A note on the infamous geomagnetic pole reversal and its magnetohydrodynamic origins.
Last week, while scrolling through my newsfeed, I came across an article on the South Atlantic Anomaly (SAA) - the near-Earth region where the magnetic field is at its weakest. The article raised concerns about the potential implications of splitting of the SAA into two daughter regions, one in the east of South America and the other in the ocean southwest of Africa. Moreover, it has been observed that SAA is drifting westward at a rate of roughly 12 miles a year [1]. Not only that, historical data on geomagnetic field evolution based on the paleomagnetic record shows that Earth’s magnetic field has reversed its polarity many times in the past with a mean period of 200,000 years between the reversals. These reversals in the polarity of Earth’s magnetic field are dubbed as pole shifts, and are often considered to be apocalyptic in popular culture.
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| Figure 1: A snapshot of the region (yellow) where the fluid flow is the greatest. Core-mantle boundary = blue mesh; inner core boundary = red mesh. Source: Prof. Gary A Glatzmaier, UC Santa Cruz. |
The research on the origin(s) of Earth’s magnetism has a history of intrigue. From the perspective of common human experience, we may assume the earth to be like a solid rock hurtling through space that carries huge mountains, oceans, and human settlements on its back. In 1600s, the English natural philosopher W. Gilbert published ‘de Magnete’, a treatise on the origin of Earth’s magnetism, in which he concluded that the Earth is magnetic and hypothesized it to be a solid ball of magnetized material. According to him, the magnetic properties of the Earth were inherited from abundance of magnetic minerals like lodestone, hence permanent in nature. Considering magnetic material abundance as the origin of Earth’s magnetism, how do we argue for dynamic events like the SAA, pole shifts or highly varying geomagnetism itself in the first place? The variable nature of the earth’s magnetism evidenced by the events like the SAA and pole shifts argues for a mechanism that continually generates and morphs the magnetic field on varying timescales.
It was not until the 20th century when Sir Joseph Larmor, an Irish mathematician, proposed that a dynamo might be responsible for giving rise to Earth’s magnetic field. This idea was further studied and developed by W. M. Elsasser, an American physicist of German origin, often considered as the father of modern dynamo theory of earth’s magnetism. Dynamo, according to the dictionary, is ‘a machine for converting mechanical energy into electrical energy; a generator.’ That is exactly what Earth is 1800 miles below the surface, in its outer core – a machine that churns the hot molten metal to generate and maintain electrical and magnetic fields. The inner core of Earth is approximately the size of the moon but can attain temperatures as high as the surface of the sun. As we move radially outwards in the outer core, the temperature drops considerably. The thermal gradient in the radial direction together with compositional buoyancy give rise to the convection of hot molten metal in the outer core. The thermal and compositional convection aided by the Coriolis effects by virtue of Earth’s rotation, and the Lorentz force organize the convective currents of fluid metal into helical rolls. These helical rolls create circulating electric currents which in turn leads to the generation of the magnetic field - a self-sustaining feedback loop. Moreover, these helical flows are strongly dipole-dominated with the dipole axis nearly parallel to the rotation axis of the earth. See Figure 1. This explains the origin of the polarity of the Earth’s magnetic field. The equations that describe the non-linear dynamos are magnetohydrodynamic and include coupled conservation equations for mass and momentum, the magnetic induction equation, and a transport equation for heat. The equations are listed below for the interested reader. However, you can skip directly to the text below the equations and their description without the loss of continuity.
\[ \frac{\partial \mathbf{B}}{\partial t} = \eta \nabla^2 \mathbf{B} + \nabla \times (\mathbf{u} \times \mathbf{B} ) \ \ \ \ (Magnetic \ Induction \ equation) \]
\[ \nabla \cdot \mathbf{B} = 0 \ \ \ \ (Maxwell's \ 2nd \ equation)\]
\[ \nabla \times \mathbf{B} = \mu_0 \mathbf{J} \ \ \ \ (Ampere's \ law \ with \ negligible \ electric \ field)\]
\[ \nabla \cdot \mathbf{u} = 0 \ \ \ \ (Continuity)\]
\[ \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} = -\frac{1}{\rho_0}\nabla p + \nu \nabla^2 \mathbf{u} + \alpha \Delta T \mathbf{g} + 2\mathbf{\Omega} \times \mathbf{u} + \mathbf{\Omega} \times \mathbf{\Omega} \times \mathbf{R} + \frac{1}{\rho_0} \mathbf{J} \times \mathbf{B} \ \ \ \ (Navier-Stokes \ equation) \]
\[ \frac{\partial {T}}{\partial t} = \kappa \nabla^2 {T} + S \ \ \ \ (Heat-transport \ equation) \]
[Equation description: As the earth is a rotating frame, some extra terms involving the rotation rate of Earth feature in the Navier-Stokes equation (NSE). The 4th and 5th terms on the RHS of NSE represent Coriolis and centrifugal accelerations respectively, while the 3rd and the last term represent the thermal buoyancy and the acceleration caused by virtue of the magnetic forces respectively. The variables B, u, J and T represent the magnetic field strength, velocity of the flow, electric current density and temperature respectively. The remaining symbols have their usual meanings. The currents are assumed to be steady in the Ampere's law, hence dropping the time derivative of the electric field is justified. The equations are strongly coupled. In computer simulations where these equations are solved, the system, at each time step, starts out with an existing 'seed' magnetic field (in the outer core) which gives rise to the currents in the convecting fluid due to Lorentz force, which in turn generate magnetic field due to Ampere's law and the cycle repeats.]
