Horizontal Mixing in Estuaries and Coastal Seas

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 [m2 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 Basin-scale Circulation Intermediate Scales Seconds to minutes Tidal and Diurnal Variations 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. P Large Scales Intermediate Small Scales 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:

9. 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 E(k) k (= 1/l) L.F. Richardson (~25 years prior to Kolmogorov)

10. Large Scales Intermediate Mean Flow Small Scales 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: • ‘Cascade’ proceeds from smaller to larger scales

11. Batchelor-KraichnanSpectrum: 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 E(k) k (= 1/l)

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 (s2): • 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 lz ly

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 = lz(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 • 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

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 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 Actual Origin 4/3-law Virtual Origin Matching Condition Compound Analysis Scale-squared

28. 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 Compound Solution, Plume Development

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 C C shoal B B A A slope channel

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

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

37. Cross-channel velocity T6 T4 T8 T10 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 Lateral mixing Variation in bed stress Longitudinal Straining Convergences and divergences intensify or relax gradients Bed StressTerm Time Each term calculated from March 9 transect data except lateral mixing term, which is calculated as the residual of the other terms Depth Lateral position

40. Term-by-term Decomposition Time [day] Ebb inferred Flood channel slope shoal

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

42. Term-by-term Decomposition Time [day] Ebb inferred Flood channel slope shoal

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

44. 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

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 ωx measurements from the slope moorings show similar variability during other partially-stratified spring ebb tides < ωx> [s-1]

47. Lateral circulation 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 ωx> 0 ωx< 0 ωx> 0

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 1.5 hours 1.3 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