Origin Of Earth's Magnetism - Geomagnetic Excursions - Bruce A. Buffett Professor At U.C. Berkeley

Geomagnetic Reversals and Excursions: The Origin of the Earth's Magnetic Fields

Key Concepts:

• Geomagnetic Reversals: Complete flips of the Earth's magnetic poles.
• Geomagnetic Excursions: Significant deviations in the magnetic field direction that don't result in a full reversal.
• Dynamo Theory: The process by which the Earth's magnetic field is generated by the motion of liquid iron in the outer core.
• Mantle Convection: The movement of the Earth's mantle, which influences the cooling of the core.
• Inner Core Solidification: The process by which the Earth's inner core is growing as the outer core cools.
• Induction Equation: Describes how magnetic fields change due to fluid motion and diffusion.
• Curie Temperature: The temperature below which a material becomes magnetic.
• Virtual Axial Dipole Moment (VADM): An estimate of the strength of the Earth's dipole field over time.
• Stochastic Model: A model that incorporates randomness to represent processes that are not fully understood.
• Fokker-Planck Equation: A mathematical equation that describes the evolution of the probability distribution of a variable over time.
• Drift Term: In the Fokker-Planck equation, the term that describes the deterministic tendency of the variable to change.
• Diffusion Term: In the Fokker-Planck equation, the term that describes the random fluctuations of the variable.
• Mean First Passage Time: The average time it takes for a variable to reach a certain threshold.

1. Introduction and Motivation

• The Earth's magnetic field is crucial for navigation and protecting the atmosphere from solar wind.
• The absence of a magnetic field poses challenges for establishing habitats on Mars.
• The talk explores the dynamics and evolution of planetary interiors, focusing on the origin of the Earth's magnetic field and geomagnetic reversals/excursions.

2. Earth's Interior Structure and Dynamo Theory

• The Earth consists of a silicate rocky mantle and a liquid iron core.
• Mantle convection cools the Earth's interior, allowing the liquid iron core to convect.
• The liquid iron core has very low viscosity, and its convection has a helical structure due to the planet's rotation.
• This helical convection is crucial for driving the geodynamo and generating the magnetic field.

3. Recent Findings on Core Conductivity and Thermal Properties

• A study in Nature presented theoretical calculations of iron's electrical and thermal conductivity at core conditions, finding values 2-3 times higher than previous estimates.
• Higher electrical conductivity makes it easier to sustain the geodynamo.
• The dipole decay time (time for the dipole to diffuse away if convection stops) increases from 25,000 to 50,000 years with the new conductivity estimates.
• Higher thermal conductivity impacts the heat budget of the core.
• Mantle convection draws heat from the core, causing the liquid metal to solidify from the inside out, forming the solid inner core.
• Solidification releases latent heat and excludes light elements, driving convection.
• The amount of heat conducted down an adiabatic temperature profile (Q adiabatic) is comparable to the heat the mantle takes from the core.
• Higher thermal conductivity increases Q adiabatic, potentially reducing the buoyancy sources that drive convection, especially as the inner core grows smaller.
• This may explain why some terrestrial planets have dynamos (Earth, Mercury) while others don't (Venus, Mars).

4. Methods for Studying the Geodynamo

• Numerical Models: Reproduce features of the Earth's magnetic field but rely on parameters extrapolated from real values by up to 10 orders of magnitude.
• Historical Records: Compilations of magnetic field measurements from early explorers (late 1500s onwards) provide 400 years of data on the magnetic field's evolution.
  • These records show the field is mostly dipolar but with significant undulations and changes over time.
  • The South Atlantic Anomaly is a region of weak field that has emerged in historical records.
• Geological Records:
  • Lava Flows: Rocks record the orientation of the magnetic field when they cool through the Curie temperature.
  • Sediments: Magnetic grains in sediments align with the magnetic field as they settle, providing a continuous record of magnetic field direction and relative intensity.
  • The magnetic field has reversed many times in the geological past, with an average of three reversals per million years.
  • Reversals don't have a typical duration, and there's no preference for one polarity over the other.

