The mixing efficiency of interfacial waves breaking at a ridge. 1. Overall mixing efficiency


Hult, E.L.; Troy, C.D.; Koseff, J.R.

Journal of Geophysical Research 116(C2): 1-10

2011


The overall mixing efficiency of periodic, interfacial waves breaking at a Gaussian ridge is investigated through laboratory experiments. Cumulative measurements are used to investigate the fraction of the wave energy lost in the breaking event that contributes to irreversible mixing of the background density gradient. Using the tank as a control volume, the distribution of energy into reflected waves, transmitted waves, and dissipation and irreversible mixing from the breaking event is determined. The overall fraction of wave energy lost in the breaking event that is converted irreversibly to mixing is found to be 3-8%, which is low compared with typical values of around 20% for steady, parallel, stratified shear instabilities. Spatial variability in the mixing event may contribute to the relatively low overall efficiency of the event.

JOURNAL
OF
GEOPHYSICAL
RESEARCH,
VOL.
116,
CO2003,
doi:10.1029/2010JC006485,
2011
The
mixing
efficiency
of
interfacial
waves
breaking
at
a
ridge:
1.
Overall
mixing
efficiency
E. L.
Hult,
1
C.
D.
Troy,
2
and
J.
R.
Koseff
l
Received
25
June
2010;
revised
22
October
2010;
accepted
2
December
2010;
published
4
February
2011.
[1]
The
overall
mixing
efficiency
of
periodic,
interfacial
waves
breaking
at
a
Gaussian
ridge
is
investigated
through
laboratory
experiments.
Cumulative
measurements
are
used
to
investigate
the
fraction
of
the
wave
energy
lost
in
the
breaking
event
that
contributes
to
irreversible
mixing
of
the
background
density
gradient.
Using
the
tank
as
a
control
volume,
the
distribution
of
energy
into
reflected
waves,
transmitted
waves,
and
dissipation
and
irreversible
mixing
from
the
breaking
event
is
determined.
The
overall
fraction
of
wave
energy
lost
in
the
breaking
event
that
is
converted
irreversibly
to
mixing
is
found
to
be
3-8%,
which
is
low
compared
with
typical
values
of
around
20%
for
steady,
parallel,
stratified
shear
instabilities.
Spatial
variability
in
the
mixing
event
may
contribute
to
the
relatively
low
overall
efficiency
of
the
event.
Citation:
Hult,
E.
L.,
C.
D.
Troy,
and
J.
R.
Koseff
(2011),
The
mixing
efficiency
of
interfacial
waves
breaking
at
a
ridge:
1.
Overall
mixing
efficiency,
J.
Geophys.
Res.,
116,
CO2003,
doi:10.1029/2010JC006485.
1.
Introduction
[2]
Internal
wave
breaking
appears
to
contribute
signifi-
cantly
to
the
vertical
mixing
of
heat
and
mass
in
the
ocean
[e.g.,
Wunsch
and
Ferrari,
2004].
Thus,
the
efficiency
with
which
these
breaking
events
convert
wave
energy
irrevers-
ibly
to
potential
energy
determines
how
much
internal
wave
energy
may
be
available
for
mixing
the
ocean.
Understanding
what
governs
this
efficiency
is
critical
to
parameterizing
smaller-scale
internal
wave
events
within
larger-scale
ocean
models.
It
is
very
difficult
to
quantify
the
diapycnal
diffu-
sivity
or
the
efficiency
of
mixing
directly
from
ocean
observa-
tions.
Thus,
in
the
interpretation
of
field
data,
it
is
typical
to
use
a
model
that
assumes
a
constant
local
mixing
efficiency
in
order
to
infer
the
mixing
from
scalar
spectra
[Osborn,
1980;
Oakey,
1982]
despite
a
host
of
evidence
from
labo-
ratory
and
numerical
studies
that
the
local
mixing
efficiency
can
vary
significantly
based
on
the
applied
forcing
and
stratification
conditions
[Ivey
et
al.,
2008].
[3]
The
efficiency
of
internal
wave
breaking
events
is
very
difficult
to
measure
directly
in
the
ocean,
and
thus
labora-
tory
experiments
and
numerical
simulations
have
been
criti-
cal
to
studying
the
mixing
efficiency
of
stratified
flows.
For
first
mode
internal
waves
breaking
on
a
slope
in
a
linearly
stratified
fluid,
the
overall
mixing
efficiency
reaches
a
maximum
of
20%
when
the
slope
of
the
wave
characteristic
is
equal
to
the
beach
slope
[Ivey
and
Nokes,
1989].
For
interfacial
solitary
waves
breaking
on
a
slope,
the
overall
efficiency
was
found
to
have
a
maximum
of
25%,
varying
'Environmental
Fluid
Mechanics
Laboratory,
Stanford
University,
Stanford,
California,
USA.
2
School
of
Civil
Engineering,
Purdue
University,
West
Lafayette,
Indiana,
USA.
Copyright
2011
by
the
American
Geophysical
Union.
0148-0227/11/2010JC006485
with
the
length
scale
of
the
incident
wave
[Michallet
and
Ivey,
1999].
In
the
numerical
simulations
of
Slinn
and
Riley
[1996],
critical
wave
rays
focusing
on
a
slope
have
a
mixing
efficiency
of
35%.
Interfacial
wave
breaking
in
deep
water
was
considered
by
Fringer
and
Street
[2003]
who
reported
a
peak
overall
efficiency
of
36%
±
2%
for
highly
nonlinear,
progressive
interfacial
waves
in
numerical
simu-
lations
and
suggest
the
instability
is
convective,
based
on
the
high
mixing
efficiency.
Although
the
mixing
efficiency
might
be
quite
high
during
the
laminar
roll-up
of
a
Kelvin-
Helmholtz
instability
in
a
stratified
shear
flow,
most
of
the
mixing
occurs
after
the
transition
to
turbulence
at
an
effi-
ciency
near
20%,
based
on
the
results
of
numerical
simu-
lations
[Smyth
et
al.,
2001;
Peltier
and
Caulfield,
2003].
[4]
The
mixing
efficiency
in
a
stratified
flow
can
vary
with
the
instability
mechanism.
Purely
convective
instabilities
may
have
a
mixing
efficiency
of
up
to
50%
[Linden
and
Redondo,
1991],
whereas
the
mixing
efficiency
for
paral-
lel,
steady
shear
flows
is
lower,
typically
about
20%
[e.g.,
Peltier
and
Caulfield,
2003].
In
part because
of
this
signif-
icant
difference
in
efficiency,
there
has
been
much
interest
in
whether
internal
wave
breaking
is
driven
primarily
by
shear
or
convection.
Hult
et
al.
[2009]
examine
the
quali-
tative
nature
of
instability
for
periodic,
interfacial
waves
breaking
at
a
ridge
and
found
that
until
the
scaled
amplitude
over
the
ridge,
a/h,.,
is
increased
above
a
critical
value,
waves
are
initially
stable.
At
a
higher
alh,.,
the
waves
exhibit
backward
breaking,
and
at
yet
higher
a/h,.,
waves
steepen,
plunge
forward
and
then
become
gravitationally
unstable.
For
sufficiently
long,
high-amplitude
waves
shear
instabilities
can
contribute
to
wave
breaking
at
the
ridge,
as
well.
