GEO 5/6690 Geodynamics 14 Nov 2014 © A.R. Lowry 2014 Read for Wed 19 Nov: T&S 226-241 Last Time: Te and Dynamics Force balance for subduction slab angle.

GEO 5/6690 Geodynamics 14 Nov 2014

© A.R. Lowry 2014Read for Wed 19 Nov: T&S 226-241

Last Time: Te and Dynamics

Force balance for subduction slab angle (T&S 6-11!) includes slab buoyancy, (absolute) velocity of over-riding plate, aperture of wedge flow (plate thickness + subduction angle!) & viscosity of subduction wedge

Observation (Pérez-Gussinyé et al. 2008): South American flat slab (Peru, Chile) has (possibly) buoyant crust near southern edges, otherwise no strong correlation to buoyancy or trench velocity… But strong correlation to distance from trench to high Te!

Models confirm that thicker over-riding plate near the trench favors flat slab subduction (but only for idealized slab with unrealistic rheology… Breaks down for real-Earth?)

Moral?

Next Journal Article Reading:For Monday Nov 17: Karow & Hampel (2010) Slip rate variations on faults in the Basin-and-Range province caused by regression of Late Pleistocene Lake Bonneville and Lake Lahontan, Int. J. Earth Sci. 99 1941-1953.(Nick will lead)

Isostasy with a viscoelastic fluid asthenosphere (“Rebound”)Given some initial sinusoidal deflection of the Earth’s surface:

where k is wavenumber = 2/

what is the evolution of w through time?

€

w = w0 cos kx( )

Elevated beaches or“strandlines” in Hudson’sBay region of Canada…

We impose:

(1) Conservation of fluid: (2D)

(This is referred to as an incompressible fluid!)

u

€

u +∂u

∂xδx

€

v +∂v

∂zδz

v

xz

z + z

x + x

€

∂u

∂x+

∂v

∂z= 0

We impose:

(2) Elemental force balance: includes pressure forces, viscous forces, and body force (= gravity).

For pressure forces, the imbalance is givenby the pressuregradients: (2D) p(x)z

p(z)x

xz

z + z

x + x

p(z+z)x

p(x+dx)z

€

−∂p

∂xˆ x −

∂p

∂zˆ z

We impose:

(2) Cont’d: viscous forces: The net viscous forces are given by

(to first order). For a Newtonian fluid with viscosity ,

zz(z)x

xz

z + z

x + x

zz(z+z)x

xx(x)zxx(x+x)z

xz(x)z

xz(x+x)z

zx(z)x

zx(z+z)x

€

∂ xx

∂x+

∂τ zx

∂z

⎛

⎝ ⎜

⎞

⎠ ⎟ˆ x +

∂τ zz

∂z+

∂τ xz

∂x

⎛

⎝ ⎜

⎞

⎠ ⎟ˆ z

€

xx = 2η∂u

∂x;τ zz = 2η

∂v

∂z

€

zx = τ xz =η∂u

∂z

The body force is simply xz in the z direction; typically we simplify pressure by subtracting a hydrostatic:

P = p – gz

Then (after some algebra; see T&S) we have the force balance equations:

€

∂2u

∂x 2+

∂ 2u

∂z 2

⎛

⎝ ⎜

⎞

⎠ ⎟−

∂P

∂x= 0

€

∂2v

∂x 2+

∂ 2v

∂z 2

⎛

⎝ ⎜

⎞

⎠ ⎟−

∂P

∂z= 0

We can define a stream function (= potential of the flow field) such that (for an incompressible fluid):

is flow velocity in the x-direction;

is flow velocity in the z-direction (vertical).

Solutions must satisfy:

Substituting for u and v, find the biharmonic equation has (eigenfunction) solutions of the form:

€

u = −∂ψ

∂z

€

v = −∂ψ

∂x

€

∂4ψ

∂x 4+

2∂ 4ψ

∂x 2∂z 2+

∂ 4ψ

∂z 4=∇ 4ψ = 0

€

exp±2πz

λ

⎛

⎝ ⎜

⎞

⎠ ⎟

€

z exp±2πz

λ

⎛

⎝ ⎜

⎞

⎠ ⎟and

Solving (with boundary conditions) we find that:

Where decay constant

€

w t( ) = w0 exp−λρgt

4πη

⎛

⎝ ⎜

⎞

⎠ ⎟= w0 exp −

t

τ

⎛

⎝ ⎜

⎞

⎠ ⎟

€

=4πη

ρgλ=

2kη

ρg

Everybody’s favorite local example: Gilbert was the first to recognize that shorelines of former Lake Bonneville are higher in the middle than at the edges…

By a lot!