(Geyer & Traykovski, 2001)

(Geyer & Traykovski, 2001)

Modeling of Clinoforms Created By Wave/Current Supported Gravity Flows:Modeling of Clinoforms Created By Wave/Current Supported Gravity Flows:Carl Friedrichs, Virginia Institute of Marine Science, presented at CSDMS 10/19/09Carl Friedrichs, Virginia Institute of Marine Science, presented at CSDMS 10/19/09

Original motivation: Wave-supported sediment gravity flows are the dominant offshore transport Original motivation: Wave-supported sediment gravity flows are the dominant offshore transport mechanism on the Eel River shelf, CA, during floods (Ogston et al., 2000; Traykovski et al. 2000). mechanism on the Eel River shelf, CA, during floods (Ogston et al., 2000; Traykovski et al. 2000).

(i) Momentum balance:

= Bottom frictionDown-slopepressure gradient

B = depth-integrated buoyancy anomaly (i.e., suspended sediment)

cd = drag coefficient = 0.003

B = cd Umax ugrav Umax = (Uw2 + vc

2 + ugrav

2)1/2

RichardsonNumber

Buoyancy

Shear= = Critical value

z = 0

z =

z

y

x

cC =

0

c dz

(ii) Maximum turbulent sediment load:

(Friedrichs, Wright, Scully, Ma 2001 - 2009)

Wave and Current Supported Sediment Gravity Flow Analytical Model (WSGFAM):

(iii) Solve for ugrav

(iv) Deposition = – d/dx (B ugrav) – (B ugrav)/r

x = local down-slope direction

r = radius of curvature of local bathymetric contour downslope lateral

convergence convergence +=

Umax =

(waves2 +

currents2)1/2

D

porosity p = 0.9, g = 9.8 m/s2, s = (sed/water -1) = 1.6

Julian Day 1996Julian Day 1996

Application of WSGFAM to Eel River shelf, CA (Scully, Friedrichs, Wright, 2002)

(iv) Deposition = d/dx (B ugrav)

(iii) Solve for ugrav

Comparison of analytical solution to observed across-shelf velocity:

(Observations from Ogston et al. 2000)

Comparison of analytical solution to observed deposition:

(Observations from Traykovski et al. 2000)

(neglecting lateral convergence in this case)

(Friedrichs & Scully, 2007)(Wheatcroft et al., 2006)

Comparison WSGFAM results to observed seasonal deposition on Po subaqueous delta, NW Adriatic Sea:

Observed Observed

Modeled

(Wheatcroft & Borgeld, 2000)

(Scully, Friedrichs

Wright, 2003)

Comparison of WSGFAM results to observed seasonal deposition on Eel shelf, Northern California:

Distinct evolution of Po distributaries: Pila vs. Tolle (post 1880) Distinct evolution of Po distributaries: Pila vs. Tolle (post 1880)

-- Pila prograded as an equilibrium form, while Tolle eroded and steepened near shore-- Pila prograded as an equilibrium form, while Tolle eroded and steepened near shore-- Carminati & Martinelli (2002) suggest Po delta is subsiding by up to several cm/year -- Carminati & Martinelli (2002) suggest Po delta is subsiding by up to several cm/year

~ 35 m/yrMaestra 1%

Pila 74%

Tolle 7%

Gnocca 11%Goro 8%

Po Delta Total sediment load15 million tons/year Bathymetric Bathymetric

profile figures profile figures from Nelson from Nelson

(1970)(1970)

Century-scale modeling of subsiding deltaCentury-scale modeling of subsiding delta

0 1 2 3 4 5 6 7 8 9 10-30

-25

-20

-15

-10

-5

0

Distance (km)

De

pth

(m

)

Qriver = 0.2 kg/m/s

x 30 days/yr = 500,000 kg/m/yr

“Pila”

1860 2000

0 1 2 3 4 5 6 7 8 9 10-30

-25

-20

-15

-10

-5

0

Distance (km)

De

pth

(m

)

Qriver = 0.04 kg/m/s

x 30 days/yr = 100,000 kg/m/yr

“Tolle”

-- Sediment load overwhelms subsidence off Pila, and delta evolves toward equilibrium-- Sediment load overwhelms subsidence off Pila, and delta evolves toward equilibrium

-- Subsidence overwhelms sedimentation near shore off Tolle, steepening delta near shore-- Subsidence overwhelms sedimentation near shore off Tolle, steepening delta near shore

(Friedrichs and Scully, 2007) (Friedrichs and Scully, 2007)

Subsidenceof 2 cm/yr

~ 35 m/yr

3. Prograding Equilibrium Condition:3. Prograding Equilibrium Condition:

4. Conservation of Mass:4. Conservation of Mass:

h

S

PPDD

B = cd Umax ugrav

1. Momentum Balance:1. Momentum Balance: 2. Maximum Turbulent Sediment Load:2. Maximum Turbulent Sediment Load:

(Friedrichs and Wright, 2005) (Friedrichs and Wright, 2005)

Equilibrium bed slope Equilibrium bed slope (() is controlled by:) is controlled by:

Riverine sediment Riverine sediment supply, Qsupply, Qrr

Wave orbital velocity, uWave orbital velocity, uww

Underlying shelf Underlying shelf depth, depth,

h

S

PPDD

General Equilibrium General Equilibrium SolutionSolution

Accommodation space Accommodation space from subsidence,from subsidence,






h

S

PPDD

Depositional slope cannot be larger than Depositional slope cannot be larger than = c = cdd/Ri/Ricc ≈ 0.012 - 0.016 . ≈ 0.012 - 0.016 .

