1. Hydrosedimentary processes in the nearshore domain Elements for the physical approach Physical and Mathematical Tools for the Study of Marine Processes of Coastal Areas 26 May – 6 June 2008, Cienfuegos, CUBA Jean-Pierre Lefebvre, IRD (FRANCE)

2. FORCING Permanent stress (current) Oscillatory stress (wave) SEABEDS Non cohesive sediment (sand) Cohesive sediment (mud) Erosion, suspension, fluidization Turbulence, energy dissipation, shoaling

3. FORCINGS I. Permanent flow II. Oscillatory flow SEDIMENTS III. Cohesive sediments

4. Permanent flow (laminar) boundary layer : for a viscous flow, layer defined from the bed (non slip condition) up to the height where the flow is no longer perturbed by the wall. µ : (absolute) dynamic viscosity (Pa.s) (1.08 10 −3 Pa.s for seawater at T = 20°C and S = 35 g.kg -1 ) Newton’s law of viscosity : kinematic fluid viscosity (m².s -1 )  w : density of water (kg.m -3 ) (≈ 1.025 for seawater for T =20°C, S = 35 g.kg -1 ) z u(z)

5. u’(t,z) t 0 T Permanent flow (turbulent) turbulent flow : fluid regime characterized by chaotic property changes. This includes high frequenty variation of velocity in space and time. da Vinci sketch of a turbulent flow time averaged (steady) component Reynolds decomposition of a parameter u(t,z) t 0 T Instantaneous local velocity steady component _ fluctuating component (perturbation)

6. Permanent flow (turbulent) Reynolds stress tensor (covariance of vertical and horizontal velocities) Turbulent shear stress e : kinematic eddy viscosity (m².s -1 )

7. Permanent flow (turbulent) Turbulent outer region  influenced by the outer boundary condition of the layer,  consists of about 80-90 % of the total region,  velocity relatively constant due to the strong mixing of the flow. Intermediate region (log layer)  logarithmic profile of the horizontal velocity Innermost region (viscous sub layer)  dominated by viscosity,  linear velocity profile,  very small. Turbulent boundary layer h ∿ 0.1h δvδv OUTER REGION LOG LAYER VISCOUS SUB-LAYER

8. Permanent flow (turbulent) The characteristic velocity scale u* is a parameter of the order of magnitude of the turbulent velocity often called friction velocity since it is used as the actual turbulent velocity action on the bed Friction velocity z u ∿ 0.1h δvδv LOG LAYER VISCOUS SUB LAYER

9. Permanent flow (turbulent) Prandtl’s model of mixing-length in the turbulent boundary layer, states that the turbulence is linearly related to the averaged velocity gradient by a term L m called, mixing length von Kármán constant ( κ = 0.408 ) von Kármán assumption states that the correlation scale is proportional to the distance from the boundary the kinematic eddy viscosity must also be proportional to the height above the bed. z u ∿ 0.1h δvδv LOG LAYER VISCOUS SUB LAYER -

10. z u ∿ 0.1h δvδv LOG LAYER VISCOUS SUB LAYER Permanent flow (turbulent) Prandtl-Kármán law of wall z 0 : hydraulic roughness of the flow depends on viscous sub-layer, grain roughness, ripples and other bedforms, stratification,…

11. Permanent flow (turbulent) Nikuradse sand roughness (physical roughness) can be approximated by the median diameter of grains of sandy bed d 50 : mean particles diameter z u ∿ 0.1h δvδv LOG LAYER VISCOUS SUB LAYER

12. Permanent flow (turbulent) the viscous sub-layer is a narrow layer close to the wall where roughness of the wall and molecular viscosity dominate transport of momentum Thickness of the viscous sub-layer Ratio of inertial force to viscous force z u ∿ 0.1h δvδv LOG LAYER VISCOUS SUB LAYER

13. Permanent flow (turbulent) The relative roughness (ratio of hydraulic roughness z 0 on the physical roughness k s ) depends on the relative length scales for the viscous sub-layer and the physical roughness roughness Reynolds or grain Reynolds number

14. Permanent flow (turbulent) Hydraulically rough regime : R e* > 70  the viscous sub-layer is interrupted by the bed roughness,  roughness elements interact directly with the turbulence. Rough regime

15. Permanent flow (turbulent) Hydraulically smooth regime : R e* < 5 the viscous sub-layer lubricates the roughness elements so they do not interact with turbulence. Rough regime Smooth regime