Because
the
qualitative
mechanism
of
the
wave
instability
appears
to
depend
on
the
scaled
amplitude,
a/h,.,
it
seems
feasible
that
the
overall
efficiency
with
which
wave
energy
is
converted
irreversibly
to
potential
energy
may
depend
on
the
scaled
amplitude
as
well.
In
this
study,
the
overall
mixing
efficiency
CO2003
i
of
10
CO2003
HULT
ET
AL.:
MIXING
EFFICIENCY
OF
INTERFACIAL
WAVES
CO2003
(a)
f_ocu
scanning
mirror
-__
_sing_
lens
beam
_
1
1
-
`wavemaker
upper
layer
lower
layer
(dyed)
horsehair
beach
expander
(3x)
/
Argon-ion
laser
NS
k
(b)
488cm
P2
h
2
k
2n/k
-
zis
h
r
T
h
o
I
' T
60cm
I
>
h
1,
13
1
l<
4a
Figure
1.
(a)
The
laboratory
setup.
(b)
Relevant
wave
and
topographic
parameters.
is
investigated
for
interfacial
waves
breaking
at
a
submerged
ridge
for
a
range
of
incident
wave
amplitudes.
[5]
While
the
overall
efficiency
of
interfacial
waves
break-
ing
at
a
submerged
ridge
has
not
been
measured
previously
in
the
lab,
a
number
of
studies
have
considered
the
reflec-
tion,
transmission
and
dissipation
of
wave
energy
in
such
interactions.
One
parameter
that
has
been
used
to
distinguish
the
strength
of
wave
obstacle
interactions
in
two-layer
flows
is
the
degree
of
blocking
B
=
ho
I
hi,.,
(
1
)
where
h
1
,,,„
and
h
c
,
are
defined
in
Figure
lb.
Internal
solitary
waves
passing
over
a
triangular
obstacle
are
essentially
not
affected
by
the
obstacle
for
B
<
0.6,
whereas
waves
split
into
a
dispersive
train
when
B
<
0.8,
and
when
B
>
1.2
the
obstacle
blocks
any
transmission
passed
the
obstacle
[Wessels
and
Huffer,
1996].
Sveen
et
al.
[2002]
introduce
a
blocking
parameter,
C
=
(a
+
h
2
)10
2
+
10,
to
characterize
transmitted
internal
solitary
wave
energy
over
a
triangular
obstacle.
Chen
[2009]
reports
that
internal
solitary
wave
reflection
and
energy
loss
scale
monotonically
with
C.
Wessels
and
Huffer
[1996]
considered
the
energy
budget
for
internal
solitary
waves
interacting
with
a
ridge
and
found
that
up
to
55%
of
the
wave
energy
was
dissipated
in
the
wave-ridge
interaction,
and
the
dissipated
fraction
is
maxi-
mum
when
0.8
<
B
<
1.1.
Chen
et
al.
[2008]
investigated
the
breaking
of
internal
solitary
waves
at
two
sequential
trian-
gular
obstacles
in
the
laboratory
and
report
that
the
fraction
of
incident
wave
energy
lost
over
the
obstacles
is
between
12%
and
48%,
increasing
with
the
energy
of
the
incident
wave.
The
current
study
incorporates
measurements
of
the
change
in
background
potential
energy
in
addition
to
the
wave
energy
loss,
so
that
the
overall
event
efficiency
can
be
estimated
for
the
wave-ridge
interaction.
[6]
In
this
study,
the
change
in
potential
energy
is
mea-
sured
in
addition
to
wave
reflection,
transmission
and
energy
loss
associated
with
the
breaking
event,
so
that
the
overall
efficiency
of
the
breaking
event
can
be
estimated.
In
this
paper,
the
experimental
setup
is
presented
in
section
2,
fol-
lowed
by
results
on
reflection
and
transmission
of
wave
energy,
potential
energy
and
wave
work
measurements,
and
the
overall
efficiency
in
section
3.
Section
4
is
a
discussion
of
the
results
and
finally
conclusions
appear
in
section
5.
2.
Experimental
Setup
[7]
Experiments
were
performed
in
a
480
cm
long
by
30
cm
wide
by
60
cm
tall
tank
with
a
vertically
oscillating
semicylindrical
wavemaker
at
one
end
and
a
horsehair
beach
at
the
other
end
to
prevent
reflection
from
the
back
wall
(see
Figure
1
a).
The
stratification
consisted
of
two
homogenous
2
of
10
CO2003
HULT
ET
AL.:
MIXING
EFFICIENCY
OF
INTERFACIAL
WAVES
CO2003
Table
1.
Experimental
Wave
Conditions
a
Stroke
(cm)
a
c
,„
(cm)
a
co
rn,.
0.0
0 0
0.84
0.5
0.4
0.07
0.85
1.0
0.8
0.14
0.86
1.5
1.2
0.21
0.87
2.0
1.6
0.27
0.88
2.5
1.9
0.33
0.89
3.0
2.3
0.39
0.90
3.5
2.6
0.44
0.91
4.0
2.9
0.49
0.92
5.0
3.4
0.59
0.93
6.0
3.8
0.65
0.94
7.0
4.1
0.71
0.95
8.0
4.4
0.76
0.96
'Here,
stroke
is
the
wavemaker
stroke
and
C
=
(h2
+
a.)/(h2
+
h,.).
For
all
cases,
h
1
,
0
=
26
cm,
h
2
=
30
cm,
w
=
0.59
rad
s
-1
,
B
=
0.78.
layers
separated
by
a
thin
interface.
At
the
start
of
each
experiment,
the
interfacial
thickness
was
S
=
1.5
cm
±0.14
cm.
The
lower
layer
had
depth
h
1
,,,
=
26
cm
and
density
p
i
=
1.0097
g
cm
-3
,
and
the
upper
layer
had
depth
h
2
=
30
cm
and
density
P2
=
0.9996
g
cm
-3
.
The
two-dimensional,
Gaussian
ridge
was
20.2
cm
tall
and
extends
100
cm
or
4
standard
deviations
(a)
in
length.
The
forced
wave
fre-
quency
was
w
=
0.59rad
s
-1
and
the
incident amplitude,
a
cc
,
was
varied
between
0
and
4.4
cm,
as
shown
in
Table
1
with
various
nonlinearity
parameters
for
comparison
with
other
wave-ridge
studies.
[8]
Here,
h
0
/2o
-
=
0.4,
which
is
comparable
to
bathymetric
slopes
of
0.07
to
0.5
where
enhanced
diapycnal
diffusivities
have
been
observed
[Kunze
and
Sanford,
1996;
Toole
et
al.,
1997].
Internal
waves
are
present
in
the
ocean
at
a
range
of
spatial
scales,
with
wavelengths
from
tens
of
meters
to
kilometers
and
amplitudes
of
up
to
hundreds
of
meters
[e.g.,
Helfrich
and
Melville,
2006]. High-frequency
internal
soli-
tary
waves
have
been
observed
with
amplitudes
that
are
a
large
fraction
of
the
local
depth,
and
even
longer
internal
tides
can
have
amplitudes
on
the
order
of
20%
of
the
total
depth
[Duda
et
al.,
2005].
[9]
Planar
laser-induced
fluorescence
(PLIF)
was
used
to
image
the
flow
during
the
experiments
[e.g.,
Crimaldi,
2008].
