[PPT] - Lecture: Advection of the Earths surface The advection equation PowerPoint Presentation

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www.helsinki.fi/yliopisto Intro to Quantitative Geology

Class overview today - November 19, 2018

Lecture: Advection of the Earth’s surface
The advection equation
Application: Bedrock river incision
Exercise 4: River advection

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SLIDE 2

Intro to Quantitative Geology www.helsinki.fi/yliopisto

Introduction to Quantitative Geology

Advection of the Earth’s surface:  Fluvial incision and rock uplift

Lecturer: David Whipp david.whipp@helsinki.fi 19.11.2018

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SLIDE 3

www.helsinki.fi/yliopisto Intro to Quantitative Geology

Goals of this lecture

Introduce the advection equation
Discuss application of the advection equation to bedrock

river erosion

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SLIDE 4

www.helsinki.fi/yliopisto Intro to Quantitative Geology

What is advection?

Advection involves a lateral translation of some quantity
For example, the transfer of heat by physical movement of

molecules or atoms within a material. A type of convection, mostly applied to heat transfer in solid materials.

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PICTURE HERE

http://homepage.usask.ca/~sab248/

SLIDE 5

www.helsinki.fi/yliopisto Intro to Quantitative Geology

What is advection?

Advection involves a lateral translation of some quantity
For example, the transfer of heat by physical movement of

molecules or atoms within a material. A type of convection, mostly applied to heat transfer in solid materials.

5

PICTURE HERE

http://homepage.usask.ca/~sab248/

SLIDE 6

www.helsinki.fi/yliopisto Intro to Quantitative Geology

∂h ∂t = −1 ρ ∂q ∂x q = −ρκ∂h ∂x

Diffusion equation

Last week we were introduced to the diffusion

equation

Flux (transport of mass or transfer of energy)

proportional to a gradient

Conservation of mass: Any change in flux results in a

change in mass/energy

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www.helsinki.fi/yliopisto Intro to Quantitative Geology

Substitute the upper equation on the left into the lower

to get the classic diffusion equation

푞 = sediment flux per unit length

휌 = bulk sediment density  휅 = sediment diffusivity  ℎ = elevation  푥 = distance from divide  푡 = time

∂h ∂t = −κ∂2h ∂x2

Diffusion equation

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Diffusion

∂h ∂t = −1 ρ ∂q ∂x q = −ρκ∂h ∂x

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www.helsinki.fi/yliopisto Intro to Quantitative Geology

This week we meet the advection equation
Two key differences:
Change in mass/energy with time

proportional to gradient, rather than curvature (or change in gradient)

Advection coefficient 푐 has units of [퐿/

푇], rather than [퐿2/푇]

∂h ∂t = c∂h ∂x

Advection and diffusion equations

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Diffusion Advection

∂h ∂t = −κ∂2h ∂x2

SLIDE 9

www.helsinki.fi/yliopisto Intro to Quantitative Geology

This week we meet the advection equation
Two key differences:
Change in mass/energy with time

proportional to gradient, rather than curvature (or change in gradient)

Advection coefficient 푐 has units of [퐿/푇],

rather than [퐿2/푇]

Advection and diffusion equations

9

∂h ∂t = c∂h ∂x

Diffusion Advection

∂h ∂t = −κ∂2h ∂x2

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www.helsinki.fi/yliopisto Intro to Quantitative Geology

(a) (b)

bedrock channel alluvial channel

t1 t2 t3 t1 t2 t3

Fig. 1.7, Pelletier, 2008
This week we meet the advection equation
Two key differences:
Change in mass/energy with time

proportional to gradient, rather than curvature (or change in gradient)

Advection coefficient 푐 has units of [퐿/푇],

rather than [퐿2/푇]

Advection and diffusion equations

10

Advection Diffusion

River channel profiles

t1 t2 t3 t1

∂h ∂t = c∂h ∂x

Diffusion Advection

∂h ∂t = −κ∂2h ∂x2

SLIDE 11

www.helsinki.fi/yliopisto Intro to Quantitative Geology

(a) (b)

bedrock channel alluvial channel

t1 t2 t3 t1 t2 t3

Fig. 1.7, Pelletier, 2008
This week we meet the advection equation
Two key differences:
Change in mass/energy with time

proportional to gradient, rather than curvature (or change in gradient)