16. Permanent flow (turbulent) hydraulically transitional regime : 5 ≤ R e* ≤ 70 For 0.26 < k s /  v < 8.62 the near-wall flow is transitional between the hydraulically smooth and hydraulically rough regimes Rough regime Smooth regime Transitional regime +

17. Permanent flow (turbulent) Bottom shear stress friction factor Friction factor for current (rough turbulent regime) turbulent outer layer log layer transition layer viscous sub-layer

18. Permanent flow (turbulent) FORCINGSEABEDS Velocities at some elevations near the bed Sediment granular distribution MeasurementsQuantification Friction velocity and hydraulic roughness Physical roughness Description Turbulent shear stress at the bed Hydraulic turbulent regime Prandtl-Kármán law of wall Nikuradse approximation

19. Oscillatory flow Waves can be defined by their superficial properties  wave height (distance between its trough and crest)  wave length (distance between two crests)  wave period (duration for the propagation of two successive extrema at a given location) wave period (s) angular velocity (rad.s -1 ) wavelength (m) wave number (rad.m -1 ) wave amplitude (m) wave height (m)

20. Oscillatory flow Airy wave : model for monochromatic progressive sinusoidal waves Wave with multispectral components

21. Oscillatory flow velocity potential ∅ Assuming an oscillatory flow V of an inviscid, incompressible fluid, with no other motions interfering (i.e. no currents)  irrotational flow (i.e. no curl between the water particles trajectories) :  V = 0  satisfying the continuity equation : . V = 0 For a sinusoidal wave field, it exists an ideal potential flow solution:  ∅ = V from which we can derivate the expressions of all the pressure and flow fields.

22. Oscillatory flow BOUNDARY CONDITIONS Dynamic boundary condition : the pressure along an iso-potential line is constant (Bernoulli ) Kinematic boundary condition : a parcel of fluid at the surface remains at the surface Boundary condition : the bottom is not permeable to water Equation of Laplace for the inviscid, uncompressible flow

23. Oscillatory flow For small amplitude gravity wave (wave amplitude a << wavelength λ)

24. Oscillatory flow SIMPLIFIED BOUNDARY CONDITIONS Simplified dynamic boundary condition Simplified Kinematic boundary condition Simplified boundary condition Linearization (only the first order terms of the Taylor series) Laplacian equation

25. Oscillatory flow General form of ∅ for a sinusoidal wave

26. Oscillatory flow VELOCITY FIELD from  ∅ = V SURFACE ELEVATION From = -g η at z = 0 ∂∅ ∂t ___ PRESSURE From p = - ρ w ∂∅ ∂t ___

27. Oscillatory flow DISPERSION EQUATION The relation between the angular velocity ω and the wave number (from the simplified Laplace equation) WAVE CELERITY velocity of the wave crest ( m.s -1 )

28. Oscillatory flow DISPERSION EQUATION WAVE CELERITY DEEP WATER DOMAIN The water height is much greater than the wavelength (h >> λ)

29. Oscillatory flow DISPERSION EQUATION WAVE CELERITY SHALLOW WATER DOMAIN The wavelength is much greater than the water height (λ >>h)

30. Oscillatory flow INTERMEDIATE DOMAIN

31. Oscillatory flow DEEP WATER INTERMEDIATE ZONE SHALLOW WATER Limit of lower orbital motions Slight erosion of the seabed No erosion of the seabed shoaling Wave breaking Strong erosion SWL ∿ λ __ 2 h ∿ λ 20 h

32. Oscillatory flow Orbital velocity at the bed Stokes’ drift

33. Oscillatory flow (turbulent) Wave boundary layer thickness Turbulent wave shear stress maximum shear velocity

34. Oscillatory flow (turbulent) Law of wall (Grant and Madsen) Phase lead

35. Oscillatory flow (turbulent) Shear stress generated by the oscillatory flow where

36. Oscillatory flow (turbulent) Maximum shear stress friction factor for wave Friction factor for wave (rough turbulent regime)

37. Nikuradse approximation FORCING SEABEDS Oscillatory flow (turbulent) Sediment granular distribution Maximum shear velocity and hydraulic roughness Physical roughness Measurements Maximum shear stress at the bed Hydraulic turbulent regime Surface wave parameters and wave height Grant-Madsen Law of wall DescriptionQuantification

38. Potential Energy Oscillatory flow Kinetic Energy Energy density (J.m -2 )

39. Oscillatory flow Flux of energy (J) Group Velocity (m.s -1 ) In deep water domain (kh→∞) and Cg = C/2 In shallow water domain (kh→0) and Cg = C

