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Experimental investigation of large-scale vortices in a freely spreading gravity
current
Yeping Yuan, and Alexander R. Horner-Devine
Citation: Physics of Fluids 29, 106603 (2017);
View online: https://doi.org/10.1063/1.5006176
View Table of Contents: http://aip.scitation.org/toc/phf/29/10
Published by the American Institute of Physics
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PHYSICS OF FLUIDS 29, 106603 (2017)
Experimental investigation of large-scale vortices in a freely
spreading gravity current
Yeping Yuan1,2,a) and Alexander R. Horner-Devine3,b)
1 Ocean
College, Zhejiang University, Hangzhou 310058, China
Key Laboratory of Satellite Ocean Environment Dynamics, Second Institute of Oceanography,
State Oceanic Administration, Hangzhou 310012, China
3 Department of Civil and Environmental Engineering, University of Washington, Seattle,
Washington 98195, USA
2 State
(Received 12 January 2017; accepted 20 September 2017; published online 11 October 2017)
A series of laboratory experiments are presented to compare the dynamics of constant-source buoyant gravity currents propagating into laterally confined (channelized) and unconfined (spreading)
environments. The plan-form structure of the spreading current and the vertical density and velocity
structures on the interface are quantified using the optical thickness method and a combined particle
image velocimetry and planar laser-induced fluorescence method, respectively. With lateral boundaries, the buoyant current thickness is approximately constant and Kelvin-Helmholtz instabilities are
generated within the shear layer. The buoyant current structure is significantly different in the spreading case. As the current spreads laterally, nonlinear large-scale vortex structures are observed at the
interface, which maintain a coherent shape as they propagate away from the source. These structures
are continuously generated near the river mouth, have amplitudes close to the buoyant layer thickness,
and propagate offshore at speeds approximately equal to the internal wave speed. The observed depth
and propagation speed of the instabilities match well with the fastest growing mode predicted by linear stability analysis, but with a shorter wavelength. The spreading flows have much higher vorticity,
which is aggregated within the large-scale structures. Secondary instabilities are generated on the
leading edge of the braids between the large-scale vortex structures and ultimately break and mix on
the lee side of the structures. Analysis of the vortex dynamics shows that lateral stretching intensifies
the vorticity in the spreading currents, contributing to higher vorticity within the large-scale structures
in the buoyant plume. The large-scale instabilities and vortex structures observed in the present study
provide new insights into the origin of internal frontal structures frequently observed in coastal river
plumes. Published by AIP Publishing. https://doi.org/10.1063/1.5006176
I. INTRODUCTION
Freshwater discharged from rivers contributes momentum
and buoyancy to the coastal ocean dynamics. In addition, rivers
carry terrestrial materials including sediment, carbon, nutrients, and contaminants, which impact the coastal ecosystem.1,2
Thus, the dynamics of river discharge after it leaves an estuary
mouth plays an important role in coastal ocean environments.
Recently, considerable effort has been made to understand the
dynamics of river plumes and their role in transport and dispersal of river-borne matter in the coastal zone using laboratory,
theoretical, and numerical model studies.3–8
River plumes can be modeled as gravity currents in which
less dense freshwater from the river flows on top of the ambient salty denser ocean water. They are typically described in
terms of four dynamical regions: the source, near-field, midfield, and far-field regions.6 In the rotating reference frame, the
current flows out of the source region into the energetic nearfield region starting from the lift-off point and then forms a
growing anticyclonic bulge (in the Northern hemisphere) near
the mouth and a coastal current downstream of the bulge.3,9,10
a) Electronic
b) Electronic
mail: yyping@zju.edu.cn
mail: arhd@uw.edu
1070-6631/2017/29(10)/106603/17/$30.00
In this study, we focus on the near-field region, the jet-like
region which is characterized by supercritical Froude numbers,
enhanced mixing, and rapid water mass modification. Flow in
the near-field region is typically dominated by local advection,
buoyancy, and interfacial turbulent stress, compared with the
far-field plume which is strongly influenced by earth’s rotation
and local wind stress.6 Thus, simulation of the near-field flow
can be achieved by a simple model consisting of river outflow
from an estuary mouth into a large and quiescent basin and
ignoring earth’s rotation and wind. At the estuary mouth, the
buoyant water flows outward and expands laterally owing to
the horizontal baroclinic pressure gradients. The dynamics in
the near-field region are critical to the mid- and far-field evolution of the plume because it is the near-field region that mixing
is most intense, thus determining the buoyancy anomaly available to drive the flow in subsequent regions of the plume.
A. Background
The classical view of the near-field flow is based on
early lock-exchange experiments conducted in a narrow
flume. von Karman,11 Benjamin,12 and Shin, Dalziel, and
Linden13 described the current propagation speed
p as a function of the internal gravity wave speed c (c = g 0h), where
g 0 = (∆ρ/ρ0 )g is the reduced gravity, g represents gravity,
29, 106603-1
Published by AIP Publishing.
106603-2
Y. Yuan and A. R. Horner-Devine
∆ρ is the density difference between two water masses, ρ0 is
the ambient water density, and h is the mean current thickness.
At the interface, mixing occurs due to shear instabilities in
the form of Kelvin-Helmholtz (KH) instabilities.14 However,
this traditional view assumes that the flow is two-dimensional
and does not account for the lateral spreading of the gravity
current.
After leaving the estuary mouth, the near-field plume
spreads laterally, which causes the plume to shoal, by continuity, and thereby accelerate, as predicted by the Bernoulli
equation. Previous work has suggested that the lateral spreading and shoaling process would enhance shear mixing because
the increased flow speed and stretching of KH billows at the
interface.4 Using laboratory experiments, Yuan and HornerDevine15 found that the total mixing in the near-field region of
the river plume was higher in a spreading plume than a channelized plume. However, they found that the local mixing was
the same in both cases, indicating that the increase in mixing was a consequence of the change in the geometry of the
process rather than the turbulence.
Previous researchers have re-examined lock-exchange
experiments using sector tanks or cylindrical configurations
to study the differences in structure between two-dimensional
and three-dimensional gravity currents. The first and the most
notable difference is the appearance of a multiple-front structure in these diverging flows, as observed by Chen.16 Later,
Simpson14 also reported that dense fluid spreading in a sector tank formed multiple fronts, with a relatively thin layer
between fronts. This multiple-front structure has also been
observed in cylindrical release laboratory experiments17 and
numerical simulations.18,19
Patterson et al.18 proposed a possible mechanism for the
formation of the multiple-front structure in an axisymmetric
experiment: a double vortex at the front. They suggested that
KH vortices are originally generated at the interface, then move
to the bottom boundary, and finally form a double vortex. As
a result, the vorticity is concentrated in multiple fronts and
is intensified due to vortex stretching. A similar process was
described by Cantero, Balachandar, and Garcia.19 Stretching
of the vortex rings in the azimuthal direction stabilizes the
KH structure and slows the transition into fully developed
three-dimensional turbulence. A cyclic process of vortex pairing forms in each front head; a counterclockwise vortex forms
first at the leading edge due to baroclinic vorticity, followed by
the formation of a clockwise vortex along the bottom surface,
Phys. Fluids 29, 106603 (2017)
and convection of heavy fluid forward by the vortex pair. Alahyari and Longmire20 suggested that the vortex pair may also
explain the fact that the frontal head is significantly larger in
axisymmetric gravity currents than in two-dimensional gravity
currents.
Several researchers have reported similar multiple-front
structures in field observations, such as a natural buoyant
discharge into Trondheim Fjord21 and the Quinault River discharging into the Pacific Ocean.22 Garvine22 suggested that
the structure may be associated with the interaction of the
leading front with the nonlinear internal waves present in
radially spreading gravity currents. Luketina and Imberger 23
observed concentric foam lines in the Koombana Bay plume,
which may be the surface manifestation of multiple fronts.
They revealed a sub-frontal feature with large horizontal density gradients approximately 90 m behind the leading edge
of the plume in one Conductivity-Temperature-Depth (CTD)
section. The sub-front has a slightly higher propagation speed
than the leading front; it moved forward and was pulled into the
frontal bore structure. Recently, Halverson and Pawlowicz24
have also reported similar multiple instances of steep salinity gradient structures in CTD transects in the Fraser River
plume. Although there is no direct comparison between in
situ observed multiple-front or sub-front structures and those
observed in the axisymmetric gravity current experiments, we
hypothesize here that both of them may be associated with
lateral spreading of the propagating layer.
B. Present experiments
We conducted laboratory experiments of gravity currents
in laterally confined and unconfined environments in order
to investigate the generation mechanism of the multiple-front
structure in river plumes. Fluids of different densities initially
at rest were separated by a vertical gate in a tank. During the
experiment, the source tank provided constant flow into the
basin after a vertical gate was opened and the buoyant water
was allowed to flow out of the domain over a weir on the
far end of the basin. Figure 1 shows two configurations of
the experiment: a channelized experiment [or laterally confined, Fig. 1(a)] in which fluids on both sides of the gate were
constrained within lateral boundaries and a freely spreading
experiment [or laterally unconfined, Fig. 1(b)] in which the
width of the dense ambient fluid tank is approximately eight
times that of the light fluid filling basin.