Advection coefficient 푐 has units of [퐿/푇],

rather than [퐿2/푇]

Advection and diffusion equations

11

Advection Diffusion t1 t2 t3 t1 t2

River channel profiles

∂h ∂t = c∂h ∂x

Diffusion Advection

∂h ∂t = −κ∂2h ∂x2

SLIDE 12

www.helsinki.fi/yliopisto Intro to Quantitative Geology

(a) (b)

bedrock channel alluvial channel

t1 t2 t3 t1 t2 t3

Fig. 1.7, Pelletier, 2008
This week we meet the advection equation
Two key differences:
Change in mass/energy with time

proportional to gradient, rather than curvature (or change in gradient)

Advection coefficient 푐 has units of [퐿/푇],

rather than [퐿2/푇]

Advection and diffusion equations

12

Advection Diffusion t1 t2 t3 t1 t2 t3

River channel profiles

∂h ∂t = c∂h ∂x

Diffusion Advection

∂h ∂t = −κ∂2h ∂x2

SLIDE 13

www.helsinki.fi/yliopisto Intro to Quantitative Geology

(a) (b)

bedrock channel alluvial channel

t1 t2 t3 t1 t2 t3

Fig. 1.7, Pelletier, 2008
This week we meet the advection equation
Two key differences:
Change in mass/energy with time

proportional to gradient, rather than curvature (or change in gradient)

Advection coefficient 푐 has units of [퐿/푇],

rather than [퐿2/푇]

Advection and diffusion equations

13

t1 t1 t2 t3 Advection Diffusion

River channel profiles

∂h ∂t = c∂h ∂x

Diffusion Advection

∂h ∂t = −κ∂2h ∂x2

SLIDE 14

www.helsinki.fi/yliopisto Intro to Quantitative Geology

(a) (b)

bedrock channel alluvial channel

t1 t2 t3 t1 t2 t3

Fig. 1.7, Pelletier, 2008
This week we meet the advection equation
Two key differences:
Change in mass/energy with time

proportional to gradient, rather than curvature (or change in gradient)

Advection coefficient 푐 has units of [퐿/푇],

rather than [퐿2/푇]

Advection and diffusion equations

14

Advection Diffusion t1 t2 t1 t2 t3

River channel profiles

∂h ∂t = c∂h ∂x

Diffusion Advection

∂h ∂t = −κ∂2h ∂x2

SLIDE 15

www.helsinki.fi/yliopisto Intro to Quantitative Geology

(a) (b)

bedrock channel alluvial channel

t1 t2 t3 t1 t2 t3

Fig. 1.7, Pelletier, 2008
This week we meet the advection equation
Two key differences:
Change in mass/energy with time

proportional to gradient, rather than curvature (or change in gradient)

Advection coefficient 푐 has units of [퐿/푇],

rather than [퐿2/푇]

Advection and diffusion equations

15

Advection Diffusion t1 t2 t3 t1 t2 t3

River channel profiles

∂h ∂t = c∂h ∂x

Diffusion Advection

∂h ∂t = −κ∂2h ∂x2

SLIDE 16

www.helsinki.fi/yliopisto Intro to Quantitative Geology

(a) (b)

bedrock channel alluvial channel

t1 t2 t3 t1 t2 t3

Fig. 1.7, Pelletier, 2008
Diffusion: Rate of erosion depends on change

in hillslope gradient (curvature)

Advection: Rate of erosion is directly

proportional to hillslope gradient

Also, no conservation of mass (deposition)

Advection and diffusion equations

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Advection Diffusion t1 t2 t3 t1 t2 t3

River channel profiles

∂h ∂t = c∂h ∂x

Diffusion Advection

∂h ∂t = −κ∂2h ∂x2

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www.helsinki.fi/yliopisto Intro to Quantitative Geology

∂h c ∂t ∂h ∂t = c∂h ∂x

Advection at a constant rate 푐

Surface elevation changes in direct proportion to surface slope
Result is lateral propagation of the topography or river channel

profile

Although this is interesting, it is not that common in nature

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푐

Elevation change Displacement

River channel profile

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Advection of the Earth’s surface: An example

