Geotherms Reading: Fowler Ch 7 Equilibrium geotherms One layer model (a) Standard model: k = 2.5 W m -1 C -1 A = 1.25 x 10-6 W m -3 Q moho = 21 x 10-3 W m -2 shallow T-gradient: 30 C km -1 deep T-gradient: 15 C km -1 Conductivity reduce T-grad increases (b) Heat generation increase T-grad increases (c) Basal heat flow increase T-grad increases (d) 1
Timescales long Increase basal heat from (a) Q moho = 21 x 10-3 W m -2 to (d) Q moho = 42 x 10-3 W m -2 Consider rock at 20 km depth t = 0 567 C t = 20 Ma 580 C t = 100 Ma 700 C t = 734 C melting and intrusion are important heat transfer mechanisms in the lithosphere Timescales From the diffusion equation we can define the characteristic timescale the amount of time necessary for a temperature change to propagate a distance l thermal diffusivity characteristic thermal diffusion distance the distance a change in temperature will propagate in time thermal diffusivity of granite: 8.5 x 10-7 m 2 s -1 l = 10 m = 4 years l = 1 km = 37,000 years l = 100 km = 370 Ma 2
Instantaneous cooling T = 0 T = T 0 Semi-infinite half-space at temperature T 0 Allow to cool at surface where T = 0 No internal heating, use diffusion equation The solution is the error function time t 1 calc error func T = 0.9T 0 time t 2 calc error func T = 0.6T 0 time Oceanic heat flow observations Higher for younger crust (mostly) Greater variability for younger crust hydrothermal circulation at mid-ocean ridges Stein & Stein, 1994 3
Oceanic heat flow observations Mid- Atlantic Ridge Black Smokers 400 C water The Blue Lagoon Ocean basins Sediment thickness cuts off hydrothermal circulation 10 10 1 0 10 0.5 1 1 0.5 0.5 0 Thickest sediments found at the base of the continental slope landslides Thinnest at the ridge no time for deposition 4
Depth distribution The ocean basins Depth distribution is related to age ie the time available for cooling Good approximation to observation out to ~70 Ma squares: North Atlantic circles: North Pacific Depth and heat flow observations Depth works best till ~70 Ma for greater ages depth decreases more slowly Stein & Stein, 1994 Heat flux initially for greater ages Q decreases more slowly 5
A simple half-space model ridge T = T a T = 0 x 3D convection and advection equation z Assume: temperature field is in equilibrium advection of heat horizontally is greater than conduction Also, t = x/u i.e. distance and time related by the spreading rate We have already seen the solution. A simple half-space model ridge T = 0 x Temperature gradient z T = T a Surface heat flow differentiate T-gradient The observed heat flux was: this simple model provides the t 1/2 relation 6
A simple half-space model ridge T = 0 x Temperature gradient z T = T a Estimate the lithospheric thickness T at base of lithosphere: 1100 C and T a = 1300 C look up inverse error function if = 10-6 m 2 s -1 L in km, t in Ma 10 Ma L = 35 km 80 Ma L = 98 km reasonable? A simple half-space model Ocean depth apply isostasy Column of lithosphere at the ridge = Rearrange Need (z) density as a function of T coefficient of thermal expansion and T as a function of age Substitute 7
A simple half-space model Ocean depth apply isostasy Approximate L Rearrange Appropriate values: w = 1.0 x 10 3 km m -3 a = 3.3 x 10 3 km m -3 = 3 x 10-5 C -1 = 10-6 m 2 s -1 T a = 1300 C t in Ma and d in km Observed The simple half-space cooling model matches ocean depths out to ~70 Ma i.e. lithosphere cools, contracts and subsides The plate model The lithosphere has a fixed thickness at the ridge and cools with time The asthenosphere below is constant temperature ridge T = 0 x Simple half-space model ridge T = 0 x z T = T a T = T a z T = T a asymptotic values of Q, depth etc. cools and thickens for ever 8
Depth and heat flow observations Which model(s) fit the data? HS Half-space model GDH1 plate model PSM plate model The GDH1 plate model does a better job of fitting the depth data (which is better constrained) Stein & Stein, 1994 All fit the heat flow data (within error) Depth distribution The ocean basins Depth distribution is related to age ie the time available for cooling Good approximation to observation out to ~70 Ma Plate model: There is a limit to the lithospheric thickness available for cooling squares: North Atlantic circles: North Pacific 9
A hybrid? crust crust lithosphere Plate model fits depth and Q best but there is other geophysical evidence for a thickening lithosphere increasing elastic thickness increasing depth to low velocity asthenosphere lithosphere thermal boundary layer with small-scale convection Continents thicker crust similar lithosphere (cratons?) The mantle geotherm convection rather than conduction more rapid heat transfer Adiabatic temperature gradient Raise a parcel of rock If constant entropy: lower P expands larger volume reduced T This is an adiabatic gradient Convecting system close to adiabatic 10
The adiabatic temperature gradient Need the change of temperature with pressure at constant entropy, S using reciprocal theory Some thermodynamics Maxwell s thermodynamic relation coefficient of thermal expansion specific heat Substitute adiabatic gradient as a function of pressure The adiabatic temperature gradient adiabatic gradient as a function of pressure but we want it as a function of depth For the Earth Substitute adiabatic gradient as a function of radius Temperature gradient for the uppermost mantle at greater depth 0.4 C km -1 0.3 C km -1 using T = 1700 K = 3 x 10-5 C -1 g = 9.8 m s -2 due to reduced cp = 1.25 x 103 J kg C -1 11
Adiabatic temperature gradients Models agree that gradient is close to adiabatic, particularly in upper mantle why would it not be adiabatic? greater uncertainty for the lowest 500-1000 km of the mantle big range of estimated T for CMB 2500K to ~4000K This is the work of Jeanloz and Bukowinski in our dept Melting in the mantle 100 km 200 km Different adiabatic gradient for fluids: ~ 1 C km -1 Potential temperature: T of rock at surface if rises along the adiabat 12