FIG. 1. Schematic of (a) channelized and (b) lateral
spreading experiments of gravity currents experiments.
Light and dark gray areas represent lighter inflow fluid
(with lower density ρi ) and heavier ambient fluid (with
higher density ρ0 ), respectively. Solid thick lines indicate vertical boundaries, while dashed thick lines are
the removable gate which initially separates two fluids.
Arrows indicate the flow direction.
106603-3
Y. Yuan and A. R. Horner-Devine
The channelized configuration simulates two-layer flows
in estuaries and the spreading configuration simulates the discharge of buoyant fluid into the coastal ocean. We expect to
observe the multiple-front structure in the spreading case, similar to the three-dimensional lock-exchange experiment.17–19
There are two unique features in the present experiment. First,
the experiment has a continuous inflow rather than the fixed
volume release as in previous laboratory experiments. In previous fixed volume experiments, the gravity current never
reaches steady-state and must be analyzed in the coordinate
frame that moves with the front. In contrast, the continuous
inflow source allows the gravity current to reach a steady-state
with respect to the control point.25,26 In the present experiments, the control point occurs at the river mouth where
the buoyant flow is forced to lift-off and separated from the
entrance channel bottom. Also, the laterally unconfined configuration allows the width and lateral expansion to adjust naturally in the cross-stream direction. Yuan and Horner-Devine15
showed that the lateral spreading rate is strongly dependent
on the inflow condition via the inflow Froude number. Rocca
et al.27 also found that the structure of freely spreading gravity
currents is different from the axisymmetric gravity currents.
The freely spreading configuration better represents gravity
current propagation into the open ocean than experiments with
pre-determined configurations such as a sector or a cylindrical
tank.
Phys. Fluids 29, 106603 (2017)
II. EXPERIMENTAL SETUP
To examine the differences in structure between channelized and freely spreading gravity currents, experiments
were conducted in a rectangular water tank (hereafter called
“plume basin”) described in detail in the work of Yuan and
Horner-Devine.15 The plume basin, illustrated in Fig. 2, has
dimensions of 4 m × 2.5 m × 1 m, and the estuary tank is
0.8 m × 0.3 m × 0.5 m with a gate that is 0.3 m wide and 0.05
m deep. The source of the buoyant water is a 500 l constant head
tank located above the level of the plume basin. To create a uniform inflow, the buoyant water was introduced into the plume
basin through a small estuary tank, containing a diffuser board
and a honeycomb. Each experiment was started by opening
the estuary gate and the buoyant water inflow valve simultaneously. When the gravity current propagated across the plume
basin, mixed fluid exited the system over an adjustable weir
at the downstream end of the basin. After the initial frontal
propagation period, the system reached a quasi-steady-state
by balancing the inflow from the source tank and the outflow
through the weir.
We used two measurement techniques in the present
experiments, which were previously described in Yuan and
Horner-Devine15 and are presented again here. A plan-view
experiment [Fig. 2(a)] was conducted in the lateral spreading case. The freshwater was dyed with colored food dye,
FIG. 2. Schematic of (a) the plan-view
dye experiments and (b) cross-sectional
view of laser experiments. Flow is from
right to left. The plan-view imaging was
conducted to get general plume structures only in the spreading experiments
[Fig. 1(a)], while the cross-sectional
view of laser experiments measured
density and velocity fields for both
spreading [Fig. 1(a)] and channelized
[Fig. 1(b)] experiments. The light gray
triangle area indicates the laser sheet,
and the box within it shows the field-ofview of the PIV-PLIF measurement. A
conductivity-temperature probe located
downstream of the laser sheet is used to
calibrate the PLIF density field.
106603-4
Y. Yuan and A. R. Horner-Devine
Phys. Fluids 29, 106603 (2017)
15 cm < y < 15 cm) at one end of the plume basin.
TABLE I. The parameters of the experiments.
Expt. name
g00 (cm2 s 1 )
Q0 (gpm)
U 0 (cm s 1 )
Fr i
Re
1.74
2.34
15
15
6.31
6.31
2.14
1.84
18 920
18 920
SP
CH
providing detailed plume freshwater thickness fields using the
Optical Thickness Method (OTM).28,29 The dyed water was
illuminated by a point light source located above the plume
basin and a sequence of images were acquired using a digital
camera mounted directly above the water surface. The images
were calibrated before each experiment using a wedge-shaped
cuvette. The resulting freshwater thickness fields were used
to generate a qualitative description of the plan-view plume
structure and the lateral spreading rate.15
A subsequent set of experiments were conducted with
similar inflow conditions for channelized and lateral spreading
geometries. These experiments were done using a combined
Particle Image Velocimetry and Planar Laser Induced Fluorescence (PIV-PLIF) technique30 in a cross-sectional view configuration [Fig. 2(b)]. PIV-PLIF measures velocity and density
fields at short time intervals from a sequence of image triplets
taken with a digital camera fitted with a wavelength cutoff
filter. The field-of-view (FOV) of this method is a small vertical plane (14 cm × 12 cm) along the plume center line (y = 0).
The velocity field is obtained from the first two images, in
which small PIV particles are illuminated by a Nd:YAG laser,
using the matPIV code.31 The third image in the triplet is illuminated by the argon ion laser, which excites florescent dye
that has been added to the buoyant fluid. The density field was
calculated based on the dye concentration measured with the
PLIF technique32 and calibrated with a profiling conductivitytemperature probe mounted at the downstream end of the laser
field-of-view.
The inflow conditions for both spreading and channelized experiments were characterized in terms of the inflow
Froude number, Fri = √U00 , where U 0 , g00 , and H 0 are
g0 H0
the inflow velocity, inflow reduced gravity, and inflow depth,
respectively. Supercritical conditions (i.e., Fri > 1) are representative of energetic gravity current flows such as a coastal
river inflow during ebb tides. We tried to match the inflow Fr i
for the spreading and channelized experiments as closely as
possible for all of our experiments; however, a perfect match
was not possible, primarily due to challenges in obtaining the
intended density anomaly. In this manuscript, we compare a
pair of supercritical cases: Fr i = 2.14 for the spreading run
(hereafter called SP) and Fr i = 1.84 for the channelized run
(hereafter called CH). The two runs discussed in this paper are
a subset of the runs in the work of Yuan and Horner-Devine.15
A summary of experimental parameters is shown in Table I.
The inflow Reynolds numbers [Re = ( ρi U0 H0 )/µ] are listed in
the last column, where µ is the dynamic viscosity of the water.
III. RESULTS
A. Variability in current thickness
Each run is initiated by opening the gate, at which time
lighter fluid flows into the plume basin from the estuary (x = 0;
The development of a freely spreading supercritical gravity current (run SP: Fr i = 2.14) is shown in Fig. 3(a),
presenting snapshots of the freshwater thickness field. The
inflowing buoyant water is observed to spread laterally as
it propagates offshore, forming a cone-shaped surface layer.
The structure and evolution of this surface layer depend
strongly on Fr i .15 The freshwater thickness was only measured in the lateral spreading configuration because the geometry of the channelized configuration prohibited good OTM
measurements.
Shortly after the flow is initiated (t = 6 s), a frontal bore
is observed as a thick dark line at the plume front [labeled
as “fb” in Fig. 3(a), left]. The plume front entrains the fluid
behind it, forming a thick frontal bore and leaving a somewhat thinner trailing layer. The front in the present case is
almost parallel to the gate. This is different from the cylindrical lock-exchange experiment by Patterson et al.18 and the
subcritical inflow case shown in the work of Yuan and HornerDevine.15 In those cases, the frontal bore is curved and the
curvature depends on the inflow condition. In the present case,
the plume with supercritical inflow forms a jet-like current with
the offshore velocity much higher than the cross-stream velocity. Subsequently, another thick dark line is generated behind
the frontal bore near the river mouth [called internal band,
labeled as “b1” in Fig. 3(a), left]. Similar multi-front structures have been observed in previous laboratory simulations14
and field observations.23 The cross-shore and alongshore
length scales of the internal bands are similar to the frontal
bore.
The frontal bore and internal bands both propagate offshore. To illustrate the multi-front propagation, the freshwater
thickness profiles along the plume center line were plotted
in Fig. 3(b) at 2 s intervals. To remove small-scale variations,
freshwater thickness profiles were averaged in the cross-stream
direction over 1 cm (0.5 cm < y < 0.5 cm). The first (t = 6 s)
and last (t = 14 s) profiles correspond to the left and right
panels in Fig. 3(a), respectively. As the plume propagates offshore, the plume thickness near the front decreases due to the
mass conservation. At the same time, the frontal bore thickness also decreases at a higher rate. As a result, the depth of
the frontal bore is similar to the following fluid after t = 10 s.
