Horizontal Mixing in Estuaries
and Coastal Seas
Mark T. Stacey
Warnemuende
Turbulence Days
September 2011
The Tidal Whirlpool
•
Zimmerman (1986) examined the
mixing induced by tidal motions,
including:
–
Chaotic tidal stirring
–
Tides interacting with residual flow
eddies
–
Shear dispersion in the horizontal
plane
•
Each of these assumed
timescales long compared to the
tidal cycle
–
Emphasis today is on
intra

tidal
mixing in the horizontal plane
–
Intratidal
mixing may interact with
processes described by Zimmerman
to define long

term transport
Mixing in the Horizontal Plane
•
What makes analysis of
intratidal
horizontal mixing hard?
–
Unsteadiness and variability at a wide
range of scales in space and time
–
Features may not be tied to specific
bathymetric or forcing scales
–
Observations based on point
measurements don’t capture spatial
structure
Mixing in the Horizontal Plane
•
Why is it important?
–
To date, limited impact on modeling due to
dominance of numerical diffusion
•
Improved numerical methods and resolution
mean numerical diffusion can be reduced
•
Need to appropriately specify horizontal
mixing
–
Sets longitudinal dispersion (shear
dispersion)
Unaligned
Grid
Aligned
Grid
Numerical Diffusion [m
2
s

1
]
Holleman
et al., Submitted to IJNMF
Mixing and Stirring
•
Motions in horizontal plan may produce kinematic straining
–
Needs to be distinguished from actual (irreversible) mixing
•
Frequently growth of variance related to diffusivity:
•
Unsteady flows
–
Reversing shears may “undo” straining
•
Observed variance or second moment may diminish
–
Variance variability may not be sufficient to estimate mixing
•
Needs to be analyzed carefully to account for reversible and irreversible
mixing
Figures adapted from
Sundermeyer
and
Ledwell
(2001); Appear in
Steinbuck
et al.
in review
Candidate mechanisms for lateral mixing
•
Turbulent motions (dominate vertical mixing)
–
Lengthscale
: meters; Timescale: 10s of seconds
•
Shear dispersion
–
Lengthscale
: Basin

scale circulation; Timescale: Tidal or diurnal
•
Intermediate scale motions in horizontal plane
–
Lengthscales
: 10s to 100s of meters; Timescales: 10s of minutes
•
Wide range of scales:
–
Makes observational analysis challenging
–
Studies frequently presume particular scales
1

10 meters
Seconds to minutes
Basin

scale Circulation
Tidal and Diurnal Variations
Intermediate Scales
Turbulence
Shear Dispersion
Motions in Horizontal Plane
Turbulent Dispersion
Solutions
•
Simplest models assume
Fickian
dispersion
–
Fixed dispersion coefficient, fluxes
based on scalar
gradients
•
For
Fickian
model to be valid,
require scale separation
–
Spatially, plume scale must exceed
largest turbulent
lengthscales
–
Temporally
,
motions lead to both
meandering and
dispersion
•
Long Timescales =>
Meandering
•
Short Timescales =>
Dispersion
•
Scaling based on largest scales (dominate dispersion):
–
If plume scale is intermediate to range of turbulent scales, motions
of comparable scale to the plume itself will dominate dispersion
Structure of three

dimensional turbulence
•
Turbulent cascade of energy
–
Large scales set by mean flow
conditions (depth, e.g.)
–
Small scales set by molecular
viscosity
•
Energy conserved across
scales
–
Rate of energy transfer between
scales must be a constant
–
Dissipation Rate:
Large Scales
Intermediate
Small Scales
P
Kolmogorov
Theory
–
3d Turbulence
•
Energy
density,
E(k)
,
scaling for different
scales
–
Large scales:
E(k) = f(Mean flow,
e
, k)
–
Small scales:
E(k) = f(
e
,
n
,k)
–
Intermediate scales:
E(k) = f(
e
,k)
•
Velocity scaling
–
Largest scales:
u
t
= f(
U,
e
,
l
t
)
–
Smallest scales:
u
n
= f(
e
,
n
)
–
Intermediate:
u
*
= f(
e
, k)
•
Dispersion Scaling
k (= 1/
l)
E(k)
L.F. Richardson (~25 years prior to
Kolmogorov
)
•
Two

