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Outline of Presentation : Richardson number influence on coastal/estuarine mixing

Damping of Turbulence b y Suspended Sediment in the Bottom Boundary Layer Carl Friedrichs, Virginia Institute of Marine Science. Outline of Presentation : Richardson number influence on coastal/estuarine mixing Derivation of stratified “overlap” layer structure

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Outline of Presentation : Richardson number influence on coastal/estuarine mixing

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  1. Damping of Turbulence by Suspended Sediment in the Bottom Boundary Layer Carl Friedrichs,Virginia Institute of Marine Science • Outline of Presentation: • Richardson number influence on coastal/estuarine mixing • Derivation of stratified “overlap” layer structure • Under-saturated (weakly stratified) sediment suspensions • Critically saturated (Ricr-controlled) sediment suspensions • Hindered settling, over-saturation, and collapse of turbulence 10 30 100 mg/l Time-series of suspended sediment in York River estuary (Friedrichs et al. 2000) 1 meter 2 hours

  2. density stratification Shear instabilities occur for Ri < Ricr “ “ suppressed for Ri > Ricr Gradient Richardson Number (Ri) = velocity shear Hans Paerl et al. (2004) Neuse River Estuary observations from 2003 Ricr ≈ 1/4 Density & Chl stratified when Ri > 1/4, mixed when Ri < 1/4 OUTLINE: 1) Ri # importance; 2) Overlap layer; 3) Under-saturation; 4) Critical Saturation; 5) Over-saturation

  3. 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) OUTLINE: 1) Ri # importance; 2) Overlap layer; 3) Under-saturation; 4) Critical Saturation; 5) Over-saturation

  4. 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. OUTLINE: 1) Ri # importance; 2) Overlap layer; 3) Under-saturation; 4) Critical Saturation; 5) Over-saturation

  5. 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 OUTLINE: 1) Ri # importance; 2) Overlap layer; 3) Under-saturation; 4) Critical Saturation; 5) Over-saturation

  6. Damping of Turbulence by Suspended Sediment in the Bottom Boundary Layer Carl Friedrichs,Virginia Institute of Marine Science • Outline of Presentation: • Richardson number influence on coastal/estuarine mixing • Derivation of stratified “overlap” layer structure • Under-saturated (weakly stratified) sediment suspensions • Critically saturated (Ricr-controlled) sediment suspensions • Hindered settling, over-saturation, and collapse of turbulence 10 30 100 mg/l Time-series of suspended sediment in York River estuary (Friedrichs et al. 2000) 1 meter 2 hours

  7. Dimensionless analysis of bottom boundary layer in the absence of stratification: Variables du/dz, z, h, n, u* h = thickness of bottom boundary layer, n = kinematic viscosity, u* = (tb/r)1/2 = shear velocity ~ 1 cm/s ~ 10-6m2/s  h ~ 10 m n/(zu*) << 1 , but z/h not small Outer Layer ~ 1 m “Overlap” Layer: z/h << 1 & n/(zu*) << 1 n/(zu*) << 1 z/h << 1 (a.k.a. “log-layer”) velocity u(z) ~ 1 cm z/h << 1 , but n/(zu*) not small Viscous Layer (Dyer, 1986) OUTLINE: 1) Ri # importance; 2) Overlap layer; 3) Under-saturation; 4) Critical Saturation; 5) Over-saturation

  8. Dimensionless analysis of bottom boundary layer in the absence of stratification: Variables du/dz, z, h, n, u* h = thickness of bottom boundary layer, n = kinematic viscosity, u* = (tb/r)1/2 = shear velocity ~ 1 cm/s ~ 10-6m2/s  h ~ 10 m n/(zu*) << 1 , but z/h not small Outer Layer ~ 1 m = 1/k Const. “Overlap” Layer: z/h << 1 & n/(zu*) << 1 k ≈ 0.41 i.e., (a.k.a. “log-layer”) ~ 1 cm z/h << 1 , but n/(zu*) not small Viscous Layer (Dyer, 1986) OUTLINE: 1) Ri # importance; 2) Overlap layer; 3) Under-saturation; 4) Critical Saturation; 5) Over-saturation