40. Oscillatory flow Shoaling Wave height (m) Water depth (m) Wave period : 8 s h dw = 49.6 mh sw = 0.8 2.8 2.3 ∿ h __ 0.8 H

41. many empirical expressions exist for coupling permanent and oscillatory stresses Combined current and wave stresses (Soulsby, 1995)

42. seabed Bed-load transport The rolling, sliding and jumping grains in almost continuous contact with the bed. Intergranular collision forces play an important role Suspended-load transport Grains are almost continuously suspended in the water column The turbulence mixing processes are dominant Sheet flow a layer with a thickness of several grain layers (10 – 100) and large sediment concentrations is transported along the bed. Transport mode for marine sediments

43. 2µ 4 8 16 32 64 125 250 500 1mm 2 4 8 16 32 64 seabed Sediment cohesion : domination of interparticle forces or the gravitational force in the behavior of sediment. CLAYSILTSAND GRAVEL COBBLES very fine very fine medium coarse very coarse very coarse pea gravel cobbles clay Cohesive sediments : material with strong interparticle forces due to their surface ionic charges Non cohesive sediments : granular material dominated by the gravitational force

44. seabed

45. Erosion In situ sampling of unperturbed seabedextraction of the unperturbed interface Measurements of the erosion (erodimeter, IFREMER) fully controled flow (flow, chenal dimensions, fixed bottom roughness) Erosion and transport ( bedload and suspension) Non cohesive sediment trapping (gravitation) Suspended fine sediments measured with OBS Grain size spectrum of defloculated material critical shear stress

46. Flocculation Mud flocs are characterized by four main physical properties:  size (diameter) D f  density ρ floc  settling velocity W s  floc strength F c  turbulent motions will cause particles, carried by the eddies to collide and form flocs Mud floc properties are governed by four mechanisms:  Brownian motions cause the particles to collide to form aggregates  particles with a large settling velocity will overtake particles with a low settling velocity and aggregate  turbulent shear may disrupt the flocs again, causing floc breakup

47. Flocculation clay < 4µm fine silt (4 ∿ 10µm) flocculus microfloc Microfloc (< 100µm) Macrofloc ( ∿ O(2) µm up to ∿ O(1) mm) strong interparticle forces due to surface ionic charges strong bound by sticky material produced by biological organisms loosely bound and very fragile Self similarity

48. Flocculation Floc size The fractal dimension n f is obtained from the description of a growing object with linear size αL and volume V(αL) α ( linear size of the primary object (seed) (arbitrary = 1) number of seeds in estuarine and coastal environments 1.7 < n floc < 2.2

49. Flocculation Floc excess density floc diameter Sediment density ρ s for clay ∿ 1390 kg.m -3 defloculated particles diameter

50. Flocculation Floc limitation by turbulence the cut-off floc diameter is determined by the local balance of floc growth and rupture within a turbulent fluid regime. Rate of turbulent shear volume average value of ε (J) the energy dissipation rate per unit mass ε expresses the process of energy transfer

51. Flocculation Taylor microscale The Taylor microscale λ is representative of the energy transfer from large to small scales. For large Reynolds numbers, the structure of turbulence tends to be approximately isotropic Normalized Taylor microscale

52. Flocculation Kolmogorov microscale At very small length scales, viscosity becomes effective in smoothing out velocity fluctuations preventing the generation of infinitely small scales by dissipating small-scale energy into heat. The smallest scale of motion automatically adjusts itself to the value of the viscosity. The Kolmogorov length defines the smallest length scale of turbulent motion and is location dependent thru λ(z)

53. Flocculation Kolmogorov microscale Turbulent mixing induces aggregation and, at the same time, subjects aggregates to higher shear stresses causing breakup for flocs of diameter greater than d max

54. Settling velocity of a spherical object settling through a fluid when the flow around the object is laminar Stokes settling velocity

55. Settling  gravity  flocculation  hindered settling The expression of the settling velocity for flocs must combine three effects:  turbulence, shear or bottom shear stress  salinities  floc strength  fractal structure  concentration  sediment composition  time spent in an equilibrium state (residence time of flocs ) The settling velocity of estuarine mud flocs is largely affected by some physical parameters:

56. Settling Hindered settling velocity At high concentrations, the return flow of water around a particle may create an upward drag on neighboring particles. Volume concentration depends on grain Reynolds number

57. CURRENT SEABED Turbulent boundary layer WAVE Water height Airy model Bottom shear stress Bed roughness Turbulence within the boundary layer

58. COHESIVE SEDIMENT Bottom shear stress Turbulence within the boundary layer Turbulent boundary layer Erosion Flocculation Settlings