The Induced Fields Of Callisto And Europa And Their Lessons For Earth
Alternate models for planetary processes, part 4
This is the fourth post of seven in support of the PROM article “An integrated physical model for planetary magnetism and heat”, which proposes an alternate origin for the geomagnetic field versus the consensus geodynamo theory. Each PSI post documents an alternate model to the existing scientific consensus on planetary processes.
Post 4 deals with how the magnetic flux energy generated by the solar wind at a planet’s magnetopause (PSI Post 3) travels relatively unimpeded to its ferromagnetic core.
The prevailing theory on induced planetary magnetic fields
It is commonly believed that the magnetic fields of Earth, Jupiter, Saturn, Uranus, and Neptune are generated by a dynamo that converts thermal to magnetic energy [1], but that the fields of Europa and Callisto are induced, as their dipole fields are consistently anti-parallel to the ambient Jovian magnetospheric field [2]. The power source of these Galilean satellites’ magnetic fields is therefore demonstrably the magnetic flux generated during their orbit through Jupiter’s inclined and rotating field. The location of the satellites’ inductors is enigmatic, so workers [2] used the Galileo flyby observations to estimate the inductor’s properties, such as its conductivity. Zimmer et al. [2] derive an equation (from the Maxwell equations, see PSI Post 3) that correlates the instantaneous magnetic field strength, BA, induced in a spherical planetary shell of arbitrary conductivity (that is less than infinite conductivity) at time t, to the induced field of a perfectly conducting sphere:
Whereby A is a scaling factor, f is the frequency of Jupiter’s field (BJ), and ϕ is a phase shift (the induced field always lags behind BJ). Flyby data indicate that both Callisto and Europa can be accurately modeled as perfectly conducting spheres (A ≃ 1; ϕ ≃ 0) [2], indicating their inductors have a very high conductivity. An important yet obscure role is played by the skin depth, the travel distance through the Galilean satellite’s mantle at which the electromagnetic energy of BA or BJ – through absorption – falls to 1/e of its original value: if the inductor lies within the satellite, then both the primary and induced field will suffer energy losses (A < 1) while traveling through the satellite’s mantle.
The flyby data, therefore, indicate that either the magnetic fields are induced in a high-conductivity shallow layer – the authors hypothesize a low-conductivity shallow saltwater ocean [2] – or that the Galilean satellites’ magnetic fields are induced in a deep high-conductivity layer, but that their mantles do not significantly absorb the magnetic flux energy. The second option is far more likely, as the conductivity of mantle layers (e.g. a saltwater inductor) is low, so any mantle inductor will lose large amounts of energy as waste heat in the induction process (A<<1; see Posts 3).
The shallowest high-conductivity layer is in fact the outermost shell of the satellite’s ferromagnetic core. The observations indicate both Callisto and Europa can be accurately modeled as perfectly conducting spheres (A ≃ 1), so the only model that makes physical sense is that the Galilean satellites’ fields are being induced in their highly-conductive ferromagnetic cores and that their mantles do not absorb significant amounts of magnetic flux energy.
Commonly-accepted skin depth models
The main issue with the Galilean satellites’ ferromagnetic cores acting as magnetic inductors is that the bulk of the magnetic flux energy generated during the satellites’ path through the Jovian magnetosphere must pass through their atmospheres and mantles without significant energy losses before it causes induction in their ferromagnetic cores.
The amount of magnetic flux energy lost to the satellites’ atmosphere is estimated to be fairly minor [2]. The absorption losses, A’, for an electromagnetic (EM) wave traveling through a shield (such as a satellite’s mantle) are commonly estimated using equations of the form [3]:
Whereby T is the thickness of the shield, f is the frequency of the wave, and µr and σr the relative magnetic permeability and electric conductivity resp. of the shield material. Note that hypothesized skin depth model for Callisto and Europa is also of this form [2].
In general, Eqn. 2 predicts that lower-frequency waves will suffer fewer absorption losses than higher-frequency waves when traveling through shields. The Earth’s magnetic field is commonly believed to be spawned in the Outer Core [4], so its outgoing magnetic energy likely travels as a low-frequency wave through its Mantle: any high-frequency geomagnetic field energy would be shielded by the thick overlying Mantle and would not be observable at the surface.
