PHYS3070 Physics of the Earth:from seismic structure to geodynamics

‘Geophysics … has the rigour of physics and thevigour of geology’

C. M. R. Fowler

Goal: to connect the seismological investigation of theEarth’s internal structure (Hrvoje Tkalcic’s segment)

with Paul Tregoning’s component concerning geodeticobservations of surface deformations

Ian.Jackson@anu.edu.au

WebCT

http://rses.anu.edu.au/people/jackson_i/PHYS3070/

Part II Earth’s thermal regime &geodynamics

Introduction to geodynamics: plate tectonics evidence ofinternal dynamical processes

Accretional energy & heat transport mechanisms

Conductive cooling of oceanic lithosphere

Radioactive heat production & continental geotherms

Heat transport by advection

Brittle behaviour & faulting (near-surface)

Ductile behaviour & flow (@ depth)

Lithospheric flexure & glacial rebound

Plate & plume modes of mantle convection

Earth’s thermal evolution

Plate tectonics: the surface expression ofinternal dynamical processes

Fowler Fig. 2.1

Global seismicity maps localised brittle failure

The tectonic plates & their boundaries

Fowler Fig. 2.2

Thermal regime:gravitational energy of accretion

Gravitational potential energy dissipated during accretion of incrementalshell of radius R & thickness dR is

dE(R) = ∫ F(R,r)dr = GM(R)dM(R)∫ (∞,R)(1/r2)dr = GM(R)dM(R)/R

So that the gravitational energy of accretion is

E(R) = G∫ M(R)dM(R)/R = (3/5)GM2(R)/R (Lowrie, eq. 4.1)

The Earth’s thermal regime: a hot start

Gravitational energy of accretionEG = (3/5)GM2/R = 2 × 1032 J

with M = 6.0 × 1024 kg

R = 6.4 × 106 m

G = 6.7 × 10-11 m3 kg-1 s-2

Temperature rise ΔT = E/MCP = 4 × 104 K

CP = 1000 J kg-1 K-1

Heat capacity = MCP = 6 × 1027 J K-1

Melting?Latent heat of melting L = 4 × 105 J kg-1

EM = ML = 2.4 × 1030 J (< EG/100)

Heat transport by radiation

Radiated power density R = σT4

Stefan-Boltzmann constant σ = 5.7 × 10-8 Wm-2 K-4

Excess (90%) of accretional energy (1.8 × 1032 J) radiatedfrom present surface area of 5 × 1014 m2 @ T = 2000 K, in

only 104 y!

Transport of heat by conduction

Involves transport of kinetic energy by phonons (insulators)& conduction electrons (metals)

Fowler Fig. 7.2Heat budget: ρCP∂T/∂ t = - ∂q/∂ z + A

For k ≠ k(z), ∂T/∂ t = κ(∂ 2T/∂ z 2) + A/ρCP

Thermal diffusivity κ = k/ρCP

Thermal diffusion timescale τ for body of dimension D:

T/ τ ~ κT/(D/2)2 τ ~ D2/ 4κ

Heat conduction: qz = -k∂ T /∂ z

Heat production rate/unit volume : A Fowler Fig. 7.1

Fowler Fig. 7.1

The age of the Earth:Kelvin’s famous calculation

Conductive cooling of half-space initially at T = T0:

T(z,t) = T0erf {z/[2(κt)1/2]} with erf(x) = 2π-1/2 ∫ (0,x) exp(-β2) dβ

Near-surface gradient @ time t: (∂ T /∂ z )(0,t) = T0/(πκt)1/2

Thermal diffusivity κ = k/ρCP = 10-6 m2 s-1

Present sub-surface temperature gradient dT/dz = 30 K km-1

Initial (melting?) temperature T0 = 4000 K

Implied age of the Earth ~ 200 My

Radioactive heat generation

K:Th:U = 104:4:1 by massStrong concentration of these geochemically incompatible elements

by melting responsible for formation of crust

A ~ 3, 30 and 2600 nW m-3 for upper mantle, oceanic crust, andupper continental crust

Impact on thermal regime: Continental heat flow &

Earth’s overall thermal evolution

Radiometric dating based on measurement of concentrations ofparent P & daughter D isotopes:

P(t) = P0exp(-λt) and D(t)=P0-P(t) so that

t = (1/λ) ln[1 + D(t)/P(t)]

