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Geodynamics Lecture 10 The forces driving plate tectonics Lecturer: David Whipp! [email protected]! ! 2.10.2014 Geodynamics www.helsinki.fi/yliopisto 1 Goals of this lecture • Describe how thermal convection operates according to basic physical principles! ! • Quantify the basic plate-driving forces! • • • Slab pull Ridge push Drag force 2 NEWS & VIEWS Mantle convection Compositional stratification around 660 km Secondary plumes Mid-ocean ridge Primary plume Deflected slab Penetrating slab Enriched piles Continent Slab grave yards 660 km discontinuity • • Core–mantle boundary Tackley, 2008 Plate tectonics is ultimately driven by convection in the Figure 1 Leaky layers. The emerging model of mantle convection suggests that some relatively cool subducting mantle reflecting produced and(dashed lostblack as line) thewhereas Earth slabs of oceanic plate (blue) are deflheat ected at the 660 km discontinuity otherscools! penetrate all the way to the core–mantle boundary (solid black line), forming slab graveyards. Piles of material that are enriched in incompatible elements compared with the expected mantle average (orange) are pushed around at the core–mantle boundary by incoming slab material, and plumes form from their edges and tops (red). Some plumes penetrate below 660 km, whereas others are deflected and may produce secondary upper-mantle plumes. An average compositional stratification exists either side of 660 km. How convection operates in the mantle is somewhat controversial, but there is clear evidence of fossil plates in the upper and lower mantle most of the mantle could be similar to the depleted upper mantle, with only small sense, this does not have much effect on the evolution of the mantle and 3 mate trace at ho stud both melt Univ thou from this abov Th mixt conv inter a glo strat of ph subs man mate grav perm disti leng may may in tr of tr Layered versus whole-mantle convection • Layered and whole-mantle convection are endmember models of mantle convection! ! • Layered convection is based on the strong seismic discontinuity at 670 km, among other lines of evidence, with the idea that this discontinuity separates upper and lower mantle circulation 4 http://instruct.uwo.ca Thermal convection A Rayleigh-Taylor instability occurs when a denser fluid overlies a less dense fluid, which is gravitationally! unstable (the denser fluid wants to sink, and the less dense fluid wants to rise due to its buoyancy) Fig. 6.21, Turcotte and Schubert, 2014 • A fluid heated from beneath will undergo thermal expansion and may become less dense than the cooler overlying fluid! • • This scenario is gravitationally unstable! If the denser fluid is able to sink into the less dense fluid, thermal convection will occur 5 Thermal convection Fig. 6.21, Turcotte and Schubert, 2014 • What is going to happen to these two fluids with time? 6 Thermal convection Time evolution of a Rayleigh-Taylor instability 7 Thermal convection • Thermal convection in the Earth results from buoyancy forces owing to thermal expansion of mantle rocks! • For an incompressible viscous fluid, the force balance of pressure, gravity and viscous forces in 2D is ✓ 2 ◆ 2 @p @ v @ v 0= + ⇢g + ⌘ + 2 2 @y @x @y where ! is pressure, " is density, # is viscosity and $ is velocity in the % direction! • To account for the buoyancy forces from thermal expansion, the density term must be variable ⇢ = ⇢0 + ⇢0 where "0 is a reference density and "ʹ is a density perturbation much smaller than "0 8 Thermal convection • • If the variable density is substituted into the 2D force balance, and the hydrostatic pressure based on the reference density is eliminated by using ' = ! - "0(% we find ✓ 2 ◆ 2 @P @ v @ v 0 0= +⇢g+⌘ + 2 2 @y @x @y ! A change in density as a result of thermal expansion is ⇢0 = ⇢0 ↵v (T T0 ) where )$ is the volumetric coefficient of thermal expansion and *0 is a reference temperature corresponding to the reference density "0! • The vertical force balance including thermal buoyancy