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Geotherms Reading: Fowler Ch 7 EPS 122: Lecture 19 – Geotherms 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 Qmoho = 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) EPS 122: Lecture 19 – Geotherms 1 Timescales …long Increase basal heat from (a) Qmoho = 21 x 10-3 W m-2 to (d) Qmoho = 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 EPS 122: Lecture 19 – Geotherms 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 m2 s-1 l = 10 m = 4 years l = 1 km = 37,000 years l = 100 km = 370 Ma EPS 122: Lecture 19 – Geotherms 2 Instantaneous cooling T=0 Semi-infinite half-space at temperature T0 Allow to cool at surface where T = 0 T = T0 No internal heating, use diffusion equation The solution is the error function time t1 calc error func T = 0.9T0 time t2 calc error func T = 0.6T0 time EPS 122: Lecture 19 – Geotherms Oceanic heat flow – observations • Higher for younger crust (mostly) • Greater variability for younger crust Stein & Stein, 1994 hydrothermal circulation at mid-ocean ridges EPS 122: Lecture 19 – Geotherms 3 Oceanic heat flow – observations MidAtlantic Ridge Black Smokers 400°C water The Blue Lagoon EPS 122: Lecture 19 – Geotherms Ocean basins Sediment thickness cuts off hydrothermal circulation 0 0.5 10 10 10 1 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 EPS 122: Lecture 19 – Geotherms 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 EPS 122: Lecture 19 – Geotherms – observations Depth …works best till ~70 Ma for greater ages depth decreases more slowly Stein & Stein, 1994 Depth and heat flow Heat flux initially… for greater ages Q decreases more slowly EPS 122: Lecture 19 – Geotherms 5 A simple half-space model ridge x T=0 T = Ta 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…. EPS 122: Lecture 19 – Geotherms A simple half-space model ridge x Temperature gradient T = Ta T=0 z Surface heat flow …differentiate T-gradient The observed heat flux was: this simple model provides the t1/2 relation EPS 122: Lecture 19 – Geotherms 6 A simple half-space model ridge x Temperature gradient T = Ta T=0 z Estimate the lithospheric thickness… T at base of lithosphere: 1100 °C and Ta = 1300 °C look up inverse error function if = 10-6 m2 s-1 L in km, t in Ma 10 Ma L = 35 km 80 Ma L = 98 km reasonable? EPS 122: Lecture 19 – Geotherms 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… EPS 122: Lecture 19 – Geotherms 7 A simple half-space model Ocean depth …apply isostasy Approximate L Rearrange… Appropriate values: w = 1.0 x 103 km m-3 a = 3.3 x 103 km m-3 = 3 x 10-5 °C-1 = 10-6 m2 s-1 Ta = 1300 °C Observed… t in Ma and d in km The simple half-space cooling model matches ocean depths out to ~70 Ma i.e. lithosphere cools, contracts and subsides EPS 122: Lecture 19 – Geotherms The “plate” model The lithosphere has a fixed thickness at the ridge and cools with time The asthenosphere below is constant temperature ridge ridge x z x T = Ta T=0 T = Ta T=0 Simple half-space model T = Ta …asymptotic values of Q, depth etc. z …cools and thickens for ever EPS 122: Lecture 19 – Geotherms 8 Depth and heat flow – observations Which model(s) fit the data? HS – Half-space model GDH1 – plate model PSM – plate model Stein & Stein, 1994 The GDH1 “plate” model does a better job of fitting the depth data (which is better constrained) All fit the heat flow data (within error) EPS 122: Lecture 19 – Geotherms 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 EPS 122: Lecture 19 – Geotherms 9 crust crust lithosphere lithosphere A hybrid? Continents • thicker crust • similar lithosphere “Plate” model fits depth and Q best (cratons?) but there is other geophysical evidence for a thickening lithosphere • increasing elastic thickness • increasing depth to low velocity asthenosphere thermal boundary layer with small-scale convection EPS 122: Lecture 19 – Geotherms 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 EPS 122: Lecture 19 – Geotherms 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 EPS 122: Lecture 19 – Geotherms 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 0.4 °C km-1 at greater depth using 0.3 °C km-1 due to reduced T = 1700 K = 3 x 10-5 °C-1 g = 9.8 m s-2 cp = 1.25 x 103 J kg °C-1 EPS 122: Lecture 19 – Geotherms 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 EPS 122: Lecture 19 – Geotherms 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 EPS 122: Lecture 19 – Geotherms 12