Math 2.0

Geophysics Definition

Mantle convection

Builds on Stokes flowBoussinesq approximation for buoyancyRayleigh numberPrandtl numberViscosityAdvection–diffusion equation

Definition

Let Ω\Omega be the region occupied by the Earth's mantle, under gravity of strength gg pointing locally along −𝐳^-\hat{\mathbf{z}}. Let the rock have reference density ρ0\rho_0, thermal expansion coefficient α\alpha, thermal diffusivity κ\kappa and viscosity μ\mu. Let HH be the rate of internal heating per unit mass divided by the specific heat, so that HH has units of temperature per time. Mantle convection is the buoyancy-driven flow of the rock in Ω\Omega. In the Boussinesq approximation with infinite Prandtl number, its velocity 𝐮\mathbf{u}, dynamic pressure pp and temperature TT satisfy:

  1. ∇⋅𝐮=0\nabla \cdot \mathbf{u} = 0;
  2. 0=−∇p+μ∇2𝐮+ρ0αg(T−T0)𝐳^0 = -\nabla p + \mu \nabla^2 \mathbf{u} + \rho_0 \alpha g (T - T_0) \hat{\mathbf{z}}, a Stokes flow with buoyancy, where T0T_0 is a reference temperature;
  3. ∂tT+𝐮⋅∇T=κ∇2T+H\partial_t T + \mathbf{u} \cdot \nabla T = \kappa \nabla^2 T + H, an advection–diffusion equation with a source.

Remarks

The vigour of the flow is measured by the Rayleigh number Ra=ρ0αgΔTd3/(κμ)Ra = \rho_0 \alpha g \Delta T d^3/(\kappa \mu), with dd the depth and ΔT\Delta T the temperature contrast. Example: take ρ0=4000\rho_0 = 4000 kg/m³, α=3×10−5\alpha = 3 \times 10^{-5} K⁻¹, g=10g = 10 m/s² and ΔT=3000\Delta T = 3000 K. With d=3×106d = 3 \times 10^6 m, κ=10−6\kappa = 10^{-6} m²/s and μ=1021\mu = 10^{21} Pa s, one gets Ra≈108Ra \approx 10^8. This is far above the critical value of order 10310^3, so the mantle convects. Non-example: a conductive layer with RaRa below the critical value has the solution 𝐮=0\mathbf{u} = 0 with a stable temperature profile; nothing flows. The infinite Prandtl number means inertia is dropped: the mantle's Prandtl number μ/(ρ0κ)\mu/(\rho_0 \kappa) is of order 102310^{23}. In real models μ\mu depends strongly on temperature and pressure, and the top layer forms rigid plates.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

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Advection–diffusion equation, concept, no evidence levelAdvection–diffusion e…conceptBoussinesq approximation for buoyancy, concept, no evidence levelBoussinesq approximat…conceptPrandtl number, concept, no evidence levelPrandtl numberconceptRayleigh number, concept, no evidence levelRayleigh numberconceptStokes flow, concept, no evidence levelStokes flowconceptViscosity, concept, no evidence levelViscosityconceptMantle convection, definition, no evidence levelMantle convectiondefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Advection–diffusion equation concept
  2. Boussinesq approximation for buoyancy concept
  3. Prandtl number concept
  4. Rayleigh number concept
  5. Stokes flow concept
  6. Viscosity concept
  7. Mantle convection definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Mantle_convection
About this page
Tag58G1MX92KB
URIm2://58G1MX92KB
Kinddefinition
WritingW0 stub
MSC86A60, 76R10, 76D07
DefinesMantle convection
UsesStokes flow, Boussinesq approximation for buoyancy, Rayleigh number, Prandtl number, Viscosity, Advection–diffusion equation
Changed3 November 2026

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