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Geodynamics Plate-driving forces Lecture 10.3 - Onset of thermal convection Lecturer: David Whipp [email protected] Geodynamics www.helsinki.fi/yliopisto 1 Goal of this lecture • Introduce the Rayleigh number as a means for estimating the conditions under which mantle convection will occur 2 Onset of thermal convection • For a fluid heated to temperature 𝑇1 at its base and cooled to temperature 𝑇0 at its upper surface thermal buoyancy will drive convection if the viscous resistance to fluid flow is overcome • When the temperature difference 𝑇1 - 𝑇0 is small, convection will not occur and the fluid velocities will be 𝑢 = 𝑣 = 0, and temperature will not change with time (𝜕/𝜕𝑡 = 0) or along the 𝑥 axis (𝜕/𝜕𝑥 = 0), reducing the heat transfer equation to Fig. 6.38, Turcotte and Schubert, 2014 d2 Tc =0 2 dy where 𝑐 indicates conductive heat transfer 3 Onset of thermal convection • For the thermal boundary conditions 𝑇 = 𝑇0 at 𝑦 = -𝑏/2 and 𝑇 = 𝑇1 at 𝑦 = 𝑏/2, the solution to the heat transfer equation is T1 + T0 (T1 T0 ) Tc = + y 2 b where 𝑏 is the thickness of the fluid layer Fig. 6.38, Turcotte and Schubert, 2014 • Convection will begin when the temperature 𝑇 just begins to exceed the conductive temperature 𝑇𝑐 • In this case, the temperature difference 𝑇ʹ is extremely small T0 ⌘ T Tc 4 Onset of thermal convection • The temperature difference 𝑇ʹ 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 = ⌘ 5 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 𝜆 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 6 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 𝐻 is the heat production per unit mass and 𝑘 is the thermal conductivity • The equation above is appropriate for the Earth’s mantle 7 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 𝛼𝑣 = 3 ⨉ 10-5 K-1, 𝜌0 = 4000 kg m-3, 𝐻 = 9 ⨉ 10-12 W kg-1, 𝑘 = 4 W m-1 K-1, 𝜂 = 1 ⨉ 1021 Pa s, and 𝜅 = 1 mm2 s-1 8 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? 1.3 x 106 • • • Should it convect? Yes What about the whole mantle? 1.76 x 109 Assume 𝛼𝑣 = 3 ⨉ 10-5 K-1, 𝜌0 = 4000 kg m-3, 𝐻 = 9 ⨉ 10-12 W kg-1, 𝑘 = 4 W m-1 K-1, 𝜂 = 1 ⨉ 1021 Pa s, and 𝜅 = 1 mm2 s-1 9 Let’s see what you’ve learned… • If you’re watching this lecture in Moodle, you will now be automatically directed to the quiz! 10