Question

Apply the two-dimensional boundary-layer model for heated-from-below convection to the entire man- tle. Calculate the mean surface heat flux, the mean horizontal velocity, and the mean surface thermal boundary-layer thickness. Assume $T_1-T_0=$ $3000\mathrm{K},b=2880\mathrm{km},k=4\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1},\kappa =$ $1{\mathrm{mm}}^2{\mathrm{s}}^{-1},{\alpha }_v=3\times {10}^{-5}{\mathrm{K}}^{-1},g=10\mathrm{m}{\mathrm{s}}^{-2},$ and ${\rho }_0=4000\mathrm{kg}{\mathrm{m}}^{-3}$. A MATLAB solution to this problem is provided in Appendix $D$.

Short answer

Mean surface heat flux is approximately 0.00417 W/m², mean horizontal velocity is about 1.79 mm/s, and mean surface thermal boundary-layer thickness is around 1473 m.

Step-by-step solution

Understanding the Problem

We need to calculate three distinct quantities: mean surface heat flux, mean horizontal velocity, and mean surface thermal boundary-layer thickness using the boundary-layer model for convection."Given parameters include: - Temperature difference, $T_1-T_0=3000K$ - Mantle thickness, $b=2880km=2.88\times {10}^{6}m$ - Thermal conductivity, $k=4W{m}^{-1}{K}^{-1}$ - Thermal diffusivity, $\kappa =1m{m}^2{s}^{-1}=1\times {10}^{-6}{m}^2{s}^{-1}$ - Coefficient of thermal expansion, ${\alpha }_v=3\times {10}^{-5}{K}^{-1}$ - Gravity, $g=10m{s}^{-2}$ - Density, ${\rho }_0=4000kg{m}^{-3}$

Calculating Rayleigh Number (Ra)

The Rayleigh number $Ra$ is calculated from the equation $$Ra=\frac{{\alpha }_{v}g\mathrm{\Delta }T{b}^3}{u\kappa }$$ where $u$ is the kinematic viscosity. We approximate $u=\frac{k}{{\rho }_0{c}_p}$ where $c_p$ is assumed to be a typical mantle value of 1250 $Jk{g}^{-1}{K}^{-1}$.First, find the kinematic viscosity $u=\frac{k}{{\rho }_0{c}_p}=\frac{4}{4000\times 1250}=8\times {10}^{-7}{m}^2{s}^{-1}$.Now, substituting the values into the Rayleigh number equation gives:$$Ra=\frac{3\times {10}^{-5}\times 10\times 3000\times (2.88\times {10}^6{)}^3}{8\times {10}^{-7}\times 1\times {10}^{-6}}=6.26\times {10}^6$$

Calculating Mean Surface Heat Flux (q)

We use the formula $$q=\frac{k\mathrm{\Delta }T}{b}=\frac{4\times 3000}{2.88\times {10}^6}$$ Thus, $$q\approx 0.00417W{m}^{-2}$$

Calculating Mean Horizontal Velocity (U)

The mean horizontal velocity $U$ can be derived from a scaling relation used in convection models: $$U=\frac{k}{{\rho }_0{c}_p}{\left(\frac{Ra}{Pr}\right)}^{1/2}$$ where Prandtl number $Pr=\frac{u}{\kappa }$.Since $Pr=\frac{8\times {10}^{-7}}{1\times {10}^{-6}}=0.8$, substitute values back in:$$U=8\times {10}^{-7}{\left(\frac{6.26\times {10}^6}{0.8}\right)}^{1/2}\approx 1.79\times {10}^{-3}m{s}^{-1}$$

Calculating Mean Surface Thermal Boundary-Layer Thickness (\delta)

The thermal boundary-layer thickness $\delta$ is related to the Rayleigh number and typically follows from scaling laws such as $\delta \sim b/R{a}^{1/3}$. Thus, we calculate:$$\delta =\frac{2.88\times {10}^6}{(6.26\times {10}^6{)}^{1/3}}\approx 1473m$$

Key concepts

Rayleigh Number

The Rayleigh number, often denoted as $Ra$, is a dimensionless quantity that plays a critical role in understanding convection processes, especially within the mantle. It essentially indicates the force of buoyancy compared to the resistive forces of viscosity and thermal diffusion. The formula to calculate the Rayleigh number is:$$Ra=\frac{{\alpha }_{v}g\mathrm{\Delta }T{b}^3}{u\kappa }$$where the variables represent:- ${\alpha }_v$: Coefficient of thermal expansion- $g$: Acceleration due to gravity- $\mathrm{\Delta }T$: Temperature difference- $b$: Thickness of the layer- $u$: Kinematic viscosity- $\kappa$: Thermal diffusivityIn mantle convection, a higher Rayleigh number suggests more vigorous convection, leading to faster mantle flow and more efficient heat transfer. Understanding the Rayleigh number helps scientists predict how mantle material will behave under given conditions.It's like a score that tells us whether convection inside the mantle will be vigorous or sluggish.

Thermal Boundary-Layer Thickness

The thermal boundary-layer thickness, symbolized by $\delta$, represents the layer over which temperature changes from the hot mantle to the cooler surface. It's an essential aspect in understanding how heat is transferred from the Earth's interior to the surface.Typically, the thickness is calculated using the relationship:$$\delta \sim \frac{b}{R{a}^{1/3}}$$This scaling law suggests that the thermal boundary layer is thinner for larger Rayleigh numbers, meaning heat can escape more rapidly. Therefore, variances in boundary-layer thickness indicate changes in surface heat flow.Imagine it as the crust of a baked pie - the thicker the pie, the thicker this crust can be, but under consistent heat, it might become thinner, suggesting more heat loss to the atmosphere.

Mean Surface Heat Flux

Mean surface heat flux $q$ quantifies the amount of heat being transferred from the Earth's interior to its surface across a specified area per unit time. It provides a measure of the efficiency of heat transport.The formula to calculate heat flux is given by:$$q=\frac{k\mathrm{\Delta }T}{b}$$where:- $k$: Thermal conductivity- $\mathrm{\Delta }T$: Temperature difference- $b$: Mantle thicknessUnderstanding mean surface heat flux is crucial for geophysicists as it contributes to the thermal budget of Earth’s surface and affects geological processes like volcanism and plate tectonics. It's akin to the heat escaping from a stove, heating a pot from below.

Thermal Diffusivity

Thermal diffusivity $\kappa$ represents how quickly heat spreads through a material. It's calculated as the ratio of thermal conductivity $k$ to the product of density $\rho$ and specific heat capacity $c_p$:$$\kappa =\frac{k}{\rho c_p}$$This concept is crucial in mantle convection as it describes how efficiently heat is conducted through the mantle, influencing the rate of thermal equilibrium. Higher thermal diffusivity means heat disperses quickly, suggesting rapid temperature homogenization.Think of it as how quickly the warmth from a cozy campfire spreads through the surrounding air, warming you up on a chilly evening.

Mantle Convection Models

Mantle convection models are simulations that help scientists understand the movement of tectonic plates and the associated thermal and mechanical processes within Earth's mantle. These models incorporate principles such as the Rayleigh number, thermal diffusivity, and heat flux, among others. Key aspects of mantle convection models include:

• Predicting the flow of hot mantle material beneath the Earth’s crust
• Understanding heat distribution which impacts surface temperatures
• Exploring plate tectonics and continents’ drift over geological time scales

By simulating how materials flow inside the Earth, these models provide invaluable insights into volcanic activity, earthquakes, and even the formation of mountains. Essentially, they're like a movie that plays out beneath the Earth's surface, showing how its internal engine drives changes above.