1. 1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 1 CHAPTER 13: THE QUASI-STEADY APPROXIMATION The conservation equations governing 1D morphodynamics can be summarized as For many applications in morphodynamics, however, it is possible to neglect the time derivatives in the first two equations, retaining it only in the Exner equation of conservation of bed sediment. That is, the flow over the bed can be approximated as quasi-steady. This result, first shown by de Vries (1965), is often implicitly used in morphodynamic calculations without justification. A demonstration follows. Conservation of flow mass Conservation of flow momentum Conservation bed sediment (sample form using total bed material load)

2. 1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 2 NON-DIMENSIONALIZATION USING A REFERENCE STATE The essence of morphodynamics is in the interaction between the flow and the bed. The flow changes the bed, which in turn changes the flow. Consider a reference mobile-bed equilibrium state with constant flow velocity U o flow depth H o, bed slope S o and total volume bed material transport rate per unit width q to. For the sake of simplicity the bed friction coefficient C f is assumed to be constant. The analysis easily generalizes, however, to the case of varying friction coefficient. The application of momentum balance to the equilibrium flow imposes the conditions where  u is the value of  at x = 0. In a problem of morphodynamic evolution, the flow and bed can be expected to deviate from this base state. In general, then, The following non-dimensionalizations are introduced:

3. 1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 3 NON-DIMENSIONALIZATION USING A CHARACTERISTIC HYDRAULIC TIME SCALE Note that the non-dimensionalization of time involves the “hydraulic” time scale H o /U o, which physically corresponds to the time required for the flow to move a distance equal to one depth in the downstream direction. Substituting the non- dimensional variables into the balance equations yields the results where

4. 1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 4 HOW LARGE IS  Bed porosity p is typically in the range 0.25 ~ 0.45 for beds of noncohesive sediment. The parameter thus scales the ratio of the volume transport of solids to the volume transport of water by a river. For the great majority of cases of interest this ratio is exceedingly small, even during floods. A case in point is the Minnesota River near Mankato, Minnesota, a medium-sized sand-bed stream. Some sample calculations follow. Minnesota River at the Wilmarth Power Plant just downstream of Mankato, Minnesota, USA. Flow is from left to right.

5. 1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 5 HOW LARGE IS  contd. Given below are 13 grain size distributions for the bed material of the Minnesota River at Mankato, along with an average of all 13. The fraction of sediment finer than 0.062 microns in the bed is negligible; such material can be treated as wash load. From http://www.usgs.gov/

6. 1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 6 HOW LARGE IS  contd. During the period 1967-1995 the highest measured suspended load concentration was 2850 mg/liter, or C = 2850/2.65*1x10 -6 = 0.001075; the discharge Q was 340 m 3 /s, so the volume total suspended load (bed material load + wash load) Q sbw = 0.366 m 3 /s. From http://www.usgs.gov/

7. 1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 7 HOW LARGE IS  contd. At a discharge of 340 m 3 /s, about 79.5% of the suspended load is wash load, giving a suspended bed material load Q s of 0.075 m 3 /s. Estimating the bedload Q b as about 15% of the total bed material load, an estimate for the highest value of Q t of 0.088 m 3 /s is obtained. From http://www.usgs.gov/

8. 1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 8 HOW LARGE IS  contd. Assuming a value of bed porosity p of 0.35, then, an estimate of the very high end of the value of  that might be attained by the Minnesota River near Mankato is The Minnesota River is by no means atypical of rivers. The largest values of  attained in the great majority of rivers is much less than unity. The exceptions include streams with slopes so high that the flows are transitional to debris flows, streams carrying lahars, or heavily sediment laden flow from regions recently covered with volcanic ash, and many streams in the Yellow River Basin of China.

9. 1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 9 NOT ALL FLOWS SATISFY THE CONDITION  << 1 Double-click on the image to see a debris flow in Japan. The volume (mass) of sediment carried by debris flows is of the same order of magnitude as the volume (mass) of water carried by such flows. The quasi-steady approximation breaks down for such flows. Video courtesy Paul Heller. rte-bookjapandebflow.mpg: to run without from relinking, download to same folder as PowerPoint presentations.

10. 1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 10 HYDRAULIC TIME SCALES Over short, or “hydraulic” times scales  t ~ H o /U o, then, when  << 1 the governing equations approximate to That is, the bed can be treated as unchanging for computations over “hydraulic” time scales, even though sediment is in motion. This is because the condition  << 1 implies lots of water flows through but very little sediment, so that the bed does not have time to change in response to the flow.

11. 1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 11 MORPHODYNAMIC TIME SCALES A dimensionless morphodynamic time t* can be defined as An order-one change  t* corresponds to a change in dimensioned time i.e. much longer than the characteristic “hydraulic” time. The governing equations thus become That is, when the time scales of interest are of “morphodynamic” scale, the flow can be treated as quasi-steady even though the bed is evolving, and thus changing the flow.

12. 1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 12 THE DIMENSIONED EQUATIONS WITH THE QUASI-STEADY APPROXIMATION According to the quasi-steady approximation, the bed changes so slowly compared to the characteristic response time of the flow that the flow can be approximated as responding immediately. The dimensioned equations thus reduce to the following forms: The quasi-steady approximation greatly simplifies morphodynamic calculations. There are, however, reasons not to use it. These include a)Cases of rapidly varying hydrographs, when it is desired to characterize the sediment transport over the entire hydrograph; b)Cases when one wishes to capture the effect of a flood wave (with a high water surface slope on the upstream side of the wave and a low water surface slope on the downstream side) on sediment transport; and c)Cases when the flow makes transitions between subcritical and supercritical flow, in which case a shock-capturing method capable of automatically locating hydraulic jumps is required.

13. 1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 13 REFERENCES FOR CHAPTER 13 de Vries, M. 1965. Considerations about non-steady bed-load transport in open channels. Proceedings, 11th Congress, International Association for Hydraulic Research, Leningrad: 381-388.