dimensional “turbulence” governed
by different constraints
–
Enstrophy
(
vorticity
squared) conserved
instead of
energy
–
Rate of
enstrophy
transfer constant across
scales
•
Transfer rate defined as
:
•
‘Cascade’
proceeds from smaller to
larger
scales
Two

dimensional
turbulent flows
Large Scales
Intermediate
Small Scales
Mean Flow
Batchelor

Kraichnan
Spectrum:
2d “Turbulence”
•
Energy density scaling changes from 3

d
–
Intermediate scales independent of mean flow, viscosity:
•
E(k) = f(
f
, k)
•
Velocity scaling
–
Across most scales:
u
*
= f(
f
, k)
•
Dispersion Scaling
k (= 1/
l)
E(k)
Solutions to
turbulent dispersion
problem
•
In each case,
diffusion coefficient approach leads to Gaussian cross

section
•
Differences
between solutions can be described by the lateral extent or variance
(
s
2
):
•
Constant diffusivity solution
•
Three

dimensional scale

dependent solution
•
Two

dimensional scale

dependent solution
t
K
b
x
y
2
)
(
2
2
s
b
Ut
e
b
x
/
2
2
2
)
(
s
y
y
K
t
2
2
s
)
constant
y
K
)
3
/
4
3
/
1
s
e
y
K
)
2
3
/
1
s
f
y
K
3
2
2
3
2
1
)
(
b
Ut
b
x
s
Okubo Dispersion Diagrams
•
Okubo (1971) assembled historical data
to consider lateral diffusion in the ocean
–
Found variance grew as time cubed within
studies
–
Consistent with diffusion coefficient
growing as scale to the 4/3
Shear Dispersion
•
Taylor (1953) analyzed dispersive effects of vertical shear
interacting with vertical mixing
–
Analysis assumed complete mixing over a finite cross

section
•
Unsteadiness in lateral means Taylor limit will not be
reached
–
Effective shear dispersion coefficient evolving as plume grows and
experiences more shear
–
Will be reduced in presence of unsteadiness
l
z
l
y
Developing Shear Dispersion
•
Taylor Dispersion assumes complete mixing over a vertical
dimension, H, with a scale for the velocity shear, U:
•
Non

Taylor limit means H =
l
z
(t):
•
Assume locally linear velocity profile:
–
Velocity difference across patch is:
•
Assembling this into Taylor

like dispersion coefficient:
z
Taylor
K
H
U
K
2
2
)
t
K
t
z
z
2
0
l
l
z
U
z
U
0
)
(
)
t
K
t
U
z
2
0
l
l
2
2
2
2
2
2
2
4
4
t
K
K
t
K
K
U
K
z
z
z
z
z
y
l
3
2
2
2
3
4
2
t
K
K
t
z
y
s
s
Okubo Dispersion Diagrams
•
Okubo (1971) assembled historical data
to consider lateral diffusion in the ocean
–
Found variance grew as time cubed within
studies
–
Consistent with diffusion coefficient
growing as scale to the 4/3
Horizontal Planar Motions
•
Motions in the horizontal plane at scales intermediate to
turbulence and large

scale shear may contribute to
horizontal dispersion
–
Determinant of relative motion, could be dispersive or ‘anti