is thus ✓ 2 ◆ 2 @P @ v @ v 0= +⌘ + 2 g⇢0 ↵v (T T0 ) 2 @y @x @y 9 Thermal convection • In order to determine the thermal buoyancy, we require a 2D equation for heat transfer via conduction and convection! • Turcotte and Schubert give a detailed derivation, arriving at the following relationship for heat conduction and convection in ✓ 2 ◆ 2D 2 @T @T @T @ T @ T +u +v = + 2 2 @t @x @y @x @y where + is the velocity in the , direction and - is the thermal diffusivity 10 Thermal convection • In order to determine the thermal buoyancy, we require a 2D equation for heat transfer via conduction and convection! • Turcotte and Schubert give a detailed derivation, arriving at the following relationship for heat conduction and convection in ✓ 2 ◆ 2D 2 @T @T @T @ T @ T +u +v = + 2 2 @t @x @y @x @y Time dependence Convection Conduction where + is the velocity in the , direction and - is the thermal diffusivity 11 Onset of thermal convection • For a fluid heated to temperature *1 at its base and cooled to temperature *0 at its upper surface thermal buoyancy will drive convection if the viscous resistance to fluid flow is overcome! • When the temperature difference *1 - *0 is small, convection will not occur and the fluid velocities will be + = $ = 0, and temperature will not change with time (././ = 0) or along the , axis (./., = 0), reducing the heat transfer equation to Fig. 6.38, Turcotte and Schubert, 2014 d2 Tc =0 2 dy where 0 indicates conductive heat transfer 12 Onset of thermal convection • For the thermal boundary conditions * = *0 at % = -1/2 and * = *1 at % = 1/2, the solution to the heat transfer equation is T1 + T0 (T1 T0 ) Tc = + y 2 b where 1 is the thickness of the fluid layer! Fig. 6.38, Turcotte and Schubert, 2014 • Convection will begin when the temperature * just begins to exceed the conductive temperature *0! • In this case, the temperature difference *ʹ is extremely small T0 ⌘ T Tc 13 Onset of thermal convection • The temperature difference *ʹ can be inserted into the heat transfer and force balance equations for a 2D incompressible viscous fluid to determine the conditions under which convection will occur! • From this, a dimensionless number known as the Rayleigh number can be defined Fig. 6.38, Turcotte and Schubert, 2014 ⇢0 g↵v (T1 T0 )b3 Ra = ⌘ 14 Onset of thermal convection Fig. 6.39, Turcotte and Schubert, 2014 • Instability and convection will occur when 2 2 4⇡ 2 (⇡ + 2b )3 Ra > 4⇡ 2 b2 2 where 2 is the wavelength of convection! Minimum at p = 2 2b • The flow is thus stable when Ra is less than the right side of the equation above! • The critical Rayleigh number when convection begins is thus 2 Ra ⌘ Racr = (⇡ + 4⇡ 2 b2 3 2 ) 4⇡ 2 b2 2 15 Onset of thermal convection • For a fluid cooled from above, heated from within and with no heat flux across its base the Rayleigh number equation is 2 5 ↵ ⇢ gHb v 0 RaH = k⌘ where 3 is the heat production per unit mass and 4 is the thermal conductivity! • The equation above is appropriate for the Earth’s mantle 16 Onset of thermal convection ! ! RaH ↵v ⇢20 gHb5 = k⌘ • The critical Rayleigh number for an internally heated fluid is Racr = 2772! • What is the Rayleigh number for the upper mantle?! • • Should it convect?! What about the whole mantle?! ! • Assume )$ = 3 ⨉ 10-5 K-1, "0 = 4000 kg m-3, 3 = 9 ⨉ 10-12 W kg-1, 4 = 4 W m-1 K-1, # = 1 ⨉ 1021 Pa s, and - = 1 mm2 s-1 17 What drives tectonic plate motions? The driving forces of plate motion Forsyth and Uyeda, 1975 Driving forces Resisting forces FDF FRP FTF FCR Downloaded from http://gji.oxfordjournals.org/ at Dalhousie University on September 1, 2013 FCD 165 FSP FDF FSR 18 What drives tectonic plate motions? The driving forces of plate motion Forsyth and Uyeda, 1975 Driving forces Resisting forces FDF FRP FTF FCR Downloaded from http://gji.oxfordjournals.org/ at Dalhousie University on September 1, 2013 FCD 165 FSP FDF FSR 19 Slab (or trench) pull FSP Forsyth and Uyeda, 1975 • Slab pull results from the gravitational body force acting on the dense, sinking oceanic lithosphere.! • It can be