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

bUtebx /222 )(

yy Kt

22

constantyK

3/43/1 yK

23/1 yK

322

3

21)(

bUtbx

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

HUK

22

tKt zz 20

zUzU 0)(

tKtU z20

2222222

44

tKK

tK

K

UK z

z

z

z

zy

3222

3

42 tKK

t zy

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

yvxv

yuxu

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-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…