dispersive’ (i.e., reducing the variance of the distribution in the
horizontal plan)
Framework for Analyzing Relative Motion
•
In a reference frame moving at the velocity of the center
of mass of a cluster of fluid parcels, the motion of
individual parcels is defined by:
–
Where (
x,y
) is the position relative to the center of mass
•
Relative motion best analyzed with
Lagrangian
data
–
For a fixed
Eulerian
array, calculation of the local velocity
gradients provide a snapshot of the relative motions experienced
by fluid parcels within the array domain
y
x
y
v
x
v
y
u
x
u
v
u
Structures of Relative Flow
•
Eigenvalues
of velocity gradient tensor determine relative
motion: nodes, saddle points, spirals, vortices
•
Real
Eigenvalues
mean nodal flows:
Stable Node:
Negative
Eigenvalues
Unstable Node:
Positive
Eigenvalues
Saddle Point:
One Positive, One Negative
Structures of Relative Flow
•
Eigenvalues
of velocity gradient tensor determine
relative motion: nodes, saddle points, spirals, vortices
•
Complex
Eigenvalues
mean vortex flows:
Stable Spiral:
Negative Real Parts
Unstable Spiral:
Positive Real Parts
Vortex:
Real Part = 0
Categorizing Horizontal Flow Structures
•
Eigenvalues
of velocity gradient tensor analyzed by Okubo
(1970) by defining new variables:
•
With these definitions,
eigenvalues
are:
Dynamics
g
Okubo, DSR 1970
•
Categorization of flow
structures can be reduced to
two quantities:
–
g
determines real part
–
determines
real v. complex
–
Relationship between and
g
differentiates nodes and saddle
points
–
Time variability of ,
g
can be
used to understand shifting
fields of relative motion
Implications for Mixing
•
Kinematic straining should be
separated from irreversible
mixing
–
Flow structures themselves may be
connected to irreversible mixing
•
Specific structures
–
Saddle point: Organize particles
into a line, forming a front
•
Anti

dispersive on short timescales,
but may create opportunity for
extensive mixing events through
folding
–
Vortex: Retain particles within a
distinct water volume, restricting
mixing
•
Isolated water volumes may be
transported extensively in
horizontal plane
McCabe et al. 2006
Summary of theoretical background
•
Three candidate mechanisms for lateral mixing, each
characterized by different scales
•
Turbulent dispersion
–
Anisotropy of motions, possibly approaching two

dimensional
“turbulence”
–
Wide range of scales means scale

dependent dispersion
•
Shear dispersion
–
Timescale may imply Taylor limit not reached
–
Unsteadiness in lateral circulation important
•
Horizontal Planar Flows
–
Shear instabilities, Folding, Vortex Translation
–
May inhibit mixing or accentuate it
Case Study
I
: Lateral Dispersion in the BBL
•
Study of plume structure in
coastal
BBL (Duck, NC)
–
Passive, near

bed, steady dye
release
–
Gentle topography
•
Plume dispersion mapped by
AUV
Plume mapping results
•
Centerline concentration and plume width vs. downstream distance
•
Fit with general solution with exponent in scale

dependency (n) as tunable
parameter
•
n=1.5 implies energy density with exponent of

2
n= 1.5
n= 1.5
Compound Dispersion Modeling
•
As plume develops, different dispersion models are
appropriate
–
4/3

law in near

field; scale

squared in far

field
4/3

law
Scale

squared
Compound Analysis
Actual Origin
Virtual Origin
Matching
Condition
Compound Solution,
Plume
Development
•
Plume scale smaller than largest
turbulent scales
–
Richardson model (4/3

law) for
rate of growth
–
Meandering driven by largest 3

d
motions and 2

d motions
•
Plume larger than 3

d turbulence,
smaller than 2

d
–
Dispersion Fickian, based on
largest 3

d motions
–
2

d turbulence defines
meandering
•
Plume scale within range of 2

d
motions
–
2

d turbulence dominates both
meandering and dispersion
–
Rate of growth based on scale

squared formulation
Spydell
and
Feddersen
2009
•
Dye dispersion in the coastal zone
–
Contributions from waves and wave

induced currents
•
Analysis of variance growth
–
Fickian
dispersion would lead to
variance growing linearly in time
–
More rapid variance growth attributed
to scale

dependent dispersion in two
dimensions
•
Initial stages, variance grows as
time

squared
–
Reaches
Fickian
limit after several
hundred seconds
Jones et al. 2008
•
Analysis of centerline
concentration and
lateral scale
–
Dispersion coefficient
increases with scale to
1.23 power
–
Consistent with 4/3 law
of Richardson and
Okubo
–
Coefficient 4

