On the origin of the Earth’s magnetic field Gregory Ryskin Robert R. McCormick School of Engineering and Applied Science Northwestern University, Evanston, Illinois 60208 ryskin@northwestern.edu ABSTRACT It is thought that the magnetic field of the Earth is generated by the hydromagnetic dynamo action in the Earth’s liquid outer core, consisting mainly of iron (the standard model). Here I propose that the magnetic field of the Earth is generated by dynamo action in the world ocean at the Earth’s surface. This hypothesis is free of the problems of the standard model; in particular, it explains the close temporal correlation between geomagnetic reversals and the stratigraphic boundaries defined by major or minor mass extinctions. Implications of this hypothesis for other terrestrial planets are briefly discussed. 1 It is thought that the magnetic field of the Earth is generated by the hydromagnetic dynamo action in the Earth’s liquid outer core, consisting mainly of iron (the standard model). Here I propose and explore an alternative hypothesis, viz., that the magnetic field of the Earth is generated by dynamo action in the world ocean at the Earth’s surface. It may appear simple to refute the present hypothesis by direct observation. However, hydromagnetic dynamo is a uniquely complex phenomenon; to directly confirm, or rule out, that a given large-scale magnetic field is produced by a given flow of an electrically conducting fluid would be extremely difficult, if not impossible (short of stopping the flow). Dynamo action begins similarly to hydrodynamic instability: in some classes of flows, infinitesimal magnetic field fluctuations grow exponentially, extracting energy from the flow. This cannot continue for long, of course; the flow is soon modified by the Lorentz force [1]. The large-scale magnetic field is eventually produced and maintained by the mechanism of inverse cascade, whereby the field growth at small scales is arrested through the back action of the Lorentz force on the flow, and magnetic energy is continuously transferred toward larger and larger scales, eventually concentrating at the largest scale allowed by the geometry of the flow [2]. The resulting magnetic field is dominated by its large-scale component, whose interaction with the flow is, of course, indistinguishable from that of an analogous but externally imposed field. A direct measurement-based proof that the dominant large-scale field is produced by dynamo action is thus out of question; only the residual field components can provide useful information – see below. It would be premature to attempt numerical simulation of dynamo action in the ocean, on the basis of some approximation to the ocean flow field; the result, whether positive or negative, would not be trustworthy. Due to severe computing power limitations, simulations of global hydrodynamic and hydromagnetic phenomena cannot resolve small scales, and must operate with parameter values that differ from the actual ones by orders of magnitude [3,4]. Such limitations are especially serious in the case of the dynamo problem because of the extreme sensitivity of dynamo action to the flow field, “often to the extent that dynamo action fails completely after a small change in flow” [4]. The physical origin of these computational difficulties is as follows. The induction equation of magnetohydrodynamics [1] ∂B/∂t = ∇ × (u × B) + (µ0σ)-1 ∇2 B (where σ is the electrical conductivity of the fluid; µ0 ≡ 4π × 10-7 H m-1 in SI) is linear in the magnetic field B, so if the velocity field u were independent of B, B would grow or decay exponentially. To maintain a steady large-scale magnetic field, u must be kept “on a knife edge” between these two outcomes through the action of the Lorentz force [3], which is extremely weak compared to other forces in the Navier-Stokes equation. (In the weak-field dynamo; “strong-field dynamos, if they exist, are intrinsically unstable” [4].) Thus, the very existence of a steady large-scale magnetic field depends on a small perturbation of the velocity field; this makes numerical simulation of the dynamo problem exceedingly difficult. Since a direct proof by either experimental or computational means is not currently feasible, indirect evidence acquires unusual importance, and will be discussed below. First, however, consider conditions that must be satisfied by any viable hypothesis. For dynamo action to be possible, intensification of the magnetic field due to stretching of its field lines by the flow must be able to balance the Ohmic decay of the 2 field. This leads to the Backus necessary condition [5]: smaxτ1 > 1, where smax is the maximum strain rate in the flow, and τ1 is the longest Ohmic decay time ([1], p. 109111). To maintain the magnetic field against Ohmic dissipation, sufficient power must be supplied to the flow; this is another necessary condition. The fluid core. There exists no evidence of hydrodynamic motion in the core independent of the belief that this motion is the raison d’etre of the geomagnetic field. In particular, its characteristic velocity