Earth's Magnetic Field

By convention, the "north-seeking" pole corresponding to that at the north end of a compass needle is called the positive pole, and the "south-seeking" pole is referred to as the negative pole The lines of force are directed outward from a positive (i.e., north) pole and inward to a negative (i.e., south) pole."  P. V. Sharma Geophysical Methods in Geology, 2nd Edition.

Note that this means the Earth's north magnetic pole is a negative pole, because the positive "north-seeking" end of a compass needle is attracted toward it. The lines of force referred to would be the force on a positive "test monopole" of unit strength.

• Gilbert (1500s) recognized Earth's field was similar to a dipole's
• 1838, Carl Friedrich Gauss proved 95% of Earth's magnetic field is internal, approx. 5% external

Earth's field is sum of 3 parts:

1. External Magnetic Field
2. Anomalous Induced Magnetic Field
3. Main Magnetic Field

1. External magnetic field

• about 1-5% of total field
• biggest secular variation is diurnal, with an amplitude of tens of nT
• also a seasonal variation, i.e., diurnal variation greatest in summer
• strongest at equator
• all suggest role of sun
  1. EM (UV and x-rays) ionizes particles in ionosphere
  2. Sun's tidal force produces cyclic wind currents in ionosphere, which in turn produces electrical currents
  • ionosphere directly produces 2/3 of observed diurnal effect; induced currents in Earth contribute about 1/3
• magnetic storms: external field occasionally variable over minutes, at hundreds of nT or more (magnetic storm)
  • from 100s to 1000s of nT
    
  • last hours to days
  • caused by sunspot activity
  • WWV reports useful to predict when not to conduct surveys
  • 27 day cycles (sunspots rotate with Sun)
  • fixed-station recording magnetometers provide data on field variability (e.g., Alaska)
  
  [3/28/20001 propagation forecast based on sunspot activity]
  • produced by electrical currents in ionosphere consisting of particles ionized by solar radiation
  • sunspot number strongly affects solar radiation
    
  • current cycle, as of November 2004:
    
  • sunspot cycles back to about 1750
    
  • Sunspots and Global Warming

2. Anomalous, induced magnetic field

• magnetization induced in crust by Main Field (and External Field) or remanent magnetization ("permanently induced")
• limited to upper crust (Curie T)
• limited to ferr(o,i)magnetic materials

3. Main magnetic field

• produced by electrical currents in outer core
• steady on time scale of days, but variable over years
• approximately 50,000 nT (0.5 Oe)
• accounts for large regional variations in intensity and direction

We showed that the potential due to a dipole is

and we obtain magnetic induction, B, or the vector field, by taking gradient of potential, and find the components of this vector field:

The best-fit dipole approximation of Earth's field, at r = a (radius of Earth):

• B₀ is the equatorial strength of best-fit dipole
• axis of dipole is currently tilted about 11 degrees to spin axis
• this gives M (dipole moment) of 7.94*10²² Am²
• because this is a best fit dipole field centered at Earth's center, the N and S poles are antipodal:
  • North best-fit pole approximately at 79^o N, 71^o W
  • South best-fit pole approximately at 79^o S, 109^o E
• Dip Poles: looking at Earth's actual field, we (currently) have two points on Earth where field is vertical:
  • North dip pole (I = +90^o), approximately at 76 deg N, 101 deg W, field strength 0.6 Oe (60,000 nT)
  • South dip pole (I = -90^o), approximately at 66 deg S, 143 deg E, field strength 0.7 Oe (70,000 nT)
  • magnetic equator: 0 deg inclination contour line; for best-fit dipole field, it is a great circle "tilted" 11^o from geographic equator; "real" magnetic equator depends on current field values but is shown schematically below

Spherical Harmonic Analysis of Earth's Field

• because magnetic field is conservative and, for a monopole, follows same 1/r² dependence as gravity, it, too, obeys Laplace's equation
• thus the potential, V, must be of the form:

• here, q and l are taken to be the geographic colatitude and geographic longitude, respectively
• in contrast to gravity, for magnetic field, S' and C' are not equal to zero (until you are outside Earth's ionosphere, where external field is sourced)
• but they are small, average out over short times, and are neglected when discussing "solid Earth" field, i.e., exclusive of external field
• l = 0 term is zero because no magnetic monopoles
• it is convenient to introduce the Gauss coefficients:
   