At t = 14 s [Fig. 3(a), right], there is no clear bore near the
plume front. Near the river mouth, five internal bands are generated subsequently within 8 s (t = 6 s to t = 14 s) [Fig. 3(b),
b1–b5]. Similar to the frontal bore, these bores only exist until
they reach approximately 60 cm offshore, where they appear
to break down into smaller instabilities. At t = 14 s, only three
bands (b3, b4, and b5) can be observed near the river mouth;
two bands (b1 and b2) that were generated earlier have already
broken down.
The dashed lines in Fig. 3 trace the offshore propagation
of the frontal bore and the following internal bands. The slope
of the lines suggests that the frontal bore propagation speed is
faster than the internal bores. A Hovmöller diagram [Fig. 4(a)]
generated from the center line of the freshwater thickness
image [Fig. 3(b)] in a sequence of 600 images (60 s) documents the propagation of the frontal and internal bores more
clearly. The slope of the upper edge [Fig. 4(a), white dashed
106603-5
Y. Yuan and A. R. Horner-Devine
Phys. Fluids 29, 106603 (2017)
FIG. 3. (a) Plan-view freshwater thickness field for a plume front at early
stage (left) and a developed plume front
(right). The frontal bore is indicated
with blue arrows and labeled “fb,” while
internal bands generated near the inflow
box are indicated with red arrows and
labeled “b1–b5.” (b) Freshwater thickness profile time series along y = 0 line.
Time increases from t = 6 s to t = 14 s
with 2 s intervals. The first and last panels in (b) correspond to the two panels in
subplot (a). Black solid lines are the best
fit to the raw data (gray dots) for each
freshwater thickness profile. The frontal
bore and internal bands are indicated by
red and blue arrows, respectively. The
dashed lines indicate the propagation of
the frontal bore and each internal band.
line] represents the frontal propagation speed (cf ≈ 10 cm s 1 ).
The diagram shows a pattern of yellow streaks (i.e., thick buoyant layer) inclined from the upper-right towards the lower-left
corner, representing the offshore propagation of the internal
bands. To further illustrate the offshore propagation of the
large-scale structures, we traced out the local maximum and
minimum as the wave crest and trough in the Hovmöller diagram [Fig. 4(b)] after applying a low-pass filter (wavelength
>1 cm) and subtracting the mean value. The wave evolution is then quantified by counting all of the zero crossings33
that occur over the region with relatively steady propagation
[x > −66 cm, to the right of the red line in Fig. 4(b)]
after the front passes (t > 12 s). The wavelength derived
from the zero crossings remains nearly constant during the
entire experiment, indicating a time-independent wavelength
λ = 17.2 ± 7 cm [Fig. 4(c)]. Similarly, the frequency evolution
is quantified using zero crossings over time 12 s < t < 60 s
[Fig. 4(d)]. As in the traces, the zero crossings show a decrease
in frequency as the current propagates offshore, from 0.2 Hz
near the gate to less than 0.1 Hz at the offshore end. The wave
deceleration may be caused by the energy dissipation through
small-scale instabilities as well as shoaling and thinning of
the surface layer due to lateral stretching. The buoyant layer
is thin beyond 60 cm; thus, the results towards the offshore
end are not as robust as the energetic region closer to the gate.
The phase speed is estimated by multiplying the wavelength
and the average frequency within this region ( f p = 0.17 s 1 ),
resulting in c ≈ 3 cm s 1 .
B. Subsurface structure and dynamics of internal
bands
We use velocity and density fields measured in a vertical slice along the center line of the current with PIV-PLIF
to investigate the subsurface structure of the internal bands,
which were observed in the plan-view plume thickness measurements. We focus on the region 44 cm < x < −30 cm
106603-6
Y. Yuan and A. R. Horner-Devine
Phys. Fluids 29, 106603 (2017)
FIG. 4. (a) Hovmöller diagram generated from the time sequence of the freshwater thickness observed in Fig. 3(b). The blue triangular area in the upper left
corner is the region before the frontal bore arrives and the white dashed line indicates the plume front propagation. Diagonal yellow streaks represent internal
bands propagating to the left (offshore). (b) Characteristics of offshore-propagating large-scale structures, with blue indicating a wave trough (negative) and
yellow indicating a wave crest (positive). Wave trough and crest data to the right of the red line were used to estimate the wavelength. (c) Temporal evolution of
the wavelength determined using the zero crossing method. The black dashed line represents the time-averaged wavelength (λ = 17.2 ± 7 cm). (d) Variation of
the frequency with respect to the offshore position determined using the zero crossing method. The black dashed line represents a linear fit. Magenta bars in (b)
and (d) indicate the field-of-view of the vertical slice PIV-PLIF experiments. The average frequency within the PIV-PLIF region is fp ≈ 0.17 s 1 .
[indicated as magenta bars in Figs. 4(b) and 4(d)], which is
the region where we observe periodic variation in the plume
freshwater thickness. A comparison of the density field in the
channelized and spreading experiments is shown in Fig. 5
(Multimedia view). Columns (a) and (c) show the plume structure during the frontal propagation period, corresponding to
the time period shown in Fig. 3(a), left. The thickness of the
frontal bore is similar in the spreading and channelized cases;
however, the structure along the interface is different. The leading edge of the frontal bore in the spreading experiment is
very sharp and two small well-defined KH billows are generated along the interface [Fig. 5(c), t = 2–4 s (Multimedia
view)]. The leading edge of the channelized current is not as
sharp and, while there is evidence of similar billows forming, they break down almost immediately and are much less
distinct.
Another pronounced difference between the two cases is
the occurrence of large-scale structures, which appear to correspond to individual fronts or bores, that form periodically in
the spreading case and not in the channelized case. Figure 5(d)
(Multimedia view) documents one of these bore structures
passing through the field-of-view. On the leading edge of
each internal bore, the buoyant layer thickens and the interface sharpens. The trailing edge is marked by intense mixing
and the buoyant layer thins after the structure passes, often
getting to nearly zero thickness between two internal bores.
This periodic oscillation of the layer thickness corresponds to
the deep bands near the river mouth observed in the plan-view
freshwater thickness fields (Fig. 3); the bands viewed from
above have the same length and time scales. Although this phenomenon is difficult to visualize in the still images [Fig. 5(d)
(Multimedia view)], it is clear in the PLIF video (see enhanced
online multi-media).
Columns (b) and (d) in Fig. 5 (Multimedia view) both
correspond to the same time during the steady-state period. In
the channelized case, the gravity current is similar to currents
generated in previous lock-exchange experiments, with a relatively uniform layer thickness and sustained mixing at the
interface between the two fluids. Mixing and entrainment of
dense ambient water into the buoyant layer is the result of shear
instabilities. However, no clear evidence of the generation,
development, and breaking of KH billows is been observed.
In contrast, the generation and evolution of small-scale KH
billows are observed more clearly in the spreading case. They
are generated along the leading edge, develop as they are carried back toward the point of maximum current thickness,
and finally break and turn into fully developed turbulence
along the trailing edge of each internal bore. We hypothesize that the small-scale billows are secondary KH instabilities generated between internal bores. We will revisit this in
Sec. IV B.
The evolution of the current structure is clearly observed
but not well quantified in the cross-sectional view experiment
since the spatial extent of the individual images is limited to
14 cm compared with the observed wavelengths of approximately 17 cm. This limitation can be overcome with a time
series constructed from a vertical density profile at a given
location in sequential images. Figure 6 shows a 40 s time series
at the center of the PLIF field-of-view. It clearly reveals the
frontal bore region and the large-scale structures described
in Sec. III A. A significant difference between channelized
and spreading currents is observed after the front has passed
106603-7
Y. Yuan and A. R. Horner-Devine
Phys. Fluids 29, 106603 (2017)
FIG. 7. Power spectrum of the plume normalized density anomaly for
the spreading (black solid) and the channelized (gray dashed) cases with
their respective 2σ confidence intervals labeled as circles. Both spectra
were calculated using the multi-taper method and averaged over 150 pixels
vertically.
FIG. 5. Normalized density anomaly sequence in the plume front region (a)
and (c) and the steady-state region (b) and (d) for the channelized (left) and
spreading (right) cases. Time increases downward in each column and the
time interval between frames is 1 s. The physical dimensions of this fieldof-view are labeled on the lower-left corner. Normalized density colorbar is
shown in the upper-right corner. The detailed comparison between the plume
evolution in channelized and spreading cases is shown in the enhanced online
multi-media. Multimedia view: https://doi.org/10.1063/1.5006176.1
(t > 10 s). The large-scale structure in the spreading case is
similar to the initial release of a lock-exchange gravity current
in a 10◦ sector tank shown in the work of Simpson14 (Fig.