divided into two components:! Downloaded from http://gji.oxfordjournals.org/ at Dalhousie University on September 1, 2013 • • • Fb1, the force resulting from the slab being colder than the surrounding mantle! Fb2, the force resulting from the elevation of the olivine→spinel phase change! Mathematically, we can say FSP = Fb1 + Fb2 20 Slab (or trench) pull FSP • Forsyth and Uyeda, 1975 For one half of a sinking mantle plume, Turcotte and Schubert show ✓ ◆1/2 u0 Fb = 2⇢0 g↵v b(Tc T0 ) v0 2⇡u0 Downloaded from http://gji.oxfordjournals.org/ at Dalhousie University on September 1, 2013 • We can simplify this relationship slightly for a sinking slab to estimate the slab pull force resulting from the relatively cold temperature of the slab " Reference mantle density * Temperature at LAB ( Gravitational acceleration + Horiz. velocity of upper layer ) Coeff. of thermal expansion $ Average vertical velocity 1 Convecting layer thickness - Thermal diffusivity * Temperature at conv. center 2 Width of 2 convection cells LAB = Lithosphere-asthenosphere boundary 21 Slab (or trench) pull FSP Forsyth and Uyeda, 1975 6.21 A Steady-State Boundary-Layer Theory 509 Downloaded from http://gji.oxfordjournals.org/ at Dalhousie University on September 1, 2013 • Here is the simplified convection cell geometry used for calculating the force acting on a sinking mantle plume! ! • Note that areas in each triangle are equal, so mass is Turcotte and Schubert, 2002 conserved, or 1 Linear velocity profiles used to model the core flow in a conv0 = u0 b l. The areas under the triangles are equal to conserve fluid. 2 22 Slab (or trench) pull FSP Forsyth and Uyeda, 1975 • Downloaded from http://gji.oxfordjournals.org/ at Dalhousie University on September 1, 2013 • Because a sinking slab is rigid, we can assume that +06=6$0, so the force resulting from the slab being colder than the surrounding mantle is ✓ ◆1/2 Fb1 = 2⇢0 g↵v b(Tc T0 ) 2⇡u0 If we use typical values for these variables, we see that 8116is6~3×1013 N m-1! • To be clear, this value is the force per meter of trench length 23 Slab (or trench) pull FSP Heat Transfer Forsyth and Uyeda, 1975 Fig. 4.59, Turcotte and Schubert, 2014 6.22 The Forces that Drive Plate Tectonics Downloaded from http://gji.oxfordjournals.org/ at Dalhousie University on September 1, 2013 • Theortransition fromseparating olivine to e Clapeyron equilibrium curve twospinel phases of al. in the mantle generally occurs at 400-500 km depth, but the transition e 4–58. Since the subducted depends on bothlithosphere pressure was andformed on the thermal structure upon subduction is given by Equation temperature! ndence of temperature upon depth prior to subduction is erm given in Figure 4–56. As the subducted lithosphere Because the descending slab is mantle, frictional heating occurs at its upper boundary. relatively cold, the olivine-spinel phase ional heating were studied in Section 4–26. As discussed transition at lower pressure ture distribution dueoccurs to frictional heating – Equation (4– (shallower depth) erimposed on the initial temperature distribution to give Figure 6.43 Elevation he slab. The result is shown in Figure 4–58. • Turcotte and Schubert, 2002 24 in of the olivine–spinel phase change Slab (or trench) pull FSP Heat Transfer Forsyth and Uyeda, 1975 6.22 The Forces that Drive Plate Tectonics Warm slab Downloaded from http://gji.oxfordjournals.org/ at Dalhousie University on September 1, 2013 Fig. 4.59, Turcotte and Schubert, 2014 Cold slab • Theortransition fromseparating olivine to e Clapeyron equilibrium curve twospinel phases of al. in the mantle generally occurs at 400-500 km depth, but the transition e 4–58. Since the subducted depends on bothlithosphere pressure was andformed on the thermal structure upon subduction is given by Equation temperature! ndence of temperature upon depth prior to subduction is erm given in Figure 4–56. As the subducted lithosphere Because the descending slab is mantle, frictional heating occurs at its upper boundary. relatively cold, the olivine-spinel