Outline of Presentation : Richardson number control of saturated suspension

Interactions of Turbulence and Suspended Sediment Carl Friedrichs Virginia Institute of Marine Science • Outline of Presentation: • Richardson number control of saturated suspension • Under-saturated (weakly stratified) sediment suspensions • Critically saturated (Ricr-controlled) sediment suspensions • Hindered settling, over-saturation, and collapse of turbulence Presented at Warnemünde Turbulence Days Vilm, Germany, 5 September 2011

Stratification Shear density stratification Shear instabilities occur for Ri < Ricr “ “ suppressed for Ri > Ricr Gradient Richardson Number (Ri) = velocity shear When strong currents are present, mud remains turbulent and in suspension at a concentration that gives Ri ≈ Ricr ≈ 1/4: = c > 0.3 g/l Ri c < 0.3 g/l - 0.25 Sediment gradient Richardson number g = accel. of gravity s = (s - )/ c = sediment mass conc. s = sediment density For c > ~ 300 mg/liter Sediment concentration (grams/liter) Ri ≈ Ricr ≈ O(1/4) Amazon Shelf (Trowbridge & Kineke, 1994)

Are there simple, physically-based relations to predict c and du/dz related to Ri? Large supply of easily suspended sediment creates negative feedback: density stratification Shear instabilities occur for Ri < Ricr ““ suppressed for Ri > Ricr Gradient Richardson Number (Ri) = velocity shear (b) (a) Height above bed Height above bed Ri > Ric Ri = Ric Ri = Ric Ri < Ric Sediment concentration Sediment concentration (a) If excess sediment enters bottom boundary layer or bottom stress decreases, Ri beyond Ric, critically damping turbulence. Sediment settles out of boundary layer. Stratification is reduced and Ri returns to Ric. (b) If excess sediment settles out of boundary layer or bottom stress increases, Ri below Ric and turbulence intensifies. Sediment re-enters base of boundary layer. Stratification is increased in lower boundary layer and Ri returns to Ric.

Consider Three Basic Types of Suspensions Height above bed Ri > Ricr 3) Over-saturated -- Settling limited Ri< Ricr Ri = Ricr 1) Under-saturated -- Supply limited 2) Critically saturated load Sediment concentration

Dimensionless analysis of bottom boundary layer in the absence of stratification: Variables du/dz, z, h, n, u* h = thickness of boundary layer or water depth, n = kinematic viscosity, u* = (tb/r)1/2 = shear velocity Outer Layer n/(zu*) << 1 = 1/k Const. Overlap Layer: z/h << 1 & n/(zu*) << 1 i.e., (a.k.a. “log-layer”) z/h << 1 (Dyer, 1986)

Bottom boundary layer often plotted on log(z) axis: Boundary layer - current log layer z/h << 1 “Overlap” layer n/(zu*) << 1 zo = hydraulic roughness (Wright, 1995)

Dimensionless analysis of overlap layer with (sediment-induced) stratification: Additional variable b = Turbulent buoyancy flux s = (rs – r)/r ≈ 1.6 c = sediment mass conc. w = vertical fluid vel. u(z) Height above the bed, z Dimensionless ratio = “stability parameter”

Deriving impact of z on structure of overlap (a.k.a. “log” or “wall”) layer Rewrite f(z) as Taylor expansion around z = 0: ≈ 0 ≈ 0 = a = 1 If there is stratification (z > 0) then u(z) increases faster with z than homogeneous case. From atmospheric studies, a ≈ 4 - 5

Eq. (1) (i) well-mixed -- Case (i): No stratification near the bed (z = 0 at z = z0).Stratification and z increase with increased z. -- Eq. (1) gives u increasing faster and faster with z relative to classic well-mixed log-layer. (e.g., halocline being mixed away from below) -- Case (ii): Stratified near the bed (z> 0 at z = z0). Stratification and zdecreases with increased z. -- Eq. (1) gives u initially increasing faster than u, but then matching du/dz from neutral log-layer. (e.g., fluid mud being entrained by wind-driven flow) stratified  as z z0 (ii) well-mixed stratified Log elevation of height above bed  as z z0 -- Case (iii): uniform z with z. Eq (1) integrates to (iii) stratified well-mixed -- u remains logarithmic, but shear is increased buy a factor of (1+az)  is constant in z z0 (Friedrichs et al, 2000) Current Speed

Effect of stratification (via z) on eddy viscosity (Az) Overlap layer scaling modified by buoyancy flux Definition of eddy viscosity Eliminate du/dz and get -- As stratification increases (larger z), Az decreases -- If z = const. in z, Az increases like u*z , and the result is still a log-profile. Connect stability parameter, z, to shape of concentration profile, c(z): Rouse balance (Reynolds flux = settling): Definition of z: z = const. in z if Combine to eliminate <c’w’> : (Assuming ws is const. in z)

z = const. in z if (i) Fit a general power-law to c(z) of the form well-mixed stratified Then A < 1  as z If A < 1, c decreases more slowly than z-1 z increases with z, stability increases upward, u is more concave-down than log(z) If A > 1, c increases more quickly than z-1 z decreases with z, stability becomes less pronounced upward, u is more concave-up than log(z) If A = 1, c ~ z-1 z is constant with elevation stability is uniform in z, u follows log(z) profile z0 (ii) well-mixed stratified A > 1 Log elevation of height above bed  as z z0 (iii) stratified well-mixed A = 1  is constant in z If suspended sediment concentration, C ~ z-A Then A <,>,= 1 determines shape of u profile z0 (Friedrichs et al, 2000) Current Speed

