7.1 Thermal Convection Theory
Convection as the Mantle Engine
Earth's mantle, despite being solid rock, convects over geological timescales because its viscosity is finite and the temperature difference between the core-mantle boundary (~4000 K) and the surface (~300 K) is enormous. This thermally driven circulation is the fundamental mechanism that transports heat from the deep interior to the surface, powering plate tectonics, volcanism, and the geodynamo indirectly through core cooling.
The physics of mantle convection is governed by the same equations that describe convection in any viscous fluid heated from below and cooled from above — the Rayleigh-Bénard problem. However, the mantle introduces complications: strongly temperature-dependent viscosity, phase transitions at 410 km and 660 km depth, internal radioactive heating, and chemical heterogeneity. Understanding the idealized theory is essential before tackling these complexities.
Rayleigh-Bénard Convection
The classical Rayleigh-Bénard problem considers a horizontal layer of fluid of depth d, heated uniformly from below (temperature Tbottom) and cooled from above (temperature Ttop), with ΔT = Tbottom − Ttop. When the temperature difference is small, heat is transported purely by conduction, and the fluid remains static. As ΔT increases, the buoyant lower fluid eventually overcomes viscous resistance and thermal diffusion, and convective overturn begins.
The onset of convection is controlled by the dimensionless Rayleigh number, which measures the ratio of buoyancy forces to the dissipative effects of viscosity and thermal diffusion:
\(\mathrm{Ra} = \frac{\alpha \rho g \, \Delta T \, d^3}{\kappa \eta}\)
where α is the thermal expansion coefficient (~2 × 10−5 K−1), ρ is density (~4000 kg/m³), g is gravitational acceleration (~10 m/s²), ΔT is the superadiabatic temperature drop across the layer, d is the layer depth, κ is the thermal diffusivity (~10−6 m²/s), and η is the dynamic viscosity (Pa·s).
Critical Rayleigh Number
For rigid (no-slip) top and bottom boundaries, the critical value is Racr ≈ 657.5 (precisely 27π&sup4;/4). For free-slip boundaries, Racr ≈ 1100. Below the critical value, the fluid conducts heat without moving; above it, convective rolls develop spontaneously.
Earth's Mantle
With d ≈ 2890 km, ΔT ≈ 2500 K (superadiabatic), and η ≈ 1021 Pa·s, the mantle Rayleigh number is Ra ~ 106–107. This is thousands of times above the critical value, meaning convection is vigorous, time-dependent, and turbulent in the sense of chaotic flow patterns with plumes and downwellings.
The Boussinesq Approximation
The Boussinesq approximation simplifies the equations of convection by treating density as constant everywhere except in the buoyancy term of the momentum equation. This is valid when the density variations due to temperature are small compared to the reference density — a condition well satisfied in the mantle where αΔT ~ 0.05, or about 5%.
Under this approximation, the non-dimensional governing equations for infinite Prandtl number flow (appropriate for the mantle, where Pr = η/(ρκ) ~ 1023) reduce to three coupled equations:
\(\nabla \cdot \mathbf{v} = 0 \quad \text{(incompressibility)}\)
\(-\nabla P + \nabla^2 \mathbf{v} + \mathrm{Ra} \, T \, \hat{\mathbf{z}} = 0 \quad \text{(Stokes flow with buoyancy)}\)
\(\frac{\partial T}{\partial t} + \mathbf{v} \cdot \nabla T = \nabla^2 T \quad \text{(heat advection-diffusion)}\)
The first equation enforces mass conservation for an incompressible fluid. The second is the Stokes equation (inertia neglected at infinite Prandtl number) with the buoyancy force Ra·T acting vertically. The third describes temperature evolution through advection by the flow and diffusion. These equations are non-dimensionalized using d for length, d²/κ for time, κ/d for velocity, and ΔT for temperature.
The infinite Prandtl number limit means that fluid inertia is negligible: the mantle cannot “coast” or develop turbulent eddies in the classical fluid-mechanical sense. Instead, it responds instantaneously to changes in buoyancy through a balance of viscous and pressure forces. Despite this, the nonlinear advection term in the energy equation still generates complex, time-dependent behavior.
Boundary Layer Theory & Heat Transfer Scaling
At high Rayleigh number, convection organizes into a pattern of thin thermal boundary layers at the top and bottom of the convecting layer, with a nearly isothermal (well-mixed) interior. The boundary layers are where the temperature changes rapidly from the interior value to the surface or basal value. In Earth, the top thermal boundary layer is the lithosphere.