Age of the oceanic lithosphere

Fowler Plate 20 160 Ma

Ocean-floor topography

Fowler Plate 8

Oceanic lithosphere: observations

Sea-floor depth & heat flow ~ (age)-1/2

with δd ~ 4 km & q ~ 50 mW m-2 @ 100 Ma

Davies Fig. 4.6 Davies Fig. 4.7

Conductive cooling of oceanic lithosphere

Conductive cooling of half-space initially at T = T0 for all depths z:T(z,t) = T0 erf {z/[2(κt)1/2]} with erf(x) = 2π-1/2 ∫ (0,x)

exp (-u2) du

satisfies 1-D heat conduction equation with A = 0i. e. ∂T/∂ t = κ(∂ 2T/∂ z 2)

Depth to T-isotherm:z = 2 erf -1(T/T0) (κt)1/2

Fowler Fig. 7.8a

Fowler Fig. 7.5 Fowler Fig. 7.9

Plate model: T(L) = 1350°C

Plate model: T(L) = 1450°C

Half space model

erf(x)

1 - erf(x)

1

0

20

Conductively cooling oceanic lithosphereThickness of oceanic lithosphere ~ depth L to ~1050°C isotherm:

L = 2(κt)1/2 = 112 km

Heat flux q(z=0,t) = -k∂ T /∂ z |0 = -kT0/(πκt)1/2 = 39 mW m-2

Sea-floor subsidence

Isostasy c.f. Fowler pp. 292-3 (d + L)ρm = dρw + ∫(0,L) ρl(z,t) dz

with ρl(z,t) = ρm{1 + α[T0-T(z,t)]}

d(t) = αT0ρm/(ρm-ρw) ∫(0,L)[1 - erf(x)] dz with x = (z/2)(κt)-1/2

d(t) ≈ 2ρmαT0(κt/π)1/2 /(ρm-ρw) = 3.5 km

(∫ (0,L)[1 - erf(x)] dx ≈ ∫ (0,∞) [1 - erf(x)] dx = π-1/2;

T0=1300°C, κ=10-6 m2s-1, k=3 Wm-1K-1 & t=100 My)

Conductive geotherms for thecontinental lithosphere

(i) No radioactive heating

A=0, T(0)=T0, dT/dz(0)=T’0T(z)=T0 + T’0z

(ii) Radioactive heating A = A0e-z/h

T(z) = T0 + (T’0- A0h/k)z + (A0h2/k) (1-e-z/h )

= T0 + T’mz + Th (1-e-z/h ) approaches

T(z) = T0 + Th + T’mz at depth

Davies Fig. 7.6

Ancient lithosphere: assumesteady state, ∂ T /∂ t = 0

d2T/dz2 = -A/k

slope: T’0

slope: T’m

Continental lithosphere: basal heat flux & radioactive heat production

q0 = - 60 mWm-2 & k = 3 Wm-1K-1

T’0 = 20 K km-1

Heat production (A0 = 2.5 µW/m3, h =10 km)

T’m = T’0 - A0h/k = 11.7 K km-1

qm = -kT’m = -35 mWm-2, Th = A0h2/k = 83 K

Lithospheric thickness & thermal blanketing

qm = -kT’m ~ -k(Tm-Th)/D =14-36 Wm-2 for D = 100-250 km

(4-5 TW for mantle heat loss through continents

c.f. 42-44 TW globally)

Davies Fig. 7.6

Heat transport by advection

Efficiency of advection:qadv = vρCPΔT

e.g. mantle plume with v = 1 ma-1 & ΔT = 100 K, qadv ~ 10 W m-2

∂ T /∂ t ∂ T /∂ t + (∂ T /∂ z )dz/dt = ∂ T /∂ t + vz(∂ T /∂ z )∂T/∂ t + vi(∂ T /∂ xi) = κ ∂ 2T/∂ xi∂ xi + A/ρCP

(summation convention: sum over all repeated values i = 1, 2, 3 of index)

Mechanics of plate tectonics I: elastic-brittlebehaviour under near-surface conditions

At low temperature, elastic deformation gives way at high stress to brittlefailure described byσs = µfσn + C0

for frictional sliding on a prexisting fault (Byerlee’s law) or fracture of intactrock (Mohr-Coulomb criterion for failure)

where σs, σn & µf are respectively the shear stress, normal stress &coefficient of friction

Davies, Fig. 6.15

Optimal conditions for brittle failureunder near-surface conditions

For given σmax & σmin, σn & σs vary with angle θ betweenfault normal & σmax

(σmax,0)

(σmax,0)

Mohr circle construction for principal axes of stress tensor

Intersection of Mohr circle & failure envelope: failure onmost favourably oriented fault!

