A multi-layered kinematic dynamo model: implications of a stratified upper layer in the Earth's core

It has been suggested that there exists a stably stratified electrically conducting layer at the top of the Earth's outer fluid core and that lateral temperature gradients in the lower mantle is capable of a driving thermal-wind-type flow near the core–mantle boundary. We investigate how such a flow in a stable layer could influence the geomagnetic field and the geodynamo using a very simple two-dimensional kinematic dynamo model in Cartesian geometry. The dynamo has four layers representing the inner core, convecting lower outer core, stable upper core, and insulating mantle. An α² dynamo operates in the convecting outer core and a horizontal shear flow is imposed in the stable layer. Exact dynamo solutions are obtained for a range of parameters, including different conductivities for the stable layer and inner core. This allows us to connect our solutions with known, simpler solutions of a single-layer α² dynamo, and thereby assess the effects of the extra layers. We confirm earlier results that a stable, static layer can enhance dynamo action. We find that shear flows produce dynamo wave solutions with a different spatial structure from the steady α² dynamos solutions. The stable layer controls the behavior of the dynamo system through the interface conditions, providing a new means whereby lateral variations on the boundary can influence the geomagnetic field.     

GUIDED LOW-FREQUENCY NOISE FROM AIRGUN SOURCES*

The presence of the water layer in marine seismic prospecting provides an effective waveguide for acoustic energy trapped between the sea-bed and the sea-surface. This energy persists to large ranges and can be the dominant early feature on far-offset traces. On airgun records, there is commonly a lower frequency set of arrivals following the water-trapped waves. These arrivals are not as obvious with higher frequency watergun sources. By using a combination of intercept-time/slowness (τ—p) mapping on observational data and theoretical modelling, we are able to identify the origin of the events. If a very rapid increase in a seismic wavespeed occurs beneath the sea-bed sediments, a new waveguide is formed bounded by the sea surface and this transition zone. The low frequency waves are principally guided within this thicker waveguide. Numerical filtering in the τ—p domain followed by trace reconstruction is very effective in removing the low frequency noise.     

THE EFFECT OF A SUPERPARAMAGNETIC LAYER ON THE TRANSIENT ELECTROMAGNETIC RESPONSE OF A GROUND*

A thin superparamagnetic layer on the earth's surface greatly affects the transient electromagnetic response of a conducting ground. The effect of the layer is most evident for singleloop transient electromagnetic data where transient voltages decay as 1/t. Even when a separate transmitter and receiver are used, the effect of the superparamagnetic layer is still pronounced. In this case the effect of the 1/t term in the equation is much less. More dominant now is a 1/t² term. The effect of the superparamagnetism can readily be seen in the analytical expressions for the apparent resistivities. If the presence of the superparamagnetic layer is not recognized, then the apparent resistivities decrease with time rather than approach the true value of the host rock.     

On the spin-up of a stably stratified,electrically conducting fluid. Part 1: Spin-up controlled by the hartmann layer

Abstract

The linear spin-up of a stably stratified, electrically conducting fluid within an electrically insulating cylindrical container in the presence of an applied axial magnetic field is analyzed for those cases in which electric currents generated within the steady Hartmann boundary layer control the fluid interior. It is shown how to obtain the known spin-up times for a homogeneous, nonconducting fluid (τ = E ^-½), a stably stratified, nonconducting fluid (τ = (σS/E, E ^?1) and a homogeneous conducting fluid (τ = α^?1 E ^-½) from the present formulation where τ = v/ωt, E = v/ωL ², σS = vN²/κω² and 2α² = σB²/pω. The problem is solved in the parameter range E?α²?1, α²/E?σS using the Laplace transform and two new spin-up times are obtained. Combined into one expression, they are τ = (1 + δ)α^?1E^-½ where δ = σμv. The spin-up mechanism is investigated and it is found that, in contrast to the homogeneous, conducting case, torsional Alfvén waves may be instrumental in the spin-up of a stratified conducting fluid. The effects of viscous and ohmic diffusion on the torsional Alfvén wave fronts are studied and the following regimes are identified: 0 < δ ?E/α², spin-up by meridional circulation of electric current with no Alfvén waves; E^-½/α ? δ ? 1, spin-up by meridional circulation of electric current with transient Alfvén waves; α/E^½ ? δ ? α²/E, spin-up by meridional circulation of current with weak Alfvén waves; 1 ? δ ? α/E^½, spin-up by strong Alfvén waves; α^½/E ? δ ? α²/E, spin-up by viscous diffusion with transient Alfvén waves; α/E ? δ < ∞, spin-up by viscous diffusion with no Alfvén waves.     

