1. Horizontal Mixing in Estuaries and Coastal Seas Mark T. Stacey Warnemuende Turbulence Days September 2011

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. Kolmogorov Theory – 3d Turbulence Energy density, E(k), scaling for different scales – Large scales: E(k) = f(Mean flow, , k) – Small scales: E(k) = f( , ,k) – Intermediate scales: E(k) = f( ,k) Velocity scaling – Largest scales: u t = f(U, , t ) – Smallest scales: u = f( , ) – Intermediate: u * = f( , k) Dispersion Scaling k (= 1/  E(k) L.F. Richardson (~25 years prior to Kolmogorov)

10. 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

11. Batchelor-Kraichnan Spectrum: 2d “Turbulence” Energy density scaling changes from 3-d –Intermediate scales independent of mean flow, viscosity: E(k) = f( , k) Velocity scaling –Across most scales: u * = f( , k) Dispersion Scaling k (= 1/  E(k)

12. 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 (  2 ): Constant diffusivity solution Three-dimensional scale-dependent solution Two-dimensional scale-dependent solution

13. 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

14. 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 z y

15. 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 = z (t): Assume locally linear velocity profile: –Velocity difference across patch is: Assembling this into Taylor-like dispersion coefficient:

16. 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

17. 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)

18. 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

19. 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

20. 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

21. Categorizing Horizontal Flow Structures Eigenvalues of velocity gradient tensor analyzed by Okubo (1970) by defining new variables: With these definitions, eigenvalues are:

22. Dynamics  Okubo, DSR 1970 Categorization of flow structures can be reduced to two quantities: –  determines real part – determines real v. complex –Relationship between and  differentiates nodes and saddle points –Time variability of,  can be used to understand shifting fields of relative motion

23. 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

24. 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

25. 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

26. 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

27. 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

28. 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

29. 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

30. 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

31. 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

32. 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?

33. 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

34. 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

35. Decelerating Ebb, Along-channel Velocity Colorscale: -1 to 1 m/s T4 T6 T8 T10

36. Salinity T6 T8 T10 T4 Colorscale 23-27 ppt

37. Cross-channel velocity T6 T8 T10 T4 Colorscale: -.2 to.2 m/s

38. 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:

39. 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

40. Term-by-term Decomposition inferred Ebb Floo d Time [day] channelslopeshoal

41. Ebb Flood Time [day] Convergences and lateral structure Convergence evident in late ebb –Intensifies shear, will be found to compress mixing POSITION ACROSS INTERFACE ACROSS CHANNEL VELOCITY ALONG CHANNEL VELOCITY

42. Term-by-term Decomposition inferred Ebb Floo d Time [day] channelslopeshoal

43. Lateral eddy viscosity: estimate From Collignon and Stacey (2011), under review, J. Phys. Oceanogr. Linear fit Background:Contours: Ebb Flood channelslopeshoal

44. 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

45. Lateral Shear Dispersion Analysis v [m/s] s [psu] Lateral Circulation over slope consists of exchange flows but with large intratidal variation

46. Repeatability Depth-averaged longitudinal vorticity ω x measurements from the slope moorings show similar variability during other partially-stratified spring ebb tides [s -1 ]

47. Lateral circulation 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

48. 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

49. Horizontal Shear Layers Basak and Sarkar (2006) simulated horizontal shear layer with vertical stratification Horizontal eddies of vertical vorticity create density perturbations and mixing

50. 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

51. 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 FloodEbb

52. Implications for Lateral Mixing Fischer (1979) Measurements in unstratified channel flow: Basak & Sarkar (2006) DNS of stratified flow with lateral shear: Bottom generated turbulenceShear instabilities Observations show that lateral mixing at the shoal-channel interface is dominated by lateral shear instabilities rather than bottom-generated turbulence.

53. 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

54. 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

55. 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…