[Paper Review] Analytical estimation of the Earth's magnetic field scale

Mauro Bologna, Bernardo Tellini|arXiv (Cornell University)|Dec 8, 2012

Geomagnetism and Paleomagnetism Studies13 references3 citations

TL;DR

This paper analytically estimates the magnetic field scale of Earth-like planets using thermal electrical currents in rotating, electrically conducting fluid cores. By modeling the Peltier-Seebeck effect in a rotating sphere, it derives a magnetic field strength proportional to conductivity, temperature, and Fermi energy, yielding estimates of ~0.3–0.5 gauss for Earth and 10–14 gauss for Jupiter—remarkably close to observed values.

ABSTRACT

In this paper we analytically estimate the magnetic field scale of planets with physical core conditions similar to that of Earth from a statistical point of view. We evaluate the magnetic field on the basis of the physical parameters of the center of the planet, such as density, temperature, and core size. We look at the contribution of the Peltier-Seebeck effect on the magnetic field, showing that an electrical thermal current can exist in a rotating fluid sphere. Finally, we apply our calculations to Earth and Jupiter. In each case we show that the thermal generation of currents leads to a magnetic field scale comparable to the observed fields of the two planets.

Motivation & Objective

• To develop an analytical framework for estimating the intrinsic magnetic field scale of planets with Earth-like core conditions.
• To investigate whether thermal gradients alone can generate a magnetic field of observable magnitude in planetary cores.
• To assess the contribution of the Peltier-Seebeck effect to planetary magnetism in a rotating fluid sphere.
• To validate the model by comparing predicted field strengths with observed values for Earth and Jupiter.
• To express the magnetic field scale in terms of fundamental physical parameters and constants, minimizing reliance on empirical fitting.

Proposed method

• Derives a dimensionless form of the magnetohydrodynamic (MHD) equations using scaling with core radius R, angular velocity Ω, and characteristic velocity.
• Introduces a thermal current model based on the Peltier-Seebeck effect, where a temperature gradient drives an electrical current in a rotating fluid.
• Applies the Fermi-Dirac statistics to model electron behavior in high-density, high-temperature fluid cores, particularly for metallic hydrogen in Jupiter.
• Derives a key analytical expression for the magnetic field scale: $ B_T = \mu_0 \sigma \frac{(k_B \bar{T})^2}{e \varepsilon_F} $, where σ is conductivity, T is temperature, and ε_F is Fermi energy.
• Uses dimensionless parameters (Reynolds, magnetic, Prandtl numbers) to reduce complexity and isolate the dominant physical mechanisms.
• Applies the derived formula to Earth and Jupiter using literature-based estimates of density, temperature, conductivity, and core size.

Experimental results

Research questions

• RQ1 Can a magnetic field scale be analytically derived from core physical parameters such as density, temperature, and size?
• RQ2 To what extent can the Peltier-Seebeck effect contribute to the generation of planetary magnetic fields in rotating fluid cores?
• RQ3 Is the observed magnetic field strength of Earth and Jupiter consistent with a thermal current-driven mechanism?
• RQ4 Can the magnetic field scale be expressed in terms of fundamental constants and measurable physical parameters?
• RQ5 How does the magnetic field strength vary with core radius and temperature gradient in a rotating, conducting fluid?

Key findings

• The analytical model predicts a magnetic field scale of approximately 0.3–0.5 gauss for Earth, which is in excellent agreement with the observed dipole field of ~0.3 gauss.
• For Jupiter, the model estimates a magnetic field strength of 30 gauss within the metallic hydrogen layer, yielding a surface field of 10–14 gauss, consistent with observations.
• The derived formula $ B_T = \mu_0 \sigma \frac{(k_B \bar{T})^2}{e \varepsilon_F} $ provides a physically grounded estimate of the magnetic field scale using only fundamental constants and core parameters.
• The thermal current contribution via the Peltier-Seebeck effect is shown to be a dominant or significant factor in generating planetary magnetic fields, independent of dynamo action.
• The model remains robust across parameter variations: even with different temperature profiles or core size estimates, the predicted field remains within the order of magnitude of observed values.
• The dependence of the magnetic field on radius is implicitly captured through temperature gradients, supporting the model's consistency with planetary core structure.