On the spin-up of a stably stratified,electrically conducting fluid. Part 2: Spin-up controlled by the mac layer

Abstract

The linear spin-up of a stably stratified, electrically conducting fluid within an electrically insulating cylindrical container in the presence of an applied axial magnetic field is analyzed for those cases in which electric currents generated within the steady MAC layer control the fluid interior, The MAC layer is a new boundary layer first studied by Loper (1976a) which controls the fluid in the parameter range E/α² ? σS ? α²/E, α² ? 1 Where E = vωL², 2α² = σB²/pω and σS = vN²/κω;². The problem is solved using the Laplace transform and four new spin-up times are obtained. Combined into one expression they are t = ω;^?1E-½[1+(σSE/α⁶)^½ + δα^-2] [1+(σSE/α^{6 1/4}]^?1 where δ = σμv. The internal spin-up mechanisms for this problem are shown to be very similar to those discussed in part 1 (Loper, 1976b). The ten known spin-up times are summarized and their inter-relationships are investigated. It is shown how to obtain the seven hydromagnetic spin-up times from a simple torsional Alfvén wave model involving a single parameter which measures the strength of the boundary layer dissipation. Finally, the present theory is applied to the solar spin-down problem and it is found that if the magnetic field in the solar interior is at least as strong as the interplanetary field of 10^-5 gauss, then the hydromagnetic spin-down time is much shorter than the Eddington-Sweet time and is comparable to the age of the sun.     

Core motions,electromagnetic core-mantle coupling and variations in the earth's rotation: New constraints from geomagnetic secular variation impulses

Recently observed secular acceleration impulses (SAI) of the geomagnetic field are interpreted in terms of organized motions of the outer core layers. Such motions have planetary dimensions (5000 km) and a large amplitude (3 × 10^?4 m s^?1) and are established in very short times (less than one year). The correlation of SAI observed in the Northern Hemisphere with minima in the Earth's rotation rate (around 1840, 1905 and 1970) is shown to be consistent with a simple model involving electromagnetic coupling of the weakly conducting (of the order of 100 ω^?1 m^?1) mantle, of a coherent outer core layer (thickness 100 to a few hundred kilometres) and of the rest of the core. The magnitude of the torque which acts suddenly on both parts of the core at the time of the impulses is estimated.     

The existence of a thin low-viscosity layer beneath the lithosphere

The horizontal temperature gradient at the base of the lithosphere at an oceanic fracture zone, where plate of different ages is juxtaposed, is expected to drive a local circulation, the characteristics of which can be constrained by the amplitude, wavelength and age-dependence of the geoid. Two-dimensional numerical models of convection in a fluid layer overlain by a solid conducting lid have been used to generate theoretical geoid profiles at right angles to the fracture zone. Only a thin, low-viscosity layer provides a reasonable fit to the data. The best model so far obtained has a fluid layer 150 km thick with viscosity 1.5 × 10¹⁹ Pa s under a 75 km lid. Such a layer, which is incapable of transmitting strong horizontal shear stresses, could provide the decoupling mechanism between plate and deep mantle flow required to balance the forces on the plates.     