Estimates of Earth’s Mantle absorption therefore often focus on calculating the highest frequency that will not be screened. Most workers (for example [2,4]) assume a constant conductivity (σr) and magnetic permeability (µr) to estimate the absorption of EM waves in their skin depth models.
For example, Merrill et al.[4] estimate Mantle skin depth 𝛅, the depth at which the amplitude of an EM wave has fallen to 1/e of its original value, by assuming a constant value of σ = 1 S/m for the Mantle. The lowest frequency wave period that will not be screened is then estimated – assuming 𝛅=2,000 km – as approximately half a year[4], indicating that the Mantle acts as a band-pass filter that roughly shields all frequencies greater than 10-7 Hz from entering or leaving the Outer Core. A number of observations indicate that this cut-off limit is at least several orders of magnitude too low:
- The magnetosphere size varies in real-time with solar wind pressure [1], indicating a real-time interaction between the solar wind and the geomagnetic field is taking place, and not one that is delayed by 6 months.
- The geomagnetic field recovers from its daily deformation in 6 hours and solar storm events – short, high-energy events similar to the solar wind – in “2-3 days” [1,4].
The frequency dependency of magnetic flux absorption
Eqn. 2 indicates significant absorption can only occur in materials with high electrical conductivity (σ) and magnetic permeability (µ). These material properties are however frequency-dependent. Eqn. 2 is an engineering equation that somewhat masks reality. The properties of the medium, not the frequency of an EM wave itself, are important in determining the amount of absorption. For example, in space, no absorption of any EM energy of any frequency occurs because σr = 0. The conductivity and magnetic permeability of all materials changes as a function of EM frequency [3]. Eqn. 2 is therefore more scientifically re-written as:
The constant conductivity and magnetic permeability assumed in commonly-accepted skin depth models are therefore inappropriate simplifications when determining the Mantle’s absorption of EM waves, as the EM properties of the Earth layers – µr and σr –vary greatly with frequency [3]. The shielding effectiveness of all materials decreases radically below 100 kHz, and both absorption and reflection losses are low for a low-frequency EM pulse [3]. This well-known frequency-dependency has numerous applications, ranging from ground-penetrating planetary radar to EM Sounding of planetary interiors (thousands of km of ground penetration [5]) to submarine and underground mine communication systems [3].
For frequencies below 100 kHz the √(µ_r (f)*σ_r (f)) absorption term in Eqn. 3 decreases rapidly with EM frequency for non-ferromagnetic materials, such as Mantle silicates, but decreases much less rapidly for ferromagnetic materials as their µr increases more than their σr decreases [3]. For example, the relative magnetic permeability of steel increases three orders of magnitude between 10 GHz and 100 Hz [3]. This implies that many commonly-used skin depth models [2,4] greatly overestimate Mantle absorption for low-frequency EM waves: at these low frequencies, most materials cannot interact with the electric field component of the wave.
However, ferromagnetic materials can interact with the magnetic field of a low-frequency EM wave due to their increasing µr. EM-sounding applications utilize this frequency dependency: between 1MHz and 1 µHz the Earth’s physical response changes from “wavelike” to “inductive” [5], and studies of planet interiors use frequencies on the order of 10-3 Hz [5]. Therefore:
- High EM frequency band > 1 kHz: the non-ferromagnetic Atmosphere and Mantle potentially absorb significant amounts of EM energy
- Low-frequency band < 1 kHz: only the ferromagnetic core absorbs significant amounts of EM energy
Mantle absorption in Earth and the Galilean Satellites
The geomagnetic flux energy caused by the solar wind deformation of the magnetosphere propagates as an EM wave with an extremely low frequency roughly equal to (1/24 h * 3600 s.h-1)-1 ≈ 10-5 Hz. The magnetic flux generated during the Galilean satellite’s orbit through Jupiter’s inclined and rotating field has the (synodic) frequency of Jupiter (11.23 h for Europa and 10.18 h for Callisto [2]), corresponding to EM wave frequencies roughly double that of Earth’s. EM frequencies of 10-5 Hz (Earth) and ~2 x 10-5 HZ (Galilean Satellites) lie well within the range of frequencies whose EM wave energy is largely absorbed by their ferromagnetic cores.