Dimensional analysis and matched asymptotic expansions yield the scaling for the thermal boundary layer thickness:
\(\delta_T \sim d \cdot \mathrm{Ra}^{-1/3}\)
For Ra ~ 107 and d = 2890 km, this gives δT ~ 130 km, remarkably close to the observed thickness of old oceanic lithosphere. The heat flux through the boundary layer is characterized by the Nusselt number, the ratio of total heat flux (convective + conductive) to the purely conductive flux:
\(\mathrm{Nu} = \frac{q_{\text{total}}}{q_{\text{conductive}}} = \frac{d}{\delta_T} \sim \mathrm{Ra}^{1/3}\)
For Ra ~ 107, Nu ~ 200, meaning convection is about 200 times more efficient at transporting heat than conduction alone. Without convection, the mantle would cool purely by conduction on a timescale of d²/κ ~ 1011 years — far longer than the age of the Earth. Convection reduces this to the order of 109 years.
δT ~ 130 km
Top boundary layer = lithosphere
Nu ~ 200
Convection >> conduction
Ra ~ 107
Vigorous, time-dependent flow
Convection Cell Geometry & Time Dependence
At the onset of convection (Ra just above Racr), the preferred pattern consists of steady-state convective rolls with an aspect ratio (width/height) close to 1 for rigid boundaries and ~2.8 for free-slip boundaries. As Ra increases, the convection transitions through a series of increasingly complex regimes.
Steady-State Regime (Ra ~ Racr)
Simple, stable convection cells with predictable geometry. The flow pattern is time-independent and can be described analytically. Upwellings and downwellings are symmetric.
Time-Dependent Regime (Ra ~ 105–106)
Boundary layers become unstable and shed thermal plumes (both hot upwellings and cold downwellings). The convective pattern fluctuates in time, though it may retain some statistical regularity. Plumes detach periodically from the boundary layers.
Chaotic/Turbulent Regime (Ra > 106)
At the Rayleigh numbers relevant to Earth (106–107), the flow is fully time-dependent and chaotic, with a rich spectrum of scales. Thin plume conduits, sheet-like downwellings (analogous to subducting slabs), and complex interactions between flow structures dominate.
For the Earth, seismic tomography reveals that the actual convection pattern does not consist of simple rolls. Instead, we observe slab-like curtains of cold material descending into the deep mantle, broad hot upwelling regions beneath Africa and the Pacific (the Large Low-Shear-Velocity Provinces), and narrow plume conduits. The aspect ratio of “cells” varies widely, from ~1 beneath the Pacific to much larger values in the Atlantic hemisphere.
Internal Heating vs Basal Heating
The classical Rayleigh-Bénard problem considers only basal heating (hot bottom, cold top). The Earth's mantle, however, is heated by two sources: heat flowing in from the core below (basal heating), and radioactive decay of uranium, thorium, and potassium distributed throughout the mantle (internal heating). The relative importance of these two sources profoundly affects the convective style.
For internally heated convection, a separate Rayleigh number is defined:
\(\mathrm{Ra}_H = \frac{\alpha \rho g \, H \, d^5}{k \kappa \eta}\)
where H is the rate of internal heat production per unit volume (W/m³) and k is the thermal conductivity. The key difference is that internally heated convection produces strong, cold downwellings but weak, diffuse upwellings — because the heat is added volumetrically rather than concentrated at the base.
Basal Heating Dominates
- ▶Symmetric upwellings and downwellings
- ▶Strong bottom boundary layer produces vigorous plumes
- ▶Active upwellings carry significant heat flux
Internal Heating Dominates
- ▶Strong cold downwellings, weak passive upwellings
- ▶Interior temperature close to bottom temperature
- ▶Downwellings dominate heat transport (like subducting slabs)
Current estimates suggest the mantle is ~60–70% internally heated and ~30–40% basally heated from the core. This “mixed heating” mode explains why subducting slabs (cold downwellings) are the dominant features of mantle circulation, while only a handful of deep-seated plumes (active upwellings from core heating) exist — consistent with the ~10 hotspots that may be rooted at the core-mantle boundary.
Key Concepts Summary
Racr ≈ 658
Critical onset for rigid boundaries
Nu ~ Ra1/3
Heat flux scaling law
60–70%
Mantle heat from internal sources