Rheology of the lower mantle

The rheology of the lower mantle of the Earth is examined from the viewpoint of solid state physics. Recent developments in high-pressure research suggest that the lower mantle contains a considerable amount of (Mg, Fe)O with Fe/Mg + Fe = 0.2–0.3. The pressure and temperature dependences of diffusion in (Mg, Fe)O are estimated by the theory of diffusion in ionic solids. Of the materials composing the lower mantle, (Mg, Fe)O may be the “softest”, and therefore the rheology of the lower mantle may be that of (Mg, Fe)O, unless the framework effect is important.Temperatures in the lower mantle are inferred from the depths of phase transitions and the melting temperatures of the core materials. A thermal boundary layer at the base of the mantle is suggested. The physical mechanisms of creep are examined based on a grain size-stress relation and non-Newtonian flow is shown to be the dominant flow mechanism in the Earth's mantle.The effective viscosity for the temperature models, with and without the thermal boundary layer, is calculated for constant stress and constant strain rate (with depth). For constant strain rate, which may be appropriate for discussing the mechanics of descending slabs, the increase in effective viscosity with depth is smaller than for the constant-stress case, which may be appropriate for discussing the flow induced by the surface motion of plates.The relatively small depth gradient of viscosity, for constant strain rate, suggests that the lower mantle could also participate in convection. The effective viscosity increases with depth, however, by at least 10² to 10³ from the top to the bottom of the lower mantle, for a reasonable range of activation volumes and temperatures. There will be a low-viscosity layer at the base of the mantle, in contrast to the high-viscosity layer at the top of the mantle (plates), if a thermal boundary layer is present. The constant Newtonian viscosity inferred from rebound data may be an apparent feature resulting from the difference in deformation mechanisms between isostatic rebound and large-scale flow.     

Study on the rheology of subducting slabs

We calculate thermal and phase structures of subducting slabs for different subducting velocities by a modified coupling code of the kinetic phase-transformation equations and the heat-diffusion equation with latent-heat release. Whereafter, we estimate their rheology structures based on the thermal and phase structures from the mineral physical point of view. At shallow depth, the upper layer has a high effective viscosity greater than 10³⁴Pa · s; while the lower layer has a relatively low effective viscosity, which is greater than 10²⁶Pa · s nevertheless. The effective viscosities below the kinetic phase boundary of olivine to wadsleyite decrease obviously, and reach a minimum of 10²²Pa · s. Small areas with higher effective viscosities exist above the depth of about 700 km in subducting slabs, which are produced by lower temperatures that are related with endothermic phase transformation of spinel to perovskite and magnesiowustite. The 1% and 99% isograds of spinel proportion delineate tortuous belts with low effective viscosities, which would affect the geodynamic behavior of subducting slabs.     

A layer of an unstable seismotectonic strain as an analog of a waveguide at depths of 12–20 km in the Earth’s crust of the Tajik depression

The existence of a layer of unstable seismotectonic strain, which spatially coincides with the waveguide previously detected here at depths of 12–20 km, is established in the central part of the Tajik Depression [Lukk et al., 1970; Nersesov and Chepkunas, 1970; 1971]. This crustal layer is assumed to be weakened and saturated by fluid. The latter is supposedly achieved due to the supply of the liquid component from the bottom crust or upper mantle in the cracks and pores of the waveguide material. A near vertical pillar-like seismogenic body revealed in the work [Shevchenko et al., 2011], which penetrates the waveguide in the depth interval 0–40 km, is considered as one of the possible channels of such inflow. The detected loosened layer (the waveguide) is characterized by a signficant reduction in seismic activity. However, no such reduction is observed within the pillar-like seismogenic body at these depths. Moreover, compared to the nearest ambient environment, the upper 10–15 km of this body feature considerably increased seismic activity that manifested itself in a series of 13 strong earthquakes (with M ≥ 4.7) that occurred within the past approximately 100 years.     

Electromagnetic propagation in an idealized earth crust waveguide,part II

Summary Further calculations for the mode characteristics for an idealized model of the earth crust waveguide have been carried out. Specifically, we consider a homogeneous waveguide region bounded below by a homogeneous transition layer whose conductivity is intermediate between that of the main waveguide region and the bottom conductive region. It is shown that the presence of the transition layer usually increases the attenuation rate of the dominant modes although there are some interesting exceptions.     