The Maxwell equations predict that this absorbed magnetic flux will induce currents that counterbalance the solar wind and ambient Jovian field-generated magnetic flux resp. This outgoing, low-frequency, counter-flux can travel relatively unimpeded through the mantle back to the surface.
Both theory and observations, therefore, indicate that on Earth a significant part of the solar wind-generated geomagnetic flux is absorbed by the Outer Core (OC). The solar wind generates a 5-6 TW magnetic pulse of roughly constant frequency (10-5 Hz) through a roughly constant thickness of Mantle rocks that do not significantly absorb its energy. This low-frequency geomagnetic flux enters a flux tube through the Mantle via a flux patch D (Fig. 1a), travels relatively unimpeded through the Mantle due to the Mantle’s low σr and µr at low frequencies, and enters the ferromagnetic Outer Core through flux patch D’ (Fig. 1a), where its energy will start to decrease exponentially as it penetrates deeper into the relatively high µr ferromagnetic OC. The largest absorption of solar wind-generated magnetic flux, therefore, occurs at the Core Mantle Boundary (CMB; Fig. 1).
A reverse direction reasoning indicates the bulk of the geomagnetic field must be generated in the outer shell of the OC: any low or high-frequency magnetic flux generated deep in the Core would be re-absorbed by an overlying high σr and µr ferromagnetic OC shell. The outer shell of the OC is therefore likely the main location of the geomagnetogenic process.
Significant heating is demonstrably occurring in the outer shell of the OC at the CMB (Fig. 1b), as some heating process must be sustaining the abnormally high 10-12 °C.km-1 [6] geothermal gradient immediately above it. This heating locally raises the outer OC temperature above its adiabatic Core melting curve, resulting in a fluid OC and a lower-than-adiabatic temperature gradient through the Core, which at increased depth and pressure results in a solid (but soft) Inner Core [4].
In this regard, it is also highly relevant that the estimated Earth-incident solar wind power of 5-6 TW (PROM Article) is roughly equal to the estimated total amount of 4 TW of heat lost by the OC to the Mantle [6], as well as the estimated 3.6-10 TW [6] necessary to power the geomagnetic field. Note that this last estimate assumes a (low) geodynamo efficiency of 0.1-0.2, but that a magnetogenic process with double that efficiency only requires half the power.
Figure 1: a) Idealized quarter-section through the Earth. D and D’ represent Mantle and OC flux patches resp.; JC is an electric current induced in the Core; Not to scale. b) geothermal gradient for 1a (dashed line) with the estimated temperature ranges (after [6])
[1] Kivelson, M.G., Bagenal, F., 2014, Planetary Magnetospheres In: Encyclopedia of the Solar System (Third Edition), Academic Press; ISBN: 978-0-12-415845-0, p. 137-157
[2] Zimmer, C., Khurana, K.K., Kivelson, M.G., 2000, Subsurface Oceans on Europa and Callisto: Constraints from Galileo Magnetometer Observations. Icarus, 147, 329-347.
[3] Ott H.W. (2009) Electromagnetic Compatibility Engineering. John Wiley & Sons Inc., Hoboken. http://dx.doi.org/10.1002/9780470508510
[4] Merrill, R.T., McElhinny, M. W., McFadden, P. L., 1998, The magnetic field of the earth: paleomagnetism, the core, and the deep mantle. Academic Press. ISBN 978-0-12-491246-5.
[5] Banerdt, W.B., Dehant, V., Grimm, R., GrOtt M., Lognonné, P., Smrekar, S.E., 2014, Probing the Interiors of Planets with Geophysical Tools. In: Encyclopedia of the Solar System (Third Edition), Academic Press ISBN: 978-0-12-415845-0, p. 1185-1203
[6] Verhoogen, J., 1980, Energetics of the Earth. National Acad. of Sciences Collection, doi: 10.17226/9579
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