12.9). Five large-scale structures within 30–35 s are observed
in the spreading case, with periods T ≈ 6−7 s. By assuming an
advective velocity of U m /2, the wavelength of the large-scale
structures is approximately 21–24 cm, slightly larger than the
wavelength estimated in Sec. III A using the zero crossing
method.
Finally, we determine the frequency of the internal bands
using the power spectrum of the normalized density anomaly.
Individual power spectra were computed for each pixel at the
density interface and then an average spectrum was computed
by averaging these spectra (Fig. 7). The confidence interval
(2σ) was derived from the chi-squared approach (open circles).
There is a distinct broad peak at 0.1–0.2 Hz in the spreading case, which is absent in the channelized case. This peak
is consistent with the average frequency f p = 0.17 Hz from
the frequency evolution [Fig. 4(c)] over the magenta bar in
Sec. III A.
IV. ANALYSES
We observe significant differences between channelized and spreading gravity currents with similar inflow
Froude numbers. Bore-like bands are observed in plan-view
FIG. 6. Temporal sequence of the vertical density profile at the center of the
PLIF field-of-view for (a) channelized
and (b) spreading cases.
106603-8
Y. Yuan and A. R. Horner-Devine
experiments with lateral spreading, which propagate offshore
at near-critical conditions (i.e., U/c ≈ 1). Detailed imaging of
the vertical structure of the spreading current reveals that these
bands correspond to periodic large-scale structures and also
that small billows form along the braids between these structures. Spectral analyses show that the bands observed in the
plan-view experiments and large-scale undulations in crosssectional experiments have the same frequency and propagation speed, confirming that they are the same phenomenon.
Similar large-scale structures have been previously observed
in several cylindrical release or sector tank laboratory experiments14,16–18 and numerical simulations.18,19 However, there
is no clear conclusion on the mechanism that generates such
large-scale structures, nor a definitive explanation for why
they are more prevalent in currents with lateral spreading.
We hypothesize that as the plume spreads laterally, there is
a mechanism that forces the generation of Kelvin-Helmholtz
instabilities at the interface to form well arranged large-scale
structures. Here, we conduct three analyses to investigate the
dynamics introduced by lateral spreading.
A. Linear stability analysis
Figure 8 shows the time-averaged density anomaly
[∆ρ(z)] and velocity profiles [U(z)] for the channelized and
spreading currents. The profiles are averaged over 100 s, which
is long relative to the time scale of the observed instabilities,
and smoothed using a low-pass filter to remove remaining
Phys. Fluids 29, 106603 (2017)
small-scale variations. Density and velocity profiles are noticeably different in the two cases. In the channelized case, the
density and velocity are uniform in the surface layer and in the
ambient water layer beneath the current [Figs. 8(a) and 8(b)],
with a mixing layer in between where the velocity and density vary linearly. This profile is similar to that of the observed
estuary channels in the field.8,34,35 In the spreading case, the
near-surface layer disappears, and the linearly varying mixing
layer extends to the water surface, which is similar to profiles
observed in river plumes.36 Yuan and Horner-Devine15 suggested that this difference in the structure is a result of the fact
that lateral spreading rate is highest at the water surface. It
may also explain why lateral spreading does not modify the
local mixing efficiency since the region of most intense mixing is at the base of the current, where lateral spreading is
reduced.
Squared buoyancy frequency [N 2 (z) = −(g/ρ0 )(d ρ/dz)],
squared shear [(dU/dz)2 ], and gradient Richardson number
Rig = N 2 (dU/dz)−2 profiles were calculated from the density
anomaly and velocity profiles [Figs. 8(c)–8(e) and 8(h)–8(j)].
The gradient Richardson number less than a quarter
(Rig ≤ 1/4) is typically used as a condition for instability
in density stratified flows;37 a decrease in Rig below this critical threshold is often accompanied by mixing in estuaries.38
In the channelized case, Rig has a value of 1/4 in the interfacial layer (−0.02 < z < −0.04), corresponding to the region
where KH instabilities and mixing are observed. Rig is also
low at the base of the layer in the channelized case where the
FIG. 8. Density anomaly (a) and (f),
velocity (b) and (g), squared buoyancy
frequency (c) and (h), squared shear (d)
and (i), and gradient Richardson number
(e) and (j) for the high Fr i channelized
case (top) and spreading case (bottom).
The shaded areas in (e) and (j) indicate
regions where Rig ≤ 1/4.
106603-9
Y. Yuan and A. R. Horner-Devine
density anomaly is very low. In the spreading case, Rig is well
below the critical value (Rig ≈ 0.1) in the entire upper layer
(z ≥ −0.06 m), indicating that the spreading gravity current is
more susceptible to instability and mixing than the channelized
case.
We use spatial linear stability analysis following Taylor 39
and Goldstein40 to assess the stability of the flow. For a sufficiently small perturbation from the background state, the
perturbation may grow in time or take the form of stable internal gravity waves. This small disturbance may be expanded
into normal modes as ψ(x, z, t) = ψ̂(z)eik(x−ct) , where k = 2π/λ
is the horizontal wavenumber and c = cr + ici is the complex
phase speed. If we further assume that the flow is incompressible, Boussinesq, inviscid, and non-diffusive, the structure
function ψ̂(z) is then a solution of the Taylor-Goldstein (TG)
equation,
#
"
N2
d 2 U/dz2
d 2 ψ̂
2
− k ψ̂ = 0,
(1)
+
−
U −c
dz2
(U − c)2
where ρ(z) and U(z) are the background density and velocity
profiles, respectively.
It is important to emphasize that linear stability analysis is most applicable in the region where instabilities
are generated, in this case, near the river mouth. It is not
ideally suited for regions where instabilities are already welldeveloped. However, due to limitations of the basin configuration, it was not possible to acquire data immediately adjacent
to the discharge point and the field-of-view marked in Fig. 4
was the closest we could get to the mouth. Our stability
analyses need to be considered with that qualification in
mind.
We used the numerical method described by Moum et al.41
to solve the TG equation based on averaged velocity and
density profiles (Fig. 8). The location where the data are
obtained is indicated with a magenta bar in Fig. 4. Although
the frequency, and thus wave speed, has already decreased
slightly from its expected value at the river mouth, we
hypothesize that it still provides useful information when
comparing with the observed wave characteristics and the
Phys. Fluids 29, 106603 (2017)
prediction based on the TG solution. We further assume that
the instabilities are two-dimensional, indicating that they do
not vary in the lateral direction (y). This is also not a perfect assumption since the spreading cases may have spanwise
velocity and density gradients. However, this two-dimensional
analysis will provide an expected range of wavenumbers
and growth rates for instabilities of interest and extension
of the TG equations to include lateral variation proved
intractable.
For each wavenumber k, the solution of the TG equation
provides an eigenfunction-eigenvalue set {ψ(z), c}. The background flow is said to be unstable if any mode exists for which
ci , 0, where the growth rate is then defined as σ = kci . In
general, the fastest-growing mode (FGM) is the one with maximum growth rate that is most likely to be observed. Although
the TG analysis is based on linear analysis, the predicted
wave properties (k and c) typically match with the observed
instabilities in the laboratory33,42,43 and field observations.35,41
FIG. 9. Growth rate versus wavenumber for (a) channelized and (b) spreading
high Fr i runs using the data shown in Fig. 8. Only the fastest growing mode is
shown in the figure. Three local prominent maxima (one for the channelized
case and two for the spreading case) are highlighted by arrows and labeled
with their maximum values and corresponding wavenumbers.
FIG. 10. Taylor-Goldstein solution of displacement
eigenfunction η̂ for the spreading case. (a) Mean velocity
profile normalized with respect to the maximum mean
velocity. (b) Mean density anomaly profile normalized
with respect to the maximum density anomaly; red lines
are the displacement eigenfunction of the fastest-growth
mode (FGM, solid line) and second fastest-growth mode
(2nd-FGM, dashed line). Gray dashed lines shown in (a)
and (b) correspond to the maximum displacement location of two unstable modes. (c) Temporal sequence of the
vertical density profile at the center of the PLIF field for
the same run.
106603-10
Y. Yuan and A. R. Horner-Devine
The solutions of the TG equation result in predictions
of the wavenumber, growth rate, and propagation speed of the
fastest growing mode (Fig. 9). For the channelized case, the TG
analysis yields one FGM with a wavenumber, peak growth rate,
and propagation speed of 35.8 m 1 , 0.038 s 1 , and 0.009 ms 1 ,
respectively. The wavelength corresponding to this mode is
17 cm. The solution for the spreading case results in two unstable modes [Fig. 9(b)]. The wavenumber, peak growth rate, and
propagation speed of the FGM are 21.0 m 1 , 0.076 s 1 , and
0.04 ms 1 , respectively. This wavenumber corresponds to a
wavelength of 30 cm, larger than the observed wavelength
from the Hovmöller diagram (Sec. III A) or the spectral analysis (Sec. III B). We suspect that TG analysis might predict a
larger wavelength because the analysis is applied to the region
where instabilities are already well-developed, instead of the
location where they are generated.