Analysis of a guided wave along a conducting structure in a borehole

A guided wave along a borehole is often observed in borehole radar measurements. These guided waves deform the antenna pattern and can cause artefacts in radar measurements. A water-filled borehole or a conducting logging cable can function as a waveguide for electromagnetic waves under some conditions. We describe the theoretical characteristics of such a guided wave in a borehole and compare them with our experiments. The measured signal discussed was obtained with a directional borehole radar. This radar uses a cylindrical conformal array antenna as receiver and is a model of a conducting structure in a borehole. The induced field around the borehole was compared with the theory. The most fundamental symmetrical and asymmetrical modes were TM₀₁ and HE₁₁, and they were identified in the measured signals using time–frequency distribution analysis and by observation of the azimuthal field distribution of the magnetic field.     

Convection and gravity waves in two layer models. I. Overstable modes driven in conducting boundary layers

Abstract

Stability analysis is formulated for a two-layer fluid model in which the upper and lower layers are convectively stable and unstable, respectively. With discontinuities in viscosity and conductivity at the interface, the exchange of stability does not generally hold and overstability is possible. A detailed analytical treatment is presented for the case of small viscosity and conductivity in which viscous and conducting boundary layers are formed at the interface.

The usual damping effect due to the energy dissipation by viscosity and thermal conductivity exists irrespective of whether the mode is the convection or the gravity wave, but, for larger horizontal wave lengths, the effect of the boundary layer can become more important. The jump in the thermal conductivity in the boundary layer can give rise to overstability of the gravity wave in agreement with Souffrin and Spiegel (1967). The jump in the viscosity provides a self-catalytic action for the unstable flow if the viscosity is assumed to be the nonlinear turbulent viscosity due to the motion itself. The effect, however, is not strong enough to overcome the usual viscous damping.     

Convective instabilities in a variable viscosity fluid cooled from above

Whether in the mantle or in magma chambers, convective flows are characterized by large variations of viscosity. We study the influence of the viscosity structure on the development of convective instabilities in a viscous fluid which is cooled from above. The upper and lower boundaries of the fluid are stress-free. A viscosity dependence with depth of the form ν₀ + ν₁ exp(?γ.z) is assumed. After the temperature of the top boundary is lowered, velocity and temperature perturbations are followed numerically until convective breakdown occurs. Viscosity contrasts of up to 10⁷ and Rayleigh numbers of up to 10⁸ are studied.For intermediate viscosity contrasts (around 10³), convective breakdown is characterized by the almost simultaneous appearance of two modes of instability. One involves the whole fluid layer, has a large horizontal wavelength (several times the layer depth) and exhibits plate-like behaviour. The other mode has a much smaller wavelength and develops below a rigid lid. The “whole layer” mode dominates for small viscosity contrasts but is suppressed by viscous dissipation at large viscosity contrasts.For the “rigid lid” mode, we emphasize that it is the form of the viscosity variation which determines the instability. For steep viscosity profiles, convective flow does not penetrate deeply in the viscous region and only weak convection develops. We propose a simple method to define the rigid lid thickness. We are thus able to compute the true depth extent and the effective driving temperature difference of convective flow. Because viscosity contrasts in the convecting region do not exceed 100, simple scaling arguments are sufficient to describe the instability. The critical wavelength is proportional to the thickness of the thermal boundary layer below the rigid lid. Convection occurs when a Rayleigh number defined locally exceeds a critical value of 160–200. Finally, we show that a local Rayleigh number can be computed at any depth in the fluid and that convection develops below depth z_r (the rigid lid thickness) such that this number is maximum.The simple similarity laws are applied to the upper mantle beneath oceans and yield estimates of 5 × 10¹⁵?5 × 10¹⁶ m² s^?1 for viscosity in the thermal boundary layer below the plate.     