  9. 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) OUTLINE: 1) Ri # importance; 2) Overlap layer; 3) Under-saturation; 4) Critical Saturation; 5) Over-saturation

  10. Additional variable b = Turbulent buoyancy flux Dimensionless analysis of overlap layer with (sediment-induced) stratification: 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” OUTLINE: 1) Ri # importance; 2) Overlap layer; 3) Under-saturation; 4) Critical Saturation; 5) Over-saturation

  11. 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 OUTLINE: 1) Ri # importance; 2) Overlap layer; 3) Under-saturation; 4) Critical Saturation; 5) Over-saturation

  12. Damping of Turbulence by Suspended Sediment in the Bottom Boundary Layer Carl Friedrichs,Virginia Institute of Marine Science • Outline of Presentation: • Richardson number influence on coastal/estuarine mixing • Derivation of stratified “overlap” layer structure • Under-saturated (weakly stratified) sediment suspensions • Critically saturated (Ricr-controlled) sediment suspensions • Hindered settling, over-saturation, and collapse of turbulence 10 30 100 mg/l Time-series of suspended sediment in York River estuary (Friedrichs et al. 2000) 1 meter 2 hours

  13. Eq. (1) (i) well-mixed stratified -- 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)  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 OUTLINE: 1) Ri # importance; 2) Overlap layer; 3) Under-saturation; 4) Critical Saturation; 5) Over-saturation

  14. 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) OUTLINE: 1) Ri # importance; 2) Overlap layer; 3) Under-saturation; 4) Critical Saturation; 5) Over-saturation

  15. z = const. in z if (i) well-mixed Fit a general power-law to c(z) of the form stratified A < 1 Then  as z z0 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 (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 OUTLINE: 1) Ri # importance; 2) Overlap layer; 3) Under-saturation; 4) Critical Saturation; 5) Over-saturation

  16. 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: STRATAFORM mid-shelf site, Northern California, USA (Friedrichs & Wright, 1997; Friedrichs et al, 2000) Inner shelf, Louisiana USA OUTLINE: 1) Ri # importance; 2) Overlap layer; 3) Under-saturation; 4) Critical Saturation; 5) Over-saturation

  17. 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) 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 (Friedrichs et al, 2000) -- 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. OUTLINE: 1) Ri # importance; 2) Overlap layer; 3) Under-saturation; 4) Critical Saturation; 5) Over-saturation

  18. 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. OUTLINE: 1) Ri # importance; 2) Overlap layer; 3) Under-saturation; 4) Critical Saturation; 5) Over-saturation

  19. 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 OUTLINE: 1) Ri # importance; 2) Overlap layer; 3) Under-saturation; 4) Critical Saturation; 5) Over-saturation

  20. Damping of Turbulence by Suspended Sediment in the Bottom Boundary Layer Carl Friedrichs,Virginia Institute of Marine Science • Outline of Presentation: • Richardson number influence on coastal/estuarine mixing • Derivation of stratified “overlap” layer structure • Under-saturated (weakly stratified) sediment suspensions • Critically saturated (Ricr-controlled) sediment suspensions • Hindered settling, over-saturation, and collapse of turbulence 10 30 100 mg/l Time-series of suspended sediment in York River estuary (Friedrichs et al. 2000) 1 meter 2 hours

  21. 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). OUTLINE: 1) Ri # importance; 2) Overlap layer; 3) Under-saturation; 4) Critical Saturation; 5) Over-saturation

  22. (i) well-mixed z and Ri const. in z if stratified A < 1 Define then and Ri as z z0 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 (ii) well-mixed stratified A > 1 Log elevation of height above bed and Ri as z z0 (iii) stratified well-mixed A = 1  and Riare constant in z If suspended sediment concentration, C ~ z-A then A <,>,= 1 determines shape of u profile and also the vertical trend in z and Ri z0 (Friedrichs et al, 2000) Current Speed OUTLINE: 1) Ri # importance; 2) Overlap layer; 3) Under-saturation; 4) Critical Saturation; 5) Over-saturation

  23. 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 OUTLINE: 1) Ri # importance; 2) Overlap layer; 3) Under-saturation; 4) Critical Saturation; 5) Over-saturation

  24. 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 OUTLINE: 1) Ri # importance; 2) Overlap layer; 3) Under-saturation; 4) Critical Saturation; 5) Over-saturation