Laser
fluorescent
dye
(Rhodamine
6G)
was
added
to
the
more
dense,
lower
layer
at
a
concentration
of
50ppb.
An
argon
ion
laser
and
a
scanning
mirror
were
used
to
gen-
erate
a
light
sheet
which
was
imaged
by
a
1024
x
1024
pixel
CCD
camera.
The
PLIF
images
were
analyzed
to
track
the
movement
of
the
interface,
as
this
method
was
found
to
be
more
reliable
than
using
ultrasonic
interfacial
wave
gages,
with
the
additional
benefit
of
providing
a
record
of
the
interfacial
displacement
across
the
field
of
view
of
the
cam-
era,
in
this
case
19
cm
wide.
For
additional
details
of
the
imaging
setup
and
lab
facility,
see
Troy
and
Koseff
[2005].
[10]
To
assess
the
efficiency
of
wave
breaking
at
con-
verting
wave
energy
irreversibly
to
potential
energy
through
mixing,
the
change
in
wave
energy
and
background
poten-
tial
energy
due
to
the
event
must
be
calculated.
In
the
lab-
oratory,
the
entire
wave
tank
can
be
used
as
a
control
volume
for
internal
wave
mixing
experiments
[Ivey
and
Nokes,
1989;
Michallet
and
Ivey,
1999].
In
the
experiments
in
this
study,
energy
enters
the
control
volume
via
the
oscil-
lating
wave
maker
which
transfers
energy
to
the
fluid
in
the
form
of
internal
waves.
There
is
both
kinetic
and
potential
energy
associated
with
the
periodic
waves
propagating
along
the
density
interface.
As
the
waves
break
at
the
ridge,
some
of
the
wave
energy
is
converted
to
turbulent
kinetic
and
potential
energy,
while
some
of
the
wave
energy
con-
tinues
to
propagate
as
waves,
either
reflected
back
from
or
transmitted
past
the
ridge.
Of
the
energy
in
the
turbulence,
some
will
be
converted
irreversibly
to
potential
energy
via
molecular
mixing
hastened
by
turbulent
straining.
The
rest
of
the
turbulent
energy
eventually
will
be
dissipated
as
heat.
Energy
propagating
away
from
the
ridge
as
waves
will
eventually
be
primarily
dissipated
as
heat,
as
the
waves
are
damped
by
the
horsehair
beach
and
the
tank
walls.
Over
time,
some
fluid
will
also
tend
to
diffuse
across
the
density
interface,
thickening
this
interface
and
converting
the
heat
(internal
energy)
of
the
fluid
irreversibly
to
potential
energy.
Once
the
fluid
in
the
control
volume
of
the
tank
has
come
to
rest,
the
fmal
potential
energy
can
be
compared
with
the
initial
potential
energy
to
indicate
how
much
mixing
occurred
during
the
experiment.
Viscous
effects
and
background
dif-
fusion
are
accounted
for
in
the
calculations
of
efficiency.
[ii]
The
procedure
for
the
overall
event
efficiency
experi-
ments,
adapted
from
Troy
[2003],
is
as
follows.
First,
an
initial
density
profile
was
established
at
t
=
Os,
using
conductivity-
temperature
probe
plunging
vertically
at
10
cm
s
-1
.
Next,
starting
at
t=
30s
a
train
of
waves
was
generated
by
the
wavemaker.
Sinusoidal
oscillations
of
the
wavemaker
were
increased
from
zero
amplitude
to
the
maximum
amplitude
over
1
wave
period,
then
10
waves
were
generated
at
the
maximum
amplitude,
and
fmally
the
amplitude
was
reduced
to
zero
over
1
wave
period.
During
this
time,
images
were
taken
at
either
upstream
of
the
ridge
at
x
i
,
=
-65
cm
or
downstream
at
x
d
=
85
cm.
Although
ideally
measurements
would
have
been
taken
upstream
and
downstream
simulta-
neously
for
each
experimental
run,
constraints
on
the
imaging
equipment
allowed
interfacial
displacement
measurements
at
only
one
station
at
a
time.
The
variation
between
experi-
ments
was
quite
small
(see
Figures
3
and
6),
so
the
error
associated
with
comparing
upstream
and
downstream inter-
facial
displacement
measurements
from
different
experi-
ments
is
thought
to
be
minimal.
Finally,
after
motions
in
the
tank
subside,
a
final
density
profile
is
obtained
at
t
=
630s.
The
mixing
induced
by
the
motion
of
the
probe
was
negli-
gible
relative
to
background
diffusion
in
the
tank.
This
pro-
cedure
was
repeated
for
a
cc
,
between
0
and
4.4
cm,
with
2
to
6
(typically
4)
repetitions
for
each
wave
amplitude,
pro-
viding
2
to
6
pairs
of
initial
and
fmal
density
profiles
and
one
to
three
interfacial
displacement
records
at
the
upstream
and
downstream
locations.
The
wave
amplitudes
are
shown
in
Table
1.
2.1.
Wave
Energy
Measurements
[12]
The
propagating
wave
energy
is
calculated
at
loca-
tions
upstream
and
downstream
of
the
breaking
event
from
the
interfacial
displacement
time
series
(see
Figure
2).
The
wave
energy
per
unit
width
propagating
past
a
given
loca-
tion
over
the
time
interval
t
i
to
t
2
can
be
approximated
WE
=
c
p
g
Ap
f
77
2
(t)dt
,
(2)
t
i
where
c
p
,
is
the
wave
phase
speed,
Ap
=
p
l
-
p
2
,
and
n
is
the
interfacial
displacement
[Wessels
and
Huffer,
1996;
Troy,
t2
3
of
10
5
-E
U.
-
5
-
E
-
0
CD
0
CO2003
HULT
ET
AL.:
MIXING
EFFICIENCY
OF
INTERFACIAL
WAVES
CO2003
(a)
50
100
150
200
250
300
t
[s]
(b)
I
\ffirl.rwv
50
100
150
200
250
300
t
[s]
Figure
2.
Example
interfacial
height
time
series,
obtained
from
PLIF
images
(a)
upstream
and
(b)
downstream
of
the
breaking
event.
Note
that
the
times
shown
are
not
consistent
between
the
two
signals.
2003].
This
estimate
requires
that
local
changes
in
depth
be
small
relative
to
the
wavelength,
that
energy
is
propa-
gating
in
one
direction,
and
that
the
Boussinesq
approxi-
mation
is
valid.
Because
a
small
fraction
of
incident
wave
energy
does
reflect
back
from
the
ridge
as
discussed
above,
the
upstream
measurement
location
was
chosen
to
coincide
with
a
spatial
node
in
the
primary
wave
amplitude.
At
the
node,
the
reflected
component
destructively
interferes
with
the
incident
wave
signal,
so
the
amplitude
measured
at
that
location
represents
only
the
nonreflected
portion
of
the
wave
2 2
x1/2
energy,
a
cc
=
(a
inci
d
ew
areflected)
[13]
The
total
measured
change
in
wave
energy
between
the
upstream
and
downstream
measurement
stations
is
shown
in
Figure
3
and
can
be
expressed,
A
WE„1
=
AWEbreak
A
WE
visc
.