Seasonal variations in the vertical structure of pelagian water stratum in the Southern Baikal

For the first time, T,S-analysis was used to determine the specifics of seasonal variations in the vertical structure of Lake Baikal active layer. In the under-ice period, the active layer includes the under-ice, top winter, and upper intermediate water masses. The under-ice water mass, unlike other masses, shows an increase in mineralization to 100.74 mg/kg, which corresponds to a release of 71.1 g salt under 1 m² of water surface in a layer 0–40 m in the process of salt freezing out during ice cover formation and accretion. In the phases of mixing (homothermy), the water masses of the active layer transform into a surface homogeneous mass. In summer and autumn, surface and upper intermediate water masses, separated by a water mass of summer thermocline can be identified. A specific feature of the summer thermocline water mass is the increased sum of ions because of an increase in HCO ₃ ^- concentration at the decay of organic matter accumulating in the bottom part of the thermocline. The existence of the under-ice water mass and the water mass of summer thermocline was established in Lake Baikal for the first time. In the deep-water zone (>250 m), except for the bottom parts, the lower water masses (the lower intermediate and the deep) are permanent, their characteristics remaining stable during the year. The changes in the bottom water mass are due to the character of the processes of bottom water renewal.     

Effect of stratification on the frequency of bounded Rossby modes over a non-flat bottom

This work attempts to express and analyze the challenges, induced by stratification, affecting the Rossby-topographic eigenmodes of a closed domain with a general uneven bottom of arbitrary shape filled with a uniform fluid in the unperturbed configuration. The modified eigenmodes have been computed analytically: stratification is introduced in the mathematical form of a perturbation of a homogeneous fluid over a non-flat bottom. The eigenmodes lose their barotropic character and differences appear in the dynamical fields (velocity and pressure) from upper to lower layer, as expected. Expressions for the baroclinic and ageostrophic velocity components due to the perturbation are given. The analysis is carried out in the frame of linear shallow water approximation. All terms have been retained apart from nonlinear advection in the governing equations. We find that the frequencies of the eigenmodes change; an analytical expression of frequency correction as a function of layer density difference and interface depth is found. Initial results for some elementary geometrical settings with a waveguide bottom are determined and expressed in a concise, easily readable closed form. The results obtained in the shallow water approximation are expanded in series with respect to the Rossby number. Next, they are compared with the frequency correction obtained in an alternative framework in which the quasi-geostrophic approximation is used, and a purely baroclinic perturbation is imposed from the outset as the result of the introduction of stratification in the otherwise homogeneous fluid. In this scenario, reduced gravity and the ratio of upper to lower layer depth are, in turn, used as the expansion parameters in lieu of the Rossby number.     

THE INDUCTIVE RESPONSE OF A HORIZONTAL CONDUCTING CYLINDER BURIED IN A UNIFORM EARTH FOR A UNIFORM INDUCING FIELD*

The inductive response of a conducting horizontal cylinder embedded in a uniform earth is studied using numerical results obtained for an analytical solution for the problem of a conducting cylinder buried in a homogeneous earth for the case of a uniform inducing field. A check of the validity of the numerical results is made by a comparison with analogue model measurements for a number of cases. Numerical results for a range of cylinder radii (a = 1–10 km), depths of burial (d= 0–4 km), conductivity contrasts (σ₂= 10^?2-10 Sm^?1), and source frequencies (f= 10^?1-10^?4 Hz) of interest in the interpretation of magnetotelluric field measurements are presented. The results indicate that for a uniform inducing field the conductivity and depth of burial of a horizontal cylindrical inhomogeneity are best determined through a measurement of the amplitudes H_y, H_z and E_x and the phases φ_y and Ψ_x.     

Thermal convection in a twisted horizontal magnetic field

The onset of convection in a layer of an electrically conducting fluid heated from below is considered in the case when the layer is permeated by a horizontal magnetic field of strength B ₀ the orientation of which varies sinusoidally with height. The critical value of the Rayleigh number for the onset of convection is derived as a function of the Chandrasekhar number Q. With increasing Q the height of the convection rolls decreases, while their horizontal wavelength slowly increases. Potential applications to the penumbral filaments of sunspots are briefly discussed.     

