nonsequitur

Science searches for the true by tying to eliminate the untrue.

“It is a process of separating the demonstrably false from the probably true.” [Lynton Caldwell]

Authentic science operates on the assumption that a concept can be shown to be false. Falsification occurs when a concept either is shown to be logically inconsistent or is demonstrated to run counter to direct observations.

Pressure, Isostasy, and Horizontal Forces

• Each column isostatic equilibrium has the same weight of overburden at its base (equal pressure)

• If the mean densities and heights of each column are different, there will be a net horizontal force on the material (isostasy only reflects a vertical force balance)

Isostasy (same overburden at base) leads to net horizontal force Ph/2

2

h

h/2

P

Frhs = Ph/4Flhs =

P

2Ph

How big?

• Continental crust is ~40km thick• Continental shelves and their adjacent oceanic abyssal plains differ in

elevation by ~5-6km (5000m)• Mean density of continental crust is 2800 kg/m3 (2.8Mg/m3)• Pressure at base of continents (compensation depth) is 2800 kg/m3 x

10 m/s2 x 40,000m = 1.1GPa (1.1GN/m2)• Net horizontal force F =1.1GPa x 5000m / 2 = 2.75TN• Net horizontal stress associated with isostasy = F/40000m = 0.07GPa

i.e. = ~2,500/40,000 (1/16) of the pressure at the depth of compensation

• Implication: Lithosphere can elastically support stresses at least of order 70 MPa (atmospheric pressure = 0.1 MPa, 700 times less). In other words, crustal rocks do not typically creep under differential stresses of order 700 atmospheres (and may be even stronger)

Earth’s Rheology: Visco-elastic

• Rock becomes viscous at depth (below lithosphere)• Rock is elastic/brittle when cold (lithosphere)

F/A u

D

Analogy to rock deformation: Bragg’s bubble model

μ∂ue∂z

=FA

Elastic

η∂&uv∂z

=FAt

Viscous

ue = elastic displacement = DFA1μ⎛⎝⎜⎞⎠⎟ &uv=∂uv∂t=rate of viscous displacement = DFA1η⎛⎝⎜⎞⎠⎟ μ~1011 Pa (100GPa)

η~1021 Pa-s (1ZPa-s)

Earth’s Rheology: Visco-elastic

F/A u

D

Analogy to rock deformation: Bragg’s bubble model

μ∂ue∂z

=FA

Elastic

η∂&uv∂z

=FAt

Viscous

ue = elastic displacement = DFA1μ⎛⎝⎜⎞⎠⎟ &uv=∂uv∂t=rate of viscous displacement = DFA1η⎛⎝⎜⎞⎠⎟

u = ue +uv

u=DFA1μ+tη⎛⎝⎜⎞⎠⎟,

Main mechanisms for creep: Movement of imperfections in crystal lattice (dislocations &

vacancies)

How do we know Earth’s mantle is viscous?

• Isostasy exists (implies underlying mantle must flow to balance changes in crustal thickness/load) — but no direct inference for how viscous

• Post-glacial rebound (Areas covered by icesheets ~12,000 years ago are still rising ~m/100yr. Also see rebound from removal of large lakes and mountain glaciers)

• Uniform plate motions measured by GPS (no ‘jerks’ in motion of plate interiors, implies ‘fluid’ movement with underlying mantle)

• At ~1200°C+ temperatures, rocks exhibit viscous creep (flow) in laboratory experiments

Modern ice sheets (Antarctica)

Modern ice sheets (Greenland)

Pleistocene ice sheets (North)

Evidence for Pleistocene ice sheets (N.A.)

Pleistocene lakes (western US)

Evidence for Pleistocene lakes (western US)

Shoreline of Lake Bonneville

Postglacial Rebound

Shoreline of Lake Bonneville

PGR

Finland and Sweden are rising today

Holmström, 1869, pointed out that Sweden is rising with respect to the Baltic (comparison of historic water levels in harbors).

In the 1800s it was noticed that Lakes and Canels in Finland rose (and rose faster near the W. Coast

than inland). Raised beaches were also remarked upon (e.g. Gotland)

Gotland

Finnland (1)

Sweden (1)

Sweden (2)

Isostasy - PGR

• Nansen, 1928 - quantified the uplift rates of beachfronts in Gotland (100 m in 10.000 years)

• Daly, 1934 - suggested that postglacial rebound was the cause of this• Haskell, 1937 - quantified the uplift over an infinite halfspace mantle Van

Bemmelen & Berlage (1935) (Vening-Munesz, 1937)used uplift history to suggest a weak asthenosphere layer above a more viscous lower mantle

• 300m of uplift have already occurred in Fennoscandia, ~20m remains to be uplifted to reach isostatic equilibrium.

Angermann Uplift

Ice vs. Time

Sealevel Rise

Daly

Postglacial Rebound (uplift rates)

Shoreline of Lake Bonneville

Postglacial Rebound

Shoreline of Lake Bonneville

Why we know Earth can be viscous (postglacial rebound)

postglacial rebound (cont.)

wd

Velocity Gradient: W/d

Stress:

Upward ‘buoyancy’:

≈ηWd

=ηd

δhδt

≈Δgh

ηd

δhδt

= Δρgh or δhδt

= Δρgdη h = h

τ h=h0e−tτ

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

τ = ηΔgd

Horizontal Velocity Component

(at kx=Pi/2)

DepthVertical Velocity Component

(at kx=0)

Depth kz

Depth kz

Flow Solutions

Flow due to sinusoidal loading (view 1)

50%

80%

90%

kz

kx

Flow due to sinusoidal loading (view 2)

50%

80%

90%

kx

kz

DepthVertical Velocity Component

(at kx=0)

Depth kz

Depth kz

Depth of Flow

Flow Depth Depends on wavelength of load (k = 2/):

50% of flow above kz = 1.68, i.e. z < 0.27

80% of flow above kz = 3, i.e. z < 0.48

What happens to hot rising

blobs in mantle convection?

Once formed, what removes an asthenosphere layer?

• plate accretion&subduction

• Slab dragdown

asthenosphere

asthenosphere flow & entrainment

Pressure, Isostasy, and Horizontal Forces

• Each column isostatic equilibrium has the same weight of overburden at its base (equal pressure)

• If the mean densities and heights of each column are different, there will be a net horizontal force on the material (isostasy only reflects a vertical force balance)

Isostasy (same overburden at base) leads to net horizontal force Ph/2

2

h

h/2

P

Frhs = Ph/4Flhs =

P

2Ph

Dynamic Isostasy — Compensation of Pressure effects of lateral flow

h

d

P= ρa

-ρw

⎛⎝⎜

⎞⎠⎟

gδh

P = ρm

-ρa

⎛⎝⎜

⎞⎠⎟

gδd

Pa

w

m

Good depth vs. age fit of seafloor relief means that dynamic topography must have a relatively small

effect on seafloor depth

All oceans depth vs. ageRavine et al., in prep.

(J. Sclater & coworkers, 1980s)

Effects of buoyant asthenosphere

Lab Models

Numerical/Lab Comparison