Here
AFVE
ota
i
=
WE
upstrean
,
WEdownstream,
and
A
WEbreak
refers
to
the
wave
energy
change
due
to
the
breaking
event.
A
WE
visc
is
the
wave
energy
lost
to
viscous
decay
not
associated
with
the
breaking
event,
i.e.,
through
viscous
damping
at
the
tank
boundaries
as
well
as
within
the
wave
interface
and
interior,
away
from
the
breaking
event
[Troy
and
Koseff,
2006].
In
order
to
calculate
L
---
W
-
gbreab
the
A
WE
visc
term
must
be
estimated.
Several
methods
are
used
here
to
estimate
the
nonbreaking,
viscous
losses
in
between
the
upstream
and
downstream
stations.
One
method
is
to
remove
the
ridge
from
the
tank
and
repeat
the
experiments
so
that
z
WE
visc
A
WE„
/
since
there
is
no
breaking.
The
change
in
wave
energy
for
the
ridge-less
case
is
shown
in
Figure
3.
It
is
important
to
note
that
the
ridge-less
case
pro-
vides
an
underestimate
of
the
nonbreaking
losses,
because
the
absence
of
the
ridge
increases
the
lower
layer
depth.
Thus
the
wave
velocities
near
the
bottom
are
decreased,
>,
0.03
rn
a)
Lu
a)
0.02
0.01
Ox
0
0.01
0.02
0.03
0.04
0.05
wave
amplitude
[m]
Figure
3.
Change
in
wave
energy
as
a
function
of
incident
wave
amplitude.
L
WE
total
is
shown
for
breaking
waves
(open
circles),
nonbreaking
waves
(solid
circle),
and
the
no
ridge
case
(crosses).
The
three
estimates
for
A
WE
visc
are
the
quadratic
fit
(solid
curve),
and
viscous
loss
is
esti-
mated
from
Troy
and
Koseff
[2006]
with
the
ridge
(dotted
curve)
and
without
the
ridge
(dashed
curve).
resulting
in
decreased
viscous
losses
at
the
bottom
and
side-
walls.
The
ridge-less
case
results
are
shown
for
reference
to
illustrate
the
magnitude
of
losses
away
from
the
ridge.
[14]
A
second
method
to
estimate
A
WE
visc
is
to
use
the
theory
developed
by
Troy
and
Koseff
[2006]
to
estimate
the
viscous
losses
due
boundary
damping,
internal
dissipation,
and
interfacial
dissipation.
Figure
4
shows
the
expected
var-
iation
in
the
wave
amplitude
over
the
ridge
assuming
linear
wave
shoaling
and
viscous
decay
[Troy
and
Koseff,
2006].
Away
from
the
ridge,
the
dominant
contribution
to
viscous
wave
decay
is
sidewall
damping,
and
only over
the
very
crest
of
the
ridge
does
bottom
boundary
layer
damping
becomes
dominant.
The
linear,
viscous
decay
theory
predicts
the
wave
energy
will
decay
by
18%
without
the
ridge
or
22%
with
the
1.1
1.05
E
1
1
5
0.95
CV
7
Ca
0.9
as
0.75
-3a
-2a
-a
0
a
2a
3a
4a
x
Figure
4.
Expected
amplitude
variation
between
upstream
and
downstream
measurement
locations
from
the
theory
of
Troy
and
Koseff
[2006]
for
the
inviscid
case
(dashed
curve);
the
viscous,
ridge-less
case
(dotted
curve);
and
the
viscous,
with
ridge
case
(solid
curve).
Here,
f
=
0.09375Hz,
h
i
,
co
=
26
cm.
0.05
0.04
0.85
0.8
4
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HULT
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MIXING
EFFICIENCY
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CO2003
so
the
change
in
potential
energy
between
the
initial
and
final
density
profiles,
APE,
is
APE
=
gA
f
z(pft
na
i(z)
Pinitiai(Mdz,
(4)
N
-3
1000
1003
1006
1009
p
[kg/m
3
]
Figure
5.
Sample
profiles
of
density,
before
(solid
curve)
and
after
(dashed
curve)
a
train
of
breaking
waves
where
=
0.59.
ridge
between
the
upstream
and
downstream
measurement
stations.
The
viscous
damping
estimates
are
also
shown
in
Figure
3,
where
the
viscous
theory
agrees
reasonably
well
with
A
WEtotai
measured
in
the
ridge-less
case.
[15]
A
final
method
for
estimating
A
WE
v4sc
was
devel-
oped
by
Troy
[2003].
In
this
method,
AWE
EIso
is
extrapo-
lated
from
the
total
change
in
wave
energy
for
nonbreaking
waves.
In
Figure
3,
a
quadratic
curve
is
fit
to
the
non-
breaking
wave
points
using
least
squares.
Because
viscous
wave
energy
losses
scale
with
the
square
of
the
wave
velocity,
(aw)
2
,
a
quadratic
curve
with
an
a
2
dependence
is
chosen.
This
quadratic
curve
is
then
extrapolated
to
give
an
estimate
of
viscous
losses
for
the
breaking
waves.
In
Figure
3,
WE„
1
increases
much
more
steeply
for
the
breaking
wave
cases.
The
estimate
of
WE
vi
from
this
extrapolation
method
is
slightly
higher
than
either
the
ridge-
less
case
estimate
or
the
linear
viscous
decay
theory
estimate.
[16]
The
three
methods
for
estimating
WE
vi
give
similar
results,
as
shown
in
Figure
3.
Because
viscous
losses
increase
where
the
ridge
restricts
the
lower
layer
depth,
it
is
expected
that
the
no
ridge
case
would
lead
to
a
lower
WE„
i
„.
The
viscous
decay
theory
is
linear,
and
thus
it
is
not
surprising
that
the
associated
estimate
of
WE„
i
is
slightly
less
than
the
extrapolation
method
estimate.
For
these
reasons,
the
extrapolation
method
is
used
in
fmal
estimate
of
Ri
o
,
and
it
is
shown
later
that
the
viscous
theory
estimate
yields
very
similar
results.
The
reason
for
including
all
three
methods
is
simply
to
support
the
extrapolation
method
and
offer
alternative
methods
for
use
when
extrap-
olation
is
not
feasible.
2.2.
Change
in
Potential
Energy
Measurements
[17]
If
the
tank
is
quiescent,
the
potential
energy
at
an
instant
can
be
calculated
from
the
density
profile
PE
=
gA
f
zp(z)dz,
(
3
)
where
A
is
the
cross-sectional
area
of
the
tank.
Sample
profiles
of
p
initiat
and
p
ow
,
/
are
shown
in
Figure
5.
As
dis-
cussed
by
Troy
[2003],
it
can
be
quite
difficult
to
get
a
reliable
estimate
of
APE
directly
by
numerically
integrating
equation
(4)
over
a
portion
of
the
tank,
due
to
drift
in
the
conductivity
probe
and
the
extreme
sensitivity
of
APE
to
noise
in
the
measured
density
profile
away
from
the
den-
sity
interface.
Here,
Troy's
alternative
method
of
fitting
an
error
function
to
the
measured
density
profile
is
used.
The
density
profiles
can
be
approximated
by
an
error
function
of
the
form
P(Z)
=
PO
2
erf(,Qz),
P1
P2
(
5
)
where
po
=
Co
l
+
p
2
)/2
and
the
length
scale
0
-1
can
be
related
to
the
99%
interfacial
thickness,
6
=
3.64//3.
