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Hydrodynamic and Hydromagnetic Stability

Boundary Layer Climates.

The observed variation of the seismic velocities with depth, below the crust, is examined with reference to the variation to be expected in a homogeneous medium. A general equation is derived for the variation of the quantity, , in a homogeneous gravitating layer with an arbitrary gradient of temperature. The parameters of this equation are then discussed in terms of the experimental and theoretical relations for solids. The principal parameter is (∂KT/∂P)T, the rate of change of isothermal incompressibility with pressure, which can be found for large compressions from Bridgman's measurements. Comparison of observed and expected rates of variation of ϕ throughout the Earth's interior leads to conclusions regarding homogeneity and, with a larger uncertainty, to estimates of temperature.

A shadow zone at a depth of about 100 km, as suggested by Gutenberg, may be accounted for by a gradient of temperature of about 6°/km in a homogeneous layer of ultrabasic rock. Between depths of about 900 and 2,900 km, the mantle appears to be substantially uniform, and at a relatively uniform temperature of the order of several thousand degrees. Between about 200 and 900 km, the rate of rise of velocity is too great for a homogeneous layer, and indicates a gradual change of composition, or of phase, or both. New phases are required to account for the high elasticity of the deeper part of the mantle (below 900 km), and it is suggested that, beginning at about 200 to 300 km, there is a gradual shift toward high-pressure modifications of the ferro-magnesian silicates, probably close-packed oxides, with the transition complete at about 800 to 900 km. There may also be a concentration of alumina, lime, and alkalis toward the upper part of the mantle, in and above the transitional layer but below the crust, existing in minerals of high elasticity such as garnets and jadeites. The transitional layer appears to hold the key to a number of major geophysical problems.

The velocities in the core and inner core are also reviewed. The inner core is most simply interpreted as crystalline iron, the outer part as liquid iron, perhaps alloyed with a small fraction of lighter elements. The density and compressibility of iron at high pressures are estimated with the aid of the experimental compressions of the alkali metals; the central density is found to be about 15. Several other recent proposals regarding the crust are discussed.

Elasticity and constitution of the Earth's interior

Jaynes's principle of maximum entropy and Kullbacks principle of minimum cross-entropy (minimum directed divergence) are shown to be uniquely correct methods for inductive inference when new information is given in the form of expected values. Previous justifications use intuitive arguments and rely on the properties of entropy and cross-entropy as information measures. The approach here assumes that reasonable methods of inductive inference should lead to consistent results when there are different ways of taking the same information into account (for example, in different coordinate system). This requirement is formalized as four consistency axioms. These are stated in terms of an abstract information operator and make no reference to information measures. It is proved that the principle of maximum entropy is correct in the following sense: maximizing any function but entropy will lead to inconsistency unless that function and entropy have identical maxima. In other words given information in the form of constraints on expected values, there is only one (distribution satisfying the constraints that can be chosen by a procedure that satisfies the consistency axioms; this unique distribution can be obtained by maximizing entropy. This result is established both directly and as a special case (uniform priors) of an analogous result for the principle of minimum cross-entropy. Results are obtained both for continuous probability densities and for discrete distributions.

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Axiomatic derivation of the principle of maximum entropy and the principle of minimum cross-entropy

Previously proposed models for the evolution of the Tyrrhenian basin-Apenninic arc system do not seem to satisfactorily explain the dynamic relationship between extension in the Tyrrhenian and compression in the Apennines. The most important regional plate kinematic constraints that any model has to satisfy in this case are: (1) the timing of extension in the Tyrrhenian and compression in the Apennines, (2) the amount of shortening in the Apennines, (3) the amount of extension in the Tyrrhenian, and (4) Africa-Europe relative motion. The estimated contemporaneous (post-middle Miocene) amounts of extension in the Tyrrhenian and of shortening in the Apennines appear to be very similar. The extension in the Tyrrhenian Sea is mostly accomplished in an E-W direction, and cannot be straightforwardly related to the calculated N-S Africa-Europe convergence. A model of outward arc migration fits all these constraints. In a subducting system, the subduction zone is expected to migrate outward due to the sinking of the underthrusting plate into the mantle. The formation of a back-arc or internal basin, i.e. of a basin internal to the surrounding belt of compression, (in this case the Tyrrhenian Sea) is then expected to take place if the motion of the overriding plate does not compensate for the retreat of the subduction zone. The sediment cover will be stripped from the underthrusting plate by the outward migrating arc of the overriding plate, and will accumulate to form an accretionary wedge. This accretionary body will grow outward in time, and will eventually become an orogenic belt, (in this case the present Apennines) when the migrating arc collides with the stable continental foreland on the subducting plate. An arc migration model satisfactorily accounts for the basic features of the Tyrrhenian-Apennine system and for its evolution from 17 Ma to the present, and appears to be analogous to the tectonic evolution of other back-arc settings both inside and outside the Mediterranean region. An interesting implication of the proposed accretionary origin of the Apennines is that the problematic “Argille Scagliose” (scaly clays) melange units might have been emplaced as overpressured mud diapirs, as observed in other accretionary prisms, and not by gravity slides from the internal zones.