If suspended sediment concentration, C ~ z-A A < 1 predicts u more concave-down than log(z) A > 1 predicts u more concave-up than log(z) A = 1 predicts u will follow log(z) Eckernförde Bay, Baltic Coast, Germany Testing this relationship using observations from bottom boundary layers: STATAFORM mid-shelf site, Northern California, USA (Friedrichs & Wright, 1997; Friedrichs et al, 2000) Inner shelf, Louisiana USA

If suspended sediment concentration, C ~ z-A A < 1 predicts u more convex-up than log(z) A > 1 predicts u more concave-up than log(z) A = 1 predicts u will follow log(z) STATAFORM mid-shelf site, Northern California, USA, 1995, 1996 Inner shelf, Louisiana, USA, 1993 A ≈ 1.0 A ≈ 3.1 A ≈ 0.73 A ≈ 0.35 A ≈ 0.11 -- Smallest values of A < 1 are associated with concave-downward velocities on log-plot. -- Largest value of A > 1 is associated with concave-upward velocities on log-plot. -- Intermediate values of A ≈ 1 are associated with straightest velocities on log-plot.

Eckernförde Bay, Baltic Coast, Germany, spring 1993 (Friedrichs & Wright, 1997) -- Salinity stratification that increases upwards cannot be directly represented by c ~ z-A. Friedrichs et al. (2000) argued that this case is dynamically analogous to A ≈ -1.

Observations showing effect of concentration exponent A on shape of velocity profile Normalized log of sensor height above bed Normalized burst-averaged current speed Observations also show: A < 1, concave-down velocity A > 1, concave-up velocity A ~ 1, straight velocity profile (Friedrichs et al, 2000)

Relate stability parameter, z, to Richardson number: Definition of gradient Richardson number associated with suspended sediment: Original definition and application of z: Relation found for eddy viscosity: Definition of eddy diffusivity: Assume momentum and mass are mixed similarly: Combine all these and you get: So a constant z with height also leads to a constant Ri with height. Also, if z increases (or decreases) with height Ri correspondingly increases (or decreases).

(i) z and Ri const. in z if well-mixed stratified Define then A < 1 and Ri as z If A < 1, c decreases more slowly than z-1 z and Ri increase with z, stability increases upward, u is more concave-down than log(z) If A > 1, c decreases more quickly than z-1 z and Ri decrease with z, stability becomes less pronounced upward, u is more concave-up than log(z) If A = 1, c ~ z-1 z and Ri are constant with elevation stability is uniform in z, u follows log(z) profile z0 (ii) well-mixed stratified A > 1 and Ri as z Log elevation of height above bed z0 (iii) stratified well-mixed A = 1 If suspended sediment concentration, C ~ z-A then A <,>,= 1 determines shape of u profile and also the vertical trend in z and Ri  and Riare constant in z z0 (Friedrichs et al, 2000) Current Speed

Now focus on the case where Ri = Ricr (so Ri is constant in z over “log” layer) Height above bed Ri > Ricr 3) Over-saturated -- Settling limited Ri< Ricr Ri = Ricr 1) Under-saturated -- Supply limited 2) Critically saturated load Sediment concentration

Connection between structure of sediment settling velocity to structure of “log-layer” when Ri = Ricr in z (and therefore z is constant in z too). Rouse Balance: Earlier relation for eddy viscosity: Eliminate Kz and integrate in z to get But we already know when Ri = const. when Ri = Ricr and So

when Ri = Ricr . This also means that when Ri = Ricr :

STATAFORM mid-shelf site, Northern California, USA Mid-shelf site off Waiapu River, New Zealand (Wright, Friedrichs et al., 1999; Maa, Friedrichs, et al., 2010)

1 10 Ricr= 1/4 0 10 -1 10 10 - 40 cm 40 - 70 cm -2 10 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 (a) Eel shelf, 60 m depth, winter 1995-96 (Wright, Friedrichs, et al. 1999) Sediment gradient Richardson number (b) Waiapu shelf, NZ, 40 m depth, winter 2004 (Ma, Friedrichs, et al. in 2008) Ricr= 1/4 18 - 40 cm Velocity shear du/dz (1/sec)

Application of Ricr log-layer equations fo Eel shelf, 60 m depth, winter 1995-96 (Souza & Friedrichs, 2005)

Now also consider over-saturated cases: Height above bed Ri > Ricr 3) Over-saturated -- Settling limited Ri< Ricr Ri = Ricr 1) Under-saturated -- Supply limited 2) Critically saturated load Sediment concentration

More Settling (Mehta & McAnally, 2008) Starting at around 5 - 8 grams/liter, the return flow of water around settling flocs creates so much drag on neighboring flocs that ws starts to decrease with additional increases in concentration. At ~ 10 g/l, ws decreases so much with increased C that the rate of settling flux decreases with further increases in C. This is “hindered settling” and can cause a strong lutecline to form. Hindered settling below a lutecline defines “fluid mud”. Fluid mud has concentrations from about 10 g/l to 250 g/l. The upper limit on fluid mud depends on shear. It is when “gelling” occurs such that the mud can support a vertical load without flowing sideways.

(Van Maren, Winterwerp, et al., 2009) (g/liter) ws Saturated flow

(Winterwerp, 2011) -- 1-DV k-e model based on components of Delft 3D -- Sediment in density formulation -- Flocculation model -- Hindered settling model

(Ozedemir, Hsu & Balachandar, in press) U ~ 60 cm/s C ~ 10 g/liter LES model Fixed sediment supply ws= 0.45 mm/s ws = 0.75 mm/s

U ~ 60 cm/s C ~ 10 g/liter LES model (Ozedemir, Hsu & Balachandar, in press) Profiles of flux Richardson number at time of max free stream U ws = 0.45 mm/s ws = 0.75 mm/s

Outline of Presentation : Richardson number control of saturated suspension