). These discrepancies are ascribed to the significant shear-induced mixing
observed in the S0 = 1.7 runs, a process that is not incorporated in either of the models.
Furthermore, some of the divergence between model predictions and experimental data arises
from the selection of the reference isopycnal to represent the measured interface elevation.
7. Concluding Remarks
Laboratory results and simple analytical modelling demonstrate that the qualitative aspects of
the dense outflows are captured well by the theoretical approaches adopted. Inviscid, rotating
hydraulics theory for critical flow conditions at the exit section is shown to underestimate the
7th Int. Symp. on Stratified Flows, Rome, Italy, August 22 - 26, 2011
8
deep-layer depth hi and the cross-channel interface slope
i, but this discrepancy may be
attributed primarily to the absence in the model of frictional effects at the channel boundaries
and the layer interface. The incorporation of Ekman dynamics improves agreement between
theoretical predictions of the parametric dependence of the outflow characteristics, though the
absence of any interfacial mixing processes in either of the theoretical models results in
quantitative discrepancies between theory and experiment.
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0 0.1 0.2 0.3 0.4 0.5
Measured hi/H
Predicted h i/H
EDFM - S4
EDFM - S6
IRHM - S6
(a)
0
5
10
15
20
25
0 5 10 15 20 25
Measured i (deg.)
Predicted i (deg.)
EDFM - S4
EDFM - S6
IRHM - S6
(b)
Figure 7: Experimental measurements and model predictions (EDFM and IRHM) of (a) maximum
normalised outflow layer thicknesses hi/H and (b) cross-channel interface slopes
i.
Acknowledgements
This work has been supported by the EU HYDRALAB III Programme (Contract no. 022441).
The authors would like to thank Dr. Thomas McClimans and his colleagues at the Norwegian
University of Science & Technology (SINTEF) for their hospitality and technical assistance.
References
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