DIMENSIONLESS BANKFULL HYDRAULIC RELATIONS FOR EARTH AND TITAN

1. DIMENSIONLESS BANKFULL HYDRAULIC RELATIONS FOR EARTH AND TITAN European Space Agency Gary Parker Dept. of Civil & Environmental Engineering and Dept. of Geology University of Illinois

2. But a thick shroud of smog produced by the breakdown of methane under ultraviolet light prevented any surface visualization. UNTIL RECENTLY TITAN WAS SHROUDED IN MYSTERY • What we knew or could reasonably infer: • Larger than Mercury • Atmospheric pressure ~ 1.5 Earth atmospheres near surface • ~ 95 K near surface • Atmosphere of nitrogen (mostly), methane, ethane • Crustal material of water/ice • Near triple point of methane/ethane: possibility of • a. methane/ethane oceans • b. methane/ethane precipitation as liquid/solid • 7. Possibility of rivers of liquid methane carrying sediment of solid water ice!

3. This and other images of Titan courtesy European Space Agency and NASA Cassini/Huygens Mission: very strong evidence for rivers of liquid methane carrying sediment of water ice AND THEN JANUARY 14, 2005 ARRIVED! I was glued to the internet! I had waited for years!

4. MARS VERSUS TITAN Mars shows evidence of ancient rivers of flowing water that carried sediment similar to that of the Earth’s crust.

5. MARS VERSUS TITAN contd. But the era of flowing rivers was a long time ago, as evidenced by the fairly intense impact cratering of Mars, and may not has lasted very long as compared to Earth.

6. MARS VERSUS TITAN contd. Tectonic ridges? Titan shows evidence of active tectonics, vulcanism, aeolian and fluvial reworking, and has very few impact craters: so its surface is likely active in modern geological time!

7. MARS VERSUS TITAN contd. Volcano? Titan shows evidence of active tectonics, vulcanism, aeolian and fluvial reworking, and has very few impact craters: so its surface is likely active in modern geological time!

8. MARS VERSUS TITAN contd. Aeolian dunes? Titan shows evidence of active tectonics, vulcanism, aeolian and fluvial reworking, and has very few impact craters: so its surface is likely active in modern geological time!

9. MARS VERSUS TITAN contd. River drainage basin? Titan shows evidence of active tectonics, vulcanism, aeolian and fluvial reworking, and has very few impact craters: so its surface is likely active in modern geological time!

10. MARS VERSUS TITAN contd. Impact crater Titan shows evidence of active tectonics, vulcanism, aeolian and fluvial reworking, and has very few impact craters: so its surface is likely active in modern geological time!

11. ALLUVIAL GRAVEL-BED RIVERS ON TITAN? The evidence suggests that at least near where Huygens touched down, there is a plethora of alluvium in the gravel and sand sizes. The gravel presumably consists of water ice and appears to be fluvially rounded.

12. CAN OUR KNOWLEDGE OF ALLUVIAL GRAVEL-BED RIVERS ON EARTH HELP US MAKE INFERENCE ABOUT TITAN?

13. IF WE KNEW THE PHYSICS BEHIND RELATIONS FOR BANKFULL GEOMETRY HERE ON EARTH • Bankfull Depth Hbf ~ (Qbf)0.4 • Bankfull Width Bbf ~ (Qbf)0.5 • Bed Slope S ~ (Qbf)-0.3 • where Qbf = bankfull discharge we might be able to extend the relations to Titan.

14. WE BEGIN WITH EARTH The Parameters: Qbf = bankfull discharge (m3/s) QbT,bf = volume bedload transport rate at bankfull discharge (m3/s) Bbf = bankfull width (m) Hbf = bankfull depth (m) S = bed slope (1) D = surface geometric mean or median grain size (m)  = density of water (kg/m3) s = density of sediment (kg/m3) R = (s/ ) – 1 = submerged specific gravity of sediment ~ 1.65 (1) g = gravitational acceleration (m/s2)  = kinematic viscosity of water (m2/s) The forms sought: dimensionless versions of Why dimensionless? In order to allow scaling between Earth and Titan!