Dipole and non-Dipole Components

• Earth's Main Field can be thought of as consisting of dipolar and non-dipolar components

Dipole Field

• if Earth's field were purely dipolar, and aligned with geographic north (i.e., with coordinate system for spherical harmonic analysis), then g₁⁰ would be only non-zero term
  • Q: Why not just use, say a Fourier series to represent Earth's field? I.e., what is advantage of using spherical harmonic analysis, Legendre polynomials, yada yada...
• if purely dipolar, but tilted, g₁⁰, g₁¹, h₁¹ would be only non-zero terms
  

• B₀ is strength of best-fit dipole at magnetic equator
  
• as mentioned above axis of dipole is tilted about 11 degrees to spin (geographic) axis, dipole moment is 8 x 10²² Am², N pole approximately at 79^o N, 71^o W, S pole approximately at 79^o S, 109^o E

Non-Dipole Field

• Non-dipole field is just Earth's total (main) field minus tilted dipole field
• shows four major anomalies (quadrupolar)
• can be constructed from spherical harmonic coefficients by setting g₁⁰, g₁¹, h₁¹ = 0

Secular variation

• one of the few "solid Earth" phenomena that change significantly over a human lifetime
• data go back to at least 1500s!  Rregular observations of magnetic field made since 1540 at London, later Paris (figure from McElhenny)
  
• data go back to at least 1500s (London Observatory)
• change about 50 - 150 nT/year most places (see g_s, h_s) mostly related to non-dipole field
• westward drift: non-dipole field drifts about 0.2 deg longitude per year
  • produces decreases and increases at some localities
  • indicates that core rotating more slowly than mantle/crust
• dipole field changing also; decreasing 1/2000 per year

International Geomagnetic Reference Field (IGRF)

• IGRF is just the main field as defined by spherical harmonic coefficients (external field excluded)
• it is a reference field in the sense that in exploration work, which is concerned with the shallow crustal induced field, we subtract IGRF from measured field values in a region, to remove a regional field
• to express the secular variation in the Earth's main field, a new set of constants in the spherical harmonic expansion of the field are made from the latest magnetic data, at 5 year intervals, and time-rate-of-change in field at end of the 5 year interval is computed for extrapolating into the future.
• for exploration field work, it is important to subtract the IGRF for the time period the data were collected

9th Generation IGRF

Geomagnetic Models and Software

My FORTRAN IGRF Program

• example: to find field in June, 2003 (2003.5), extrapolate by using IGRF2000 values for 2000.0 plus secular terms for 2000.0 times 3.5; to find field in 1994, interpolate between 1990.0 and 1995.0 coefficients
• example:

Model: USGS90 Latitude : 35 N
Date : 10/6/93 Longitude: 97 W Elevation: 300.000 m
   D        I      H     X    Y      Z    F
deg  min deg  min  nT     nT   nT     nT   nT
---  --- --- ---- ----- ----- ---- ----- -----
  6  3.0  64 14.1 22914 22787 2414 47474 52715

Model: USGS90 Latitude : 35 N
Date : 10/6/83 Longitude: 97 W Elevation: 300.000 m
   D        I      H     X    Y      Z    F
deg min deg  min   nT    nT   nT     nT   nT
--- --- --- ---- ----- ----- ---- ----- -----
  6 56.3 64 16.5 23199 23029 2802 48149 53447

Field Elements

• Earth's field can be separated into vector components:
  
• where F is the total field intensity, and X, Y, Z and H are the north, east, vertical and horizontal components, respectively
• inclination: angle field makes with horizontal ("dip")
  
  where q_m is magnetic colatitude; f_m is magnetic latitude
• declination: angle horizontal field makes with true north
• X, Y and Z can be found from |F| (magnitude of total field) and declination, D, and inclination, I
• and vice versa:
   

Magnetic field of Earth based on IGRF 1990 (Blakely):

(a) Isodynamic map showing total intensity, contour interval 2,500 nT

(b) Isoclinic map showing constant inclination, contour interval 10^o

(c) Isogonic map showing constant declination, contour interval 10^o

Present-day total magnetic field

total field

declination

inclination

Non-dipole field

Secular variation in vertical field

Weakness in Earth's Field Damages Spacecraft

Location of Source of Magnetic Field

• we assume that, at the depth where the field is created, all wavelengths would be of equal strength ("white" spectrum)
• note that terms in spherical harmonic expansion of internal part of Earth's field drops of like:
  