is inferred from the velocity, ~ 1 mm s-1, of the “westward drift” of the magnetic field’s non-dipole features, observed on the Earth’s surface [3]. If motion does occur in the fluid core, it is likely to satisfy the Backus condition and the power requirements [3,6]; however, at this stage one can neither confirm nor rule out its presence. The ocean. The winds ensure that the ocean is always in motion. In the largescale wind-driven currents velocities are typically ~ 1 m s-1, can be as high as 2 m s-1, and vary substantially over vertical distances ~ 100 m, so a typical maximum strain (shear) rate is smax = 2 × 10-2 s-1. Comparable maximum strain rates occur in the internal gravity waves [7]. (Much higher strain rates occur in small-scale currents and turbulence, but even the fastest small-scale motions will appear frozen on the timescale of the magnetic field diffusion across their spatial dimensions, and thus cannot contribute to dynamo action.) The inverse of smax yields the shortest timescale of the magnetic field amplification by the flow, ~ 1 min. The longest Ohmic decay time for a thin spherical shell with radius R and thickness d is ~ µ0σRd [8]. With R = 6370 km and d = 4 km for the ocean, and σ = 3.2 S m-1 for seawater, this yields ~ 1 day. The Backus condition requires that smaxτ1 > 1, where τ1 is the longest decay time. Even if the dynamo action per se is confined to the ocean, the core may influence the Ohmic decay of the field, and for the ocean-plus-core system τ1 is unknown (it would depend on how much of the field could reach the core); however, it cannot be less than the above estimate for the ocean alone. Neglecting the influence of the core and using τ1 = 1 day yields smaxτ1 = O(103), so the Backus condition is satisfied by a wide margin. The inverse cascade mechanism concentrates magnetic energy at the largest scale compatible with the spatial extent of the motion [2]. Thus, the structure of the large-scale field generated by the ocean dynamo, or by the fluid-core dynamo, must be entirely determined by the gross geometry and symmetry of the generating flow. This suggests that the dominant field should be a dipole (there is no magnetic monopole), centered near the center of the Earth, its dipole moment roughly aligned with the Earth’s axis of rotation. The total energy of the dipole part of the field (outside of the Earth) is easily calculated; the result is ~ 0.8 × 1018 J. This value is very close to the total kinetic energy of the large-scale ocean currents: the estimates for the latter range from ~ 0.4 × 1018 J to ~2.8 × 1018 J, depending on whether the mesoscale eddies are included [9]. Such equipartition of energy is predicted theoretically [2], and has been observed repeatedly in astrophysics: in rotating galaxies, including ours, the energy of the large-scale magnetic field is approximately equal to the kinetic energy of the flow that maintains the field via dynamo action [10]. For the fluid-core dynamo, however, numerical simulations yield magnetic energy that is greater than the kinetic one by a factor ~103 [3]; the meaning of this result is subject to debate [4]. 3 When the large-scale field is steady, the total rate of dissipation by Ohmic heating can be calculated as σ -1∫ J2 dV = σ { ∫ (u × B)2 dV – ∫ E2 dV } , where J is the electric current density, E is the electric field, and integration is over the volume of the fluid [11]. This is less than σ ∫ (u × B)2 dV, which is less than σ Bmax2 ∫ u2 dV = (2σ Bmax2 /ρ)K, where Bmax is the maximum value of B, ρ is the fluid density, and K is the total kinetic energy of the flow. For the ocean, the latter includes the energy of the large-scale currents, ~2.8 × 1018 J [9], and the energy of the internal gravity waves, ~1.4 × 1018 J [12], plus relatively small contributions from the small-scale currents, surface waves, and turbulence. Using Bmax = 0.6 gauss, one obtains σ Bmax2 /ρ ≈ 10-11 s-1, so that the total rate of Ohmic dissipation in the ocean is less than ~108 W. The rate at which the ocean is supplied with kinetic energy by the wind is ~1012 W [13]; contribution by the tides is of the same order [7]. A small fraction of this power supply will be sufficient to maintain the geomagnetic field (though a significant fraction may be required to produce it anew). Thus far, both the fluid-core dynamo and the ocean dynamo appear viable, but neither is confirmed. Only the history of the Earth –a single long-running experiment contains the data that may help to resolve this issue. Consider the following Problems of the standard model: a) If motion does occur in the fluid outer core, it is driven by natural convection caused by rejection of latent heat and the light component at the boundary of the growing solid inner core [3,6]. However, this mechanism could not operate before the solid inner core appeared ~ 1 Ga (billion years ago) [14], whereas the paleomagnetic evidence indicates that the magnetic field has existed, at about the same strength as today, since at least ~ 3.5 Ga [3]. b) The non-dipole features of the main field show surprising