15. Meet my friends the DIMENSIONLESS PARAMETERS Particle Reynolds number Dimensionless bankfull discharge Dimensionless bankfull depth Dimensionless bankfull width Down-channel bed slope Dimensionless bankfull bedload transport rate Bankfull Shields number Shields number at threshold of motion

16. DATA SETS FOR GRAVEL-BED RIVERS ON EARTH • Alberta streams, Canada1 • Britain streams (mostly Wales)2 • Idaho streams, USA3 • Colorado River, USA (reach averages) • 1 Kellerhals, R., Neill, C. R. and Bray, D. I., 1972, Hydraulic and • geomorphic characteristics of rivers in Alberta, River Engineering • and Surface Hydrology Report, Research Council of Alberta, Canada, • No. 72-1. • 2 Charlton, F. G., Brown, P. M. and Benson, R. W., 1978, The • hydraulic geometry of some gravel rivers in Britain, Report INT 180, • Hydraulics Research Station, Wallingford, England, 48 p. • 3 Parker, G., Toro-Escobar, C. M., Ramey, M. and Beck S., 2003, • The effect of floodwater extraction on the morphology • of mountain streams, Journal of Hydraulic Engineering, 129(11), • 2003. • 4 Pitlick, J. and Cress, R., 2002, Downstream changes in the channel of a • large gravel bed river, Water Resources Research 38(10), 1216, • doi:10.1029/2001WR000898, 2002.

17. WHAT THE DATA SAY: WIDTH, DEPTH, SLOPE The four independent sets of data form a coherent set!

18. REGRESSION RELATIONS BASED ON THE DATA To a high degree of approximation, Remarkable, no?

19. WHAT DOES THIS MEAN?

20. WHAT THE DATA SAY: BANKFULL SHIELDS NUMBER

21. THE PHYSICS BEHIND IT ALL Assume the following relations. Manning-Strickler resistance relation Parker-Einstein bedload relation Relation for bankfull Shields number Channel form relation of type of Parker (1978) “Gravel yield” relation

22. THE RELATIONS OF THE PREVIOUS SLIDE YIELD PRECISELY THE OBSERVED DIMENSIONLESS RELATIONS!

23. GENERALIZATION FOR OTHER PLANETS/SATELLITES Manning-Strickler resistance relation Parker-Einstein bedload relation Relation for bankfull Shields number Channel form relation of type of Parker (1978) “Gravel yield” relation (volume to mass) The presence of g and R allow us to go from to

24. BACK-CALCULATED DIMENSIONALLY HOMOGENEOUS BANKFULL HYDRAULIC RELATIONS FOR ALLUVIAL GRAVEL RIVERS ON ARBITRARY HEAVENLY BODIES The presence of g and R allow us to go from to

25. FROM TO

26. CONSIDER A STREAM WITH THE SAME BANKFULL DISCHARGE Qbf AND CHARACTERISTIC GRAIN SIZE D HOW SHOULD TITAN COMPARE WITH EARTH? E = Earth, T = Titan From to

27. CONSIDER A STREAM WITH THE SAME BANKFULL DISCHARGE Qbf AND CHARACTERISTIC GRAIN SIZE D HOW SHOULD TITAN COMPARE WITH EARTH? E = Earth, T = Titan = 1.48 x 0.83 = 1.23 = 1.57 x 1.56 = 2.46 = 0.72 x 0.80 = 0.57

28. SO FOR THE SAME BANKFULL DISCHARGE Qbf AND CHARACTERISTIC GRAIN SIZE D A gravel-bed river on might be 1.23 x the bankfull depth, 2.46 x the bankfull width and 0.57 x the down-channel slope of a gravel-bed river on Could braiding be more common on Titan?

29. BUT WAIT A MINUTE! IS “GRAVEL” ON TITAN GRAVEL ON EARTH? For dynamic similarity in grain Reynolds number or or So the answer is “yes” to a reasonable approximation!

30. GRAIN REYNOLDS INVARIANCE Besides, the dynamics of sediment transport becomes approximately invariant to particle Reynolds number for or D >~ 8.8 mm on Earth or D >~ 10.6 mm on Titan based on the condition c*/c,asymp*  0.90 using

31. WHAT ABOUT AEOLIAN PROCESSES ON TITAN? Let Ua = wind velocity, a = atmospheric density, Cf = drag coefficient, s = sediment density, D = grain size. Scaling for mobility of grain size D: Atmospheric density Earth 293K 1 E-atmo, a = 1.21 kg/m3 Titan (nitrogen) 95K 1.5 E-atmo, a = 5.39 kg/m3 Assuming Reynolds invariance (Cf  constant), critical velocity Uac to blow around size D scales as: Much easier to blow sediment around on Titan! But much less solar heating to drive meteorology!

32. QUESTIONS OR COMMENTS?