In addition to the wavelength, phase speed, and growth
rate of each unstable mode, the vertical structure of each growing mode is also given by the TG solution as the eigenfunction
[η̂(z) = −ψ̂/(U − c)]. The instabilities are predicted to occur at
the maximum of η̂. As shown in Fig. 10, the FGM occurs in the
middle of the buoyant layer 2.6 cm below the surface. This corresponds to the location of local maxima in N 2 and shear [Figs.
8(h) and 8(i)] and also the center of the large-scale structures
visible in the density field [Fig. 10(c)]. The average current
speed at this depth U1 = 0.04 ms 1 is approximately equal to
the propagation speed cr , suggesting again that these instabilities propagate at critical speeds. The calculated wavelength is
larger than the large-scale structure observed in the plan-view
x t diagram (Sec. III A), while the propagation speed is consistent. The second unstable mode corresponding to the higher
wavenumber has wavenumber, peak growth rate, and propagation speed of 83.4 m 1 , 0.057 s 1 , and 0.002 ms 1 , respectively.
This mode occurs deeper in the water column compared to the
Phys. Fluids 29, 106603 (2017)
FGM, where the current speed and propagation speed are one
order of magnitude smaller than the FGM.
Although the wavelength predicted by the stability analysis is larger than that observed, the similar propagation speed
and the vertical location of the unstable mode supports the
conclusion that the large-scale structures observed in spreading
currents are KH instabilities. These structures are continuously
generated at the river mouth and have propagation speeds that
match the local velocity.
B. Secondary instabilities
Small billows are observed on the braid between the largescale structures in the spreading case [Fig. 5 (Multimedia
view)]. We hypothesize that these are secondary instabilities
along the sharp interface with high stratification and high shear.
Using direct numerical simulation (DNS), Smyth44 showed
that secondary instabilities occur along the braids between
the cores of the primary shear instabilities in high Reynolds
number flows. Geyer et al.45 observed similar structures in
field observations in a strongly stratified estuary and showed
that the secondary instabilities are responsible for most of
the mixing. They suggested that the majority of previous laboratory studies and numerical simulation did not reach the
Reynolds number criteria at which the secondary instabilities
could be observed. Atsavapranee and Gharib46 conducted a
laboratory study of a stratified plane mixing layer at Re ≈ 2000
and showed that secondary instabilities do occur along the
braids of planar shear instabilities. For stratified turbulent
flow, the buoyancy Reynolds number (I = ε/νN 2 , where ε
is the turbulent dissipation rate and ν is the viscosity) is often
used to evaluate the ratio of destabilizing effects of turbulent
stirring to stabilizing effects of buoyancy and viscosity.47 In
the present experiment, although Re is low compared with
FIG. 11. (a) Current interface showing
braid coordinates. zf is the interface
between the two layers, and x0 − z0 is the
braid coordinate tilted with angle ϕ. The
braid density (ρ0 ), along-braid velocity
(u0 ), and cross-braid velocity (w0 ) are
measured along lines perpendicular to
the along-braid axis x0 and shown in (b)–
(d), respectively. (e) Ratio of braid shear
ω 0 to strain rate γ overlaid on temporal
sequence of the vertical density profile
at the center of the PLIF field-of-view
shown in Fig. 6(b).
106603-11
Y. Yuan and A. R. Horner-Devine
geophysical flows, I is comparable to values observed in field
observations.15
One way to evaluate whether the braid region of the largescale structures in the spreading case has the potential to generate secondary instabilities is the ratio of braid shear to strain,
i.e., ω0 /γ 44 along the braid coordinate x 0 − z 0 [Fig. 11(a)]. The
braid coordinate is defined such that the axis is tilted with the
interface between the two layers.44 The tilt angle ϕ is identified
as the local slope of the interface [zf (x)], i.e., tan ϕ = dzf /dx.
The objective is to identify a central portion of the braid in
a given set of flow fields and to average along the braid to
obtain profiles of ρ0(z 0), u 0(z 0) and w 0(z 0) [Figs. 11(b)–11(d)],
where u 0(z 0) is the along-braid velocity and w 0(z 0) is the braidperpendicular velocity. Stability of the flow along the braid is
determined based on the ratio of two braid parameters: shear
ω0 is the maximum value of du 0/dz 0 and strain rate γ is the
value of dw 0/dz 0 at z 0 = 0 [Fig. 11(e)]. Staquet 48 first proposed
that the critical value of ω0 /γ for secondary instability was 54.
Phys. Fluids 29, 106603 (2017)
Later, Smyth44 suggested that a value in the range of 35–40 is
sufficient to produce secondary instability. We need to note that
along the front side of primary KH structures, the braid coordinates are easily determined, and the values of braid parameters
are robust. At other times, however, the braid can be far from
its equilibrium state; thus, there are many places with no values at the lee sides. The maximum values of ω0 /γ observed
in the spreading current are around 20, below the critical values presented previously in the literature. However, the highest
values correspond to distinct spikes that occur at the same time
that the smaller KH billows are observed to form on the leading edge of the larger structures [Fig. 11(e)]. Thus, while the
established criteria are not quite met, the smaller billows seem
likely to be secondary instabilities. This is also consistent with
the vortex dynamics analysis, which shows that vortex stretching intensifies spanwise vorticity to re-stratify the plume layer.
Thus, the re-stratification on the leading edge of the structures
may contribute to the occurrence of secondary instabilities.
FIG. 12. Channelized cases of phase-averaged dimensionless density (a) and velocity (b) fields averaged based on the wavelength solved from the TG equation.
Gradient Richardson number (c) and vorticity (d) are subsequently calculated from the phase-averaged density and velocity fields. The contours of ρ̃ = 0.5 and
ũ = 0 are overlaid as black lines in (a) and (b), respectively. The contour of the critical gradient Richardson number (Ri = 1/4) is overlaid as the black line and
the region below the critical number is indicated as the blue region in (c). Four terms in the vortex balance equation: advection (e), stretching and tilting (f),
baroclinic production (g), diffusion (h) and their sum (i) are calculated subsequently. Note that different color scales are used in each vortex dynamics term to
better illustrate the information.
106603-12
Y. Yuan and A. R. Horner-Devine
C. Phase-averaged vortex dynamics
In this section, we investigate whether lateral spreading
of the plume influences the dynamics of the observed vortex
structures. One disadvantage of the PIV-PLIF method is that
it only gives us fixed-frame observations with 14 cm horizontal field-of-view (FOV), less than the wavelength of the
large-scale structures (20 cm–35 cm). To obtain a record with
longer duration, we average density and velocity fields with
the frame of reference moving with the gravity current front.
In order to achieve the average, we assume that the gravity
current propagates offshore at a constant speed. Although a
decrease in the frequency (i.e., wave speed) was observed in
Fig. 4(d), this assumption still holds because the decrease in
the wave speed is small within the FOV. The constant sheared
inflow provides a continuous source of vorticity; thus, we can
assume that the gravity current in the present study is in a quasisteady-state. Therefore, the flow characteristics are stationary
in the moving frame of reference and the turbulence is approximately in steady-state. With this assumption, small-scale turbulent fluctuations are smoothed out in the time-averaged field,
but other information (such as large-scale band structures) is
preserved.49 The time averaging process eliminates the smallscale fluctuations and secondary instabilities along the braids
Phys. Fluids 29, 106603 (2017)
discussed in Sec. IV B, leaving relatively smooth density and
velocity fields for the subsequent phase-averaging calculation.
In the following analysis, we use non-dimensional variables
by adopting the inflow fluid depth H 0 and velocity U 0 as the
length and velocity scales, respectively. Consequently, the time
scale is (τ = H 0 /U 0 ). The density anomaly is normalized by
the inflow density anomaly.
For the vortex dynamics analysis, we generate phaseaveraged fields by averaging over multiple instances of the
structures. The wavelengths and phases in the spreading case
correspond to large-scale structures observed by the eyes since
the wavelength is slightly different in each case. Regions with
no clear large-scale structures are excluded in the averaging
process. There are no large-scale structures in the channelized case, so we used the wavelength from the channelized
TG solution (λ = 17.5 cm). We tried several averaging wavelengths ranging from 10 cm to 20 cm and the results were
insensitive to this choice. Phase-averaged density and velocity
fields are shown in Figs. 12 and 13 for the channelized and
spreading cases, respectively. The gradient Richardson number is then calculated based on these averaged fields and shown
in Figs. 12(c) and 13(c). The vertical distribution of the Ri field
shows similar patterns as in Fig. 8. In the channelized case, Ri
is below the critical value of 1/4 at the water surface and the
FIG. 13. Similar to Fig. 12 but for the spreading case. The wavelengths used in the phase average are based on the large-scale structure observed by the eyes.