The exceptional magnitude and intensity of the Toba eruption,sumatra: An example of the use of deep-sea tephra layers as a geological tool

The eruption of Toba (75,000 years BP), Sumatra, is the largest magnitude eruption documented from the Quaternary. The eruption produced the largest-known caldera the dimensions of which are 100 × 30 km and which is surrounded by rhyolitic ignimbrite covering an area of over 20,000 km². The associated deep-sea tephra layer is found in piston cores in the north-eastern Indian Ocean covering a minimum area of 5 × 10⁶ km². We have investigated the thickness, grain size and texture of the Toba deep-sea tephra layer in order to demonstrate the use of deep-sea tephra layers as a volcanological tool. The exceptional magnitude and intensity of the Toba eruption is demonstrated by comparison of these data with the deep-sea tephra layers associated with the eruptions of the Campanian ignimbrite, Italy and of Santorini, Greece in Minoan time. The volume of ignimbrite and distal tephra fall deposit produced in the Toba eruption are comparable, a total of at least 1000 km³ of dense rhyolitic magma. In contrast the volume of dense magma produced by the Campanian and Santorini eruptions are approximately 70 and 13 km³ respectively. Thickness versus distance data on the three deep-sea tephra layers show that eruptions of smaller magnitude than Santorini are unlikely to be preserved as distinct tephra layers in most deep-sea cores. In proximal cores all three tephra layers show two distinct units: a lower coarse-grained unit and an upper fine-grained unit. We interpret the lower unit as a plinian deposit and the upper unit as a co-ignimbrite ash-fall deposit, indicating two major eruptive phases. The Toba tephra layer is coarser both in maximum and median grain size than the Campanian and Santorini layers at a given distance from source. These data are interpreted to indicate a very high cruption column, estimated to be at least 45 km. We have applied a method for estimating the duration of the Toba eruption from the style of graded-bedding in deep-sea tephra layers. Studies of two cores yield estimates of 9 and 14 days. The eruption column height and duration estimates both indicate an average volume discharge rate of approximately 10⁶ m³/sec. The Toba eruption therefore was not only of exceptional magnitude, but also of exceptional intensity.     

The effect of screening of the magnetotelluric field by high resistivity layers: Three-dimensional numerical modeling constraints

A 3-D numerical modeling approach is applied to the effect of screening of the magnetotelluric (MT) field by high resistance (≥10⁶ Ω m²) layers, which up to now has mainly been studied in terms of 2-D heterogeneous models of geological sections [Berdichevsky and Yakovlev, 1990]. Three-layer models above a poorly conducting basement (ρ=10⁴ Ω m) are used. The resistance of the screening layer was generally taken equal to 2×10⁶ Ω m² and its thickness was varied from 200 to 2000 m. The resistivity of the host medium was set at 10 Ω m. Heterogeneities of a low and a high resistivity (ρ=10^?3 and 10⁴ Ω m, respectively) ranging in horizontal size from 4 to 40 km and having a height of 1–2.8 km (a protrusion on the basement) were examined. Based on calculations of these models, 2-D and 3-D screening effects were compared. The 3-D modeling determined values of model parameters at which the screening properties are preserved both in the case of a screen of limited horizontal dimensions and in the presence of a rupture in the screen comparable in horizontal size with the 3-D heterogeneity. As follows from the modeling results, the screening effect of the high resistivity model layers seriously complicates the use of MT soundings for the identification of a local heterogeneity of both a low and a high resistivity (ρ=10^?3 and 10⁴ Ω m, respectively) if its lateral size is smaller than the 4-km thickness of the studied three-layer section in question and its height is 1–1.5 km. The regular patterns of the screening effect revealed in this work are of interest in electromagnetic sounding applications.