Extension in the Tyrrhenian Sea and Shortening in the Apennines as Result of Arc Migration Driven by Sinking of the Lithosphere

The paper deals with the electromagnetic effect of motions in the earth's core, considered as a fluid metallic sphere. On the basis of simple estimates the electric conductivity of the core is assumed of the same order of magnitude as that of common metals. The mathematical treatment follows Hansen and Stratton: three independent vector solutions of the vectorial wave equation are introduced; two of these have vanishing divergence, and they are designated as toroidal and poloidal vector fields. The vector potential and electric field are toroidal, whereas the magnetic field is poloidal. These vectors, expressed in terms of spherical harmonics and Bessel functions, possess some notable properties of orthogonality which are briefly discussed. The theory of the free, exponentially decaying current modes is then given, leading to decay periods of the order of some tens of thousands of years. Next, the field equations in the presence of mechanical motions of the conducting fluid are set up. The field is developed in a series of the fundamental, orthogonal vectors, and the field equations are transformed into a system of ordinary differential equations for the coefficients of this development. The behavior of the solutions depends on the symmetry of the "coupling matrix" that arises from the term of the field equations expressing the induction effects. In order to evaluate this matrix the velocity field is developed into a series of the fundamental vectors similar to the series for the electromagnetic field. It is then shown that when the velocity is a toroidal vector field the coupling matrix is antisymmetrical. When the velocity field is poloidal, the coupling matrix is neither purely symmetrical nor purely antisymmetrical. For stationary fluid motion the linear differential equations can be integrated in closed form by a transformation to new normal modes, whenever the matrix of the system is either symmetrical or antisymmetrical. In the latter case the eigenvalues are purely imaginary and the coefficients of the new normal modes are harmonic functions of time, representing oscillatory changes in amplitude of the field components. For a symmetrical matrix the eigenvalues are real and the time factors of the new normal modes are real exponentials representing amplification or de-amplification as the case may be, depending on the sign of the velocity. For a matrix without specific symmetry, normal modes do not, as a rule, exist but similar, somewhat less stringent results can be derived in special cases. In the case of toroidal flow, in particular, the oscillatory changes of the field components are superposed upon the slow exponential decay characteristic of the free modes.

Induction Effects in Terrestrial Magnetism Part I. Theory

The basis of this theoretical paper is the assumption that the asthenosphere is ‘soft’ enough so that it does not transmit significant horizontal shear stress to the overlying lithosphere. It is pointed out that any thermal mantle convection is likely to have its origin in the cooling of the top layers of the oceanic mantle rather than being due to any heating that may arise farther down. Next, there is geological evidence for the existence of regions that exhibit widespread horizontal tension, e.g., in East Africa. Submarine ridges can justifiably be considered as regions of horizontal tensile stress. In our model, the choice of mechanisms that can generate such stresses is severely limited; the one found to be satisfactory qualitatively and quantitatively arises through the disappearance of pieces of lithosphere, which slide down along the Gutenberg-Benioff fault zones. The lithosphere spreads to fill the gaps created. It is shown that this motion is likely to represent the surficial part of a thermally driven conveetive circulation in the mantle. The nature of the ‘return flow’ farther down is discussed.

Sea‐Floor Spreading as Thermal Convection

The Earth's Interior and Geomagnetism

Mean Absorption and Equivalent Absorption Coefficient of a Band Spectrum

In Part I a method has been developed for the integration of the electromagnetic field equations in the presence of fluid motions in a spherical conductor. This analysis is here applied to an interpretation of the secular geomagnetic variations. A very brief survey of some of the observed features of the secular variation is first given. It is pointed out that not only the phases but also the magnitudes of the harmonic components, including the main dipole, are subject to large changes at the present time. There follows a brief study of the skin effect which indicates that the observed variations of the dipole terms originate in a layer adjacent to the core's boundary several hundred kilometers deep; those of the higher terms originate in a layer no more than 200 km deep. Next, the "coupling matrix" introduced in Part I is evaluated in form of a table of all matrix elements that contain vectors of dipole and quadrupole type but no higher harmonics. It is shown that a zonal fluid motion (zonal toroidal flow in the terminology adopted here) produces rotation of the tesseral magnetic dipole terms and also oscillatory changes in amplitude of these terms. There is one and only one type of matrix element that represents an interaction of the main magnetic dipole with itself; the corresponding fluid motion is a meridional flow (poloidal flow) of quadrupole symmetry. With this term amplification or de-amplification occurs, depending on the sign of the velocity. The theory thus can account for all the observed components of the secular variation.

Walter M. Elsasser

Papers

Induction Effects in Terrestrial Magnetism Part I. Theory

Sea‐Floor Spreading as Thermal Convection

The Earth's Interior and Geomagnetism

Mean Absorption and Equivalent Absorption Coefficient of a Band Spectrum

Induction Effects in Terrestrial Magnetism Part II. The Secular Variation