  25. when Ri = Ricr . This also means that when Ri = Ricr : OUTLINE: 1) Ri # importance; 2) Overlap layer; 3) Under-saturation; 4) Critical Saturation; 5) Over-saturation

  26. 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) OUTLINE: 1) Ri # importance; 2) Overlap layer; 3) Under-saturation; 4) Critical Saturation; 5) Over-saturation

  27. 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) OUTLINE: 1) Ri # importance; 2) Overlap layer; 3) Under-saturation; 4) Critical Saturation; 5) Over-saturation

  28. Application of Ricr log-layer equations fo Eel shelf, 60 m depth, winter 1995-96 (Souza & Friedrichs, 2005) OUTLINE: 1) Ri # importance; 2) Overlap layer; 3) Under-saturation; 4) Critical Saturation; 5) Over-saturation

  29. Damping of Turbulence by Suspended Sediment in the Bottom Boundary Layer Carl Friedrichs,Virginia Institute of Marine Science • Outline of Presentation: • Richardson number influence on coastal/estuarine mixing • Derivation of stratified “overlap” layer structure • Under-saturated (weakly stratified) sediment suspensions • Critically saturated (Ricr-controlled) sediment suspensions • Hindered settling, over-saturation, and collapse of turbulence 10 30 100 mg/l Time-series of suspended sediment in York River estuary (Friedrichs et al. 2000) 1 meter 2 hours

  30. 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 OUTLINE: 1) Ri # importance; 2) Overlap layer; 3) Under-saturation; 4) Critical Saturation; 5) Over-saturation

  31. More Settling 1 (Mehta & McAnally, 2008) ws (mm/s) .1 .01 10 100 1 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(vertical sediment gradient) to form. A lutecline with hindered settling can cause turbulent collapse. The sediment can’t leave the water column, so dC/dz keeps increasing, creating positive feedback. Ri increases further above Ricr, and more sediment to settles. Then there is more hindered settling and a stronger lutecline, increases Ri further. OUTLINE: 1) Ri # importance; 2) Overlap layer; 3) Under-saturation; 4) Critical Saturation; 5) Over-saturation

  32. (Winterwerp, 2011) Observed Ri >> Ricr around slack Ri ≤ Ricr during flood Ri ≈ Ricr at peak ebb Modeled Ri >> Ricr around slack -- 1-DV k-e model based on components of Delft 3D -- Sediment in density formulation -- Flocculation model -- Hindered settling model Ri≤Ricr during flood Ri ≈ Ricr at peak ebb OUTLINE: 1) Ri # importance; 2) Overlap layer; 3) Under-saturation; 4) Critical Saturation; 5) Over-saturation

  33. (Ozedemir, Hsu & Balachandar, in press) U ~ 60 cm/s C ~ 10 g/liter “large eddy simulation” model Fixed sediment supply ws= 0.45 mm/s ws = 0.75 mm/s OUTLINE: 1) Ri # importance; 2) Overlap layer; 3) Under-saturation; 4) Critical Saturation; 5) Over-saturation

  34. U ~ 60 cm/s C ~ 10 g/liter “large eddy simulation” model (Ozedemir, Hsu & Balachandar, in press) Profiles of flux Richardson number at time of max free stream U 3.4 cm 2.9 cm Non-dimensional elevation Non-dimensional elevation ws = 0.45 mm/s ws = 0.75 mm/s The Richardson number is of “order” critical (relatively close to 0.25) near top of suspended layer OUTLINE: 1) Ri # importance; 2) Overlap layer; 3) Under-saturation; 4) Critical Saturation; 5) Over-saturation

  35. Damping of Turbulence by Suspended Sediment in the Bottom Boundary Layer Carl Friedrichs,Virginia Institute of Marine Science • Outline of Presentation: • Richardson number influence on coastal/estuarine mixing • Derivation of stratified “overlap” layer structure • Under-saturated (weakly stratified) sediment suspensions • Critically saturated (Ricr-controlled) sediment suspensions • Hindered settling, over-saturation, and collapse of turbulence 10 30 100 mg/l Time-series of suspended sediment in York River estuary (Friedrichs et al. 2000) 1 meter 2 hours

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