Rheology of the Earth and a thermoconvective mechanism for sedimentary basin formation

A power-law non-Newtonian fluid is usually assumed to model slow flows in the mantle and, in particular, convective flows. However, the power-law fluid has no memory, in contrast to a real material. A new non-linear integral (having a memory) model is proposed to describe the rheology of rocks. The model is consistent with the theory of simple fluids with fading memory and with laboratory studies of rock creep. The proposed model reduces to the power-law fluid model for stationary flows and to the Andrade model for flows associated with small strains. Stationary convection beneath continents has been studied by Fleitout & Yuen (1984 ), who used the power-law fluid model and obtained the cold immobile boundary layer (continental lithosphere). In a stability analysis of this layer, the Andrade model must be used. The analysis shows that the lithosphere is overstable (the period of oscillation is about 200  Ma). In the present study, it is suggested that these thermoconvective oscillations of the lithosphere are a mechanism for sedimentary basin formation. The vertical crustal movement in sedimentary basins can be considered as a slow subsidence on which small-amplitude oscillations are superimposed. The longest period of oscillatory crustal movement is of the same order of magnitude as the period of convective oscillation of the lithosphere found in the stability analysis. Taking into account the difference between depositional and erosional transport rates we can explain the permanent subsidence as well as the oscillations.     

Contourites associated with pelagic mudrocks and distal delta-fed turbidites in the Lower Proterozoic Timeball Hill Formation epeiric basin (Transvaal Supergroup), South Africa

Five genetic facies associations/architectural elements are recognised for the epeiric sea deposits preserved in the Early Proterozoic Timeball Hill Formation, South Africa. Basal carbonaceous mudrocks, interpreted as anoxic suspension deposits, grade up into sheet-like, laminated, graded mudrocks and succeeding sheets of laminated and cross-laminated siltstones and fine-grained sandstones. The latter two architectural elements are compatible with the Te, Td and Tc subdivisions of low-density turbidity current systems. Thin interbeds of stromatolitic carbonate within these first three facies associations support photic water depths up to about 100 m. Laterally extensive sheets of mature, cross-bedded sandstone disconformably overlie the turbidite deposits, and are ascribed to lower tidal flat processes. Interbedded lenticular, immature sandstones and mudrocks comprise the fifth architectural element, and are interpreted as medial to upper tidal flat sediments. Small lenses of coarse siltstone–very fine-grained sandstone, analogous to modern continental rise contourite deposits, occur within the suspension and distal turbidite sediments, and also form local wedges of inferred contourites at the transition from suspension to lowermost turbidite deposits. Blanketing and progressive shallowing of the floor of the Timeball Hill basin by basal suspension deposits greatly reduced wave action, thereby promoting preservation of low-density turbidity current deposits across the basin under stillstand or highstand conditions. A lowstand tidal flat facies tract laid down widespread sandy deposits of the medial Klapperkop Member within the formation. Salinity gradients and contemporaneous cold periglacial water masses were probably responsible for formation of the inferred contourites. The combination of the depositional systems interpreted for the Timeball Hill Formation may provide a provisional model for Early Proterozoic epeiric basin settings.     

Spotless flares and type II radio bursts

Flares accompanied by type II meter radio bursts that occurred in plages with no visible spots are examined in this paper. There have been found 12 such spotless flares observed in the period of January 1981–August, 1990. Six out of all the flares may be said to have not been associated with any filament activation or disruption. A few of these flares have shown features of major events. The study suggests that a filament activation seems not to be the crucial factor for the occurrence of major flares in regions with no visible spots.     

A theory of libration

It is shown here that many problems of libration in celestial mechanics can be reduced to a perturbation of anintermediary defined by the Hamiltonian $$F = B\left( y \right) + 2\mu ^2 A\left( y \right)f\left( x \right).$$ This generalization of the Ideal Resonance Problem, with a periodic functionf(x) replacing sin² x, is solved here toO(μ ²) by an algorithm that is essentially the same as the one used in the original formulation. The solution is of the formx=x(u), u=u(t), y=y(x), with the functionx(u) commonly involving the inversion of a hyperelliptic integralu(x), evaluated by quadrature. Libration may be simple or multiple, depending on the nature of the functionf(x) and on the initial conditions. Double libration is illustrated here by the horseshoe-shaped orbits enclosing two libration centers.     

