DIMENSIONLESS BANKFULL HYDRAULIC RELATIONS FOR EARTH AND TITAN

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DIMENSIONLESS BANKFULL HYDRAULIC RELATIONS FOR EARTH AND TITAN

Gary ParkerDept. of Civil & Environmental Engineering and Dept. of Geology

University of Illinois

European Space Agency

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UNTIL RECENTLY TITAN WAS SHROUDED IN MYSTERYWhat we knew or could reasonably infer:1. Larger than Mercury2. Atmospheric pressure ~ 1.5 Earth atmospheres near surface3. ~ 95 K near surface4. Atmosphere of nitrogen (mostly), methane, ethane5. Crustal material of water/ice6. Near triple point of methane/ethane: possibility of

a. methane/ethane oceansb. methane/ethane precipitation as liquid/solid

7. Possibility of rivers of liquid methane carrying sediment of solid water ice!

But a thick shroud of smog produced by the breakdown of methane under ultraviolet light prevented any surface visualization.

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AND THEN JANUARY 14, 2005 ARRIVED!

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

I was glued to the internet! I had waited for years!

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MARS VERSUS TITAN

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

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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.

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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!

Tectonic ridges?

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Volcano?


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Aeolian dunes?


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River drainage basin?


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Impact crater


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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.

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

TITAN?

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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.

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WE BEGIN WITH EARTHThe 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 ofbTsbh n

bfbf,bTnbf

nbfbf

nbfbf Q~Q,Q~S,Q~B,Q~H

Why dimensionless?In order to allow scaling between Earth and Titan!

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Meet my friends the DIMENSIONLESS PARAMETERS

5/2bf

bf5/1

QHgH~

Particle Reynolds number

2bf,bT

T DgDQ

Q̂

5/2bf

bf5/1

QBgB~

2bf

DgDQQ̂

DRgDpRe

RDSHbf

bf

Dimensionless bankfull discharge

Dimensionless bankfull depth

Dimensionless bankfull width

Dimensionless bankfull bedload transport rate

Bankfull Shields number

Shields number at threshold of motion]1006.022.0[5.0 )7.7(6.0

pc

6.0p ReRe

S Down-channel bed slope

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DATA SETS FOR GRAVEL-BED RIVERS ON EARTH

1. Alberta streams, Canada1

2. Britain streams (mostly Wales)2

3. Idaho streams, USA3

4. 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 morphologyof mountain streams, Journal of Hydraulic Engineering, 129(11), 2003.4 Pitlick, J. and Cress, R., 2002, Downstream changes in the channel of alarge gravel bed river, Water Resources Research 38(10), 1216,doi:10.1029/2001WR000898, 2002.

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WHAT THE DATA SAY: WIDTH, DEPTH, SLOPEThe four independent sets of data form a coherent set!

0.0001

0.001

0.01

0.1

1

10

100

1.0E+02 1.0E+03 1.0E+04 1.0E+05 1.0E+06 1.0E+07

Qhat

Btil

de, H

tilde

, S

Britain widthAlberta widthIdaho widthColorado widthBritain depthAlberta depthIdaho depthColorado depthBritain slopeAlberta slopeIdaho slopeColorado slope

H~

B~

S

S,H~,B~

Q̂

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y = 0.3785x4E-05

y = 4.6977x0.0661

y = 0.1003x-0.3438

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07

Qdim

Bdi

mtil

de, H

dim

tilde

, S

BdimtildeHdimtildeSPower (Hdimtilde)Power (Bdimtilde)Power (S)

REGRESSION RELATIONS BASED ON THE DATA3438.00661.000004.0 Q̂1003.0S,Q̂698.4B~,Q̂3785.0H~

To a high degree of approximation,

3785.0H~H~ c Remarkable, no?

Q̂

S,H~,B~

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WHAT DOES THIS MEAN?

4.0bf5/1bf

4.0bfbf

Qg3785.0H

orQ~H

4661.0bf

5/10661.02/5bf

4661.0bfbf

QgDg698.4B

orQ~B

3438.0bf

3438.02/5

3438.0bf

QDg1001.0S

orQ~S

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WHAT THE DATA SAY: BANKFULL SHIELDS NUMBER

62.1r,03.0c

bfc

)average(0486.0~bf

Shields Diagram with Threshold for Motion, Threshold for Significant Suspension and Bankfull Shields Number for Gravel-bed Streams

0.001

0.01

0.1

1

10

1 10 100 1000 10000 100000 1000000

Rep

*

suspensionmotionAltaBritIdaColoAverage

threshold of motion (modified Shields curve)

threshold for significant suspension

]1006.022.0[5.0 )7.7(6.0pc

6.0p ReRe

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THE PHYSICS BEHIND IT ALLAssume 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

2645.0bf

bfbfbf

bf

DH732.3

gHHBQCz

5.4

bf

c2/3bf

bf

bf,bT 12.11DRgDB

Q

0562.0bf Q̂02301.0

62.1rc

bf

5504.0T Q̂003176.0Q̂

1

10

100

1 10 100 1000

Hhat

Cz

CzFitCz

D/Hbf

0.001

0.01

0.1

1

1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07

Qhat

taus

bf tausbfFitQ

Q̂

bf

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THE RELATIONS OF THE PREVIOUS SLIDE YIELD PRECISELY THE OBSERVED DIMENSIONLESS

RELATIONS!

