Rayleigh number (for geophysical applications - related to bottom heating of the mantle from the core)

Description

In fluid mechanics, the Rayleigh number (Ra) for a fluid is a dimensionless number associated with buoyancy-driven flow, also known as free convection or natural convection. When the Rayleigh number is below a critical value for that fluid, heat transfer is primarily in the form of conduction; when it exceeds the critical value, heat transfer is primarily in the form of convection.

The Rayleigh number is named after Lord Rayleigh and is defined as the product of the Grashof number, which describes the relationship between buoyancy and viscosity within a fluid, and the Prandtl number, which describes the relationship between momentum diffusivity and thermal diffusivity. Hence the Rayleigh number itself may also be viewed as the ratio of buoyancy and viscosity forces multiplied by the ratio of momentum and thermal diffusivities.

For most engineering purposes, the Rayleigh number is large, somewhere around 10^6 to 10^8.

In geophysics, the Rayleigh number is of fundamental importance: it indicates the presence and strength of convection within a fluid body such as the Earth’s mantle. The mantle is a solid that behaves as a fluid over geological time scales.

A Rayleigh number for bottom heating of the mantle from the core can be defined by the formula shown here.

High values for the Earth’s mantle indicates that convection within the Earth is vigorous and time-varying, and that convection is responsible for almost all the heat transported from the deep interior to the surface.

Related formulas

Variables

RaTRayleigh number for bottom heating of the mantle from the core (dimensionless)
ρ0density (kg/m3)
gStandard gravity
βthermal expansion coefficient (equals to 1/T, for ideal gases, where T is absolute temperature) (1/K)
ΔTsasuperadiabatic temperature difference between the reference mantle temperature and the core–mantle boundary (K)
Ddepth of the mantle (m)
CPspecific heat capacity at constant pressure (J/K*kg)
ηdynamic viscosity (Pa*s)
κthermal conductivity (W/m*K)