The regularizing function in resonance problems

The regularizing function ψ(x) in the theory of resonance removes the singularities discovered by Poincaré (1893), of the form 1/x, at theturning points x ₁ andx ₂ of thecritical argument x, librating in the rangex ₁≤x≤x ₂. This function has been explicitly introduced into the HamiltonianF ₀ of the Ideal Resonance Problem in the author's recent paper (1977) in order to remove the singularities in the second-order perturbations. It is shown here that this procedure can be extended toall orders. ThenF ₀ can be put into the form $$F_0 = B(y) + \Psi (x)$$ where ψ is thecomplete regularizing function, removing theclassical singularity of thesmall divisor, in addition to the singularities of Poincaré.     

A three-dimensional hydrodynamic model of estuarine circulation with an application to Southampton Water, UK

A three-dimensional hydrodynamic model has been developed to simulate water mass circulation in estuarine systems. This model is based on the primitive equation in Cartesian coordinates with a terrain-following structure, coupled with a Mellor–Yamada 2.5 turbulence scheme. A fractional-step method is applied and the subset of equations is solved with finite volume and finite element methods. A dry–wet process simulates the presence of the tidal flat at low water. River inputs are introduced using a point-source method. The model was applied to a partially mixed, macrotidal, temperate estuary: Southampton Water, UK. The model is validated by comparisons with sea surface elevation, ADCP measurements and salinity data collected in 2001. The mean spring range 2(M₂ + S₂) and the mean neap range 2(M₂ − S₂) are modelled with an error relative to observation of 12 and 16%, respectively. The unique tidal regime of the system with the presence of the ‘young flood stand’ corresponding to the slackening conditions occurring at mid flood and ‘double high water’ corresponding to an extension of the slackening conditions at high tide is accurately reproduced in the model. The dynamics of the modelled mean surface and bottom velocity closely match the ADCP measurements during neap tides (rms of the difference is 0.09 and 0.01 m s⁻¹ at the bottom and at the surface, respectively), whereas at spring the difference is greater (rms of the difference is 0.25 and 0.20 m s⁻¹ at bottom and surface, respectively). The spatial and temporal variation of the degree of stratification as indicated by salinity distributions compares well with observations.     

Recent progress in the theory of Trojan asteroids

In previous publications the author has constructed a long-periodic solution of the problem of the motion of the Trojan asteroids, treated as the case of 1:1 resonance in the restricted problem of three bodies. The recent progress reported here is summarized under three headings:

1. The nature on the long-periodic family of orbits is re-examined in the light of the results of the numerical integrations carried out by Deprit and Henrard (1970). In the vicinity of the critical divisor $$D_k \equiv \omega _1 - k\omega _2 ,$$ not accessible to our solution, the family is interrupted by bifurcations and shortperiodic bridges. Parametrized by the normalized Jacobi constant α², our family may, accordingly, be defined as the intersection of admissible intervals, in the form $$L = \mathop \cap \limits_j \left\{ {\left| {\alpha - \alpha _j } \right| > \varepsilon _j } \right\};j = k,k + 1, \ldots \infty .$$ Here, {α_j(m)} is the sequence of the critical α_j corresponding to the exactj: 1 commensurability between the characteristic frequencies ω₁ and ω₂ for a given value of the mass parameterm. Inasmuch as the ‘critical’ intervals |α?α_j|<ε_j can be shown to be disjoint, it follows that, despite the clustering of the sequence {α_j} at α=1, asj→∞, the family extends into the vicinity of the separatrix α=1, which terminates the ‘tadpole’ branch of the family.
2. Our analysis of the epicyclic terms of the solution, carrying the critical divisorD _k , supports the Deprit and Henrard refutation of the E. W. Brown conjecture (1911) regarding the termination of the tadpole branch at the Lagrangian pointL ₃. However, the conjecture may be revived in a refined form. “The separatrix α=1 of the tadpole branch spirals asymptotically toward a limit cycle centered onL ₃.”
3. The periodT(α,m) of the libration in the mean synodic longitude λ in the range $$\lambda _1 \leqslant \lambda \leqslant \lambda _2$$ is given by a hyperelliptic integral. This integral is formally expanded in a power series inm and α² or \(\beta \equiv \sqrt {1 - \alpha ^2 }\) .

