Introduce the advection equation Discuss application of the - PowerPoint PPT Presentation

Introduction to Quantitative Geology Advection of the Earth’s surface: 
 Fluvial incision and rock uplift Lecturer: David Whipp david.whipp@helsinki.fi 20.11.2017 Intro to Quantitative Geology www.helsinki.fi/yliopisto 3

Goals of this lecture • Introduce the advection equation • Discuss application of the advection equation to bedrock river erosion Intro to Quantitative Geology www.helsinki.fi/yliopisto 4

What is advection? PICTURE HERE http://homepage.usask.ca/~sab248/ • 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. Intro to Quantitative Geology www.helsinki.fi/yliopisto 5

What is advection? PICTURE HERE http://homepage.usask.ca/~sab248/ • 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. Intro to Quantitative Geology www.helsinki.fi/yliopisto 5

Diffusion equation • Last week we were introduced to the diffusion q = − ρκ∂ h equation ∂ x • Flux (transport of mass or transfer of energy) proportional to a gradient ∂ h ∂ q ∂ t = − 1 • Conservation of mass: Any change in flux results in a ρ ∂ x change in mass/energy Intro to Quantitative Geology www.helsinki.fi/yliopisto 6

Diffusion equation Diffusion ∂ t = − κ∂ 2 h ∂ h ∂ x 2 • Substitute the upper equation on the left into the lower q = − ρκ∂ h to get the classic diffusion equation ∂ x • 푞 = sediment flux per unit length 
 휌 = bulk sediment density 
 ∂ h ∂ q ∂ t = − 1 휅 = sediment diffusivity 
 ρ ∂ x ℎ = elevation 
 푥 = distance from divide 
 푡 = time Intro to Quantitative Geology www.helsinki.fi/yliopisto 7

Advection and diffusion equations Diffusion Advection ∂ t = − κ∂ 2 h ∂ h ∂ t = c ∂ h ∂ h ∂ x 2 ∂ x • 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 [ 퐿 2 / 푇 ] Intro to Quantitative Geology www.helsinki.fi/yliopisto 8

Advection and diffusion equations Diffusion Advection ∂ t = − κ∂ 2 h ∂ h ∂ t = c ∂ h ∂ h ∂ x 2 ∂ x • 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 [ 퐿 2 / 푇 ] Intro to Quantitative Geology www.helsinki.fi/yliopisto 9

Advection and diffusion equations Diffusion Advection River channel profiles ∂ t = − κ∂ 2 h ∂ h ∂ t = c ∂ h ∂ h Advection ∂ x 2 ∂ x (a) bedrock • channel This week we meet the advection equation t 3 t 2 t 1 t 3 t 2 t 1 • t 1 Two key differences: (b) alluvial channel t 1 t 2 t 3 • Change in mass/energy with time proportional to gradient, rather than Diffusion curvature (or change in gradient) Fig. 1.7, Pelletier, 2008 • Advection coefficient 푐 has units of [ 퐿 / 푇 ], rather than [ 퐿 2 / 푇 ] Intro to Quantitative Geology www.helsinki.fi/yliopisto 10

Advection and diffusion equations Diffusion Advection River channel profiles ∂ t = − κ∂ 2 h ∂ h ∂ t = c ∂ h ∂ h Advection ∂ x 2 ∂ x (a) bedrock • channel This week we meet the advection equation t 3 t 2 t 1 t 3 t 2 t 1 • t 1 Two key differences: (b) alluvial t 2 channel t 1 t 2 t 3 • Change in mass/energy with time proportional to gradient, rather than Diffusion curvature (or change in gradient) Fig. 1.7, Pelletier, 2008 • Advection coefficient 푐 has units of [ 퐿 / 푇 ], rather than [ 퐿 2 / 푇 ] Intro to Quantitative Geology www.helsinki.fi/yliopisto 11

Advection and diffusion equations Diffusion Advection River channel profiles ∂ t = − κ∂ 2 h ∂ h ∂ t = c ∂ h ∂ h Advection ∂ x 2 ∂ x (a) bedrock • channel This week we meet the advection equation t 3 t 2 t 1 t 3 t 2 t 1 • t 1 Two key differences: (b) alluvial t 2 channel t 1 t 2 t 3 • t 3 Change in mass/energy with time proportional to gradient, rather than Diffusion curvature (or change in gradient) Fig. 1.7, Pelletier, 2008 • Advection coefficient 푐 has units of [ 퐿 / 푇 ], rather than [ 퐿 2 / 푇 ] Intro to Quantitative Geology www.helsinki.fi/yliopisto 12

