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Electrical Conductivity Varies Over 5 Orders of Magnitude in Common

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Electrical Conductivity Varies Over 5 Orders of Magnitude in Common Electrical conductivity varies over 5 orders of magnitude in common Earth materials, and provides the ideal mechanism for studying fluids, water, and melt in the crust and mantle Conductivity, S/m 10 1 0.1 0.01 10-3 10-4 20°C 0°C water seawater fresh water sediments marine land 1400°C 1200°C 1100°C mantle lower mantle peridotite basalt, salt, carbonates 10% 3% 1% basaltic melt partial melt ice, hydrate hydrocarbons 0.1 1 10 100 1,000 10,000 Resistivity, Ωm CSEM: Controlled Source ElectroMagnetic sounding has been used since the 1930’s. It is used for mining exploration, groundwater, archeology, and geological structure for oil exploration. Depths to a few km. Transmitters Receivers Bz Bx I By grounded wire (E) V I Ey V horizontal loop (B) Ex MT: MagnetoTelluric (MT) sounding, which uses similar receivers to measure Earth’s natural magnetic field variations and the induced electric currents. Depths to a few hundred km. Natural-source Magnetotelluric fields Transmitters Receivers Bz Bx I By grounded wire (E) V I Ey V horizontal loop (B) Ex Primary field Secondary field Transmitter coil B secondary B total B primary Induced currents Conductive body Secondary fields are a) In a different direction b) Are a different magnitude c) Are a different phase A timeline. 1820: Oersted. Currents deflect magnets. Ampere. 1830: Fox maps earth potentials. 1831: Faraday makes current from moving magnet. 1838: Magnetic storm seen on telegraph cables. 1839: Gauss develops the spherical harmonic expansion. 1864: Maxwell’s equations. 1889: Schuster noted the relationship between magnetic field variations and earth potentials. 1939: Lahiri and Price model earth conductivity to 1,000 km depth. Barlow, 1849, currents in UK telegraph cables. Earth’s electromagnetic environment Earth’s magnetic field varies on all time scales: Grand Spectrum Reversals Cryptochrons? Secular variation ? ? Annual and semi-annual Solar rotation (27 days) Daily variation Amplitude, T/√Hz Amplitude, Storm activity Quiet days Powerline noise Radio Schumann resonances 1 year 10 million years 1 thousand years 1 month 1 day 1 hour 1 minute 1 second 10 kHz Frequency, Hz The high frequencies come from external sources ... Grand Spectrum Reversals Cryptochrons? Secular variation ? ? Annual and semi-annual Solar rotation (27 days) Daily variation external origin Amplitude, T/√Hz Amplitude, Storm activity Quiet days Powerline noise Radio Schumann resonances 1 year 10 million years 1 thousand years 1 month 1 day 1 hour 1 minute 1 second 10 kHz Frequency, Hz and the low frequencies are internal. Both can be used to estimate conductivity. Grand Spectrum Reversals Cryptochrons? Secular variation ? ? Annual and semi-annual Solar rotation (27 days) Daily variation external origin Amplitude, T/√Hz Amplitude, Storm activity Quiet days internal origin Powerline noise Radio Schumann resonances 1 year 10 million years 1 thousand years 1 month 1 day 1 hour 1 minute 1 second 10 kHz Frequency, Hz bearing in mind that between about one year and ten years there can be overlap between internal and external fields. Grand Spectrum Reversals Cryptochrons? Secular variation ? ? Annual and semi-annual Solar rotation (27 days) Daily variation external origin Amplitude, T/√Hz Amplitude, Storm activity Quiet days internal origin Powerline noise Radio Schumann resonances 1 year 10 million years 1 thousand years 1 month 1 day 1 hour 1 minute 1 second 10 kHz Frequency, Hz A spatial view of Earth’s magnetic field: A spatial view of Earth’s magnetic field: The solar wind drives changes in Earth’s magnetic field 1.2 1 Magnetic Field (N-S) 0.8 pT 0.6 0.4 0.2 1 pT = 10-12 Tesla : Earth’s field is 30-60 µT (10-6 T) 0 So, this signal is one hundred millionth of the main field. -0.2 0 400 800 1200 1600 2000 seconds These changes generate an electric field in the ground through induction 1.2 1 Magnetic Field (N-S) 0.8 pT 0.6 0.4 Electric Field (E-W) 0.2 0 x10 µV/m -0.2 0 400 800 1200 1600 2000 seconds In practice we measure both north and south fields, and maybe Bz Particles from the sun and magnetic storms inject particles into the ring current. The size of the ring current generated by solar activity is measured using the Dst index (“disturbed storm time”). Because the effect of the ring current is to reduce Earth’s main field, storms go negative. Magnetic storms are tied to the 11-year solar cycle, but not too strongly. Dst index (blue, nT) and 10.7 cm radio flux (red, 10−22 J/s/m2/Hz) 400 300 200 100 0 −100 −200 −300 −400 −500 −600 1960 1970 1980 1990 2000 2010 Year The ionosphere is electrically conductive. 