1. 1 LECTURE 12 MORPHODYNAMICS OF 1D SUBMARINE/SUBLACUSTRINE FANS CEE 598, GEOL 593 TURBIDITY CURRENTS: MORPHODYNAMICS AND DEPOSITS As the Colorado River flows into Lake Mead, USA, it forms a delta.

2. 2 STRUCTURE OF A DELTAIC DEPOSIT Rivers flowing into lakes and reservoirs deposit their sediment in deltas. In a delta, the coarser material (e.g. sand and coarser) material is fluvially emplaced in a topset deposit and emplaced by avalanching in a foreset deposit. Finer material (e.g. mud) is preferentially emplaced in deep water beyond the toe of the foreset. A major mechanism for this deep-water emplacement consists of turbidity currents associated with plunging.

3. 3 PLUNGING IN LAKE MEAD Image from USBR Logjam near plunge point plunge line

4. 4 DEPOSITION WITHIN LAKE MEAD, COLORADO RIVER, USA

5. 5 SOME DETAIL OF THE DEPOSIT Sand deposits in the topset and foreset. Mud deposits in the bottomset. Two moving boundaries: topset-foreset break and foreset-bottomset break.

6. 6 MOST DELTAS SPREAD OUT LATERALLY TO MAKE FAN-DELTAS Delta of the Selenga River as it flows into Lake Baikal, Russia

7. 7 WHEN LAKE MEAD BEGAN TO FILL, HOWEVER, THE DELTA WAS CONFINED TO A NARROW CANYON FOR MANY YEARS So the somewhat abstract case of a 1D delta prograding into a zone of constant width is not entirely without field analogs. Besides, if we can understand 1D delta, the treatment can then be generalized to 2D deltas.

8. 8 HERE WE SIMPLIFY THE PROBLEM TO A 1D SUBAQUEOUS FAN The upstream feed point of the 1D subaqueous fan is set at x = 0. The vertical position of the feed point is, however, allowed to adjust with the evolution of the bed.

9. 9 FURTHER SIMPLIFICATIONS 1.The flow is assumed to be continuous. 2.Only a single grain size is considered. 3.The flow is assumed to be supercritical everywhere, so that the “backwater” equations for the flow can be integrated downstream from the feed point. 4.Erosion is neglected in a first formulation, so that the turbidity current is treated as purely depositional. All of these assumptions can be relaxed in a more elaborate model.

10. 10 THE GOVERNING EQUATIONS Here we use a 3-equation formulation for the flow. Since the flow is assumed to be dilute and continuous, the relations: can be simplified to:

11. 11 PARAMETERS AND BOUNDARY CONDITIONS In the model presented here, r o, C f, v s, R (and of course g, because we just might not be working on Earth) must be specified. We are assuming supercritical flow, i.e. Ri < 1, so that the governing equations are integrated in the downstream direction. Upstream boundary conditions for C, U and H must be specified at x = 0. Here we specifiy the upstream water discharge per unit width q wo = U o H o, the suspended sediment discharge per unit width q so = U o H o C o and the upstream Richardson number Ri o = (RgC o H o /U o 2 ) = (Rgq so /U o 3 ). Thus

12. 12 BED EVOLUTION The Exner equation of bed sediment continuity takes the form That is, here we neglect both sediment entrainment and bedload transport: An initial condition must be specified for the bed profile: To simplify things here, we assume that the initial bed has a constant slope S bI and an initial bed elevation of 0 at x = 0.

13. 13 FLOW OF THE CALCULATION At any given time t, solve the equations below over the existing bed from x = 0 to x = L, where L is some domain length. Use the results of this solution to find the bed some time t +  t later:

14. 14 SPATIAL DISCRETIZATION The problem is solved over a domain extending from x = 0 to x = L, where L is a specified parameter. This domain is discretized to M intervals of length  x bounded by M + 1 points:

15. 15 SOLUTION FOR THE FLOW The simplest method you can use to solve for the flow is the Euler Step Method. (Yes, you can use a more accurate method if you know it). We rewrite the equations of the flow as where for example The equations discretize to where for example H i denotes the value of H at x i. For any given bed profile (which specifies S), the equations can be solved stepwise downstream from i = 1 to i = M + 1.

16. 16 Solve for the new bed elevations from Exner: and obtain the new bed slopes S i as And with these new bed slopes, it is possible to solve for the flow over the new bed! SOLUTION FOR THE BED EVOLUTION

17. 17 WELCOME TO THE EXCEL WORKBOOK WITH IMBEDDED CODE IN VISUAL BASIC FOR APPLICAITIONS: Rte-book1DSubaqueousFan.xls

18. 18 If the flow were morphodynamically active for only fraction  of time (i.e. a constant flow that is maitained for only a few days per year), the Exner equation must be modified to The code allows for this possibility through the parameter  = Inter. The coefficient of bed friction C f is related to C zs via the relation The fall velocity v s is computed from the Dietrich (1982) relation introduced earlier. The water entrainment relation used is that of Parker et al. (1987): NOTES ON THE PARAMETERS

19. 19 THE CODE The code is found in the Visual Basic Editor. From the Excel menu, select Tools, Macro, Visual Basic Editor If the code is (macros are) not enabled when you open the Excel file, you will have to go to the Excel Menu, select Tools, Macro, Security and set Security no higher than “medium”. You then have to close and open Excel in order to have the code enabled when you open the file.

20. 20 SAMPLE CALCULATION: BASE CASE These are the base input parameters. Note that D = 0.02 mm (mud) S bI = initial bed slope = 0.05 Simulation time = 0.6 years of continuous flow

21. 21 D = 0.02 mm, S bI = initial bed slope = 0.05

22. 22 D = 0.005 mm, S bI = initial bed slope = 0.05

23. 23 D = 0.05 mm, S bI = initial bed slope = 0.05 Bed gets too steep because the flow is purely depositional!

24. 24 D = 0.02 mm, S bI = initial bed slope = 0.05 (again)

25. 25 D = 0.02 mm, S bI = initial bed slope = 0.02

26. 26 D = 0.02 mm, S bI = initial bed slope = 0.065

27. 27 MATERIAL TO BE ADDED: EROSIONAL CASE NEED TO ADD. RTe-book1DSubaqueousFanWErosTRY.xls

28. 28 REFERENCES Ashida, K. and M. Michiue, 1972, Study on hydraulic resistance and bedload transport rate in alluvial streams, Transactions, Japan Society of Civil Engineering, 206: 59-69 (in Japanese). UNDER CONSTRUCTION