With
the
density
profile
in
this
form,
equation
(4)
is
evaluated
ana-
lytically
to
give
APE
in
terms
of
initial
and
final
interfacial
length
scales
13
initial
and
13.finai
APE
gA(Pi
192)
1
3
;nal
(6)
2
2Q2
Using
this
method,
the
measured
APE
is
very
consistent
between
repeated
experiments.
The
change
in
potential
energy
between
the
initial
and
final
density
profiles
is
shown
in
Figure
6,
where
the
maximum
and
minimum
values
of
APE
are
shown
by
the
error
bars.
[18]
To
estimate
the
changes
in
background
potential
energy
that
are
not
associated
with
the
breaking
event,
the
experimental
procedure
was
repeated
for
each
amplitude
after
removing
the
ridge
from
the
tank,
as
discussed
above
X10
3
cb
x
x
0
0.01
0.02
0.03
0.04
0.05
wave
amplitude
[m]
Figure
6.
Change
in
potential
energy
as
a
function
of
wave
amplitude,
a
cc
.
APE
totai
is
shown
for for
breaking
waves
(open
circles),
nonbreaking
waves
(solid
circles),
and
the
no
ridge
case
(crosses).
APE
nonbreak
is
extrapolated
using
a
quadratic
fit
(solid
curve).
2.5
2
w
1.5
0_
1
0.5
5
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MIXING
EFFICIENCY
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INTERFACIAL
WAVES
CO2003
t/T
=
0.06
t/T
=
0.19
t/T
=
0.31
t/T
=
0.44
t/T
=
0.56
t/T
=
0.69
t/T
=
0.81
IMP
(b)
1;1
Pil
Figure
7.
PLIF
images
of
three
breaking
events
where
(a)
a
co
1
h
r
=
0.44,
(b)
a
co
lh
r
=
0.59,
and
(c)
a
cc
,
1
h
r
=
0.76.
The
field
of
view
in
the
images
is
5.7
cm
wide
by
12.2
cm
tall,
and
the
top
of
the
ridge
is
visible
at
the
bottom
of
the
images.
(c)
for
the
estimation
of
A
WE
vi
„.
The
change
in
potential
energy
for
the
ridge-less
cases
is
shown
in
Figure
6.
These
results
indicate
that
wave
trains
propagating
in
the
absence
of
the
ridge
do
not
cause
significant
irreversible
mixing
above
the
level
for
the
zero
amplitude
wave
case.
There-
fore,
mixing
at
the
wavemaker
and
the
horsehair
beach
must
be
minimal
and
the
measured
APE
in
the
no
ridge
case
is
assumed
to
be
caused
primarily
by
background
diffu-
sive
processes
over
the
time
of
the
experiment.
For
unknown
reasons,
the
variation
between
experiments
appears
to
be
greater
in
the
ridge-less
cases
than
when
the
ridge
was
in
place,
as
indicated
by
both
the
APE
and
A
WE
measurements.
[19]
In
the
cases
where
the
ridge
is
in
place,
not
all
of
the
measured
APE
is
thought
to
be
associated
with
the
breaking
event.
A
certain
portion
of
the
measured
APE
is
associated
with
background
diffusive
processes,
as
mentioned
above.
There
is
also
expected
to
be
a
portion
of
APE
resulting
from
enhanced
mixing
due
to
the
increased
wave
velocities
over
the
ridge,
as
the
wave
amplitude
is
increased.
For
example,
as
the
wave
velocity
increases,
mixing
is
also
expected
to
increase
where
the
density
interface
intersects
the
boundary
layer
of
the
tank
wall.
This
trend
is
shown
in
the
slight
increase
of
APE
for
nonbreaking
wave
cases.
As
in
the
A
WE
measurements,
the
associated
change
in
potential
energy
from
boundary
effects
is
expected
to
scale
with
a
2
.
The
total
measured
change
in
potential
energy
can
be
sepa-
rated
into
breaking
and
nonbreaking
components:
APE
totai
=
APEbreak
+
APE
nonb
w
k.
To
estimate
APEnonbreak,
a
qua-
dratic
curve
with
an
a
dependence
is
fit
using
least
squares
to
the
APE
measurements
for
the
nonbreaking
cases
(e.g.,
a
cc
<
2
cm).
APE
break
is
estimated
by
extrapolating
the
qua-
dratic
curve
to
higher
amplitudes,
and
then
subtracting
the
nonbreaking
portion
of
APE
from
APE
2.3.
Overall
Event
Efficiency
Calculation
[20]
From
the
change
in
wave
energy
associated
with
the
breaking
event,
A
WEbreak
=
(WEu
WEd)
A
WE
A
„,
and
the
change
in
potential
energy
associated
with
the
breaking
event,
APE
break
=
APE
I
APE
non
b
rea
k,
the
overall
event
efficiency
can
be
calculated
as
APEbreak
Rf
0
=
A
WEbreak
The
overall
efficiency
results
are
discussed
in
section
3.2.
3.
Results
[21]
Figure
7
shows
density
fields
calculated
from
PLIF
images
for
breaking
events
of
three
incident
wave
ampli-
tudes.
As
the
incident
amplitude
is
increased,
breaking
becomes
more
vigorous
and
the
resulting
turbulent
patch
becomes
larger
in
size.
After
the
initial
wave
instability,
the
flow
transitions
to
turbulence,
and
then
eventually
decays.
The
mixed
fluid
resulting
from
the
turbulent
event
eventually
spreads
away
from
the
point
of
breaking
along
the
interface.
In
Figure
7a,
the
wave
is
just
past
the
breaking
threshold
and
breaks
mildly
backward,
whereas
Figure
7b
shows
a
more
vigorous
forward
breaking
wave,
and
Figure
7c
shows
an
even
more
vigorous
breaking
event.
From
the
density
fields
in
Figure
7,
it
is
clear
that
the
higher
the
amplitude,
the
more
mixing
occurs
during
the
breaking
event,
as
the
local
(
7
)
6
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ET
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INTERFACIAL
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CO2003
0.5
0.1
(a)
m
0
(b)
0
-0
y=
,
a
0.3
0.2
cc
0
0.1
2
k
h
4
0.2
0.4
a
/
h
r
0.6
Figure
8.
(a)
The
reflection
coefficient
as
a
function
of
the
scaled
wave
number
k
oc
h
i
,
a)
.
The
incident
amplitude
is
fixed
a
cc
=
1.3
cm.
(b)
The
reflection
coefficient
as
a
function
of
the
scaled
wave
ampli-
tude,
a
co
lh,..
Stable
(squares)
and
breaking
(circles)
are
shown.
Here,
the
wave
frequency
is
fixed
at
w
=
0.59
rad
s
-1
and
k
a
iz
i
,
a)
=
1.9
where
k
and
w
can
be
related
through
the
two
layer
dispersion
relation.
interfacial
thickness
increases
substantially
in
Figure
7c
com-
pared
with
sequence
(Figure
7a).
Quantitative
evidence
of
this
trend
can
be
seen
in
section
2.2.
3.1.
Wave
Reflection
and
Transmission
[22]
For
each
incident
wave,
some
energy
is
reflected
back
from
the
ridge.