• hence, higher degree terms drop off faster with distance from source
• by examining spectral content (relative amplitude of different wavelengths (orders) of Earth's field, one can estimate depth to source
• define a quantity (analogous to power in seismology) called mean square field per harmonic degree:
  
• gotten by adding up square of terms of all orders, m, for each degree l.
• for lower order terms (e.g., 10 or less, as in IGRF), use IGRF coefficients
• for very high order (short wavelength) terms (), use Fourier analysis of field strength from aeromagnetic data (great-circle airborne magnetic survey at 3 km altitude)
  
  
• note that above illustration based on Magsat data
• Another illustration of spectral roll-off
• strong roll-off of low-degree terms indicates a deep source
• downward continuation of low degree terms (1-8) indicates the spectrum would be white (no roll-off) at 0.47a
• therefore, long wavelength terms generated in outer core
• further evidence for outer core source:
  • dipole and non-dipole components change in direction and magnitude with time
  • motion is of order 0.2^o per year (~20 km/yr)
  • major non-dipole features move westward, hence called westward drift
  • this is ~10⁵ times rate of mantle convection
  • thus implies origin in fluid core
• less rapid roll-off indicates a much shallower source, out ~10 km of crust

Physics of Magnetism

Three sources of magnetism:

1. Current loops 
    
2. Permanent magnets; remanent magnetization (primarily magnetite and related minerals; hematite)
   1. TRM: thermal remanent magnetization
   2. CRM: chemical remanent magnetization
   3. DRM: detrital remanent magnetization
      "detritus: material produced by disintegration and weathering of rock that has been moved from its site of origin"
3. Induced magnetization (same minerals)

It is empirically found that the intensity of magnetization, I, is proportional to ambient field, H:

The total magnetic induction, B, is

Remanence (2.) and induced magnetization (3.) result from these phenomena:

• paramagnetism:  weak susceptibility possessed by most materials
• diamagnetism:  weak negative susceptibility (e.g., halite, anhydrite, H₂O)
• ferro- and ferrimagnetism:  magnetite, pyrrhotite, ilmenite, hematite, etc.)

Curie temperature

• above about 500-700 ^oC, minerals cease being ferromagnetic (about what depth is this?)
• since para- and diamagnetism are weak, must attribute low degree components of field to electric currents
• mantle is poor electrical (and thermal) conductor, so currents are in Fe/Ni outer core

Source of High Degree/Order Components of Magnetic Field

• example of application of geomagnetism to "exploration"
• remanent and induced magnetism resides in crust
• concentration of magnetic minerals in igneous and metamorphic rocks, especially mafic rocks
• "basement" usually higher in magnetic minerals than sedimentary rocks (sed rx are "transparent")

• since higher order terms tend to be sourced in the basement rocks in crust, often depth to basement can be found
• involves measuring slopes or widths of first and and second derivatives of anomalies (e.g., Peter's half-slope method), or spectral methods

Core Magnetism: Self-Exciting Dynamo

Faraday Induction Law: time-varying magnetic field induces currents in a conductor (e.g., car's alternator)

Biot-Savart Law:  moving charges (current) create a magnetic field

• must have had external field to kick this off (Sun, especially in T-Tauri phase), but now self-sustaining
• magnetohydrodynamics:  especially difficult because flow of conducting core in presence of its own field gives rise to the currents causing the field, but then we have:

Lenz's Law:  the induced current will appear in such a direction that it opposes the change that produces it

• this means the core must do work to maintain the field
• the work turns into Joule heating in core (I²R) ==> P = IE, E = IR, ==> P = I²R
• where does work (energy) come from?
  • estimated 10⁹ - 10¹¹ W required; for comparison:
    • earthquakes: 10¹² W
    • Earth's total heat flux: 4x10¹³ W
  • precessional torque exerted on core by mantle probably insignificant
  • latent heat of fusion from solidifying core: crystallization drives convection
  • radioactivity drives thermal convection
• thermal convection must provide significantly more heat than
  
  in order to be efficient
• ⁴⁰K is only radioactive isotope compatible with core materials
• would require 0.1% K in core (equivalent to 67% of K in a chondritic Earth)

Modeling the Self-Exciting Dynamo by Lee J. Harper

One of Maxwell's Equations is

Since

then

or