For For > 0.012 - 0.016, turbidity current will be auto-suspending and erosional. > 0.012 - 0.016, turbidity current will be auto-suspending and erosional.






First consider waves on landward, bypassing portion of profile with no subsidence.First consider waves on landward, bypassing portion of profile with no subsidence.

For wind waves, substitute in uFor wind waves, substitute in uww = = (H/2) (sinh kh) (H/2) (sinh kh)-1-1 , where H is wave height, , where H is wave height, is wave is wave

radian frequency and k is wave number.radian frequency and k is wave number.

h

S

PPDD


General Equilibrium General Equilibrium SolutionSolutionWave orbital velocity, uWave orbital velocity, uww



Equilibrium bypassing solution:Equilibrium bypassing solution:

(Friedrichs & Wright, 2004)

(Friedrichs & Wright, 2004)

Model trends for bathymetry:Model trends for bathymetry: Observed bathymetries:Observed bathymetries:

-- Only variable quantities are wave height, river -- Only variable quantities are wave height, river discharge (Hdischarge (H33/Q/Qrr) and wave period (T).) and wave period (T).

-- H and T tend to be positively correlated, so H-- H and T tend to be positively correlated, so H33/Q/Qrr is is

the dominant environmental parameter. the dominant environmental parameter.




h

S

PPDD







h

S

PP

Next consider a delta profile prograding across a flat shelf with no subsidence.Next consider a delta profile prograding across a flat shelf with no subsidence.

DD





0 1 2 3 4 5 6 7 8 9 100

5

10

15

20

Distance (km)

Dep

th (

m)

0 1 2 3 4 5 6 7 8 9 10-30

-25

-20

-15

-10

-5

Equilibrium Profile

Hsig = 1 m, T = 6 s

Qriver = 0.2 kg/m/sBathymetry from

Correggiari & Trincardi

Dep

ositi

on (

cm)

Equilibrium Annual Deposition

Flood duration = 3 weeksBed porosity = 0.6

Progradation rate = 12 m/year

= 28 m

12.4 12.5 12.6 12.7˚ E44.8

44.9

45˚N

4128

1620

2428

Pila

Pila mouth of Po: Present day slope and theoretical equilibriumPila mouth of Po: Present day slope and theoretical equilibrium

Pila subaqueous delta is close to equilibrium solution for reasonable model parametersPila subaqueous delta is close to equilibrium solution for reasonable model parameters


Effects of (a) longer wave period (waves to tides) and (b) increased tidal Effects of (a) longer wave period (waves to tides) and (b) increased tidal range on equilibrium bathymetric profiles:range on equilibrium bathymetric profiles:

1 10 100-30

-25

-20

-15

-10

-5

10 100 1000-40

-35

-30

-25

-20

-15

-10

-5

Distance (km)

Dep

th (

m)

Dep

th (

m)

Equilibrium Profiles

Hsig = 1 m,

Qriver = 0.2 kg/m/s= 28 m

T = 6 sT = 12 s

T = 12 hr(tide)

Equilibrium Tidal

Profiles

Qriver = 1 kg/m/s= 40 m

Tidal range = 1 m

Tidal range = 2 m

Tidal range = 4 m

(a)(a)

(b)(b)

Pila case

Width of equilibrium subaqueous delta increases with wave period or tidal rangeWidth of equilibrium subaqueous delta increases with wave period or tidal range

(c.f. Walsh et al. 2003; Vakarelov et al. 2008) (c.f. Walsh et al. 2003; Vakarelov et al. 2008)






h

PP

D = SD = S

-- Deposition is determined by subsidence (i.e., shape of accommodation space).-- Deposition is determined by subsidence (i.e., shape of accommodation space).

-- Subsidence reduces profile slope but equilibrium slope is independent subsidence details. -- Subsidence reduces profile slope but equilibrium slope is independent subsidence details.

S(x)S(x)

e.g., Mid-shelf e.g., Mid-shelf depocenter,depocenter,

Eel shelfEel shelf



(Orange, 1999)(Scully et al., 2003, showing 95-97 depocenter)

-- Deposition lines up with syncline (subsidence)-- Deposition lines up with syncline (subsidence) and shallower mid-shelf slopeand shallower mid-shelf slope-- Bypassing lines up with anticline (uplift)-- Bypassing lines up with anticline (uplift) and steeper mid-shelf slopeand steeper mid-shelf slope

Eel River shelf, CA:Eel River shelf, CA:

(Geyer & Traykovski, 2001)

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