106603-13
Y. Yuan and A. R. Horner-Devine
bottom of the plume layer, while in the spreading case, Ri is significantly below its critical value within the entire plume layer.
The horizontal distribution of density, velocity, and Ri fields
clarify the differences in the plume spatial structure between
the spreading and channelized flows. In the spreading case, the
density field shows a sinusoidal wave-like structure with the
thickest plume layer at the center. The velocity field lags
the density field by approximately one quarter phase, leading to
a similar lag in the Ri field. Nevertheless, the Richardson number is low (Ri < 1/4) in the entire upper layer except regions
where the stratification is extremely high. The major region
with a high Ri is along the braid of the large-scale structure,
associated with the region where secondary instabilities are
generated.
Vortex dynamics are controlled by a balance between production, stretching, tilting, transport, and dissipation.19 Following Cantero, Balachandar, and Garcia,19 the circulation at
the interface Γ is the integral of the vorticity in the crossplume direction in an area surrounding the interface. Thus
the net circulation (or, net normal vorticity within a fixed surface S) is controlled according to the following dimensionless
equation:
∂
dΓ
=
ω̃ · dS = (−ũ · ∇ω̃) · dS + (ω̃ · ∇ũ) · dS
dt̃
∂ t̃ S
S
S
1
+ (∇ ρ̃ × e) · dS + ( ∇2 ω̃) · dS.
(2)
S
S Re
Here ũ = (u − c)/U0 is the dimensionless fluid velocity,
˜ × ũ is the dimensionless vorticity [Figs. 12(d) and
ω̃ = ∇
Phys. Fluids 29, 106603 (2017)
0
13(d)], ρ̃ = ρ−ρ
ρ−ρi is the dimensionless density, and e is the unit
vector pointing along the direction of gravity. The first term
on the right-hand side is the advection (A) of vorticity in the
along-flow direction. The second term is vortex stretching (S)
due to lateral spreading, which is calculated by multiplying the
vorticity by the lateral spreading rate. To estimate this term,
we assume that the plume is symmetric about the x z plane
along the plume centerline (i.e., y = 0); thus, the out-of-plane
velocity component 3 is zero and the gradient d3/dy is at its
maximum value in this plane.15 The third term is the baroclinic
production (B) due to the continuous input of freshwater. The
last term is the diffusion (D) of vorticity.
Panels (e)–(i) in Figs. 12 and 13 show the spatial distribution of each term in Eq. (2) and the sum of these terms, which
is essentially the net change in circulation. Vorticity is significantly higher overall in the spreading case and regions of high
vorticity correspond well with regions of low Ri [Fig. 13(d)].
Although we cannot investigate the time evolution of the circulation because our measurement region is too small, comparison between the two cases provides insight into the generation
of vorticity in the spreading case.
In the channelized case, there are no pronounced sources
or sinks of vorticity in the advection and stretching/tilting
terms. The spatial scatter of positive and negative values
in these terms is primarily due to the heterogeneity of the
flow fields. Baroclinic production at the base of the plume
layer serves as the most coherent source of vorticity, although
the magnitude of this term is relatively small [Fig. 12(g)].
Kelvin-Helmholtz instabilities are generated at the interface,
entraining salt water upward into the mixing layer and mixing
FIG. 14. Non-dimensional density and velocity profiles
(a) and (d), gradient Richardson number and vorticity
profile (b) and (e), vertical profile of four terms in the
vortex balance equation (c) and (f), namely, advection
(A), stretching and tilting (S), baroclinic production (B),
and diffusion (D). Profiles in top panels are channelized
cases, averaged horizontally based on Fig. 12. Profiles in
bottom panels are spreading cases, averaged horizontally
based on Fig. 13.
106603-14
Y. Yuan and A. R. Horner-Devine
freshwater downward. This leads to thickening of the mixing layer that occurs between the top and bottom layers. This
process and the resulting velocity and density structures are
consistent with the flow structure in highly stratified channel
flows, such as those observed recently in the Connecticut River
estuary8 and Fraser River Estuary.34,35 The contribution from
the diffusion term [Fig. 12(h)] is negligible.
In the spreading case, each term contributes significantly
to the total vortex balance except the diffusion term (Fig. 13).
The advection term provides a source of vorticity at the base of
the plume. Baroclinic production provides a source of vorticity in the highly stratified near-surface layer, but with a lower
magnitude than the other two terms. The most striking difference is the enhanced stretching term in the spreading case,
corresponding to the region with high shear (i.e., low Ri and
high vorticity). The intensification of vorticity due to lateral
spreading may result in the layer maintaining a marginally
unstable state similar to that observed for downslope currents
by Pawlak and Armi.26 We hypothesize that the strong lateral
stretching of the vortices in the spreading case increases the
vorticity within the high shear region and maintains the coherence of large-scale KH vortices. This process is not observed
in the channelized case.
To more clearly quantify the contributions of the four
mechanisms discussed above, we average each term horizontally to obtain vertical profiles of the four terms in the
vortex balance equation. These are plotted for comparison with
density, velocity, vorticity, and gradient Ri profiles (Fig. 14).
Diffusion and baroclinic production terms show similar patterns in the channelized and spreading cases. The diffusion
term is negligible due to the relatively high Reynolds number.
Baroclinic production is the major source of vorticity at the
interface where the stratification is high. The main difference
between two cases in the vortex balance is in the advection and
stretching terms. In the channelized case, the advection and
stretching terms are nearly balanced; the sum of the two has a
small positive value. In the spreading case, the advection term
is high along the base of the plume and the stretching is high
in the upper mixing layer. The reduction in the stretching term
with depth is consistent with the lateral spreading rate profiles,
which show that the spreading rate has a similar decrease with
depth due to the vertical variation in the baroclinic pressure
gradient.15
V. DISCUSSION
The plume is observed to evolve as follows in the present
experiments. With lateral boundaries (i.e., channelized case),
strong shear at the interface of the two layers generates
Kelvin-Helmholtz (KH) instabilities. Freshwater is entrained
downward while denser ambient water is entrained upward,
leading to the continuous growth of the mixing layer. The flow
forms a three-layer density structure with a near-surface uniform density layer at the top, a linearly varying mixing layer
beneath it, and quiescent ambient water at the bottom [Figs.
8(a), 8(b), and 14(a)]. Linear stability analysis shows that the
fastest growth mode associated with classical KH instabilities
is located at the interface. Without lateral boundaries (i.e., the
spreading case), inflowing buoyant water spreads laterally as it
Phys. Fluids 29, 106603 (2017)
propagates offshore. Although the lateral expansion does not
modify the local mixing efficiency, it dramatically changes the
plume structure. The top uniform density layer disappears so
that the mixing layer with high stratification and shear extends
to the water surface15 [Figs. 8(f), 8(g), and 14(b)]. Linear stability analysis suggests that the fastest growth mode (FGM)
has a similar propagation speed and occurs at the same vertical
location as the large-scale structures observed in the plan-view
experiments, although this analysis predicts a somewhat larger
wavelength.
The vorticity balance analysis confirms that baroclinic
production of vorticity is an important source of vorticity
throughout the stratified region of both the channelized and
spreading currents [Figs. 14(c) and 14(f)]. In the channelized
case especially, shear in this region is responsible for generating the observed small-scale interfacial instabilities. Lateral
vortex stretching in the spreading case, which is two to three
times greater than the channelized case, significantly intensifies the vorticity in the buoyant layer. We suggest that this
process maintains the top layer in a marginally unstable state,
consistent with the persistent low Ri observed in the buoyant
layer [Fig. 14(e)]. Vorticity advection is also important in the
spreading case, primarily at the base of the mixing layer. These
additional vorticity sources lead to a significant enhancement
of vorticity in the spreading case, which generates and sustains
the observed large-scale vortex structures.
Previous researchers have observed similar largescale structures in axisymmetric lock-exchange experiments.
Simpson14 described vortex ring structures, which he suggested were generated by intensified vorticity resulting from
lateral spreading. This process is most pronounced during the
rapid expansion that occurs near the source and produces a
large vortex that occupies almost the full depth of the dense
fluid. However, those experiments did not include detailed
velocity or density profiles and were thus unable to quantitatively investigate the role of vortex stretching. Alahyari and
Longmire20 described the velocity and vorticity structures in
a dense axisymmetric spreading gravity current based on PIV
measurements. They noted the appearance of a coherent pattern of large-scale vortices that formed in the dense current,
similar to those described in the present experiments. Without analogous channelized experiments, they suggested that
this vortex structure would be common to all gravity currents
due to the interaction with the non-slip bottom. Our experiments show that the periodic vortex structures are distinct to
spreading currents and are a consequence of the vortex stretching associated with lateral spreading [see comparison between
spreading and channelized experiments in the enhanced online
multi-media of Fig. 5 (Multimedia view)].
