# Visualizing the time-dependence of the Earth's magnetic field

## Summary

Visualizing the International Geomagnetic Reference Field. This activity falls under "visualization" of a geophysical field as expressed in spherical harmonics, which has been appreciated by college Freshmen, but also under "programming", targeted to and tried by college Seniors (Geosciences and Physics), since the "product" is both a series of maps that can be viewed and discussed as a movie highlighting the main time-variable features of the Earth's geomagnetic main field, and a Matlab suite of programs that ingests input files with spherical harmonic coefficients and produces the visualizations, and which can be changed by the students (e.g. to look at different reference fields, or to plot different properties than the surface geomagnetic potential.)

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## Learning Goals

The Earth's magnetic field varies on different spatial scales, and on different temporal scales. Students need to learn about the length-scales of the field itself (the difference between the very long-wavelength core-generated field and the much shorter-wavelength crustal field), they also need to learn about the relative contributions of the external versus the internal fields, and lastly, about diurnal variations and longer-term secular variations. Finally, students need to be exposed to the representation of the main field in the basis of spherical harmonics. For the visualization of spherical harmonics, we developed a separate module. As to the visualization of the main field and its variations, this module serves to plot the International Geomagnetic Reference Field (11 editions and counting), which has downloadable spherical harmonic coefficients. A Matlab program loads those coefficients and plot the field in the time-dependent rendition as shown in the animated gif that accompanies this module.

## Context for Use

Discussion of the main field and its variations is well within the grasp of the average college Freshman (undeclared majors), which is the target audience for which this module has been developed. But we have also gone into more depth by using the materials for an upper-class course (Seniors majoring in the geosciences) in Global Geophysics. There, the concepts of the Geocentric Axial Dipole can be discussed in a geological context. Also, the presence of "westward drift" is very noticeable in these graphs. And finally, having the students observe what happens to "flux patches" - best embodied by closed contour lines, leads to a theoretical discussion of the "frozen-flux" hypothesis, which underlies much of geomagnetic research today. Understanding this, then, can lead to a discussion of the fundamental equations of Geophysical MagnetoHydroDynamics - in particular, the "magnetic induction equation" which has two clearly identifiable terms, an advective and a diffusive one, with two end-member results (no advection - no diffusion) which are readily grasped by an undergraduate audience. Turn the field generating mechanism off, and Ohmic decay ruins the field (on what time scale)? Focus on advection, and the attached graphs lead into an estimate of the velocity of the convective currents deep inside the Earth.

## Description and Teaching Materials

The animations are made by Matlab computer code that is available on our own webpages and also on our GitHub channel (https://github.com/csdms-contrib/slepian_alpha/blob/master/blob.m). They are "open" in the sense that the students can access the code, rerun it (e.g. switching IGRF-10 for IGRF-11 and watching the changes), and also making different plotting choices.

## Teaching Notes and Tips

Depending on the level of the students, the focus can be either on "looking" but also on "doing". Looking is easiest: we have shown Freshmen the animated gifs and asked them what the main variability of the main field is. What can be noticed on the maps? What are the main variations? On what time scale do they vary? What does the short time scale of main field variation teach us about the inner workings of the Earth? The main field is generated by convection currents in the outer core of the Earth. What is the time-scale of these currents? How do they compare to the time-scale of variations (convection-generated, also) of the mantle (which is much slower)? Finally, we ask the students why these variations are important - e.g. for navigation, but also for the wider implications of the Geocentric Axial Dipole hypothesis - that over geological time, the main field IS well represented by a simple dipole term. Note that the dipole terms have been removed from the attached animations: What we see is everything that is commonly attributed to the main field without the dipole terms. The software that generates the graphs can be adapted to also show the dipoles of course - in which case the maps become entirely uninteresting.

## Assessment

The module shown here, per se, is about "looking" at the field. But we typically give the students an appreciation for the individual terms in the spherical harmonic expansion series by listing explicitly the values for the low-degree terms. The students are then asked to understand what the meaning is of the degree-0 term (not present here), the degree-1 term (the three coefficients that yield the dipole and that can be used in a simple calculation to derive the angle between the Earth's geomagnetic and geographic axes) and so on. Students need to understand the difference between geographic and geomagnetic coordinate systems. Often we ask students to pull out their iPhones and look at the magnetic compass, which has two settings (geographic and geomagnetic) which illustrate precisely this difference. Students are asked to make linkages between these coordinate systems and the basic concepts of inclination and declination, and when we hand out actual compasses, we discuss how they can be adjusted to reflect the local declination at the site of where the compass is being used. Exercises then ask the students to discuss inclination and declination of a simple dipole, evaluate the strength of the field at the equator versus at the pole, calculating the decay time from the derivative coefficients that are tabulated in the IGRF files, and so on. Finally, the module here is a lead-in for the study of simple magnetic anomalies, i.e. dipoles buried at depth, which are of paramount importance for exploring the shallow substructure e.g. in geoarcheology and the exploration of mineral resources.