8 times
larger than Fong/Stacey,
likely due to increased
wave influence
Dye, Drifters and Arrays
•
Each of these studies
relied on dye dispersion
–
Limited measurement of
spatial variability of velocity
field
•
Analysis of motions in
horizontal plane require
velocity gradients
–
Drifters:
Lagrangian
approach
–
Dense Instrument arrays
provide
Eulerian
alternative
Summary of Case Study
I
•
Scale dependent dispersion evident in coastal bottom
boundary layer
–
Initially, 4/3

law based on three

dimensional turbulent structure
appropriate
–
As plume grows, dispersion transitions
to
Fickian
or exponential
•
Depends on details of velocity spectra
•
Dye Analysis does not account for kinematics of local velocity
gradients
–
Future opportunity lies in integration of dye, drifters and fixed moorings
•
Key
Unknowns:
–
What is the best description of the spectrum of velocity fluctuations in
the coastal ocean? What are the implications for lateral dispersion
?
–
What role do intermediate

scale velocity gradients play in coastal
dispersion?
–
How should scalar (or particle) dispersion be modeled in the coastal
ocean? Is a
Lagrangian
approach necessary, or can traditional
Eulerian
approaches be modified to account for scale

dependent dispersion?
Recent Studies II: Shoal

Channel Estuary
•
Shoal

channel estuary provides environment to study
effects of lateral shear and lateral circulation
–
Decompose lateral mixing and examine candidate mechanisms
•
Pursue direct analysis of horizontal mixing coefficient
Shoal
Channel
All work presented in this section from
:
Collignon
and Stacey, submitted to JPO,
2011
Study site
•
ADCPs at channel/slope, ADVs on Shoals, CTDs at all
•
Boat

mounted transects along A

B

C line
–
ADCP and CTD profiles
A
B
C
A
B
C
channel
slope
shoal
Decelerating Ebb, Along

channel Velocity
Colorscale
:

1 to 1 m/s
T4
T6
T8
T10
Salinity
T6
T8
T10
T4
Colorscale
23

27
ppt
Cross

channel velocity
T6
T8
T10
T4
Colorscale
:

.2 to .2 m/s
Lateral mixing analysis
•
Interested in defining the net lateral transfer of momentum
between channel and shoal
–
Horizontal mixing coefficients
•
Start from analysis of evolution of lateral shear:
Dynamics of lateral shear
Convergences and
divergences intensify
or relax gradients
Longitudinal
Straining
Variation in
bed stress
Lateral mixing
Each term calculated from March 9 transect data
except lateral mixing term, which is calculated as the
residual of the other terms
Bed
StressTerm
Time
Lateral position
Depth
Term

by

term Decomposition
inferred
Ebb
Floo
d
Time
[day]
channel
slope
shoal
Ebb
Flood
Time [day]
Convergences and lateral structure
•
Convergence evident in late ebb
–
Intensifies shear, will be found to compress mixing
POSITION ACROSS INTERFACE
POSITION ACROSS INTERFACE
ACROSS CHANNEL VELOCITY
ALONG CHANNEL VELOCITY
Term

by

term Decomposition
inferred
Ebb
Floo
d
Time
[day]
channel
slope
shoal
Lateral eddy viscosity: estimate
From Collignon and Stacey (2011), under review,
J. Phys. Oceanogr.
Linear fit
Background:
Contours:
Ebb
Flood
channel
slope
shoal
Inferred mixing coefficient
•
Inferred viscosities around 10