phase ional heating were studied in Section 4–26. As discussed transition at lower pressure ture distribution dueoccurs to frictional heating – Equation (4– (shallower depth) erimposed on the initial temperature distribution to give Figure 6.43 Elevation he slab. The result is shown in Figure 4–58. • Turcotte and Schubert, 2002 25 in of the olivine–spinel phase change Slab (or trench) pull FSP Heat Transfer Forsyth and Uyeda, 1975 6.22 The Forces that Drive Plate Tectonics Warm slab Downloaded from http://gji.oxfordjournals.org/ at Dalhousie University on September 1, 2013 Fig. 4.59, Turcotte and Schubert, 2014 Cold slab • Theortransition fromseparating olivine to e Clapeyron equilibrium curve twospinel phases of al. in the mantle generally occurs at 400-500 km depth, but the transition e 4–58. Since the subducted depends on bothlithosphere pressure was andformed on the thermal structure upon subduction is given by Equation temperature! ndence of temperature upon depth prior to subduction is erm given in Figure 4–56. As the subducted lithosphere Because the descending slab is mantle, frictional heating occurs at its upper boundary. relatively cold, the olivine-spinel phase ional heating were studied in Section 4–26. As discussed transition at lower pressure ture distribution dueoccurs to frictional heating – Equation (4– (shallower depth) erimposed on the initial temperature distribution to give Figure 6.43 Elevation he slab. The result is shown in Figure 4–58. • Fig. 6.43, Turcotte and Schubert, 2014 26 in of the olivine–spinel phase change Slab (or trench) pull FSP Forsyth and Uyeda, 1975 • 6.22 The Forces that Drive Plate Tectonics Thus, the driving force from the phase transition depends on the location of the phase transition isotherm *os in the sinking slab! Downloaded from http://gji.oxfordjournals.org/ at Dalhousie University on September 1, 2013 • • The driving force from the phase transition is ✓ ◆1/2 2(Tc T0 ) ⇢os Fb2 = ⇢0 2⇡u0 The olivine-spinel phase transition increases density by ~270 kg m-3 @ A" Slope of Clapeyron curve Fig. 6.43, Turcotte and Schubert, 2014 27 in Gravitational acceleration Figure 6.43 Elevation of the olivine–spinel phase change Slab (or trench) pull FSP Forsyth and Uyeda, 1975 Using typical values for a sinking slab, the gravitational body force due to the olivine-spinel phase transition in the slab 812 is ~1.5×1013 N m-1, or about half of that from the difference in temperature alone! Downloaded from http://gji.oxfordjournals.org/ at Dalhousie University on September 1, 2013 • 6.22 The Forces that Drive Plate Tectonics ! • Going back to our original equation for the slab pull force, F = Fb1 + Fb2 SP we find a total slab pull force of 8E'6=6~4.5×1013 N m-1 Fig. 6.43, Turcotte and Schubert, 2014 28 in Figure 6.43 Elevation of the olivine–spinel phase change Ridge push FRP Forsyth and Uyeda, 1975 • Ridge push results from the elevation of oceanic ridges relative to the seafloor! • The difference in elevation results in a pressure head that drives the plate away from the ridge! Downloaded from http://gji.oxfordjournals.org/ at Dalhousie University on September 1, 2013 • This motion may also be viewed as gravitational sliding! ! • To calculate the ridge push force 8G' we must consider the forces acting on the top (81), bottom (82) and side (83) of the oceanic lithospheric plate: FRP = F1 F2 F3 29 Ridge push FRP 6.22 The Forces that Drive Plate Tectonics Forsyth and Uyeda, 1975 With this force balance in mind, we can see! Downloaded from http://gji.oxfordjournals.org/ at Dalhousie University on September 1, 2013 • Fig. 6.44, Turcotte and Schubert, 2014 1. The horizontal force on the base of the plate must be equal to the integrated lithostatic pressure in the mantle along RD! 2. The horizontal force on the top of the plate must be equal to the integrated hydrostatic pressure along AB! Figure 6.44 Horizontal forces acting on a section of the ocean, li and mantle at an ocean ridge. 