Advection and diffusion equations Diffusion Advection River channel profiles ∂ t = − κ∂ 2 h ∂ h ∂ t = c ∂ h ∂ h Advection ∂ x 2 ∂ x (a) bedrock • channel This week we meet the advection equation t 1 t 3 t 2 t 1 • t 1 Two key differences: (b) alluvial t 2 channel t 1 t 2 t 3 • t 3 Change in mass/energy with time proportional to gradient, rather than Diffusion curvature (or change in gradient) Fig. 1.7, Pelletier, 2008 • Advection coefficient 푐 has units of [ 퐿 / 푇 ], rather than [ 퐿 2 / 푇 ] Intro to Quantitative Geology www.helsinki.fi/yliopisto 13

Advection and diffusion equations Diffusion Advection River channel profiles ∂ t = − κ∂ 2 h ∂ h ∂ t = c ∂ h ∂ h Advection ∂ x 2 ∂ x (a) bedrock • channel This week we meet the advection equation t 2 t 1 t 3 t 2 t 1 • t 1 Two key differences: (b) alluvial t 2 channel t 1 t 2 t 3 • t 3 Change in mass/energy with time proportional to gradient, rather than Diffusion curvature (or change in gradient) Fig. 1.7, Pelletier, 2008 • Advection coefficient 푐 has units of [ 퐿 / 푇 ], rather than [ 퐿 2 / 푇 ] Intro to Quantitative Geology www.helsinki.fi/yliopisto 14

Advection and diffusion equations Diffusion Advection River channel profiles ∂ t = − κ∂ 2 h ∂ h ∂ t = c ∂ h ∂ h Advection ∂ x 2 ∂ x (a) bedrock • channel This week we meet the advection equation t 3 t 2 t 1 t 3 t 2 t 1 • t 1 Two key differences: (b) alluvial t 2 channel t 1 t 2 t 3 • t 3 Change in mass/energy with time proportional to gradient, rather than Diffusion curvature (or change in gradient) Fig. 1.7, Pelletier, 2008 • Advection coefficient 푐 has units of [ 퐿 / 푇 ], rather than [ 퐿 2 / 푇 ] Intro to Quantitative Geology www.helsinki.fi/yliopisto 15

Advection and diffusion equations Diffusion Advection River channel profiles ∂ t = − κ∂ 2 h ∂ h ∂ t = c ∂ h ∂ h Advection ∂ x 2 ∂ x (a) bedrock channel t 3 t 2 t 1 t 3 • t 2 t 1 Diffusion : Rate of erosion depends on change t 1 in hillslope gradient (curvature) (b) alluvial t 2 channel t 1 t 2 t 3 t 3 Diffusion • Advection : Rate of erosion is directly Fig. 1.7, Pelletier, 2008 proportional to hillslope gradient • Also, no conservation of mass (deposition) Intro to Quantitative Geology www.helsinki.fi/yliopisto 16

Advection at a constant rate 푐 River channel profile ∂ h 푐 Elevation change ∂ h ∂ t = c ∂ h ∂ t ∂ x c ∂ t Displacement • 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 Intro to Quantitative Geology www.helsinki.fi/yliopisto 17

Advection of the Earth’s surface: An example Athabasca Falls, Jasper National Park, Canada • Bedrock river erosion • Purely an advection problem with a spatially variable advection coefficient Intro to Quantitative Geology www.helsinki.fi/yliopisto 18

Bedrock river erosion Drainage basin • Not much bedrock being eroded here… Intro to Quantitative Geology www.helsinki.fi/yliopisto 19

Bedrock river erosion Drainage basin Kali Gandaki river gorge, central Nepal http://en.wikipedia.org/ • Rapid bedrock incision has formed a steep gorge in this case Intro to Quantitative Geology www.helsinki.fi/yliopisto 20

River erosion as an advection process (a) bedrock channel t 3 t 2 t 1 Fig. 1.7, Pelletier, 2008 • With a constant advection coefficient 푐 , we predict lateral migration of the river profile at a constant rate ( 푐 ) Intro to Quantitative Geology www.helsinki.fi/yliopisto 21

River erosion as an advection process (a) bedrock channel t 3 t 2 t 1 Fig. 1.7, Pelletier, 2008 • 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? Intro to Quantitative Geology www.helsinki.fi/yliopisto 21

River erosion as an advection process (a) bedrock channel t 3 t 2 t 1 Fig. 1.7, Pelletier, 2008 • 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? Intro to Quantitative Geology www.helsinki.fi/yliopisto 21

What affects the efficiency of river erosion? Discharge (a) bedrock Slope Bedrock channel strength t 3 t 2 t 1 Fig. 1.7, Pelletier, 2008 • The amount of water flowing in the river (discharge) and sediment • The slope of the river channel • The strength of the underlying bedrock Intro to Quantitative Geology www.helsinki.fi/yliopisto 22