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

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

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

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

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

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

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

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

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

10. Current Speed Log elevation of height above bed  as z well-mixed stratified (i) (iii) (ii)  is constant in z well-mixed stratified well-mixed stratified  as z -- Case (i): No stratification near the bed (  = 0 at z = z 0 ). Stratification and  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 (  > 0 at z = z 0 ). Stratification and  decreases 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) z0z0 z0z0 z0z0 Eq. (1) -- Case (iii): uniform  with z. Eq (1) integrates to -- u remains logarithmic, but shear is increased buy a factor of (1+  ) (Friedrichs et al, 2000)

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

12. If suspended sediment concentration, C ~ z -A Then A,= 1 determines shape of u profile (Friedrichs et al, 2000) If A < 1, c decreases more slowly than z -1  increases with z, stability increases upward, u is more concave-down than log(z) If A > 1, c increases more quickly than z -1  decreases with z, stability becomes less pronounced upward, u is more concave-up than log(z) If A = 1, c ~ z -1  is constant with elevation stability is uniform in z, u follows log(z) profile  = const. in z if Fit a general power-law to c(z) of the form Then Current Speed Log elevation of height above bed  as z well-mixed stratified (i) (iii) (ii)  is constant in z well-mixed stratified well-mixed stratified z0z0 z0z0 z0z0  as z A < 1 A = 1 A > 1

13. STATAFORM mid-shelf site, Northern California, USA Inner shelf, Louisiana USA Eckernförde Bay, Baltic Coast, Germany 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) Testing this relationship using observations from bottom boundary layers: (Friedrichs & Wright, 1997; Friedrichs et al, 2000)

14. STATAFORM mid-shelf site, Northern California, USA, 1995, 1996 Inner shelf, Louisiana, USA, 1993 -- 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. A ≈ 0.11 A ≈ 3.1 A ≈ 0.35 A ≈ 0.73 A ≈ 1.0 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)

15. Eckernförde Bay, Baltic Coast, Germany, spring 1993 -- 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. (Friedrichs & Wright, 1997)

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

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

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

19. (Friedrichs et al, 2000) Current Speed Log elevation of height above bed  and Ri as z well-mixed stratified (i) (iii) (ii)  and Ri are constant in z well-mixed stratified well-mixed stratified z0z0 z0z0 z0z0 A < 1 A = 1 A > 1  and Ri as z If suspended sediment concentration, C ~ z -A then A,= 1 determines shape of u profile and also the vertical trend in  and Ri  and Ri const. in z if then Define If A < 1, c decreases more slowly than z -1  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  and Ri decrease with z, stability becomes less pronounced upward, u is more concave-up than log(z) If A = 1, c ~ z -1  and Ri are constant with elevation stability is uniform in z, u follows log(z) profile

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

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

22. when Ri = Ri cr. This also means that when Ri = Ri cr :

23. 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)

24. (a) Eel shelf, 60 m depth, winter 1995-96 (Wright, Friedrichs, et al. 1999) Velocity shear du/dz (1/sec) Sediment gradient Richardson number (b) Waiapu shelf, NZ, 40 m depth, winter 2004 (Ma, Friedrichs, et al. in 2008) Ri cr = 1/4 18 - 40 cm

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

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

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

28. 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 w s starts to decrease with additional increases in concentration. At ~ 10 g/l, w s 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.

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

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

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

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

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