3. The horizontal force acting on the lithospheric section BC is equal to the integrated pressure in the lithosphere! • The integrated pressure force on the upper surface of the lith Note that this pressure that resulting frombecause the overlying equal toshould F4 , theinclude net pressure force on AB, the section o oceanic water must be in equilibrium. Thus we can integrate the hydrostat 30 the water to obtain ! Ridge push FRP 6.22 The Forces that Drive Plate Tectonics Forsyth and Uyeda, 1975 Fig. 6.44, Turcotte and Schubert, 2014 Downloaded from http://gji.oxfordjournals.org/ at Dalhousie University on September 1, 2013 ! • At a constant depth H P = ⇢gz assuming constant density of the overlying material! • Integrated over a Pint = Z Figure 6.44 Horizontal forces acting on a section of the ocean, li depthand range H1 to mantle at Han 2 ocean ridge. z2 The integrated pressure force on the upper surface of the lith ⇢gzdz equal to F4 , the net pressure force on AB, because the section o z1 must be in equilibrium. Thus we can integrate the hydrostat 31 the water to obtain ! Ridge push FRP 6.22 The Forces that Drive Plate Tectonics Forsyth and Uyeda, 1975 • Going back to the original force balance equation for ridge push, we see FRP = F1 • F2 Downloaded from http://gji.oxfordjournals.org/ at Dalhousie University on September 1, 2013 Fig. 6.44, Turcotte and Schubert, 2014 F3 After some mathematical substitutions and integrations we find F RP = g⇢m ↵v (T1 " " * 2 ⇢m ↵v (T1 T0 ) Figure T0 ) 6.44 1 +Horizontal forces acting on a section t of the ocean, li and mantle at an ⇡ ocean (⇢ridge. ⇢w ) m Temperature surface The integrated*pressure force onat theplate upper surface of the lith force onplate AB, because the section o Water density equal to F4 , the net Age of oceanic / pressure must be in equilibrium. Thus we can integrate the hydrostat 32 Mantle temperature the water to obtain Mantle density ! Ridge push FRP 6.22 The Forces that Drive Plate Tectonics Forsyth and Uyeda, 1975 Fig. 6.44, Turcotte and Schubert, 2014 Downloaded from http://gji.oxfordjournals.org/ at Dalhousie University on September 1, 2013 • Using typical values, we find the ridge push force is 8G'6=6~4×1012 N m-1! • Note that this is about an order Figure 6.44 Horizontal forces acting on a section of the ocean, li of magnitude smaller than the slab and mantle at an ocean ridge. pull force The integrated pressure force on the upper surface of the lith equal to F4 , the net pressure force on AB, because the section o must be in equilibrium. Thus we can integrate the hydrostat 33 the water to obtain ! Drag force FDF • Fluid Mechanics Forsyth and Uyeda, 1975 The drag force on the base of the oceanic lithosphere can both drive and resist plate tectonics, depending on the relative motion between the plate and the underlying mantle! Downloaded from http://gji.oxfordjournals.org/ at Dalhousie University on September 1, 2013 416 ! • If we assume that the underlying mantle Fig. 6.2, Turcotte and Schubert, 2014 resists or drives plate motion by viscous flow across a fixed-thickness layer, the drag force on the plate is simply FDF = ⌘as # Viscosity of asthenosphere A+ Velocity difference u L h ℎ Thickness of viscous layer J Length of the plate 34 Figure 6.2 One-dimensional channel flows of a constan Drag force FDF 416 Fluid Mechanics Forsyth and Uyeda, 1975 Downloaded from http://gji.oxfordjournals.org/ at Dalhousie University on September 1, 2013 • Again using typical values, we find the drag force is 8K86=6~1×1013 N m-1! • Fig. 6.2, Turcotte and Schubert, 2014 Note that this value is similar in magnitude to the slab pull force 35 Figure 6.2 One-dimensional channel flows of a constan Recap • Thermal convection in the mantle results from buoyancy forces as the mantle is heated and undergoes thermal expansion! ! • Slab pull and the drag force are similar and about an order of magnitude larger than the ridge push force 36 References Forsyth, D., & Uyeda, S. (1975). On the Relative Importance of the Driving Forces of Plate Motion*. Geophysical Journal International, 43(1), 163–200. doi:10.1111/j.1365-246X.1975.tb00631.x 37