correlation with the geography of the Earth’s surface: one stationary anomaly, centered on Mongolia, covers most of Eurasia, another – most of North America [15,16], whereas the field is almost perfectly dipolar over the Pacific Ocean. Paleomagnetic data show that these features have been stable over millions of years [15]. Explanations invoke different types of irregularities at the core-mantle boundary: in topography (“inverted mountains”), in electrical conductivity, or in heat flow. No explanation exists for the correlation between these hypothetical irregularities at the core-mantle boundary and the prominent features of the Earth’s surface geography. c) Some of the non-dipole features are drifting, mostly westward, with velocity ~1mm s-1. Since the field is believed to originate in the core, its secular variation is usually discussed in terms of its projection back to the core-mantle boundary. “The most prominent region of westward drift is in the [Atlantic] hemisphere, … particularly of features in a band straddling the equator. They tend to drift west until reaching the region of the core-mantle boundary below the western edge of the Atlantic, and are then deflected polewards” ([16], p. 125). The drift is thought to result from dragging of the field lines by the flow at the fluid core surface, and used to infer the main features of this flow. Using projection of the Earth’s surface geography to indicate location on the coremantle boundary, Ref. [17] finds “in the Atlantic hemisphere, … two prominent circulations of fluid: one to the north of the Equator, the other to the south. Near the Equator the fluid from both the northern and the southern circulations flows westward”. 4 The distinctive magnetic-flux features “form near Indonesia and drift westward rapidly … When they reach southern Africa, they intensify quickly. They then cross beneath the Atlantic Ocean … and move increasingly southward”. d) The rate of secular variation occasionally undergoes a sudden, sharp change, called a “geomagnetic jerk”. Such jerks have occurred in 1969, 1979, 1992, and in 1998 or 1999 [18]. A few more have been inferred for earlier times on the basis of historical records. The jerks present a problem for the standard model, not only because they imply singularities of unknown origin in the temporal behavior of the fluid-core dynamo, but also because they should have suffered much greater distortion (smoothing) by the time they arrive to the Earth’s surface, due to the small, but not negligible, electrical conductivity of the mantle. e) A similar problem is posed by the paleomagnetic evidence that “the direction of the magnetic field during a polarity transition could change at the astonishingly rapid rate of 6° per day averaged over several days” [19]. As noted in [19], “we should not observe the very fast directional changes if they are of core origin”, and a number of other explanations (rock magnetic problems, etc.) have been advanced, but the recent data [20] show that these very fast changes are not artifacts of interpretation. f) The geologic (sediment) record of variation in the geomagnetic field intensity (paleointensity) over the last million years shows strong correlation with climate change [21,22,23]. It has even been proposed that the slight variations in the Earth’s orbital parameters, i.e., eccentricity, obliquity, etc. (which, according to the Milankovitch hypothesis, determine the climate cycles via modulation of insolation), exert influence on the motion in the Earth’s core [21]. This explanation is so implausible that a great effort has been devoted to proving that the observed correlation is an artifact of faulty interpretation of data. However, the latest analyses of the data confirm that the correlation is present [22,23]; moreover, in the eccentricity band (periodicity ~ 100 kyr (thousand years)), the correlation between the paleointensity and the climate proxy (δ18O) is much stronger than that between either of these variables and the eccentricity itself [22]. Significantly, “the [paleointensity] signal for the 100 kyr period … is restricted to the interval 0-750 ka” [23], just like the ~ 100 kyr glaciation cycle in the climate record. (The variation of the eccentricity itself has no such restriction, of course.) g) The geomagnetic field has been reversing its direction at irregular intervals throughout the Earth’s history. This fact by itself can be explained within the standard model as resulting from the highly nonlinear character of hydromagnetic dynamo; indeed, numerical simulations of the core dynamo have produced field reversals [3]. In these simulations, the reversal itself (i.e., the transition process) is completed in a few thousand years. However, recent paleomagnetic evidence shows that “directional changes during a reversal can be astonishingly fast, possibly occurring as a nearly instantaneous jump from one inclined dipolar state to another in the opposite hemisphere” [20]. In particular, “the jump from a northern-hemisphere transitional state to a southernhemisphere one occurred in less than a few