20 m
2
/s
–
Turbulence scaling based on tidal velocity and depth less than
0.1 m
2
/s
–
Observed viscosity must be due to larger

scale mechanisms
Lateral Shear Dispersion Analysis
v [m/s]
s [psu]
Lateral Circulation over slope consists of exchange flows but with large
intratidal
variation
Repeatability
Depth

averaged longitudinal vorticity ω
x
measurements from
the slope moorings show similar variability during other
partially

stratified spring ebb tides
<
ω
x
> [s

1
]
Lateral circulation
ω
x
> 0
ω
x
> 0
ω
x
< 0
2
nd
circulation reversal (late
ebb): driven by lateral density
gradient,
Coriolis
, advection
1
st
circulation reversal (mid ebb):
driven by lateral density
gradient
induced by spatially variable mixing
Implications of lateral circ for dispersion
•
Interaction of unsteady shear and vertical mixing
–
Estimate of vertical diffusivity:
–
Mixing time:
•
Circulation reversals on similar timescales
–
Taylor dispersion estimate:
•
Would be further reduced, however, by reversing, unsteady, shears
1.3 hours
1.5 hours
Horizontal Shear Layers
•
Basak
and
Sarkar
(2006)
simulated horizontal shear
layer with vertical
stratification
Horizontal eddies of vertical
vorticity
create density perturbations and mixing
Lateral Shear Instabilities
•
Consistent source of shear due to
variations in bed friction
–
Inflection point and
Fjortoft
criteria for
instability essentially always met
•
Development of lateral shear
instabilities limited by:
–
Friction at bed
–
Timescale for development
Lateral eddy viscosity: scaling
From
Collignon
and Stacey (2011), under review,
J. Phys.
Oceanogr
.
Mixing length scaling based on large
scale flow properties
Characteristic velocity:
Mixing length: vorticity thickness
Linear fit:
Estimate (o)
Scaling (+)
Effect of convergence front
Flood
Ebb
Implications for Lateral Mixing
Fischer (1979)
Measurements in
unstratified channel flow:
Basak
&
Sarkar
(2006)
DNS
of
stratified
flow with
lateral shear:
Bottom generated turbulence
Shear instabilities
Observations show that lateral mixing at the shoal

channel
interface is dominated by lateral shear instabilities rather than
bottom

generated turbulence.
Summary: Case Study II
•
Lateral mixing in shoal

channel
estuary likely due to combination
of mechanisms
–
Shear dispersion due to exchange
flow at bathymetric slope
–
Lateral shear instabilities
•
Intratidal
variability fundamental to
lateral mixing dynamics
–
Exchange flows vary with
timescales of 10s of minutes
–
Lateral shear instabilities
•
Horizontal scale of 100s of meters,
timescales of 10s of minutes
•
Convergence fronts alter effective
lengthscale
•
Key Unknown: What is relative
contribution of intermediate scale
motions in non

shoal

channel
estuaries
–
Intermediate scales appear to
dominate in shoal

channel system
Summary and Future Opportunities
•
Lateral mixing in coastal ocean appears to be
characterized by scale

dependent dispersion processes
–
Could be result of turbulence or intermediate scale motions
•
Estuarine mixing in horizontal plane due to combination of
lateral shear dispersion and intermediate scale motions
–
Intratidal
variability fundamental to mixing process
–
Creates particular tidal phasing for lateral exchanges
•
Future Opportunities:
–
Clear delineation of anisotropy in stratified coastal flows and
associated velocity spectra/structure
–
Role of bathymetry in establishing lateral mixing processes
–
Parameterization for numerical models
Thanks!
•
Contributors:
Audric
Collignon
,
Rusty
Holleman
, Derek Fong
•
Funding: NSF (OCE

0751970,
OCE

0926738), California
Coastal Conservancy
•
Special Thanks to Akira Okubo
for figuring this all out long
ago…
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