Linear stability and vortex balance analyses suggest that
the formation of the large-scale structures and enhancement
of vorticity aggregated within these structures in the spreading case are a result of its density and velocity structures. The
distinct structure of the density and velocity profiles observed
in these experiments is consistent with that observed in stratified geophysical flows. In the channelized case, we observe
a three-layered structure with a uniform surface layer above a
linearly stratified mixing layer. The same structure is observed
in measurements in a strongly stratified and laterally confined
106603-15
Y. Yuan and A. R. Horner-Devine
estuarine channels.8,34,35 In the spreading case, the uniform
surface layer disappears and the mixing layer extends to the
water surface, as is also observed in the near-field region of
unconfined coastal river plumes.36
Previously, researchers have conducted several numerical
modeling experiments on the impact of azimuthal variation
in dense gravity currents which found that the stretching in
the azimuthal direction creates a multi-front structure, but different hypotheses have been suggested. The first hypothesis
was proposed by Swearingen, Crouch, and Handler,50 who
suggested that the multi-front structure is the result of the interaction between the leading ring vortex and the solid boundary.
Their DNS model of three-dimensional vortex rings suggested
that the ring vortex at the wall develops azimuthal instabilities, which grow rapidly because of vortex stretching and
tilting in the presence of the mean strain field generated by
the primary vortex ring. Cantero, Balachandar, and Garcia19
also associated multi-fronts with a vortex pair structure. They
suggested that the vorticity is concentrated in multiple fronts
and is intensified due to vortex stretching with decreasing
cross-sectional area. The stretching of the vortex rings along
the azimuthal direction stabilizes the KH structure and slows
down the transition into fully developed three-dimensional
turbulence. However, gravity currents in these experiments
are generated from fixed-volume source, so the vortex ring
structure is damped out after the initial stage. Our experiments suggest that this large-scale vortex ring structure will
be maintained in the presence of the continuous source of vorticity provided by lateral stretching. The large-scale structures
are continuously generated as a result of the later spreading, which more closely reproduces a buoyant coastal river
inflow. We find that the lateral spreading modifies the density and velocity profiles, generating different modes of KH
instabilities based on the linear stability analysis. The vorticity
generated in the shear layer is aggregated into large-scale vortex structures. Lateral stretching of the spanwise vortex tube
significantly increases the vorticity, especially in the high shear
region, and appears to delay the transition on the instability to
three-dimensional turbulence at the front side of large-scale
vortices.
Pawlak and Armi51 found similar large-scale structures
in a hydraulically controlled, arrested wedge stratified flow in
their laboratory experiments. Their experiments were designed
to simulate the flow over a sill in Knight Inlet in British
Columbia52 and consisted of stratified flow that was constrained in a rectangular channel with a constant slope sill at
the center. They observed a sequence of large-scale structures
in the low-entrainment developing region after the initial highentrainment stage. Unlike a single vortex or a vortex pair within
those large-scale structures suggested by previous axisymmetric lock-exchange laboratory experiments, they found that
the vorticity was highest along the “node” of the large-scale
structures. They also showed evidence that small entraining
scales are correlated with the large-scale structure. Corcos and
Sherman53 suggested that the secondary instabilities may provide a significant energy sink. Thus, the appearance of secondary instabilities arrests the growth of primary instabilities
and forestalls their collapse. Geyer et al.45 also reported that
there was no indication of growth, pairing, or collapse of the
Phys. Fluids 29, 106603 (2017)
primary instabilities in the Connecticut River estuary when
secondary instabilities were observed. In addition to the vortex stretching in the cross-plume direction as the main force to
retain primary instabilities as suggested in Sec. IV C, the occurrence of the secondary instabilities may also play an important
role.
VI. CONCLUSIONS
An overarching objective of gravity current and river
plume studies is to quantify the mixing of the buoyant layer
with the ambient fluid. The present experiments show that
lateral spreading modifies the plume structure dramatically
[Fig. 5 (Multimedia view)]. Previous analysis of these experiments, however, showed that the local mixing efficiency is
the same in both cases; we show that the spreading does not
enhance local mixing because of the mismatch between the
locations of maximum spreading and mixing.15 The present
study shows that the spreading intensifies vorticity to the system due to lateral stretching of the large-scale vortices, leading to a marginally unstable upper layer with a significantly
reduced Ri. The presence of large-scale vortices re-arranges
the mixing location, i.e., suppressing at the front side of largescale structures while enhancing at the lee side [see enhanced
online multi-media of Fig. 5 (Multimedia view)].
Previous researchers have suggested that the bands
observed in laboratory scale gravity currents may be similar
to the internal fronts observed in river plumes. Our experiments provide new insights into the structures observed in
these experiments and perhaps their field analog. In our laboratory experiments, vorticity generated in the shear layer is
aggregated into large-scale vortex structures only when lateral
spreading is significant. We observed a process in which lateral vortex stretching modifies existing shear layer vorticity,
generating the large-scale vortex structures. These structures
propagate at the local wave-speed c, occur at the depth of the
inflection point, and are similar to KH instabilities in that they
correspond to the fastest-growing mode predicted by the TG
equation. The structures observed in our experiments appear to
be distinct in some respects from the internal fronts observed
in the field. First, they tend to be parallel to the river mouth
instead of curved as the plume front. Second, they appear
further behind the leading front of the plume. However, the
process by which shear layer vorticity forms into large-scale
vortex structures in the presence of strong lateral spreading
that subsequently propagate as internal bores is likely to be
common to both the laboratory and field cases. It is possible that the large-scale structures we observe in the laboratory
may propagate further out into the plume in the field, but they
are dissipated earlier in the lower Reynolds number laboratory
conditions.
Although the present experiments were designed to investigate the three-dimensional structure of freely spreading gravity currents, we do not have a complete view of the plume
structure especially along the cross-plume direction. Our analysis is based primarily on the assumption that the plume is
symmetric along the plume center line and that the lateral
velocity is zero there. Detailed three-dimensional analysis is
still needed to describe the vortex dynamics more completely,
106603-16
Y. Yuan and A. R. Horner-Devine
which can be only achieved by using 3-D PIV techniques. It
is also important to note that the lift-off point is somewhat
arbitrary in the present experiment because there is no sloping
bottom. In a real river plume system, the lift-off point is controlled by bathymetry25,54 and typically occurs at or near a sill.
We have assumed that the field-of-view is downstream of the
lift-off point in the present study. However, the relationship
between the lift-off dynamics and lateral spreading still needs
further study.
ACKNOWLEDGMENTS
The authors thank the three anonymous reviewers for
their constructive comments and suggestions. We thank three
undergraduate students, Anthony Poggioli, Amanda Gehman,
and Stephanie Wei, who helped with the experiments. This
work has benefited from discussion with William D. Smyth,
Weifeng G. Zhang, Rocky Geyer, and members of the MeRMADE project team. The authors are grateful to the National
Science Foundation for support of the project through Grant
Nos. OCE-0850847 and OCE-1233068. Finally, Y.Y. acknowledges the support of Steve and Sylvia Burges Endowed Presidential Fellowship in Civil and Environmental Engineering at
the University of Washington and the support of project from
the National Natural Science Foundation of China through
Grant No. 41506101.
1 C.
A. Nittrouer, D. J. DeMaster, A. G. Figueiredo, and J. M. Rine,
“AmasSeds: An interdisciplinary investigation of a complex coastal environment,” Oceanography 4, 3–7 (1991).
2 B. M. Hickey, R. M. Kudela, J. Nash, K. W. Bruland, W. T. Peterson,
P. MacCready, E. J. Lessard, D. A. Jay, N. S. Banas, A. M. Baptista et al.,
“River influences on shelf ecosystems: Introduction and synthesis,”
J. Geophys. Res. 115, C00B17, doi:10.1029/2009jc005452 (2010).
3 A. R. Horner-Devine, D. A. Fong, S. G. Monismith, and T. Maxworthy,
“Laboratory experiments simulating a coastal river inflow,” J. Fluid Mech.
555, 203–232 (2006).
4 F. Chen, D. G. MacDonald, and R. D. Hetland, “Lateral spreading of a nearfield river plume: Observations and numerical simulations,” J. Geophys.
Res. 114, C07013, doi:10.1029/2008jc004893 (2009).
5 L. Kilcher and J. D. Nash, “Structure and dynamics of the Columbia
River tidal plume front,” J. Geophys. Res. 115, C05S90, doi:
10.1029/2009jc006066 (2010).
6 A. R. Horner-Devine, R. D. Hetland, and D. G. MacDonald, “Mixing and
transport in coastal river plumes,” Annu. Rev. Fluid Mech. 47, 569–594
(2015).