weeks”, possibly, in “only a few days” [20]. This estimate, based on the lava flow, is the most reliable one because “one lava flow has recorded both of the antipodal transitional components” [20], and thus the duration of the transition could be calculated knowing the lava’s cooling rate. A change that fast could not be produced in the fluid core: the characteristic time of the magnetic field 5 decay in the core is estimated as ~ 10 kyr [3]. Nor could it reach the Earth’s surface because of the screening by the mantle (see (e) above). h) Equally difficult to explain within the standard model is the extremely close temporal correlation in the geological record between many of the field reversals, on the one hand, and the standard stratigraphic boundaries and the sequence boundaries, on the other. (The boundaries between stratigraphic stages are defined by breaks in the fossil record, i.e., by major or minor mass extinctions [24], while sequence boundaries are defined by abrupt sea-level falls [25].) Attempts were made to explain the coincidence of reversals with mass extinctions by bolide impacts disturbing the flow in the core ([26], § 8.4), but this idea has little credibility, and over the last decade the issue appears to have slipped into oblivion. Yet the correlation is striking [27], as shown by the following examples ([25], Chart 2), listing the ages, in Ma, of a field reversal, a stage boundary, and a sequence boundary: 2.58, 2.60, 2.55; 14.80, 14.80, 14.80; 20.52, 20.52, 20.52; 23.80, 23.80, 23.80; 28.3, 28.5, 28.5; 60.9, 60.9, 60.7; etc. i) Some magnetic reversals are marked by the presence of magnesioferrite spinels, ~10 µm in size [30]. It was concluded in [30] that “The presence of … magnesioferrite is … a marker for … a minimum in the intensity of the Earth’s magnetic field”. As a mechanism, it was suggested [30] that such a minimum allows a greater flux of cosmic particles to reach the Earth’s surface. This explanation is untenable: the Earth’s magnetic field has negligible influence on the motion of cosmic particles greater than ~ 0.1 µm in size [31]. j) The close correlation between biostratigraphic boundaries and sequence boundaries (see (h) above) is a remarkable fact in itself. Some experts see a direct connection between them: “stage boundaries are sequence boundaries that reflect the global event of the sea-level fall” ([25], p. 215); others acknowledge the correlation but consider it to be a stratigraphic artifact. Moreover, “no mechanism [is known] that can cause the necessary short-wavelength, large-amplitude sea-level changes implicit in globally synchronous eustatic third-order cycles” of sequence stratigraphy [32], so these sea-level changes themselves are viewed as artifact by many researchers. Resolution of these problems by the present hypothesis: (a) The oldest sedimentary rocks providing evidence of the existence of the ocean are dated ~3.7 Ga. The large-scale ocean currents are wind-driven and must have been present ever since the ocean came into being. (b), (c) It comes as no surprise that the field generated by dynamo action in the Earth’s ocean should depend on the Earth’s surface geography. The resemblance between the flow patterns inferred for the Earth’s liquid core surface and the major ocean currents is also understandable: the secular variation must be caused by the latter. The fact that many distinctive features of the non-dipole field form near Indonesia may have to do with the extreme complexity and unsteadiness of the currents comprising the Indian-Pacific interocean exchange, etc. If anything may appear surprising, it is the fact that all these deviations from the dominant dipole are so small; e.g., it is understandable why the field be dipolar over most of the Pacific Ocean, but it seems that the continents should have a strong effect, etc. The reason why such a “linear” cause-and-effect relationship does not hold even approximately in the dynamo problem is the tendency of the inverse cascade to concentrate magnetic energy at the largest possible scale while strongly suppressing 6 higher harmonics of the field. This tendency has been predicted theoretically and demonstrated in model studies [2]. (d), (e) The issue of mantle screening does not arise in the present hypothesis. The origin of the geomagnetic jerks requires further study; it may be significant that the jerks are closely correlated in time with the major tsunamis that originated near Indonesia or Papua/New Guinea in 1969, 1979, 1992, and 1998. The correlation between the behavior of the geomagnetic field and climate, mass extinctions, sequence boundaries, etc., finds a natural explanation within the present hypothesis. In [33] I proposed the following mechanism as a cause of mass extinctions and climate perturbations: an extremely fast, explosive release of dissolved methane (and other dissolved gases such as carbon dioxide and hydrogen sulfide) that accumulated in the oceanic water masses isolated by topography and prone to stagnation and anoxia. The mechanism of the explosive release is the water-column