(Geyer & Traykovski, 2001) Modeling of Clinoforms Created By Wave/Current Supported Gravity Flows: Carl Friedrichs, Virginia Institute of Marine Science, presented at CSDMS 10/19/09 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).

(i) Momentum balance: = Bottom friction Down-slope pressure gradient B = depth-integrated buoyancy anomaly (i.e., suspended sediment) c d = drag coefficient =  B = c d U max u grav U max = (U w 2 + v c 2 + u grav 2 ) 1/2 Richardson Number Buoyancy Shear = = Critical value z = 0 z =   z y x c C = 0  c dz (ii) Maximum turbulent sediment load: (Friedrichs, Wright, Scully, Ma ) Wave and Current Supported Sediment Gravity Flow Analytical Model (WSGFAM): (iii) Solve for u grav (iv) Deposition = – d/dx (B u grav ) – (B u grav )/r x = local down-slope direction r = radius of curvature of local bathymetric contour downslope lateral convergence + =

U max = ( waves 2 + currents 2 ) 1/2 D porosity p = 0.9, g = 9.8 m/s 2, s = (  sed /  water -1) = 1.6 Julian Day 1996 Application of WSGFAM to Eel River shelf, CA (Scully, Friedrichs, Wright, 2002) (iv) Deposition = d/dx (B u grav ) (iii) Solve for u grav 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 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) -- 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 ~ 35 m/yr Maestra 1% Pila 74% Tolle 7% Gnocca 11% Goro 8% Po Delta Total sediment load 15 million tons/year Bathymetric profile figures from Nelson (1970)

Century-scale modeling of subsiding delta Distance (km) Depth (m) Q river = 0.2 kg/m/s x 30 days/yr = 500,000 kg/m/yr “Pila” Distance (km) Depth (m) Q river = 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 -- Subsidence overwhelms sedimentation near shore off Tolle, steepening delta near shore (Friedrichs and Scully, 2007) Subsidence of 2 cm/yr ~ 35 m/yr

3. Prograding Equilibrium Condition: 4. Conservation of Mass: h S P D  B = c d U max u grav 1. Momentum Balance: 2. Maximum Turbulent Sediment Load: (Friedrichs and Wright, 2005)

Equilibrium bed slope (  ) is controlled by: Riverine sediment supply, Q r Wave orbital velocity, u w Underlying shelf depth, depth, h S P D General Equilibrium Solution Accommodation space from subsidence, (Friedrichs and Wright, 2005)

Equilibrium bed slope (  ) is controlled by: Riverine sediment supply, Q r Wave orbital velocity, u w Underlying shelf depth, depth, h S P D Depositional slope cannot be larger than  = c d /Ri c ≈ For  > , turbidity current will be auto-suspending and erosional. General Equilibrium Solution Accommodation space from subsidence, (Friedrichs and Wright, 2005)

Equilibrium bed slope (  ) is controlled by: Riverine sediment supply, Q r First consider waves on landward, bypassing portion of profile with no subsidence. For wind waves, substitute in u w =  (H/2) (sinh kh) -1, where H is wave height,  is wave radian frequency and k is wave number. h S P D Accommodation space from subsidence, General Equilibrium Solution Wave orbital velocity, u w Underlying shelf depth, depth, (Friedrichs and Wright, 2005)

Equilibrium bypassing solution: (Friedrichs & Wright, 2004) Model trends for bathymetry: Observed bathymetries: -- Only variable quantities are wave height, river discharge (H 3 /Q r ) and wave period (T). -- H and T tend to be positively correlated, so H 3 /Q r is the dominant environmental parameter.

Equilibrium bed slope (  ) is controlled by: Riverine sediment supply, Q r Underlying shelf depth, depth, h S P D General Equilibrium Solution Accommodation space from subsidence, Wave orbital velocity, u w (Friedrichs and Wright, 2005)

Equilibrium bed slope (  ) is controlled by: Riverine sediment supply, Q r h S P Next consider a delta profile prograding across a flat shelf with no subsidence. D Accommodation space from subsidence, Wave orbital velocity, u w Underlying shelf depth, depth, (Friedrichs and Wright, 2005)

Distance (km) Depth (m) Equilibrium Profile H sig = 1 m, T = 6 s Q river = 0.2 kg/m/s Bathymetry from Correggiari & Trincardi Deposition (cm) Equilibrium Annual Deposition Flood duration = 3 weeks Bed porosity = 0.6 Progradation rate = 12 m/year = 28 m ˚ E ˚N Pila Pila mouth of Po: Present day slope and theoretical equilibrium Pila subaqueous delta is close to equilibrium solution for reasonable model parameters (Friedrichs and Wright, 2005)

Effects of (a) longer wave period (waves to tides) and (b) increased tidal range on equilibrium bathymetric profiles: Distance (km) Depth (m) Equilibrium Profiles H sig = 1 m, Q river = 0.2 kg/m/s = 28 m T = 6 s T = 12 s T = 12 hr (tide) Equilibrium Tidal Profiles Q river = 1 kg/m/s = 40 m Tidal range = 1 m Tidal range = 2 m Tidal range = 4 m (a) (b) Pila case Width of equilibrium subaqueous delta increases with wave period or tidal range (c.f. Walsh et al. 2003; Vakarelov et al. 2008) (Friedrichs and Wright, 2005)

Equilibrium bed slope (  ) is controlled by: Riverine sediment supply, Q r Underlying shelf depth, depth, Accommodation space from subsidence, h P D = S -- Deposition is determined by subsidence (i.e., shape of accommodation space). -- Subsidence reduces profile slope but equilibrium slope is independent subsidence details. S(x) e.g., Mid-shelf depocenter, Eel shelf Wave orbital velocity, u w (Friedrichs and Wright, 2005)

(Orange, 1999) (Scully et al., 2003, showing depocenter) -- Deposition lines up with syncline (subsidence) and shallower mid-shelf slope and shallower mid-shelf slope -- Bypassing lines up with anticline (uplift) and steeper mid-shelf slope and steeper mid-shelf slope Eel River shelf, CA: