Someone computes the whole-mantle Rayleigh number using , , , ,

PHYSICS · Multi-part · Fermi University Physics · Earth, Climate & Energy Physics · Seismology, tectonics and geodynamics · Seismology, tectonics and geodynamics

The Solid Earth, Ch Plates, creep and convection: the mantle in motion

Problem

Someone computes the whole-mantle Rayleigh number using ρ=3300 kg m−3\rho = 3300\,\mathrm{kg\,m^{-3}}, g=10 m s−2g = 10\,\mathrm{m\,s^{-2}}, αV=3×10−5 K−1\alpha_V = 3\times10^{-5}\,\mathrm{K^{-1}}, ΔT=1500 K\Delta T = 1500\,\mathrm{K}, d=2890 kmd = 2890\,\mathrm{km}, ηv=1021 Pa s\eta_v = 10^{21}\,\mathrm{Pa\,s} and κ=10−6 m2 s−1\kappa = 10^{-6}\,\mathrm{m^2\,s^{-1}} as follows: Ra=ρgαVΔT d2ηvκ=12.4,\mathrm{Ra} = \frac{\rho g\alpha_V\Delta T\,d^2}{\eta_v\kappa} = 12.4, and concludes that since 12.4<Rac=657.512.4 < \mathrm{Ra}_c = 657.5, the mantle should be conductive and not convecting. Find the error and give the correct Rayleigh number. ((a)) Identify which step is wrong. ((b)) Recompute Ra\mathrm{Ra} correctly and compare it with Rac\mathrm{Ra}_c.

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