3438.0

0661.0

o

Q̂1003.0S

Q̂698.4B~

3785.0H~H~

y = 0.3785x4E-05

y = 4.6977x0.0661

y = 0.1003x-0.3438

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07

Qdim

Bdi

mtil

de, H

dim

tilde

, S

S,H~,B~

Q̂

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GENERALIZATION FOR OTHER PLANETS/SATELLITESManning-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)

2645.0bf

bfbfbf

bf

DH732.3

gHHBQCz

5.4

bf

c2/3bf

bf

bf,bT 12.11DRgDB

Q

0562.0

2bfbf

DgDQ02301.0

RDSH

623.1rc

bf

5504.0

2bf

2bf,bT

DgDQ

R100841.0

DgDQ

The presence of g and R allow us to go from

to

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BACK-CALCULATED DIMENSIONALLY HOMOGENEOUS BANKFULL HYDRAULIC RELATIONS FOR

ALLUVIAL GRAVEL RIVERS ON

The presence of g and R allow us to go from

to

4.0bf5/1

7908.0

bf Qg

)R1(1751.0H

1653.0

4661.0bf

2331.0bf DQ

g)R1(R992.15B

3438.0bf

8595.0

7908.0

1719.0

QD

)R1(gR1314.0S

ARBITRARY HEAVENLY BODIES

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FROM TO

Parameter Earth TitanPressure E-atmo p 1 1.5Temperature K T ~ 293 ~ 95Grav. accel. m/s2 g 9.81 1.40Fluid dens. kg/m3 1000 446Sed. Dens. kg/m3 s 2650 931

(s/) - 1 R 1.65 1.09

Kin. Viscosity m2/s 1.00x10-6 4.04x10-7

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CONSIDER A STREAM WITH THE SAME BANKFULL DISCHARGE Qbf AND CHARACTERISTIC GRAIN SIZE D

HOW SHOULD TITAN COMPARE WITH EARTH?

From

to

7908.0

E

T

5/1

E

T

E,bf

T,bf

)R1()R1(

gg

HH

1

E

T

2/1

E

T

2331.0

E

T

E,bf

T,bf

)R1()R1(

RR

gg

BB

7908.0

E

T

E

T

1719.0

E

T

E

T

)R1()R1(

RR

gg

SS

E = Earth, T = Titan

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CONSIDER A STREAM WITH THE SAME BANKFULL DISCHARGE Qbf AND CHARACTERISTIC GRAIN SIZE D

HOW SHOULD TITAN COMPARE WITH EARTH?

7908.0

E

T

5/1

E

T

E,bf

T,bf

)R1()R1(

gg

HH

1

E

T

2/1

E

T

2331.0

E

T

E,bf

T,bf

)R1()R1(

RR

gg

BB

7908.0

E

T

E

T

1719.0

E

T

E

T

)R1()R1(

RR

gg

SS

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

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SO FOR THE SAME BANKFULL DISCHARGE Qbf AND CHARACTERISTIC GRAIN SIZE D

A gravel-bed river on

might be1.23 x the bankfull depth,2.46 x the bankfull width and0.57 x the down-channel slope

of a gravel-bed river on Could braiding be more common on Titan?

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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!

E,pT,p ReRe

E

EEEE

T

TTTT DDgRDDgR

20.1gg

RR

DD

3/1

E

T

3/1

E

T

3/2

E

T

E

T

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GRAIN REYNOLDS INVARIANCE

3320~p Re

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

]1006.022.0[5.0 )7.7(6.0pc

6.0p ReRe

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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 293K 1 E-atmo, a = 1.21 kg/m3

Titan (nitrogen) 95K 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!

WHAT ABOUT AEOLIAN PROCESSES ON TITAN?

Es

2af

Ts

2af

gDUC

gDUC

aa

16.0gg

UU

2/1

E

T

2/1

E,s

T,s

2/1

E,a

T,a

E,ac

T,ac

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QUESTIONS OR COMMENTS?

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DIMENSIONLESS BANKFULL HYDRAULIC RELATIONS FOR EARTH AND TITAN