200 160 Ionosphere (anisotropic) 120 80 Atmosphere Altitude, km 40 Earth’s surface 0 -14 -10 -6 -2 log(conductivity, S/m) Thermally induced currents generated on the dayside create a daily variation, or Sq, in Earth’s magnetic field. USGS via BGS Schumann resonance at 1/8 second and harmonics. Electrical conductivity of rocks and minerals. Rock conductivity (and resistivity): 1 m 1 m current = I 1 m Electrical resistivity ρ has units of Ωm. A 1m x 1m x 1m cube of 2 Ωm rock would have a series resistance of 2 Ω across the faces: ρL RA R = ρ = A L where L is the length of any prism and A is the cross-sectional area. Conductivity σ is just the reciprocal of resistivity. Units are S/m. Approximate conductivities and resistivities of various Earth materials. Ionic conduction through water 1 seawater at 20◦C 5 Sm− 0.2 Wm 1 seawater at 0◦C 3 Sm− 0.3 Wm 1 marine sediments 1–0.1 Sm− 1–10 Wm 1 land sediments 0.001–0.1 Sm− 10–1000 Wm 5 2 1 igneous rocks 10− –10− Sm− 100–100,000 Wm Good semiconductors 1 4 6 graphite 10,000–1,000,000 Sm− 10− –10− Wm 1 1 5 galena 10–100,000 Sm− 10− –10− Wm 1 3 pyrite 1–1,000 Sm− 10− –1 Wm 1 5 magnetite 20,000 Sm− 5 10− Wm ⇥ Thermally activated conduction 1 olivine at 1000◦C 0.0001 Sm− 10,000 Wm 1 olivine at 1400◦C 0.01 Sm− 100 Wm 1 theoleite melt 3 Sm− 0.3 Wm 1 silicate perovskite 1 Sm− 1 Wm Resistivity of Rocks and Minerals (Data from Telford et al.) basalts/lavas limestones OIL SANDS sandstones, quartzites wet dry clays graphite evaporites pyrrhotite intrusive igneous rocks seawater groundwater wet dry quartz -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Log 10 (Resistivity, Ωm) In semiconductors, charge carriers can be a product of doping, as well as Intrinsic thermal activation. regime The same is true of electrically charged Extrinsic regime defects in mineral crystals. Conductivity Impurity Dislocation Temperature Frenkel defect Schottky defect Now the theory... Some basic electromagnetic theory. Let’s start with E is the electric field, measured in volts/meter. B is the magnetic field, measured in tesla. H is sometimes also called the magnetic field, and has units of A/m, and then B is called magnetic induction vector, or the magnetic flux density. But here we call B the magnetic field, and H the magnetizing field, in recognition of its relationship to magnetic polarization. B and H are related through the permeability µ of a material. B = µH 7 In a vacuum, µ = µ =4⇡ 10− H/m o ⇥ Integrating over surfaces: S A surface vector, s, has a direction that is the outward normal to the surface and a magnitude proportional to the area of the surface. An infinitesimal surface element, ds, is useful for integrating things over surfaces. In particular, if we have a vector field A, then the integral A ds = A. cos ✓.ds · ZW ZW over surface W is the Flux through the surface. ✓ is the angle between A and ds. We are going to need a little calculus, and the use of the operator. r @ @ @ = , , r @x @y @z ✓ ◆ so A is just a gradient: r @Ax @Ay @Az A = , , r @x @y @z ✓ ◆ A is the divergence: r· @Ax @Ay @Az A = + + r· @x @y @z A is the curl, r⇥ @Az @Ay @Ax @Az @Ay @Ax A = , , r⇥ @y − @z @z − @x @x − @y ✓ ◆ and A is the Laplacian. r·r 2 2 2 @ Ax @ Ay @ Az A = 2A = + + r·r r @x2 @y2 @z2 Gauss’ Law: Q ⇢ E ds = E = · ✏ r· ✏ ZW o o Q is charge, C ⇢ is charge density, C/m3 12 ✏ is permittivity of free space, = 8.895 10− F/m. o ⇥ Gauss’ Law says that the electric field leaving a volume is proportional to the enclosed charge. E Q Faraday’s Law: dFB @B E dl = E = · − dt r⇥ − @t Ic FB is magnetic flux Faraday’s Law says that the electric field integrated around a loop (i.e. the voltage) is given by the time rate of change of the enclosed magnetic flux. E dB dt Gauss’ Law (magnetism): B ds =0 B =0 · r· ZW Gauss’ Law for magnetism says that there are no magnetic monopoles. Any flux entering a volume has to leave it. B Ampere’s` Law: B dl = µoI B = µoJ · r⇥ Ic I is electric current, A J is current density, A/m2 8 µ is permeability of free space, = 4⇡ 10− H/m. o ⇥ Ampere’s` Law says that an electric current will generate a circulating magnetic field.
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