For µf ~ 0.6-0.8 expect 2θ = φ + π/2 θ ~ 60-65°Geological fault types: normal, reverse, strike-slip inclined

towards maximum principal stress

Davies Fig. 6.17 Davies Fig. 6.16

(σmax,0)(σmin,0)

(σn,σs)

Mechanics of plate tectonics II:the brittle-ductile transition

pressure inhibits brittle failure, butfluids lower effective values of σn & µf?

Experimental deformation ofWombeyan marble

Paterson & Wong Figs. 74 and 75

Viscous deformation of Earth’s deep interior

Relax condition ofrecoverability: permanent

viscous deformation

Burgers modelcaptures realistic

combination of (linear)elastic, anelastic &viscous behaviour

c.f., Lowrie Fig. 6.10 (replace Lowrie’s ‘viscoelastic’ by ‘anelastic’)

Linear viscous rheology: Newtonian viscosity

Diffusional flow (creep) ofa polycrystal: dε/dt ∝ σc.f. shear of fluid withNewtonian viscosity η

Simple shear of a fluidσxz = η dvx/dz (engineering)

Relate to tensor strain in solids:σxz = σ13, vx = du1/dt &

ε13 = (1/2)(du1/dx3 + du3/dx1)σ13 = 2η dε13/dt (geophysics)

Lowrie Fig. 6.8

Frost & Ashby Figure 2.7

Non-Newtonian (power-law) creep

More generally: dε/dt = A(σ/G)n(d/b)-m exp[-(E*+PV*)/RT]

with Burgers vector b, activation energy E* and volume V*

Strongly temperature & pressure-dependent effective viscosityηeff = σ/(dε/dt) ~ σ1-n dm exp[ (E*+PV*)/RT]

E* = 400 kJ mol-1: δT = +100°C δlogη = -0.8

V* = 5 cm3 mol-1: δP = +10 GPa δlogη = +1.6

Relaxation of requirement of linearity: power-law creep

Dislocation creep dε/dt ~ σn

with n ≥ 3Lowrie Figure 6.11

Strength vs depth in the lithosphere

Fowler Fig. 10.4, c.f. Lowrie Fig. 6.7

Strength (max. differential stress) increaseswith increasing pressure in the brittle regime,but decreases with increasing temperature in

the ductile regime

60 Ma Brittle-ductile

transition in quartz-rich rocks

Mechanical behaviour & geodynamics:elastic flexure of the oceanic lithosphere

2-D static flexure of thin elastic plate under load V(x): bendingmoment M ∝ 1/R = -d2w/dx2 & d2M/dX2 = net load

Dd4w/dx4 = V(x) - (ρm-ρw)gw

Elastic restoring force = Load - buoyancyFlexural rigidity D = Eh3/12(1-ν2)

with Young’s modulus E & Poisson’s ratio ν

sensitive to plate thickness h

Fowler Fig. 5.14, c.f.Lowrie Fig. 6.13a

ρw

ρm

Flexure of oceanic lithosphere

For line load V(x) = Vδ(x), e.g. chain of volcanic islands

Balance of distributed forces (with V(x) = 0):w(x) = w0e-x/α[cos(x/α)+sin(x/α)] for x ≥ 0 with

α = [4D/(ρm-ρw)g]1/4

w0 = Vα3/8D [from D ∫(0,∞)d4w/dx4dx = D [d3w/dx3]0 = V/2]

dw/dx = 0 bulge of amplitude wb/w0= exp(-π) ~0.04 @ xb = πα

Inference of lithospheric thickness hfrom observed xb:

xb α = xb/π D = α4(ρm-ρw)g/4

h = [12(1-ν2)D/E]1/3Fowler Fig. 5.15

Intactplate

Lithospheric bending @ subduction zones

Elastic flexure with load V @ one end x = 0 & bending momentM per unit length:

w(x) = (α2/2D) e-x/α[(Vα+M) cos(x/α) - M sin(x/α)] for x ≥ 0

Fowler Fig. 5.16 c.f. Lowrie Fig. 6.16

Elastic thickness of oceanic lithosphere c.f.other geophysical observables

Elastic flexure by volcanic loads & subduction h(age) typically10-30 km c.f. seismogenic regime - but thinner than ‘seismic

lithosphere’ - reflecting viscoelastic relaxation between s & My

Lowrie Fig. 6.17

FowlerFig. 5.17

Post-glacial rebound & mantle viscosityRebound of lithosphere following melting of ice loadthrough time-dependent viscous flow of uppermost

mantle beneath elastic plate:w(t) = w0 exp(-t/τ) with τ = 4πη/ρmgλ

(scaling analysis: w/λτ ~ ε/τ ~ dε/dt = σ/η ~ ρmgw/η)

FowlerFig. 5.19

present upliftrate, mm/y

FowlerFig. 5.18

Post-glacial rebound & mantle viscosity

τ = 4400 y for λ~1000 km η ~ 1020 Pa s

LowrieFig. 6.20uplift (t)

Lowrie Fig. 6.21uplift rate (x)

Lowrie Fig. 6.19depression (time)with exponential fit

Mantle viscosity structure

Fowler Fig. 5.22viscosity (depth)

Pronounced viscosityminimum in the uppermantle, below which ηincreases 30-100-fold

The viability of mantle convection?