The large amplitude of the libration, peculiar to our solution, is made possible by the mode of the expansion of the disturbing functionR. Rather than expanding about Lagrangian pointL ₄, with the coordinatesr=1, θ=π/3, we have expandedR about the circler=1. This procedure is equivalent to analytic continuation, for it replaces the circle of convergence centered atL ₄ by an annulus |r?1|<ε with 0≤θ<2π.     

On resonance in celestial mechanics

This brief survey of the author's contribution to the theory of resonance in celestial mechanics begins with the genesis of the Small Divisor. The fundamental distinction between theshallow anddeep resonance is illustrated by the 52 Jupiter-Saturn and the 3-2 Neptune-Pluto resonances in the planetary system.The search for aglobal solution through a removal of the small divisor is put into a historical perspective through the work of Laplace, Bohlin, and Poincaré. The author's own contribution to the methodology is the formulation and the solution of the Ideal Resonance Problem. If the resonance issimple, all the singularities in the solution are removed by means of aregularizing function. On the other hand, if the resonance isdouble, the second critical divisor seems irremovable, and a global solution may be precluded.Invited paper, IAU 1979, Commission 7, Montreal, Canada.     

Prevention of well clogging during aquifer storage of turbid tile drainage water rich in dissolved organic carbon and nutrients

Well clogging was studied at an aquifer storage transfer and recovery (ASTR) site used to secure freshwater supply for a flower bulb farm. Tile drainage water (TDW) was collected from a 10-ha parcel, stored in a sandy brackish coastal aquifer via well injection in wet periods, and reused during dry periods. This ASTR application has been susceptible to clogging, as the TDW composition largely exceeded most clogging mitigation guidelines. TDW pretreatment by sand filtration did not cause substantial clogging at a smaller ASR site (2 ha) at the same farm. In the current (10 ha) system, sand filtration was substituted by 40-μm disc filters to lower costs (by 10,000–30,000 Euro) and reduce space (by 50–100 m²). This measure treated TDW insufficiently and injection wells rapidly clogged. Chemical, biological, and physical clogging occurred, as observed from elemental, organic carbon, 16S rRNA, and grain-size distribution analyses of the clogging material. Physical clogging by particles was the main cause, based on the strong relation between injected turbidity load and normalized well injectivity. Periodical backflushing of injection wells improved operation, although the disc filters clogged when the turbidity increased (up to 165 NTU) during a severe rainfall event (44 mm in 3 days). Automated periodical backflushing, together with regulating the maximum turbidity (<20 NTU) of the TDW, protected ASTR operation, but reduced the injected TDW volume by ~20–25%. The studied clogging-prevention measures collectively are only viable as an alternative for sand filtration when the injected volume remains sufficient to secure the farmer’s needs for irrigation.

     

P-wave dispersion and attenuation in fractured and porous reservoirs – poroelasticity approach

Natural fractures in hydrocarbon reservoirs can cause significant seismic attenuation and dispersion due to wave induced fluid flow between pores and fractures. We present two theoretical models explicitly based on the solution of Biot's equations of poroelasticity. The first model considers fractures as planes of weakness (or highly compliant and very thin layers) of infinite extent. In the second model fractures are modelled as thin penny-shaped voids of finite radius. In both models attenuation is a result of conversion of the incident compressional wave energy into the diffusive Biot slow wave at the fracture surface and exhibits a typical relaxation peak around a normalized frequency of about 1. This corresponds to a frequency where the fluid diffusion length is of the order of crack spacing for the first model and the crack diameter for the second. This is consistent with an intuitive understanding of the nature of attenuation: when fractures are closely and regularly spaced, the Biot's slow waves produced by cracks interfere with each other, with the interference pattern controlled by the fracture spacing. Conversely, if fractures are of finite length, which is smaller than spacing, then fractures act as independent scatterers and the attenuation resembles the pattern of scattering by isolated cracks. An approximate mathematical approach based on the use of a branching function gives a unified analytical framework for both models.