7 P. L. Mazzini and R. J. Chant, “Two-dimensional circulation and mixing
in the far field of a surface-advected river plume,” J. Geophys. Res. 121,
3757–3776, doi:10.1002/2015jc011059 (2016).
8 W. R. Geyer, D. K. Ralston, and R. C. Holleman, “Hydraulics and mixing
in a laterally divergent channel of a highly stratified estuary,” J. Geophys.
Res. 122, 4743–4760, doi:10.1002/2016jc012455 (2017).
9 R. W. Garvine, “Estuary plumes and fronts in shelf waters: A layer model,”
J. Phys. Oceanogr. 17, 1877–1896 (1987).
10 D. A. Fong and W. R. Geyer, “The alongshore transport of freshwater in a
surface-trapped river plume,” J. Phys. Oceanogr. 32, 957–972 (2002).
11 T. von Karman, “The engineer grapples with nonlinear problems,” Bull.
Am. Math. Soc. 46, 615–683 (1940).
12 T. B. Benjamin, “Gravity currents and related phenomena,” J. Fluid Mech.
31, 209–248 (1968).
13 J. O. Shin, S. B. Dalziel, and P. F. Linden, “Gravity currents produced by
lock exchange,” J. Fluid Mech. 521, 1–34 (2004).
14 J. E. Simpson, Gravity Current: In the Environment and the Laboratory,
2nd ed. (Cambridge University Press, 1997), p. 244.
15 Y. Yuan and A. R. Horner-Devine, “Laboratory investigation of the impact
of lateral spreading on buoyancy flux in a river plume,” J. Phys. Oceanogr.
43, 2588–2610 (2013).
Phys. Fluids 29, 106603 (2017)
16 J.
C. Chen, “Studies on gravitational spreading currents,” Ph.D. thesis,
California Institute of Technology, 1980.
17 M. R. Maclatchy, “Radially spreading surface flows,” Ph.D. thesis, University of British Columbia, 1999.
18 M. D. Patterson, J. E. Simpson, S. B. Dalziel, and G. J. F. van Heijst, “Vortical
motion in the head of an axisymmetric gravity current,” Phys. Fluids 18,
046601 (2006).
19 M. I. Cantero, S. Balachandar, and M. H. Garcia, “High-resolution simulations of cylindrical density currents,” J. Fluid Mech. 590, 437–469
(2007).
20 A. Alahyari and E. Longmire, “Development and structure of a gravity
current head,” Exp. Fluids 20, 410–416 (1996).
21 T. A. McClimans, “Fronts in fjords,” Geophys. Astrophys. Fluid Dyn. 11,
23–24 (1978).
22 R. W. Garvine, “Radial spreading of buoyant, surface plumes in coastal
waters,” J. Geophys. Res. 89, 1989–1996, doi:10.1029/jc089ic02p01989
(1984).
23 D. A. Luketina and J. Imberger, “Characteristics of a surface buoyant jet,” J. Geophys. Res. 92, 5435–5447, doi:10.1029/jc092ic05p05435
(1987).
24 M. Halverson and R. Pawlowicz, “Entrainment and flushing time in the
Fraser River estuary and plume from a steady salt balance analysis,”
J. Geophys. Res. 116, C08023, doi:10.1029/2010jc006793 (2011).
25 L. Armi and D. M. Farmer, “Maximal two-layer exchange through a
contraction with barotropic net flow,” J. Fluid Mech. 164, 27–51 (1986).
26 G. Pawlak and L. Armi, “Mixing and entrainment in developing stratified
currents,” J. Fluid Mech. 424, 45–73 (2000).
27 M. L. Rocca, C. Adduce, G. Sciortino, and A. B. Pinzon, “Experimental
and numerical simulation of three-dimensional gravity currents on smooth
and rough bottom,” Phys. Fluids 20, 106603 (2008).
28 C. Cenedese and S. Dalziel, “Concentration and depth field determined
by the light transmitted through a dyed solution,” in 8th International
Symposium on Flow Visualization, 1998, pp. 61.1–61.5.
29 Y. Yuan, M. E. Averner, and A. R. Horner-Devine, “A two-color optical
method for determining layer thickness in two interacting buoyant plumes,”
Exp. Fluids 50, 1235–1245 (2011).
30 A. R. Horner-Devine, “Velocity, density and transport measurements in
rotating, stratified flows,” Exp. Fluids 41, 559–571 (2006).
31 J. K. Sveen, An Introduction to MatPIV v.1.6.1 (Department of
Mathematics, University of Oslo, 2004), Eprint 2, 27 pp.
32 J. P. Crimaldi, “Planar laser induced fluorescence in aqueous flows,” Exp.
Fluids 44, 851–863 (2008).
33 E. W. Tedford, R. Pieters, and G. A. Lawrence, “Symmetric Holmboe
instabilities in a laboratory exchange flow,” J. Fluid Mech. 636, 137–153
(2009).
34 D. G. MacDonald and A. R. Horner-Devine, “Temporal and spatial variability of vertical salt flux in a highly stratified estuary,” J. Geophys. Res.
113, C09022, doi:10.1029/2007jc004620 (2008).
35 E. W. Tedford, J. R. Carpenter, R. Pawlowicz, R. Pieters, and
G. A. Lawrence, “ Observation and analysis of shear instability in the Fraser
River estuary,” J. Geophys. Res. 114, C11006, doi:10.1029/2009jc005313
(2009).
36 D. G. MacDonald, L. Goodman, and R. D. Hetland, “ Turbulent dissipation
in a near-field river plume: A comparison of control volume and microstructure observations with a numerical model,” J. Geophys. Res. 112, C07026,
doi:10.1029/2006jc004075 (2007).
37 S. A. Thorpe, The Turbulent Ocean (Cambridge University Press, 2005),
pp. 201–204.
38 W. R. Geyer and J. D. Smith, “Shear instability in a highly stratified estuary,”
J. Phys. Oceanogr. 17, 1668–1679 (1987).
39 G. Taylor, “Effect of variation in density on the stability of superprosed
streams of fluid,” Proc. R. Soc. A 132, 499 (1931).
40 S. Goldstein, “On the stability of superposed streams of fluid of different
densities,” Proc. R. Soc. A 132, 524 (1931).
41 J. N. Moum, D. M. Farmer, W. D. Smyth, L. Armi, and S. Vagle, “Structure
and generation of turbulence at interfaces strained by internal solitary wave
propagating shoreward over the continental shelf,” J. Phys. Oceanogr. 33,
2093–2112 (2003).
42 S. A. Thorpe, “Experiments on instability and turbulence in a stratified shear
flow,” J. Fluid Mech. 61, 731–751 (1973).
43 G. A. Lawrence, F. K. Browand, and L. G. Redekopp, “The stability of a
sheared density interface,” Phys. Fluids 3, 2360–2370 (1991).
44 W. D. Smyth, “Secondary Kelvin-Helmholtz instabilities in weakly stratified
shear flow,” J. Fluid Mech. 497, 67–98 (2003).
106603-17
Y. Yuan and A. R. Horner-Devine
45 W. R. Geyer, A. C. Lavery, M. E. Scully, and J. H. Trowbridge, “ Mixing by
shear instability at high Reynolds number,” Geophys. Res. Lett. 37, L22607,
doi:10.1029/2010gl045272 (2010).
46 P. Atsavapranee and M. Gharib, “Structures in stratified plane mixing layers
and the effects of cross-shear,” J. Fluid Mech. 342, 53–86 (1997).
47 G. N. Ivey, K. B. Winters, and J. R. Koseff, “Density stratification, turbulence, but how much mixing?,” Annu. Rev. Fluid Mech. 40, 169–184
(2008).
48 C. Staquet, “Two-dimensional secondary instabilities in a stratified shear
layer,” J. Fluid Mech. 296, 73–126 (1995).
49 C. Cenedese, R. Nokes, and J. Hyatt, “Lock-exchange gravity currents over
rough bottoms,” Environ. Fluid Mech. (published online 2016).
Phys. Fluids 29, 106603 (2017)
50 J.
D. Swearingen, J. D. Crouch, and R. A. Handler, “Dynamics and the
stability of a vortex ring impacting a solid boundary,” J. Fluid Mech. 297,
1–28 (1995).
51 G. Pawlak and L. Armi, “Vortex dynamics in a spatially accelerating shear
layer,” J. Fluid Mech. 376, 1–35 (1998).
52 D. M. Farmer and L. Armi, “The generation and trapping of solitary waves
over topography,” Science 283, 188–190 (1999).
53 G. M. Corcos and F. S. Sherman, “Vorticity concentration and the dynamics
of unstable free shear layers,” J. Fluid Mech. 73, 241–264 (1976).
54 D. M. Farmer and L. Armi, “Maximal two-layer exchange over a sill and
through the combination of a sill and contraction with barotropic flow,”
J. Fluid Mech. 164, 53–76 (1986).
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