eruption caused by the interplay of buoyancy forces and exsolution of dissolved gas. The eruption brings to the surface deep anoxic waters that cause extinctions in the marine realm. Terrestrial extinctions are caused by explosions and conflagrations that follow the massive release of methane, and by the eruption-triggered floods. Quasi-periodic eruptions from different depths are the likely root of the cyclical nature of the geological record, with periodicities depending on the depth and ranging from tens of thousands to a few millions of years. (g), (h) A large deep-ocean eruption would disrupt the ocean currents and could cause a temporary cessation of dynamo action. The ocean currents and dynamo action would resume after the eruption ended, but the direction of the resulting dipolar field could be different (or it could be the same) because the equations of magnetohydrodynamics are invariant with respect to the change of sign of B. This explains why many (but not all) of the stratigraphic boundaries are accompanied by simultaneous geomagnetic polarity reversals. (f) The more frequent eruptions at intermediate depths, such as the ones possibly responsible for the ~ 100 kyr glacial termination cycle [33], would be much weaker than the deep-ocean ones, and thus more likely to result in only a temporary reduction in the magnetic field intensity (though a complete reversal could occasionally occur). Over the last 0.8 Myr, such geomagnetic excursions, and the geomagnetic field intensity in general, are closely correlated in time with the ~ 100 kyr climatic cycle, in complete agreement with the present hypothesis. (i) The magnesioferrites found at reversals could have been produced in the hightemperature zone of the methane flame; close analogues of the magnesioferrites are found in combustion products of fossil fuels [34]. At some locations, magnesioferrites are directly associated with abundant methane combustion byproduct (coronene) [34]. (j) As the erupting region of the ocean “boils”, a surface elevation (a “bulge”) is created. The resulting disturbance will propagate over the surface of the ocean as a giant tsunami. The sea-level fluctuations implicit in the third-order cycles of sequence stratigraphy may have been these giant tsunamis. Khabarov et al. [35] found that “negative [carbon-isotope] excursions coincide with the boundaries of the five major sequences” that they studied; similar observations were reported in ([25], p. 252-253). Methane-driven eruptions are expected to produce such carbon-isotope excursions [33]. The last four stage boundaries, dated 3.6, 2.6, 1.8, and 0.8 Ma, coincide with geomagnetic polarity reversals ([36], p. 144). Recently, the International Commission on 7 Stratigraphy [37] has formally recognized this fact and defined placement of these stage boundaries by the corresponding polarity reversals. The last three stage boundaries also coincide with sequence boundaries [25]. If these boundaries and reversals are records of deep-ocean eruptions, the next such eruption should be expected fairly soon; large-sale efforts will be required to prevent it. Terrestrial planets and Galilean satellites. Venus has no ocean and no magnetic field. Mars has no magnetic field now, but had one in the past, when it possibly had a liquid water ocean too. A dynamo could also operate in a magma ocean that some planets (Mercury?) had in their early history [38]: magma has electrical conductivity of the same order as seawater, and buoyancy-driven convective velocities in a magma ocean are estimated as ~1 m s-1 [39]. The Moon had a strong magnetic field for ~ 200 Myr beginning at ~3.85 Ga [40]. During the same time period it also experienced the “lunar cataclysm” - extremely heavy bombardment by asteroids [41]. It is conceivable that this bombardment energized (or created anew) a magma ocean where a dynamo could operate. Europa, Callisto, and Ganymede appear to have subsurface liquid water oceans that interact inductively with the magnetic field of Jupiter, but only Ganymede has an intrinsic magnetic field [42]. The origin of the latter can be sought in the subsurface ocean, or in the fluid core; in either case, it is far from obvious that natural convection would be sufficiently vigorous for dynamo to operate. (There are no wind-driven currents in a subsurface ocean.) Acknowledgments The manuscript benefited from comments and criticism by A. B. Kostinski, L. P. Kadanoff, E. N. Parker, F. Cattaneo, F. H. Busse, D. J. Stevenson, S. I. Braginsky, B. A. Buffett, J. M. Ottino, G. D. Acton, and M. S. Robinson. [1] H. K. Moffatt, Magnetic field generation in electrically conducting fluids (Cambridge Univ. Press, 1978) [2] U. Frisch et al., J. Fluid Mech. 68, 769 (1975); A. Pouquet, U. Frish and J. Leorat, J. Fluid Mech. 77, 321 (1976); A. Brandenburg, Astrophys. J. 550, 824 (2001) [3] P. H. Roberts and G. L. Glatzmaier, Rev. Mod. Phys. 72, 1081 (2000) [4] K. Zhang and G. Schubert, Annu. Rev. Fluid Mech. 32, 409 (2000); K. Zhang and D. Gubbins, Phil. Trans. R. Soc. Lond. 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