When
a
portion
of
the
incident
wave
energy
reflects
back
from
the
obstacle,
the
result
is
a
spatial
modulation
of
the
wave
amplitude
upstream
of
the
obstacle,
as
the
reflected
wave
constructively
and
destructively
inter-
feres
with
the
incident
wave
[e.g.,
Dean
and
Dalrymple,
1984].
The
coefficient
of
reflection
is
typically
defined
as
the
ratio
of
the
reflected
wave
amplitude
and
the
incident
wave
amplitude,
C
R
=
at-elcted!
a
incident-
Here,
C
R
can
be
estimated
from
the
variation
of
the
measured
wave
ampli-
tude
in
space
a
2
)
1/2
C
R
=
(1
2
a
mean
where
a
m
i
n
and
a
mean
are
the
local
minimum
and
mean
measured
amplitudes
in
a(x).
Spectral
analysis
of
the
inter-
facial
height
time
series
from
PLIF
images
gives
the
amplitude
a(x)
across
19
cm
image
windows.
A
sine
curve
is
fit
to
a(x)
using
least
squares,
and
the
mean
amplitude
is
then
a
mmamit
.
Note,
the
relevant
amplitude
to
the
breaking
event,
a
cc
,
is
considered
to
be
the
transmitted
wave
energy.
The
transmitted
energy
is
equal
to
the
incident
energy
minus
the
reflected
portion
and
the
wave
energy
is
2
propor-
tional
to
the
square
of
the
amplitude,
thus:
a
a
,
=
(aincident
a
reectedl
\
112
In
Figure
8a,
the
reflection
coefficient,
CR,
decreases
as
the
wave
number,
kh
i
,
a
is
increased
for
waves
of
constant
incident
amplitude.
The
error
bars
reflect
vari-
ation
in
the
fitted
parameters
between
two
repetitions
of
the
experiment
The
oscillation
of
C
R
with
kh
i
,
a)
seen
in
Figure
8a
may
be
consistent
with
surface
wave
reflection
from
a
sub-
merged
obstacle
[e.g.,
Mei
and
Black,
1969].
Previous
work
indicates
that
C
R
is
also
sensitive
to
the
degree
of
blocking,
B
[Wessels
and
Hutter,
1996],
but
B
is
not
varied
in
this
study.
[23]
Somewhat
surprisingly,
the
reflection
coefficient
appears
to
be
insensitive
to
the
incident
wave
amplitude,
as
shown
in
Figure
8b,
where
C
R
=
0.30
±
0.05.
A
reflection
coefficient
of
C
R
=
0.30
corresponds
to
approximately
9%
of
the
incident
wave
energy
being
reflected
back
from
the
ridge
for
both
breaking
and
nonbreaking
waves.
The
fact
that
C
R
is
indifferent
to
whether
or
not
the
wave
is
breaking
suggests
that
wave
reflection
from
the
ridge
slope
may
take
place
upstream
of
where
the
breaking
event
occurs
at
the
ridge
crest.
It
is
possible
that
C
R
may
be
affected
by
breaking
when
waves
are
shorter
relative
to
the
depth
(higher
k
oc
h
i
,
a)
),
or
when
the
wave
slope
(h
o
/2a)
is
steeper,
either
of
which
would
compress
the
horizontal
region
over
which
breaking
and
reflection
occur.
The
range
of
C
R
for
the
periodic
waves
in
this
study
corresponds
reasonably
well
with
the
results
for
internal
solitary
waves
of
depression
reported
by
Chen
[2009],
where
0.3
<
C
R
<
0.6
for
0.8
< <
1.0
(see
Table
1
to
relate
and
a
cc
,Ih
r
in
this
study).
It
appears
that
for
periodic,
progressive
interfacial
waves,
however,
C
R
varies
with
the
length
scale
of
the
wave,
and
the
parameter
does
not
capture
this
dependence
explicitly
as
Chen
studied
solitary
waves
where
a
and
k
are
not
independent.
[24]
The
incident
wave
energy
can
be
partitioned
into
reflected,
transmitted
and
dissipated
components
based
on
information
from
the
upstream
and
downstream
interfacial
displacement
signals,
such
as
those
shown
in
Figure
2.
The
amplitude
of
the
downstream
interface
displacement
in
Figure
2b
is
noticeably
smaller
than
the
upstream
displace-
ments
in
Figure
2a.
Higher-frequency
oscillations
are
also
visible
in
the
downstream signal
in
Figure
2b.
Section
3.2
investigates
how
much
incident
energy
is
dissipated
in
the
breaking
event
and
how
much
energy
goes
to
irreversible
mixing.
3.2.
Overall
Event
Efficiency
[25]
The
overall
event
efficiency
in
Figure
9
ranges
from
3%
to
8%
±
1%.
This
assumes
that
the
nonbreaking
wave
energy
loss
is
estimated
from
the
quadratic
fit
to
the
non-
breaking
points,
AWE„
isc
-
quad
fit'
in
Figure
3.
If
the
(8)
7
of
10
%
o
f
inc
iden
t
wa
ve
energy
100
80
60
40
20
CO2003
HULT
ET
AL.:
MIXING
EFFICIENCY
OF
INTERFACIAL
WAVES
CO2003
o
0.2
0.4
0.6
0.8
a
/h
r
Figure
9.
Overall
event
efficiency
R
i
co
as
a
function
of
scaled
wave
amplitude
a
co
lh,.,
where
z
WEvisc
is
estimated
from
the
quadratic
fit
to
the
nonbreaking
points
(open
cir-
cles)
and
from
the
viscous
decay
theory
of
Troy
and
Koseff
[2006]
(solid
circles).
estimate
of
z
WE
visc
from
the
viscous
decay
theory
of
Troy
and
Koseff
[2006]
is
used
instead,
the
resulting
values
of
Rfo
are
typically
lower
by
about
0.7%
(Figure
9).
The
uncer-
tainty
is
quite
large
for
wave
amplitudes
just
large
enough
to
break
(a
co
lh,.
S
0.45
in
Figure
9),
because
the
uncer-
tainty
in
the
measured
quantities
is
on
the
order
of
the
dif-
ference,
APE
b
ak
or
A..
_
WF:
break
.
It
is
notable,
however,
that
for
a
co
lh,.
>
0.55,
at
least
half
of
6.
WE
tota
i
and
APEtotat
results
from
the
breaking
event.
Without
such
a
strong
signal
from
the
breaking
event,
this
method
is
less
effective
at
determining
the
overall
event
efficiency
[Troy,
2003].
3.3.
Wave
Energy
Partitioning
[26]
With
the
results
discussed
in
this
study
thus
far,
the
end
fate
of
all
of
the
incident
wave
energy
is
determined.
Figure
10
illustrates
how
much
of
the
incident
wave
energy
was
reflected
from
the
ridge,
dissipated
through
laminar
viscous
decay,
transferred
to
higher
harmonics,
transmitted
past
the
ridge,
converted
irreversibly
to
potential
energy
and
dissipated
in
the
breaking
event
(eb
reak
).
The
total
trans-
mitted
energy
begins
to
decrease
from
about
60%
when
waves
begin
to
break
at
a
co
lh,.
••=-:
0.4.
Once waves
begin
to
break,
the
total
incident
wave
energy
transmitted
past
the
ridge
decreases
from
about
60
to
30%
and
an
increasing
portion
is
dissipated
in
the
breaking
event.