With D = 3000 km, ρ = 4000 kg m-3, α = 2 × 10-5 K-1,

T = 1400°C, κ = 10-6 m2s-1, & η = 1022 Pa s

v = 2.8 × 10-9 m s-1 = 90 mm y-1

order-of-magnitude match to the higher plate velocities!

Buoyancy FB = VραΔTg ~ Dd ρgαT/2

Viscous resistance FV = Aηdv/dx = 2ηv

Boundary layer thickness d~(κt)1/2~(κD/v)1/2

Solution to force balance FB = FV: v = D[ρgαTκ1/2/4η]2/3

Davies Fig. 8.1

Combine understanding of conductivecooling of oceanic lithosphere & evidence

of viscous flow in the deeper mantle

Scaling relations from boundary-layermodel of mantle convection

(D/d)3 = ρgαTD3/4ηκ ~ Ra

(Rayleigh number ~106 for Earth’s mantle)

So we have the following scalings:

d/D ~ Ra-1/3 ~ 10-2

Peclet number v(D/κ) = v/V = Pe ~ Ra2/3

τconv/τcond = (D/v)/(D2/κ) = Ra-2/3 ~ 10-4

Convective timescale τconv= Ra-2/3 (D2/κ)

Heat flow q = kT/d = (kT/D)Ra1/3

Nusselt number Nu = q/qκ = q/(kT/D) = Ra1/3 ~100

efficiency of advection!

Onset of convection: marginal stability analysis

Fate of bulge in lower layer of less dense fluidBuoyancy: B = Δρwhg

Viscous resistance R = η(dε/dt)A = η(v/w)w = ηv = ηdh/dt

Force balance: (1/h)dh/dt = 1/τ with τ = η/gΔρw

Solution: h = h0 exp(t/τ)

Growth of bulge fastest for w ~ D: Rayleigh-Taylor instabilitywith τRT = η/gΔρD

For thermal convection, need τRT << τκ = D2/κ, i.e.

τκ/τRT = (gΔρD3/ ηκ) = Ra >> 1 (in fact Racrit ~ 1000)

Davies Fig. 8.2

Mathematical modelling of mantle convection:coupled fluid flow & heat transport

Flow of a viscous (incompressible) fluid:Conservation of mass ∂vi/∂xi = 0

Force balance 2∂(ηsij)/∂ xj - dP/dxi + Bi = 0

with strain-rate tensor sij = (1/2)(∂ v i/∂ xj + ∂ vj/∂ x i)c.f. strain tensor εij = (1/2)(∂ ui/∂ xj + ∂ uj/∂ x i)

Heat transport by conduction & advection:DT/Dt = ∂ T /∂ t + vi ∂ T /∂ x i = κ ∂ 2T/∂ x i

2 + A/ρCP

Solve coupled equations for flow field v(r) &temperature distribution T(r)

Heating modes & boundary layers

Davies Fig. 8.3

Development of lower boundarylayer depends on strength of

heating from below

Cold (high-η) upper thermalboundary layer dominant plate

mode of mantle convection

Hot (low- η) lower boundary layer subsidiary plume mode

Snapshots from 2-D numerical model (Davies Fig. 8.4)

Bottom heating Internal heating

Convection & the adiabatictemperature gradient

Adiabatic temperature gradient

(∂ T/∂ P)S = (∂ V/∂ S)P = (∂ V/∂ T)P/ (∂ S/∂ T)P (Maxwell relation)With α = (1/V)(∂ V/∂ T)P; CP = (∂ Q/∂ T)P = T(∂ S/∂ T)P & γ = αKSV/CP,

(∂ T/∂ P)S = αVT/CP = γT/KS

or (∂ lnT/∂ lnρ)S = − (∂ lnT/∂ lnV)S = γ

Super-adiabatic temperature gradient buoyancy convection (if Ra > Rac) reduction

of temperature gradient towards adiabat

Adiabatic temperature gradients

(∂T/∂z)S = (∂T/∂P)S (∂P/∂z)S = αgT/CP

Upper mantle: 0.4 K km-1 (α ~ 3 ×10-5 K-1, g ~ 10 m s-2,T ~ 1600 K, CP ~ 1200 J kg-1 K-1)

