Mantle Convec,on and Plate Driving Forces Ge11d Plate Mo,ons w.r.t. Africa Outline •  Oceanic lithosphere as a thermal boundary layer –  Heat Flow –  Bathymetry (ocean depth) •  Boundary Layer Theory of Convec,on •  Slab pull versus ridge Push •  Predic,on of plate veloci,es Simple coorling model (boundary layer, ‘half-­‐
space’ model appears to fit the data very well. Sclater et al. [1980] Cooling of oceanic lithosphere Ts z Tm T L Fourier's Law :
∂T
q = −k
∂z
ΔT
~ −k
= −k(Tm − Ts )(κt) −1/ 2
L
Subsidence of the Ocean Floor Apply the principle of isostasy; more like PraV isostasy because each column has a different density down to some compensa,on depth Ts distance or age cTm But how does density vary with temperature? PraV Airy Thermal expansion and contraction
δρ = − ρoαδT = − ρmα (Tm − Ts )
ρw
h ρm
ρm + δρ
L+h ρm
L
Apply "Pratt" Isostatsy
(L + h) ρm = hρw + L( ρm + δρ)
h( ρm − ρw ) = Lδρ
δρ = − ρmα (Tm − Ts )
ρmα (Tm − Ts )
h=
(κt)1/ 2
ρm − ρw
Average Bathymetry or subsidence “Plate model” Sqrt(age) Sclater et al. [1980] Mantle temperatures Fit heat flow & bathymetry versus age data
q = −k(Tm − Ts )(κt) −1/ 2
ρmα (Tm − Ts )
h=
(κt)1/ 2
ρm − ρw
In both cases
T ≈ 1500 o C
A simplified boundary layer theory of convec,on u
u
u
η
Boundary layer thickness
d ~ (κτ )
1/ 2
⎛ κD⎞
=
⎝ u⎠
1/ 2
since τ = D /u
Buoyancy force , Fb , in the slab is proportional
to T /2
Density in the slab is ραT /2
(average temperature in slabs is about T /2)
There is a conservation of buoyancy for the
slab as it descends through the mantle (i.e. the
horizontal diffusion of the slab spreads the
thermal anomaly out, but it doesn't cause it to
lose buoyancy).
Fb = byouancy for in slabs (per unit length of trench)
= gDdραT /2
D g d Viscous Resistance
u
strain rate =
= ε˙
D /2
D/2 u stress = σ = ηε˙ = 2ηu /D
FR = Dσ = 2ηu
D The system is in balance (no inertial forces)
FR = Fb
⎛ κD⎞
2ηu = gDραT /2
⎝ u⎠
⎛ κ ⎞ ⎛ gραTD ⎞
u=
⎝ D⎠ ⎜⎝ κη ⎟⎠
3
1/ 2
2/3
Ra = Rayleigh Number
gραTD3
=
κη
Buoyancy _ Forces
∝
viscous _ forces
⎛ κ ⎞ 2/3
∴u ≅
Ra
⎝ D⎠
Application of Boundary Layer Theory
gρoα v ΔTb 3
⎛κ ⎞
if Ra=
u ≈ ⎜ ⎟ Ra 2 / 3
⎝ b⎠
κη
u Quantity we wish to predict & compare Obs plate motions
b = 3 × 10 6 m, seismology
D κ = 10 −6 m 2s-1 lab
ρo = 4, 000 kg m -3 seismology
g = 10 m s-2 (constant through mantle)
α v = 2 × 10 −5 K -1 lab
T⎫
⎬ → dynamically determined by convection (must
η⎭
use values measured directly from the mantle)
η ~ 10 21 to 10 22 Pa s (post glacial rebound)
ΔT ≅ 1500 C K cooling oceanic lithosphere
Application of B.L.T.
Whole mantle convection
bD = 3000 km
η = 10 22 Pa s
u = 6 cm yr -1
'upper mantle' convection
bD = 1, 500 km
η = 10 21 Pa s
u = 8 cm yr -1
Average plate velocity on the Earth's surface 5 cm yr -1
⇒ B.L.T. is a plausible model
Plate Driving Forces Forsyth & Uyeda [1975] Plate Driving Forces (Empirical View) Forsyth & Uyeda [1975] Plate Driving Forces Forsyth & Uyeda [1975] Plate Driving Forces Forsyth & Uyeda [1975] Plate Driving Forces Forsyth & Uyeda [1975] Ridge Push: Force exerted on surface of plate due to eleva,on of an ocean ridge FR = F1 − F2 − F3
F ~ gρ α (T − T )κt
⎡ 2 ρ α (T − To ) ⎤
FR = Rgρmα v (Tm −m
To ) ⎢1 + m m v ms
⎥κ t
⎣ π ( ρ m − ρw ) ⎦
t, plate age
why FR ∝ t :
pressure ∝ t 1/2
length over which this presssure acts ∝ t 1/2
∴t 1/2t 1/2 → t
For t = 100 Myr
FR = 4.2 × 1012 N m -1
Slab Pull ~ 3 × 1013 N m -1
Ridge push ~ 4 × 1012 N m -1
Ridge Push ~ 10% Slab pull
Ridge push eliminated from B.L.T. (good assumoption)
Spherical models with radial varia,ons in viscosity can predict present day plate mo,ons Observed predicted Lithgow-­‐Bertelloni & Richards