Mantle heat production

Mantle heat production#

Most of the heat lost through the Earth’s surface originates from the interior of the planet by the decay of radioactive elements.

We can calculate the present total heat lost through the Earth’s surface if we assume the following parameters (Turcotte and Schubert 2014):

  • total continental area: \( S_{c} = 2 \times 10^{8}\) km\(^{2} \)

  • mean continental surface heat flow: \( q_{c} = 65 \) mW m\(^{2} \)

  • total oceanic area: \( S_{o} = 3.1 \times 10^{8}\) km\(^{2} \)

  • mean oceanic surface heat flow: \( q_{o} = 101 \) mW m\(^{2} \)

Heat loss through continents and oceans#

continental_area           = 2e14  # m^2
mean_continental_heat_flow = 0.065  # W/m^2
oceanic_area               = 3.1e14  # m^2
mean_oceanic_heat_flow     = 0.101  # W/m^2

# Total Earth heat loss:
heat_loss_continents = continental_area * mean_continental_heat_flow  # W
heat_loss_oceans     = oceanic_area * mean_oceanic_heat_flow          # W
total_heat_loss      = heat_loss_continents + heat_loss_oceans      # W

print('The continental heat loss is {:.2e} W \n'.format(heat_loss_continents) + \
      'The oceanic heat loss is {:.2e} W \n'.format(heat_loss_oceans) + \
      'The total Earth heat loss is {:.2e} W \n'.format(total_heat_loss))

Heat generated per unit mass#

The heat generation per unit mass can be given by:

\[ A = \frac{Q}{M} \tag{10}, \]

where Q (W) is the heat flow and M (kg) is the mass.

We know, based on geochemical studies, that the Earth’s core do not contain a significant amount of heat-producing elements. Thus, we only needs the mantle mass to compute the heat generated by the Earth per unit mass.

The mass of the mantle, \( M_{mantle} = 4 \times 10^{24} kg \) (see Turcotte and Schubert 2014 for details)

mantle_mass = 4.e24  # kg

volumetric_heat_production = total_heat_loss / mantle_mass  # Heat flow per unit mass in W/kg

print('The Earth\'s total heat generation is {:.2e} W/kg'.format(volumetric_heat_production))

Corrections to compute the mantle heat production#

1. Crustal heat generation:#

The continental heat flux value used above (\( q_{c} = 65\) mW m\(^{2} \)) includes the heat produced in the mantle and conducted through the crust and the heat generated by heat-producing elements in the crust.

In order to parameterize the geotherm calculation in terms of surface heat flow, an empirical relationship that partitions the observed surface heat flow, with 40% being attributed to upper crustal radiogenic sources and 60% to mantle sources, is often used (Pollack and Chapman 1977; Vitorello and Pollack 1980; Chapman 1986).

Based on estimates of mean concentrations of the elements in the upper and lower crust (Table 2), we can attribute \( 37 \) mW m\(^{2} \) to heat producing elements and \( 28 \) mW m\(^{2} \) to basal heating by mantle convection.

2. Secular cooling:#

We also need to correct for the contribution of the secular cooling of the Earth to the surface heat flow. It is assumed that 20% of the present day surface heat flow comes from the cooling of the Earth and 80% can be attributed to radioactively generated heat.

#----------------------------------------------------------------------
# Add corrections described above to the mantle heat production
#----------------------------------------------------------------------

# 1. Remove crust radioactive sources
heat_flow_continental_mantle = 28e-3
heat_loss_continental_mantle = continental_area * heat_flow_continental_mantle

heat_loss_mantle = heat_loss_continental_mantle + heat_loss_oceans

# Mantle total heat generation
volumetric_heat_production_mantle = heat_loss_mantle / mantle_mass

# 2. Secular cooling correction
volumetric_heat_production_mantle *=  80 / 100
print('The Earth\'s mantle total heat generation is {:.2e} W/kg'.format(volumetric_heat_production_mantle))

 


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