1. CROSS-SHORE TRANSPORT
Gradients typically related to changes in beach profile through continuity equation
Leads to things like sand bars, troughs, terraces, berms (But along shore motions can and do effect these as well).
Due mostly to waves, wave breaking and undertow
Hydrodynamic and sediment coupling still poorly understood.

2. SIMPLE ENGINEERING MODEL
Relate the transport to level of dis-equilibrium through the Energy dissipation per unit volume (similar to EBP Theory)
Where qs is the volumetric transport rate per unit width and D is the energy dissipation per unit volume
D* is found from EBP as

3. SIMPLE ENGINEERING MODEL
D&D
Example, profile is too shallow (slope less than equilibrium), sediment transport should be onshore to steepen profile.
For this case, shallow profile indicates a D<D*. Waves will break further offshore and there will be less turbulence than the equilibrium profile case for each cross-shore location

4. PROCESS-BASED MODEL Watanabe, 1982)
Φ is the dimensionless transport rate, Ψm is the magnitude of the instantaneous Shields parameter and Ψc is the critical Shields parameter
Note the massive scatter (log-log plot) and variations in the coefficients.

5. PROCESS-BASED MODEL
There are many other models that are similar to the one presented in Dean and Dalrymple
Note that since theb ed shear stress is typically related to the velocity through the quadratic drag law, it is proportional to the velocity squared.
Thus,
This proportionality is a common feature in most of the cross-shore sediment transport models (most notably bed load)

6. PROCESS-BASED MODEL: BAILARD, BAGNOLD, BOWEN
Transport is related to the fluid power that is delivered to the bed.
Originally developed for steady, uni-directional flow
Bailard (others similar):
Where :
I is the total immersed weight transport rate per unit width
ω is the fluid power
Subscripts: B= bedload, S=suspended load

7. PROCESS-BASED MODEL: BAILARD CONT
Where:
u is the fluid velocity vector (horizontal dimensions)
w is the fall velocity
tanβ is the beach slope in horizontal dimensions
tanφ is the internal friction angle of sand (angle of repose)

8. PROCESS-BASED MODEL: BAILARD CONT
Plug all the pieces in and get
The first two terms are bedload and the last two are suspended load. In each set of parentheses, the first term is transport in the direction of flow and the last term is the downslope transport (negative in the coordinate system chosen) regardless of flow velocity direction.
Gravity is persistent!

9. PROCESS-BASED MODEL: TEST
Often used in swash zone and surf zone. Typically requires calibration factor, so the friction coefficient is tuned.
Gallagher et al 1998

10. PROCESS-BASED MODEL: TEST
Many other attempts to use this or similar models to predict sand bar motion (Duck 1994 being the most widely attempted)
Simple model had success predicting the rapid offshore motion of the bar, but not the slow onshore motion of the bar (more on bars later)
It has been suggested that additional transport mechanisms exist, most notably horizontal pressure gradients and acceleration skewness.

11. ACCELERATION SKEWNESS
Solid line is sandbar location
Elgar et al., 2001

12. ACCELERATION SKEWNESS
Drake and Calantoni, 2001 used a discrete particle model to investigate these additional mechanisms
They found that the additional “push: could be represented by an additional term in the Bailard type model as
a is the fluid acceleration, Ispike becomes acceleration skewness, Ka is a coefficient and angle brackets denote averaging. Thus cannot be applied instantaneously. Other models have made mods for instantaneous a.

13. ACCELERATION SKEWNESS
Drake and Calantoni, 2001

14. ACCELERATION SKEWNESS
What about for sandbar motion?
Hoefel and Elgar, 2003
Much better, but still needs work!