As
discussed
by
Hult
et
al.
[2010],
waves
of
higher
harmonic
frequencies
can
be
excited
when
a
train
of
periodic
waves
passes
over
the
ridge
and
thus
the
transmitted
wave
energy
at
twice
the
forcing
frequency
(2w
1
)
is
shown
separately
from
the
rest
of
the
transmitted
wave
energy.
The
energy
transmitted
at
twice
the
forcing
frequency
(2w
1
)
peaks
when
accihr
0.4.
While
a
substantial
fraction
of
the
incident
wave
energy
is
dissipated
in
the
breaking
event,
only
a
small
percentage
of
the
wave
energy
goes
to
irreversible
mixing
(APE
b
ak
).
The
portion
of
energy
dissipated
during
breaking
appears
to
level
off
as
a
co
lh,.
approaches
unity.
4.
Discussion
[27]
The
overall
event
efficiency
of
3-8%
is
much
lower
than
the
36%
efficiency
reported
for
deep
water
interfacial
wave
breaking
[Fringer
and
Street,
2003;
Troy,
2003].
It
is
reasonable
that
the
overall
efficiency
of
an
event
would
be
lower
for
a
topography
induced
breaking
event
than
for
deep
water
wave
breaking.
In
a
deep
water
breaking
event,
both
density
overturning
and
turbulent
kinetic
energy
dissipation
are
focused
along
the
interface.
When
breaking
occurs
at
a
ridge
or
slope,
on
the
other
hand,
the
wave
flow
tends
to
be
focused
along
the
topographic
boundary.
A
two-layer
wave
interaction
with
a
topographic
feature
may,
however,
lead
to
separation
vortices
within
the
well-mixed
lower
layer
that
do
little
to
alter
the
background
density
gradient.
As
seen
in
Figure
12
of
Hult
et
al.
[2009],
strong
flow
over
the
ridge
in
the
vertically
constricted
lower
layer
separates
and
the
development
of
a
large
vortex
is
observed
on
the
upstream
side
of
the
ridge
as
the
wave
crest
approaches.
The
gener-
ation
of
this
separation
vortex
could
influence
the
overall
efficiency
of
the
wave-ridge
interaction,
as
energy
is
transferred
from
the
propagating
wave
to
the
vortex
in
a
region
where
the
background
density
gradient
is
extremely
weak.
This
process
may
help
to
explain
why
the
overall
event
efficiency
is
much
lower
than
the
deep
water
case.
Hult
et
al.
[2011,
hereafter
Part
2]
use
high-resolution
0.3
0.25
0.2
0.15
0.1
0
0
S
break
APE
break
transmission
(not
2w
1
)
2co
1
transmission
_AWE
w
.
reflection
0.2
0.4
0.6
0
8
a
/
h
r
Figure
10.
Partitioning
of
incident
wave
energy,
where
z
WE
vis
,
is
estimated
from
the
quadratic
fit
method.
0
0
8
of
10
CO2003
HULT
ET
AL.:
MIXING
EFFICIENCY
OF
INTERFACIAL
WAVES
CO2003
measurements
over
the
spatial
domain
to
explore
this
hypothesis
of
variation
in
local
mixing
processes.
[28]
The
range
reported
here
is
at
the
low
end,
compared
with
the
15
±
5%
efficiency
reported
by
Helfrich
[1992],
or
the
7-25%
efficiency
reported
by
Michallet
and
Ivey
[1999],
both
for
breaking
solitary
waves
at
a
slope.
Note
that
Michallet
and
Ivey
did
not
account
for
background
diffusive
effects
in
their
analysis,
which
would
conserva-
tively
lower
their
efficiency
range
of
3-22%.
If
it
is
the
interaction
with
topography
causing
spatial
variability
in
the
mixing
process,
why
would
the
overall
event
efficiency
in
this
study
be
so
low
compared
with
previous
results
for
internal
solitary
waves
breaking
at
a
slope?
While
there
is
much
in
common
with
the
slope
case,
there
are
some
key
differences
in
the
flow
for
waves
over
a
ridge.
In
the
ridge
case,
there
is
strong,
periodic
flow
over
the
ridge
crest,
including
acceleration
of
the
flow
around
the
obstacle
and
separation
from
the
boundary.
This
may
lead
to
more
energy
dissipation
within
the
lower
layer
than
in
the
slope
case
where
motion
is
concentrated
where
the
density
interface
intersects
the
slope.
The
energy
dissipation
and
mixing
within
the
lower
layer
is
discussed
in
Part
2.
[29]
Although
the
range
of
overall
mixing
efficiency
is
quite
low,
the
most
nonlinear
breaking
events
were
2.5
times
more
efficient
than
the
least
nonlinear
cases.
This
difference
may
be
due
to
the
range
of
breaking
mechanisms
that
occurred
in
this
study.
As
reported
by
Hult
et
al.
[2009],
when
0.45
<
a
cc
/h
r
<
0.55,
breaking
events
appear
to
be
primarily
driven
by
shear,
whereas
convective
instability
is
observed
when
0.55
<
a
co
lh,..
The
transition
values
depend
on
the
scaled
wavelength,
and
those
listed
here
correspond
with
1c6
=
0.11
as
in
this
study.
In
Figure
9,
when
a
cc
/h
r
exceeds
about
0.55,
the
overall
event
efficiency
increases
from
3%
to
8%.
The
dependence
of
the
overall
efficiency
on
the
topographic
slope,
wave
frequency
or
interface
height
was
not
explored
here.
[3o]
When
the
wave
amplitude
is
small
relative
to
the
lower
layer
depth
over
the
obstacle
(a
co
lh
r
<
0.2),
the
depen-
dence
of
energy
transmission
on
the
degree
of
blocking,
B,
developed
by
Wessels
and
Hutter
[1996]
appears
to
break
down.
Wessels
and
Hutter
[1996]
stated
that
waves
were
unaffected
by
the
presence
of
a
ridge
when
B
<
0.6.
How-
ever
in
this
study,
B
is
held
constant
at
0.78,
and
very
low
amplitude
waves
were
hardly
affected
by
the
ridge
whereas
in
high-amplitude
cases
significant
energy
was
dissipated
at
the
ridge
(see
Figure
10).
This
suggests
that
the
amplitude
of
the
incident
wave
affects
the
intensity
of
the
wave-ridge
interaction.
While
the
parameter
includes
a
dependence
on
amplitude,
there
is
a
discrepancy
in
the
energy
dissipated
at
the
ridge
between
the
results
in
Figure
10
and
of
Chen
et
al.
[2008],
likely
due
to
the
insensitivity
of
the
wave-ridge
interaction
to
the
upper
layer
depth
in
the
periodic
wave
case.
[31]
While
the
distribution
in
Figure
10
provides
a
over-
view
of
how
energy
is
transferred
as
a
function
of
acorn,"
the
percentages
in
the
distribution
may
vary
to
some
extent
with
additional
wave
and
topographic
parameters.
The
overall
mixing
efficiency
may
vary
to
some
extent
based
on
breaking
mechanism,
as
discussed
above.
Although
this
limit
was
not
explored
in
this
study,
it
seems
probably
that
the
mixing
efficiency
would
increase
as
h
r
is
reduced
dramatically,
and
the
geometry
approaches
the
case
of
interfacial
wave
breaking
on
a
slope
case
studied
by
Michallet
and
Ivey
[1999].
Also,
the
fraction
of
the
incident
wave
energy
that
goes
to
reflection,
transmission,
higher
harmonics,
viscous
dissipation
and
the
breaking
event
(€
+
APE)
will
vary
to
some
extent
with
the
wave
and
topographic
parameters.
The
percent
of
incident
energy
transmitted
for
internal
sol-
itary
waves
passing
over
a
ridge
[Helfrich,
1992]
has
been
shown
to
vary
with
the
length
scale
of
the
wave.
As
seen
in
Figure
8a,
the
amount
of
energy
reflected
from
the
ridge
tends
to
increase
with
the
incident
wavelength,
whereas
vis-
cous
decay
tends
to
damp
shorter
waves
more
quickly.
The
wave
nonlinearity,
the
strength
of
the
stratification
relative
to
the
wave
frequency,
and
the
wavelength
relative
to
the
topographic
length
can
affect
the
excitement
of
higher
har-
monics
[Hult
et
al.,
2010].
[32]
It
is
possible
that
the
sidewall
or
bottom
boundary
layers
may
become
turbulent
as
the
wave
amplitude
is
increased.
In
this
case,
the
choice
of
an
a
2
dependence
of
z
WE
visc
or
the
use
of
laminar
viscous
theory
to
estimate
the
nonbreaking
wave
energy
loss
may
not
be
appropriate.
Troy
and
Koseff
[2006]
suggest
the
transition
to
turbulence
occurs
when
the
sidewall
Reynolds
number
R
s
=
u
ma
yL
orb
dzi
is
200-800,
where
L
or
bu
is
the
orbital
excursion
distance.
The
conservative
approximations
for
this
case,
u
aw,
and
Lorbit
a,
give
R
s
=
300-1200
for
the
breaking
wave
cases
which
suggests
the
sidewall
boundary
layers
are
probably
turbulent
for
the
highest-amplitude
cases
in
this
study.
Over
most
of
the
tank
length,
the
dominant
contribution
to
vis-
cous
damping
is
from
the
sidewalls.
Only
as
waves
pass
over
the
very
crest
of
the
ridge
is
damping
at
the
bottom
dominant.
If
a
transition
to
turbulent
sidewall
boundary
layers
occurs,
the
viscous
damping
rate
is
expected
to
increase
compared
to
the
laminar
rate.
The
overall
event
efficiency
is
not,
however,
terribly
sensitive
to
the
sidewall
damping
coefficient.
If
a
transition
to
turbulence
doubled
the
sidewall
and
bottom
damping
coefficients
estimated
by
the
laminar
theory
of
Troy
and
Koseff
[2006],
then
Rfo
would
only
increase
by
about
2%
to
give
Rfo
=
5-10%,
assuming
no
change
in
APE.
[33]
The
Reynolds
number
of
these
experiments
may
be
modest
relative
to
oceanic
flows.
The
key
message,
how-
ever,
is
relevant
to
flows
of
a
wide
range
of
scales.
If
internal
wave
energy
is
dissipated
in
an
interaction
with
topography,
the
overall
event
efficiency
of
the
event
may
vary
depending
on
the
instability
mechanism.
The
hypoth-
esis
to
be
explored
in
Part
2
that
spatial
variability
in
the
stratification
can
reduce
the
overall
event
efficiency
from
a
typical
value
of
20-25%
is
also
expected
to
be
independent
of
Re.
The
details
of
the
separation
flow
may
depend
on
the
Reynolds
number,
however
separation
can
occur
over
a
wide
range
of
Reynolds
number
[e.g.,
Kundu
and
Cohen,
2004].
5.
Summary
[34]
Using
the
interfacial
displacement
upstream
and
downstream
of
the
wave-ridge
interaction
as
well
as
the
measured
change
in
potential
energy
during
the
experiment,
the
partitioning
of
energy
for
periodic,
progressive
interfa-
cial
waves
breaking
at
a
ridge
was
investigated.
Between
30%
and
65%
of
the
incident
wave
energy
was
transmitted
over
the
ridge,
where
the
fraction
of
energy
transmitted
9
of
10
CO2003
HULT
ET
AL.:
MIXING
EFFICIENCY
OF
INTERFACIAL
WAVES
CO2003
tends
to
decrease
as
the
amplitude
of
the
wave
is
increased.
The
reflection
coefficient
C
R
=
areflectediaincident
was
shown
to
decrease
from
0.5
to
0.2
(i.e.,
25%
to 4%
of
incident
wave
energy)
with
scaled
wave
number,
When
the
ampli-
tude
is
varied
for
a
fixed
wave
number,
kh
i
,„„
=
2.0,
the
reflection
coefficient
remains
constant
at
C
R
=
0.30
±
0.05
for
nonbreaking
and
breaking
wave
cases.
[35]
The
fraction
of
wave
energy
lost
in
the
breaking
event
that
is
converted
irreversibly
to
potential
energy,
referred
to
as
the
overall
event
efficiency
Rfo,
is
estimated
by
using
the
wave
tank
as
a
control
volume.
The
wave
energy
is
mea-
sured
upstream
and
downstream
of
the
breaking
event,
and
viscous
losses
not
associated
with
the
breaking
event
are
estimated
with
several
methods
including
extrapolating
from
nonbreaking
wave
cases
and
viscous
decay
theory
[Troy
and
Koseff,
2006].
The
overall
event
efficiency
is
between
3%
and
8%
±
1%
for
waves
of
varied
incident
amplitude.
There
is
not
a
clear
demarkation
in
overall
efficiency
between
shear
and
convective
breaking
events,
however,
the
effi-
ciency
tends
to
increase
slightly
with
the
nonlinearity
While
this
efficiency
is
consistent
with
results
for
breaking
internal
waves
at
topography
[Ivey
and
Nokes,
1989;
Helfrich,
1992;
Michallet
and
Ivey,
1999],
the
measured
efficiency
of
interfacial
waves
in
deep
water
is
as
high
as
36%
[Fringer
and
Street,
2003;
Troy,
2003].
The
relatively
low
overall
efficiency
appears
to
be
related
to
spatial
variability
in
the
mixing
processes.
The
wave-ridge
interaction
can
lead
to
separation
of
the
lower
layer
flow
over
the
ridge,
which
in
turn
can
cause
enhanced
turbulent
kinetic
energy
dissipation
as
shown
in
Part
2.
In
this
region
of
enhanced
dissipation
within
the
essentially
homogenous
lower
layer,
the
expected
mixing
efficiency
would
be
quite
low
as
there
are
no
density
gradients
to
mix,
and
this
could
reduce
the
overall
mixing
efficiency
relative
to
the
deep
water
case
where
turbulence
is
concentrated
in
the
interfacial
region.
The
impact
of
spatial
variability
on
the
overall
efficiency
is
investigated
in
Part
2.
[36]
Acknowledgments.
The
authors
acknowledge
the
help
of
two
anonymous
reviewers
which
greatly
improved
this
work.
E.L.H.
was
funded
by
a
National
Defense
Science
and
Engineering
Graduate
Fellow-
ship
as
well
as
a
Graduate
Research
Fellowship
from
the
National
Science
Foundation.
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