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

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

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

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

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

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

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

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

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

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

Rewrite f(  ) as Taylor expansion around  = 0: = 1 =  ≈ 0 From atmospheric studies,  ≈ 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 OUTLINE: 1) Ri # importance; 2) Overlap layer; 3) Under-saturation; 4) Critical Saturation; 5) Over-saturation

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

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

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

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

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

STATAFORM mid-shelf site, Northern California, USA, 1995, 1996 Inner shelf, Louisiana, USA, 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 concave-down than log(z) A > 1 predicts u more concave-up than log(z) A = 1 predicts u will follow log(z) (Friedrichs et al, 2000) OUTLINE: 1) Ri # importance; 2) Overlap layer; 3) Under-saturation; 4) Critical Saturation; 5) Over-saturation

Eckernförde Bay, Baltic Coast, Germany, spring 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) OUTLINE: 1) Ri # importance; 2) Overlap layer; 3) Under-saturation; 4) Critical Saturation; 5) Over-saturation

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

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

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

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

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

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

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

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

(a) Eel shelf, 60 m depth, winter (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/ cm OUTLINE: 1) Ri # importance; 2) Overlap layer; 3) Under-saturation; 4) Critical Saturation; 5) Over-saturation

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

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

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

More Settling Starting at around 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 (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 Ri cr, and more sediment to settles. Then there is more hindered settling and a stronger lutecline, increases Ri further w s (mm/s) OUTLINE: 1) Ri # importance; 2) Overlap layer; 3) Under-saturation; 4) Critical Saturation; 5) Over-saturation (Mehta & McAnally, 2008)

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

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

(Ozedemir, Hsu & Balachandar, in press) U ~ 60 cm/s C ~ 10 g/liter “large eddy simulation” 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 3.4 cm 2.9 cm Non-dimensional elevation 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

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