Bedrock river erosion
Purely an advection problem

with a spatially variable advection coefficient

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Athabasca Falls, Jasper National Park, Canada

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Bedrock river erosion

Not much bedrock being eroded here…

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Drainage basin

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Bedrock river erosion

Rapid bedrock incision has formed a steep gorge in

this case

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Drainage basin

Kali Gandaki river gorge, central Nepal http://en.wikipedia.org/

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www.helsinki.fi/yliopisto Intro to Quantitative Geology

River erosion as an advection process

With a constant advection coefficient 푐, we predict lateral

migration of the river profile at a constant rate (푐)

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(a)

bedrock channel

t1 t2 t3

Fig. 1.7, Pelletier, 2008

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www.helsinki.fi/yliopisto Intro to Quantitative Geology

River erosion as an advection process

With a constant advection coefficient 푐, we predict lateral

migration of the river profile at a constant rate (푐)

Do you think this works in real (bedrock) rivers?

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(a)

bedrock channel

t1 t2 t3

Fig. 1.7, Pelletier, 2008

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www.helsinki.fi/yliopisto Intro to Quantitative Geology

River erosion as an advection process

With a constant advection coefficient 푐, we predict lateral

migration of the river profile at a constant rate (푐)

Do you think this works in real (bedrock) rivers?
What might affect the rate of lateral migration?

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(a)

bedrock channel

t1 t2 t3

Fig. 1.7, Pelletier, 2008

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www.helsinki.fi/yliopisto Intro to Quantitative Geology

What affects the efficiency of river erosion?

The amount of water flowing in the river (discharge) and

sediment

The slope of the river channel
The strength of the underlying bedrock

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(a)

bedrock channel

t1 t2 t3

Fig. 1.7, Pelletier, 2008

Discharge Slope Bedrock strength

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www.helsinki.fi/yliopisto Intro to Quantitative Geology

What affects the efficiency of river erosion?

The amount of water flowing in the river (discharge) and

sediment

The slope of the river channel
The strength of the underlying bedrock
Are these constant?

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(a)

bedrock channel

t1 t2 t3

Fig. 1.7, Pelletier, 2008

Discharge Slope Bedrock strength

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www.helsinki.fi/yliopisto Intro to Quantitative Geology

Rather than being constant, the rate of lateral advection in

river systems is spatially variable        where 푘푓 is a material property of the bedrock (erodibility),  푤 is the channel width, and 푄 is discharge

∂h ∂t = c∂h ∂x ∂h ∂t = kf w Q∂h ∂x

Stream-power model of river incision

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www.helsinki.fi/yliopisto Intro to Quantitative Geology

Rather than being constant, the rate of lateral advection in

river systems is spatially variable        where 푘푓 is a material property of the bedrock (erodibility),  푤 is the channel width, and 푄 is discharge

This is known as the stream-power erosion model

∂h ∂t = c∂h ∂x ∂h ∂t = kf w Q∂h ∂x

Stream-power model of river incision

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∂h ∂t = KAmSn ∂h ∂t = kf w Q∂h ∂x

Stream-power model of river incision

If we assume precipitation is uniform in the drainage basin,

discharge 푄 will scale with drainage basin area, so we can modify our equation to read        where 퐾 is an erosional efficiency factor (accounts for lithology, climate, channel geometry, sediment supply, etc. (!)),  퐴 is upstream drainage area, 푆 is channel slope, and 푚 and 푛 are area and slope exponents

If we assume the drainage basin area increases with distance

from the drainage divide 푥, we can replace the area with an estimate 퐴 = 푥5/3

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Test your might

Based on our stream-power erosion equation, what general

form would a channel profile take?

If we assume we have reached a steady state (휕h/휕t = 0) and

푛 = 1, erosion must balance uplift U everywhere

If we further assume precipitation is constant, bedrock

erodibility is constant and 퐴 = 푥5/3, how would the channel steepness vary as you move downstream from the divide?

Think about how S would change as x increases

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h x

Initial geometry

U

∂h ∂t = U − KAmSn

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Evolution of a channel profile

A few stream-power erosion
bservations:
Stream power increases

downstream as the discharge grows

Steeper slopes occur upstream

where the discharge is low

Incision migrates upstream until

a balance is attained between erosion and uplift

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Fig. 3.23, Allen, 1997

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Recap

What is the main difference between the advection and

diffusion equations?

