OCEAN/ESS 410 15. Physics of Sediment Transport William Wilcock (based in part on lectures by Jeff Parsons)

Lecture/Lab Learning Goals Know how sediments are characterized (size and shape) Know the definitions of kinematic and dynamic viscosity, eddy viscosity, and specific gravity Understand Stokes settling and its limitation in real sedimentary systems. Understand the structure of bottom boundary layers and the equations that describe them Be able to interpret observations of current velocity in the bottom boundary layer in terms of whether sediments move and if they move as bottom or suspended loads – LAB

Sediment Characterization φ Diameter, D Type of material -6 64 mm Cobbles -5 32 mm Coarse Gravel -4 16 mm Gravel -3 8 mm -2 4 mm Pea Gravel -1 2 mm Coarse Sand 1 mm 1 0.5 mm Medium Sand 2 0.25 mm Fine Sand 3 125 μm 4 63 μm Coarse Silt 5 32 μm 6 16 μm Medium Silt 7 8 μm Fine Silt 8 4 μm 9 2 μm Clay Sediment Characterization There are number of ways to describe the size of sediment. One of the most popular is the Φ scale. φ = -log2(D) D = diameter in millimeters. To get D from φ D = 2-φ

Sediment Characterization Sediment grain smoothness Sediment grain shape - spherical, elongated, or flattened Sediment sorting % Finer Grain size

Sediment Transport Two important concepts Gravitational forces - sediment settling out of suspension Current-generated bottom shear stresses - sediment transport in suspension (suspended load) or along the bottom (bedload) Shields stress - brings these concepts together empirically to tell us when and how sediment transport occurs

Definitions

1. Dynamic and Kinematic Viscosity The Dynamic Viscosity μ is a measure of how much a fluid resists shear. It has units of kg m-1 s-1 The Kinematic viscosity ν is defined where ρf is the density of the fluid. ν has units of m2 s-1, the units of a diffusion coefficient. It measures how quickly velocity perturbations diffuse through the fluid.

2. Molecular and Eddy Viscosities Molecular kinematic viscosity: property of FLUID The molecular kinematic viscosity (usually referred to just as the ‘kinematic viscosity’), ν is an intrinsic property of the fluid and is the appropriate property when the flow is laminar. It quantifies the diffusion of velocity through the collision of molecules. (It is what makes molasses viscous). The Eddy Kinematic Viscosity, νe is a property of the flow and is the appropriate viscosity when the flow is turbulent flow. It quantities the diffusion of velocity by the mixing of “packets” of fluid that occurs perpendicular to the mean flow when the flow is turbulent Eddy kinematic viscosity: property of FLOW In flows in nature (ocean), eddy viscosity is MUCH MORE IMPORTANT! Like, 104 times more important

3. Submerged Specific Gravity, R Typical values: Quartz = Kaolinite = 1.6 Magnetite = 4.1 Coal, Flocs < 1 f

Sediment Settling

Settling Velocity: Stokes settling Settling velocity (ws) from the balance of two forces - gravitational (Fg) and drag forces (Fd)

Settling Speed Balance of Forces Write balance using relationships on last slide k is a constant Use definitions of specific gravity, R and kinematic viscosity ν k turns out to be 1/18

Limits of Stokes Settling Equation Assumes smooth, small, spherical particles - rough particles settle more slowly Grain-grain interference - dense concentrations settle more slowly Flocculation - joining of small particles (especially clays) as a result of chemical and/or biological processes - bigger diameter increases settling rate Assumes laminar settling (ignores turbulence) Settling velocity for larger particles determined empirically

Boundary Layers

Bottom Boundary Layers The layer (of thickness δ) in which velocities change from zero at the boundary to a velocity that is unaffected by the boundary δ is likely the water depth for river flow. δ is a few tens of meters for currents at the seafloor Inner region is dominated by wall roughness and viscosity Intermediate layer is both far from outer edge and wall (log layer) Outer region is affected by the outer flow (or free surface)

Shear stress in a fluid Shear stresses at the seabed lead to sediment transport force rate of change of momentum τ = shear stress = = area area

The inner region (viscous sublayer) Only ~ 1-5 mm thick In this layer the flow is laminar so the molecular kinematic viscosity must be used Unfortunately the inner layer it is too thin for practical field measurements to determine τ directly

The log (turbulent intermediate) layer Generally from about 1-5 mm to 0.1δ (a few meters) above bed Dominated by turbulent eddies Can be represented by: where νe is “turbulent eddy viscosity” This layer is thick enough to make measurements and fortunately the balance of forces requires that the shear stresses are the same in this layer as in the inner region

Shear velocity u* Sediment dynamicists define a quantity known as the characteristic shear velocity, u* The simplest model for the eddy viscosity is Prandtl’s model which states that Turbulent motions (and therefore νe) are constrained to be proportional to the distance to the bed z, with the constant, κ, the von Karman constant which has a value of 0.4

Velocity distribution of natural (rough) boundary layers From the equations on the previous slide we get Integrating this yields z0 is a constant of integration. It is sometimes called the roughness length because it is generally proportional to the particles that generate roughness of the bed (usually z0 = 30D)

What the log-layer actually looks like Slope = κ/u* = 0.4/u* lnz0 Plot ln(z) against the mean velocity u to estimate u* and then estimate the shear stress from Z0

When will transport occur and by what mechanism? Shields Stress When will transport occur and by what mechanism?

Hjulström Diagram

Shields stress and the critical shear stress The Shields stress, or Shields parameter, is: Shields (1936) first proposed an empirical relationship to find θc, the critical Shields shear stress to induce motion, as a function of the particle Reynolds number, Rep = u*D/ν

Shields curve (after Miller et al Shields curve (after Miller et al., 1977) - Based on empirical observations Transitional Sediment Transport Transitional No Transport

Initiation of Suspension If u* > ws, (i.e., shear velocity > settling velocity) then material will be suspended. Suspension Transitional transport mechanism. Compare u* and ws Bedload No Transport