Let us discuss the reversal of the poles - the infamous pole shift.
Now that the convective nature of Earth’s magnetism has been established, I will try to explain its reversal dynamics from a purely fluid dynamical analogue. Rayleigh-Bénard convection is a type of natural convection in which a fluid confined between two surfaces maintained at different temperatures. For the onset of instability, hence the convection, the gravitational body force must be directed away from the cold surface and towards the hot one. Such type of convection is ubiquitous in the outer cores of normal planets like ours, and in the convective envelopes of main sequence stars. The animation shown below is a qualitative 2D representation of this effect.
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| Animation : Bulk flow reversal in Rayleigh-Bénard convection in a fluid confined between the plates maintained at different temperatures. Red and blue represent hot and cold fluid. The instantaneous velocity vectors (tiny arrows) have been superimposed upon the temperature contour. The red (hot blobs) are seen to rise due to buoyancy while the blue (cold blobs) sink. [Simulation details: The simulation was run in leisure over the weekend using Opensource Field Operation And Manipulation (OpenFOAM) using a Steady-state solver for buoyant, turbulent flow of compressible fluids.] |
The thermal gradient imposed on the fluid by virtue of the confinement between the lower hot plate and an upper cold plate, starts to form the convective elements. These convective cells or blobs of hot and cold fluid transport momentum and heat across the domain. The main interesting feature of these turbulent thermal structures is that they exhibit many interesting properties like the large spatial, temporal correlations, and random reversals of the velocity field [2]. If you look carefully, in the animation the velocity reversal in the dominant structure in the bulk of the domain is noticeable at around 11-12 seconds after the start. For your reference, the start of the animation is apparent when the frame appears to freeze, sort of. Prior to that, the red (hot) blobs rising upwards are dominant near the right wall while as the blue (cold) blobs sinking downwards are dominant near the left wall, giving rise to a positive (counterclockwise) bulk vorticity. After the 12th second, the flow gradually begins to turn in a clockwise manner with a negative bulk vorticity. This change is evident by the red and blue blobs switching sides. For clarity, Figure 2 shows the snapshots of the flow before, during and after reversal in a similar simulation [3] as that of the animation.
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| Figure 2: Convective flow profiles in the Rayleigh-Bénard convection (a) before the reversal, (b) during the reversal and (c) after the reversal [3]. |
The velocity field reversal, although in a purely convective environment without any Coriolis or Lorentz effects, bears a strong resemblance to the reversals in the magnetic field of the Earth. It is not unreasonable to extrapolate from a purely hydrodynamic case to a magnetohydrodynamic one. The reversals in the velocity (bulk vorticity) of an electrically conducting fluid present in a magnetic field imply reversals in the induced magnetic fields. The velocity field reversal in the convection currents of the liquid metal present in the outer core of Earth, thus, is the main reason that drives the gradual geomagnetic pole shift. The continuous changes in the distribution of the turbulent liquid metal in Earth’s interior leads to the smearing and stretching of the magnetic field lines in and around the planet. These changes occur on a wide range of timescales, for example splitting of the SAA and its westward drift may occur in relatively shorter spans of time – years or decades, while as the complete reversal of the geomagnetic poles may take up to 200,000 years. The dipole-structure of Earth’s magnetic field (geomagnetic poles) is continuously morphing, implying that the pole shift is happening right now as I write this blog. It is evident that these changes are taking place all the time and the apprehensions concerned with the false description of a sudden/instantaneous pole shift are misplaced. However, large changes that weaken the magnetic field intensity at any near-Earth location may enable increased charged particle flux from the ghastly solar winds through the Earth's protective shield and hence can have drastic consequences on our satellites and communication systems. The changes are slow, gradual and controllable; not apocalyptic.
Cite as
Bader, Shujaut H., “Pole shift, geodynamo and fluid dynamics: A note on the infamous geomagnetic pole reversal and its magnetohydrodynamic origins.” Backscatter, May 27, 2020, www.backscatterblog.blogspot.com/2020/05/pole-shift-geodynamo-and-fluid-dynamics.html.References
- https://www.esa.int/Applications/Observing_the_Earth/Swarm/Swarm_probes_weakening_of_Earth_s_magnetic_field
- Eric Brown and Guenter Ahlers. Effect of the Earth’s Coriolis force on the large-scale circulation of turbulent Rayleigh-Bénard convection. Physics of Fluids, 18, 125108 (2006), 2006, American Institute of Physics. DOI: 10.1063/1.2402875.
- M. Chandra and M. K. Verma. On flow reversals in Rayleigh-Bénard convection. 13th European Turbulence Conference (ETC13), Journal of Physics: Conference Series 318 (2011) 082002 doi:10.1088/1742-6596/318/8/082002.
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