What is special about the stream power erosion model

compared to the general advection equation?

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Recap

What is the main difference between the advection and

diffusion equations?

What is special about the stream power erosion model

compared to the general advection equation?

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References

Allen, P . A. (1997). Earth Surface Processes (First edition.). Wiley-Blackwell. Pelletier, J. D. (2008). Quantitative modeling of earth surface processes (Vol. 304). Cambridge University Press.

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Class overview today - November 19, 2018

Introduction to Quantitative Geology

Advection of the Earth’s surface: Fluvial incision and rock uplift

Lecturer: David Whipp david.whipp@helsinki.fi 19.11.2018

Goals of this lecture

river erosion

What is advection?

molecules or atoms within a material. A type of convection, mostly applied to heat transfer in solid materials.

PICTURE HERE

What is advection?

molecules or atoms within a material. A type of convection, mostly applied to heat transfer in solid materials.

PICTURE HERE

∂h ∂t = −1 ρ ∂q ∂x q = −ρκ∂h ∂x

Diffusion equation

equation

proportional to a gradient

change in mass/energy

to get the classic diffusion equation

휌 = bulk sediment density 휅 = sediment diffusivity ℎ = elevation 푥 = distance from divide 푡 = time

∂h ∂t = −κ∂2h ∂x2

Diffusion equation

Diffusion

∂h ∂t = −1 ρ ∂q ∂x q = −ρκ∂h ∂x

proportional to gradient, rather than curvature (or change in gradient)

푇], rather than [퐿2/푇]

∂h ∂t = c∂h ∂x

Advection and diffusion equations

Diffusion Advection

∂h ∂t = −κ∂2h ∂x2

proportional to gradient, rather than curvature (or change in gradient)

rather than [퐿2/푇]

Advection and diffusion equations

∂h ∂t = c∂h ∂x

Diffusion Advection

∂h ∂t = −κ∂2h ∂x2

proportional to gradient, rather than curvature (or change in gradient)

rather than [퐿2/푇]

Advection and diffusion equations

Advection Diffusion

River channel profiles

t1 t2 t3 t1

∂h ∂t = c∂h ∂x

Diffusion Advection

∂h ∂t = −κ∂2h ∂x2

proportional to gradient, rather than curvature (or change in gradient)

rather than [퐿2/푇]

Advection and diffusion equations

Advection Diffusion t1 t2 t3 t1 t2

River channel profiles

∂h ∂t = c∂h ∂x

Diffusion Advection

∂h ∂t = −κ∂2h ∂x2

proportional to gradient, rather than curvature (or change in gradient)

rather than [퐿2/푇]

Advection and diffusion equations

Advection Diffusion t1 t2 t3 t1 t2 t3

River channel profiles

∂h ∂t = c∂h ∂x

Diffusion Advection

∂h ∂t = −κ∂2h ∂x2

proportional to gradient, rather than curvature (or change in gradient)

rather than [퐿2/푇]

Advection and diffusion equations

t1 t1 t2 t3 Advection Diffusion

River channel profiles

∂h ∂t = c∂h ∂x

Diffusion Advection

∂h ∂t = −κ∂2h ∂x2

proportional to gradient, rather than curvature (or change in gradient)

rather than [퐿2/푇]

Advection and diffusion equations

Advection Diffusion t1 t2 t1 t2 t3

River channel profiles

∂h ∂t = c∂h ∂x

Diffusion Advection

∂h ∂t = −κ∂2h ∂x2

proportional to gradient, rather than curvature (or change in gradient)

rather than [퐿2/푇]

Advection and diffusion equations

Advection Diffusion t1 t2 t3 t1 t2 t3

Advection of the Earth’s surface:  Fluvial incision and rock uplift

휌 = bulk sediment density  휅 = sediment diffusivity  ℎ = elevation  푥 = distance from divide  푡 = time

river systems is spatially variable        where 푘푓 is a material property of the bedrock (erodibility),  푤 is the channel width, and 푄 is discharge

river systems is spatially variable        where 푘푓 is a material property of the bedrock (erodibility),  푤 is the channel width, and 푄 is discharge