1. LECTURE 8 LAYER-AVERAGED GOVERNING EQUATIONS FOR TURBIDITY CURRENTS
CEE 598, GEOL 593
TURBIDITY CURRENTS: MORPHODYNAMICS AND DEPOSITS
LECTURE 8
LAYER-AVERAGED GOVERNING EQUATIONS FOR TURBIDITY CURRENTS

2. SOME DEFINITIONS
x = boundary-attached (seafloor-attached) streamwise coordinate
z = boundary-perpendicular upward normal coordinate
u = streamwise flow velocity (averaged over turbulence)
c = volume suspended sediment concentration (averaged over turbulence)
Now we assume that   tan() = S << 1.
Thus x is “almost” horizontal and z is “almost” vertical.

3. SOME MORE DEFINITIONS a = density of ambient water
f = density of flowing water-sediment mixture in turbidity current
R = submerged specific gravity of sediment in suspension

4. LETS’ JUMP TO THE 1D APPROXIMATE LAYER-INTEGRATED EQUATIONS GOVERNING A TURBIDITY CURRENT
Flow discharge per unit width = qf, volume suspended sediment discharge per unit width = qs, streamwise momentum discharge per unit width = qm:
Thus layer-averaged quantities can be defined as

5. SUSPENDED SEDIMENT OF UNIFORM SIZE
THE EQUATIONS:
SUSPENDED SEDIMENT OF UNIFORM SIZE
In the above equations, the bed shear stress b is given by the relation
and
vs = sediment fall velocity [L/T],
ew = dimensionless rate of entrainment of ambient water into the turbidity current,
Es = dimensionless rate of entrainment of bed sediment into the turbidity current [1]
ro = cb/C > 1, where cb is a near-bed suspended sediment concentration

6. MOMENTUM BALANCE a) b) c) d) e) What the terms mean:
time rate of change of depth-integrated momentum
inertial force term
pressure force term
downstream gravitational force
bed resistive force

7. FLUID MASS BALANCE a) b) c) What the terms mean:
time rate of change of depth-integrated mass in flow (multiply by a)
streamwise change in flow discharge per unit width
rate at which flow incorporates ambient fluid from above by mixing across interface

8. SEDIMENT MASS BALANCE a) b) c) d) What the terms mean:
a) time rate of change of depth-integrated sediment mass in flow (multiply by s
b) streamwise change in sediment mass flow per unit width
c) rate at which sediment is eroded from the bed into suspension
d) rate at which sediment settled out from the flow onto the bed

9. ENTRAINMENT OF AMBIENT WATER
A SAMPLE RELATION:
ENTRAINMENT OF AMBIENT WATER
Here Ri denotes the bulk Richardson number. It is a ratio of the gravitational force resisting mixing to the inertial force of the flow. The larger is Ri, the less mixing there is across the interface between the turbidity current and the ambient flow.
Note that subcritical turbidity currents tend to have less mixing of ambient water across their interface than supercritical turbidity currents.
The relation is from Parker et al. (1987).

10. FROM PARKER ET AL. (1987)

11. ENTRAINMENT OF SEDIMENT INTO SUSPENSION
A SAMPLE RELATION:
ENTRAINMENT OF SEDIMENT INTO SUSPENSION
The relation is from Garcia and Parker (1991).

12. RELATION OF NEAR-BED SUSPENDED SEDIMENT CONCENTRATION TO LAYER-AVERAGED VALUE
Let cb denote the suspended sediment concentration evaluated a distance z = b above the bed, where b/H << 1 (near-bed concentration).
The volume downward flux of suspended sediment onto the bed is given as vscb. In a layer-averaged model, cb must be related to the layer-averaged value C by means of a dimensionless coefficient ro:
Sample relation: Parker (1982), estimated from the Rouse () profile for rivers:

13. TURBIDITY CURRENTS AND RIVERS RIVERS: flowing water under ambient air
a = density of air << f
f = density of water
TURBIDITY CURRENTS: flowing dilute suspension of dirty water under ambient clear water
a = density of clear water
f = density of dirty water =
C = volume concentration of sediment << 1 (dilute!)

14. COMPARISON OF 1D LAYER-INTEGRATED (APPROXIMATE) GOVERNING EQUATIONS FOR TURBIDITY CURRENTS AND RIVERS
Turbidity current:
Compare with river:

15. REFERENCES
Garcia, M, and G. Parker, 1991, Entrainment of bed sediment into suspension. Journal of Hydraulic Engineering, 117(4), 414‑435.
Parker, G., 1982, Conditions for the ignition of catastrophically erosive turbidity currents. Marine Geology, 46, pp. 307‑327, 1982.
Parker, G., M. H. Garcia, Y. Fukushima, and W. Yu, 1987, Experiments on turbidity currents over an erodible bed., Journal of Hydraulic Research, 25(1), 123‑147.