horizontal mixing in estuaries and coastal seas
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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 - PowerPoint PPT PresentationTRANSCRIPT
Horizontal Mixing in Estuaries and Coastal Seas
Mark T. StaceyWarnemuende 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)
UnalignedGrid
AlignedGrid
Numerical Diffusion [m2 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 VariationsIntermediate Scales
Turbulence Shear DispersionMotions 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: ut = f(U,e ,lt)– Smallest scales: un = 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 ScalesMean 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 (s2):
• Constant diffusivity solution
• Three-dimensional scale-dependent solution
• Two-dimensional scale-dependent solution
tKbx y2)( 22 s
bUtebx /222 )( s
yy Kt
22
s
)constantyK
)3/43/1 seyK
)23/1 sfyK
322
321)(
bUtbx 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
lz
ly
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 = lz(t):
• Assume locally linear velocity profile:
– Velocity difference across patch is:
• Assembling this into Taylor-like dispersion coefficient:
zTaylor K
HUK22
) tKt zz 20 ll
zUzU 0)(
) tKtU z20 ll
2222222
44 tKKtK
KUK z
z
z
z
zy l
3222
342 tKK
t zy ss
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
yx
yvxvyuxu
vu
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-squaredCompound Analysis
Actual Origin
Virtual Origin
MatchingCondition
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
AB
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
Tim
eLateral position
Depth
Term-by-term Decomposition
inferred
Ebb
Flood
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
Flood
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 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
ωx > 0 ωx > 0ωx < 0
2nd circulation reversal (late ebb): driven by lateral density gradient, Coriolis, advection
1st 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…