Lower mantle: 0.3 K km-1

Shallow outer core: 0.8 K km-1

Much shallower temperature gradients than for near-surface conductive regime (~20 K km-1) - reflecting

greater efficiency of heat transport by advection

Mantle adiabats & melting temperatures

The Gibbs free energy changes with P & T as dG = -SdT + VdP∴For the solid & liquid phases coexisting (GS = GL) at the unique meltingpoint Tm(P) of a single-component compound:d(GS-GL) = 0 = - (SS-SL)dTm + (VS-VL)dP, i.e. dTm/dP = ΔV/ΔS

Clapeyron-Clausius relation (ΔS = L/Tm, L being the latent heat of melting

Typically dTm/dP > (∂T/∂P)S

Fowler Fig. 7.17 Jackson & Rigden, Ringwood volume, CUP, 1998

Mantle convection & stirring

Compositionalheterogeneity survivesconvective mixing to

participate in subsequentmelting events

Fowler Fig. 8.16

isotherms

flow lines

Plate mode of mantle convection: summary

Lithospheric plates are an integral part of mantle convection,comprising the upper thermal boundary layer Plates are cold & ∴ ‘rigid’ - either elastic or viscoelastic with high η

But the lithosphere as a whole is mobile because of its capacity forbrittle failure allowing fracture into plates capable of relative motion

Plates sink back into the mantle because of negative buoyancyaccumulated during conductive cooling at the surface

A passive ascending flow, ultimately emerging at mid-ocean ridges, isrequired to conserve mass The pattern of major mantle downwellings & upwellings is ∴determined by the positions of ocean trenches & ridges, respectively

> 90% of the mantle’s heat loss is associated with the cycle involvingformation of oceanic lithosphere, its cooling, subduction & reheating

Davies pp. 290-1

Evidence for mantle plumes

Age-progressive volcanism asPacific plate moves WNW overa ‘hot-spot’ fixed within mantle

Davies Fig. 11.2

Davies Fig. 11.13

Plume-associated heatresponsible for both

volcanism & buoyancyneeded to support broad

topographic high?

Plume buoyancy & heat fluxes

Rate of increase of load associated with swell topographyW = g(ρm-ρw)hwu

Setting W = b, with swell height h = 1 km, width w = 1000 km & plate speedu = 0.1 my-1 Q = 0.2 TW for largest (Hawaiian) plume

Plume heat fluxQ = (πr2v)ρmCPΔT

Thermal buoyancy fluxb = (πr2v)Δρg = πr2vρmαΔTg

Davies Fig. 11.3Global plume heat flux~ 2 TW (6% of global

heat flow) hw

u

Entrainment & plume shapes

Uniformhigh-ηplumes

Laboratory studies ofcolumnar plumesrising from the hot

lower boundarylayer

Characteristic ‘head &tail’ shape of low-η

plumes

Davies Figs. 11.4 & 5a

Numerical modelling of thegrowth of a low-η plumehead by entrainment of

surrounding fluid

Davies Fig. 11.6

Plume mode of mantle convection: summary

Mantle plumes are less directly observed than lithosphericplates

Inferred from occurrence of isolated, slow-movingvolcanic hot-spots with associated topography (swells)

Swells constrain plume buoyancy flux & hence heat flow@ < 10% mantle heat budget

Plumes are ∴ a secondary, well-established mode ofmantle convection arising from instabilities in hot, low-η

boundary layer

Mantle convection overview

Figures complement summaries for plate & plume modes of convection

Davies Fig. 12.1

Davies Fig. 13.14

Earth’s long-term thermo-mechanical evolutionThermal evolution of a layer: ∂ T /∂ t = H(t)/C + [Qin(T)- Qout(T)]/MC

RHS: radioactive heating + net impact of advection∫ numerically from specified initial conditions

Davies Fig. 14.3

Punctuated evolution resulting fromepisodic disruption of an internal(transition-zone) boundary layer?

Smooth evolutionreflecting sustained

mantle-wide convection

Davies Fig. 14.7

Part II Earth’s thermal regime &geodynamics

Introduction to geodynamics: plate tectonics evidence ofinternal dynamical processes

Accretional energy & heat transport mechanisms

Conductive cooling of oceanic lithosphere

Radioactive heat production & continental geotherms

Heat transport by advection

Brittle behaviour & faulting (near-surface)

Ductile behaviour & flow (@ depth)

Lithospheric flexure & glacial rebound

Plate & plume modes of mantle convection

Earth’s thermal evolution

Physics of the Earth Honours Projects 2009

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