Report

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…