5. Reconstructing Fluid Flow at the Core-Mantle Boundary

• The induction equation describes how the magnetic field changes due to fluid motion and diffusion.
• On historical timescales (400 years), diffusion can be neglected, allowing the inversion of magnetic field data to reconstruct velocity fields at the top of the core.
• Fluid motions in the core are on the order of tens of kilometers per year.
• One day of flow in the atmosphere is comparable to 100 years in the core, making it challenging to understand the dynamo using only 400 years of magnetic observations.

6. Constructing Time Series of Dipole Moment

• Researchers merge lava and sediment data to construct time series of the Virtual Axial Dipole Moment (VADM), estimating the strength of the dipole field over time.
• Leah Ziegler's study provides a VADM time series for the last two million years.

7. Stochastic Modeling of Dipole Field Evolution

• The time evolution of the dipole field (X) is modeled as a combination of a deterministic term (slowly varying function of field intensity) and a random term (representing unconstrained fluctuations).
• This approach is analogous to Brownian motion, where the random movement of pollen grains is caused by unseen water molecules.
• The stochastic model represents small-scale turbulent convection in the core as a random process.
• The noise term is assumed to have the following properties:
  • Average to zero (no drift).
  • Be uncorrelated (white noise).
• The model produces realizations of dipole moment evolution that drift toward the time average but vary due to the randomness of the process.

8. Fokker-Planck Equation and its Application

• The Fokker-Planck equation describes the evolution of the probability distribution of the dipole moment over time.
• The equation involves a drift term (V) and a diffusion term.
• The drift term controls the motion of the probability distribution, while the diffusion term governs its breadth.
• The drift and diffusion terms can be estimated from the VADM time series.
• The first moment (difference between dipole moments at two times) is used to estimate the drift term.
• The second moment (squared difference) is used to estimate the diffusion term.

9. Estimating Drift and Diffusion Terms from Data

• The drift term is calculated as the difference in dipole moment divided by the time difference.
• The diffusion term is calculated as the squared difference.
• The time difference (tau) needs to be large enough for the noise to be treated as uncorrelated (around 4,000 years).
• A time difference of 4,000 years may reflect the time required for sediments to acquire stable magnetization.

10. Results of Drift and Diffusion Term Estimation

• The drift term is approximately linear in the intermediate range of dipole moments.
• The average dipole moment is about 5.3 (in the usual units), corresponding to where the drift is nearly zero.
• If the dipole moment is weak, the drift is positive, making it bigger. If the dipole moment is large, the drift is negative, making it smaller.
• The drift term is assumed to be an odd function of X (goes through the origin).
• The diffusion term is assumed to be an even function of X (derivative is zero at the origin).

11. Physical Interpretation and Model Validation

• Small fluctuations in the field around the average can be modeled as exponential decay with a relaxation time scale of 29,000 years.
• Larger fluctuations can be represented using a potential, where the drift term is the gradient of the potential.
• The data suggests the existence of a potential well in which the dipole moment rattles around, with occasional large perturbations causing it to hop over the well and reverse polarity.
• The steady-state probability distribution predicted by the Fokker-Planck equation shows equal probabilities of being in either polarity, consistent with the geological record.
• Transient solutions of the Fokker-Planck equation show how the probability distribution evolves over time, starting from the time average.
• The mean first passage time to leave the positive interval (reversal) is calculated to be one million years, compared to the observed three reversals per million years.
• The mean first passage time to reach a point where an excursion takes place is calculated to be about right (8.4 per million years).
• The model seems to capture the frequency of excursions but underestimates the frequency of full reversals.

12. Connecting Dipole Moment to Core Processes

• The dipole moment is related to the current flowing through a cable or, more generally, to the circulation of current in the Earth's core.
• The dipole moment can be expressed in terms of the magnetic field.

13. Conclusion

• The stochastic model, based on the Fokker-Planck equation and constrained by geological data, provides insights into the dynamics of the Earth's magnetic field.
• The model captures some aspects of geomagnetic reversals and excursions but needs further refinement, particularly in representing the behavior of weak fields.