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
Holleman et al., Submitted to IJNMF
Aligned
Grid
Numerical Diffusion [m2 s-1]
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
Basin-scale Circulation
Seconds to minutes
Intermediate Scales
Turbulence
Motions in Horizontal Plane
Tidal and Diurnal Variations
Shear Dispersion
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
P
Large Scales
• Energy conserved across
scales
– Rate of energy transfer between
scales must be a constant
– Dissipation Rate:
Intermediate
Small Scales
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
E(k)
– Largest scales: ut = f(U,e ,lt)
– Smallest scales: un = f(e ,n)
– Intermediate: u* = f(e , k)
k (= 1/l)
• Dispersion Scaling
L.F. Richardson (~25 years prior to Kolmogorov)
Two-dimensional turbulent flows
• 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:
Large Scales
• ‘Cascade’ proceeds from smaller to
larger scales
Intermediate
Mean Flow
Small Scales
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
E(k)
– Across most scales: u* = f(f , k)
k (= 1/l)
• Dispersion Scaling
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
(s2):
2
s
t
•
y
 2K y
Constant diffusivity solution K y
 constant
)
s (x)  b  2 K y t
2
•
2
Three-dimensional scale-dependent solution
K
y
e
2

2
2
s ( x )  b 1   Ut b 
3


•
Two-dimensional scale-dependent solution
s ( x)  b e
2
2
K
2  Ut / b
y
f
1/ 3
s
s
2
3
1/ 3
)
4/3
)
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
lz
ly
Developing Shear Dispersion
• Taylor Dispersion assumes complete mixing over a vertical
dimension, H, with a scale for the velocity shear, U:
2
K Taylor 
• Non-Taylor limit means H = lz(t):
U H
Kz
l z t )  l 0 
• Assume locally linear velocity profile:
– Velocity difference across patch is:
2
2 K zt
U (z)  U 0   z
 U   l t )   l 0  
2 K zt
• Assembling this into Taylor-like dispersion coefficient:
U lz
2
Ky 
Kz
2
4 K z t
2

Kz
2 2
 4 K z t
2
2
s
t
2
 2K y  s
2

4
3
 K zt
2
3
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 ‘antidispersive’ (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:
 u   u  x
   
 v   v  x
 u  y  x 
  
 v  y  y 
– 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
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
• 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
g
Okubo, DSR 1970
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
•
n=1.5 implies energy density with exponent of -2
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
Actual Origin
4/3-law
Virtual Origin
Matching
Condition
Scale-squared
Compound Analysis
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 scalesquared formulation
Spydell and Feddersen 2009
• Dye dispersion in the coastal zone
– Contributions from waves and waveinduced 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
C
C
B
B
A
shoal
slope
A
channel
• ADCPs at channel/slope, ADVs on Shoals, CTDs at all
• Boat-mounted transects along A-B-C line
– ADCP and CTD profiles
Decelerating Ebb, Along-channel Velocity
T4
T6
T8
T10
Colorscale: -1 to 1 m/s
Salinity
T4
T6
T8
T10
Colorscale 23-27 ppt
Cross-channel velocity
T4
T6
T8
T10
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
Lateral mixing
Variation in
bed stress
Longitudinal
Straining
Convergences and
divergences intensify
or relax gradients
Time
Each term calculated from March 9 transect data
except lateral mixing term, which is calculated as the
residual of the other terms
Bed StressTerm
Depth
Lateral position
Term-by-term Decomposition
Time
[day]
Ebb
Floo
d
channel
slope
shoal
inferred
Convergences and lateral structure
ACROSS CHANNEL VELOCITY
Time [day]
Ebb
ALONG CHANNEL VELOCITY
Flood
POSITION ACROSS INTERFACE
• Convergence evident in late ebb
POSITION ACROSS INTERFACE
– Intensifies shear, will be found to compress mixing
Term-by-term Decomposition
Time
[day]
Ebb
Floo
d
channel
slope
shoal
inferred
Lateral eddy viscosity: estimate
Background:
Contours:
Ebb
Linear fit
channel
slope
shoal
Flood
From Collignon and Stacey (2011), under review, J. Phys. Oceanogr.
Inferred mixing coefficient
• Inferred viscosities around 10-20 m2/s
– Turbulence scaling based on tidal velocity and depth less than
0.1 m2/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 ωx measurements from
the slope moorings show similar variability during other
partially-stratified spring ebb tides
< ωx > [s-1]
Lateral circulation
1st circulation reversal (mid ebb):
driven by lateral density gradient
induced by spatially variable mixing
ωx > 0
2nd circulation reversal (late
ebb): driven by lateral density
gradient, Coriolis, advection
ωx < 0
ωx > 0
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
Flood
Ebb
Mixing length scaling based on large
scale flow properties
Characteristic velocity:
Mixing length: vorticity thickness
Effect of convergence front
Linear fit:
Estimate (o)
Scaling (+)
From Collignon and Stacey (2011), under review, J. Phys. Oceanogr.
Implications for Lateral Mixing
Observations show that lateral mixing at the shoal-channel
interface is dominated by lateral shear instabilities rather than
bottom-generated turbulence.
Bottom generated turbulence
Fischer (1979)
Measurements in
unstratified channel flow:
Shear instabilities
Basak & Sarkar (2006)
DNS of stratified flow with
lateral shear:
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…