STABLE AGGREGATES OF FLEXIBLE ELEMENTS : MAKING A SILK PURSE FROM SOW ’ S EARS
We discuss the application of a ray-based method for wave propagation to problems involving refraction and diffraction. The method assigns a Gaussian contribution to each ray such that their sum is insensitive to the Gaussian’s width. Diffraction is modelled by including contributions from rays scattered by the edges of obstacles. Although the Gaussian components are individually only poor approximations to physical waves, they collectively give accurate solutions even for problems where the wavelength approaches the dimensions of system components. INTRODUCTION A framework for building field estimates based on ray information was introduced in a recent series of papers [1-6]. In this framework, each ray is assigned a field contribution of Gaussian profile. These contributions, however, are not parabasal Gaussian beamlets, as each of them is not required to satisfy the wave equation. They are poor approximations to fields (the “sow’s ears” of the title), but their individual errors are forced to cancel when they are summed, leading to a field estimate of extraordinary accuracy. This estimate is also asymptotically (for small wavelength) insensitive to the width of the individual contributions. The sums are therefore referred to as a stable aggregates of flexible elements (SAFE). Here, we discuss the use of SAFE for modelling refraction and reflection at interfaces, as well as diffraction by opaque obstacles. WAVES IN INHOMOGENEOUS MEDIA Consider a scalar monochromatic field U in a two-dimensional inhomogeneous medium. This field satisfies [k∇ 2 + n(x,z)]U(x,z) = 0, (1) where k = 2π λ is the wave number, and n is the refractive index. We assume that this field propagates essentially towards larger z, so it is sufficient to specify U(x,0) in order to find U at any z > 0. Finding a closed form solution is generally not possible, so numerical schemes are necessary. Some of these schemes, based on rays, are particularly suited for problems where the wavelength is much smaller than any other characteristic length. It is natural to assume that the rays associated to the field described above also travel towards larger z. Each of these rays can then be described by an equation of the form x = X(z) . We assume that a solution to Eq. (1) is associated with a one-parameter family of rays, described collectively by a function X(ξ;z) , where ξ identifies the rays. The propagation of the rays in the inhomogeneous medium is ruled by the equations ∂ X ∂z (ξ;z) = P(ξ;z) H(ξ;z) , ∂P ∂z (ξ;z) = 1 2H(ξ;z) ∂n ∂x [X(ξ;z),z], (2) where P and H are respectively the xand z-components of a vector of length n(X,z) locally tangent to the ray. The optical path length of the rays, measured from a common phase front, is denoted by L(ξ;z). In order to find a ray-based estimate for the field at z > 0, one must first determine from U(x,0) the location and direction of the rays at z = 0, as well as their weight. (This process is discussed in [2].) The rays are then propagated by using Eqs. (2). Finally, the field estimate is constructed from the propagated rays, i.e. from X, P, H, L, and the weight function. For SAFE, this estimate is a sum of ray contributions of the form Uγ (x,z) = w(ξ ,γ;z) g(ξ ,γ; x,z) dξ ∫ , (3) where each contribution has a Gaussian profile in x and is centred at its corresponding ray: g(ξ ,γ; x,z) = k 2π exp − kγ 2 x − X(ξ;z) [ ]2 + ik L(ξ;z) + x − X(ξ;z) [ ] P(ξ;z) { } ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ . (4)
Using rays better. II. Ray families to match prescribed wave fields.
A key step in any ray-based method for propagating waves is the choice of a family of rays to be associated with the initial wave field. We develop some basic prescriptions for constructing initial ray families to match two particular types of waves. Various Gaussian and Bessel beams are separately given special treatment because of their general interest. These ideas are directly useful for a newly developed method for ray-based wave modeling. The new method expresses the wave as a superposition of ray contributions that is independent of the width of the field element associated with each ray. This insensitivity is investigated here even when the elemental width varies from ray to ray. The results increase the applicability of the new wave-modeling scheme.
O2 constraints from Paleoproterozoic detrital pyrite and uraninite
Redox-sensitive detrital grains such as pyrite and uraninite in sedimentary successions provide one of the most conspicuous geological clues to a different composition of the Archean and early Paleoproterozoic atmosphere. Today, these minerals are rapidly chemically weathered within short transport distances. Prior to the rise of oxygen, low O2 concentrations allowed their survival in siliciclastic deposits with grain erosion tied only to physical transport processes. After the rise of oxygen, redox-sensitive detrital grains effectively vanish from the sedimentary record. To get a better understanding of the timing of this transition, we examined sandstones recorded in a scientifi c drill core from the South African 2.415 Ga Koegas Subgroup, a mixed siliciclastic and iron formation–bearing unit deposited on the western deltaic margin of the Kaapvaal craton in early Paleoproterozoic time. We observed detrital pyrite and uraninite grains throughout all investigated sandstone beds in the section, indicating the rise of oxygen is younger than 2.415 Ga. To better understand how observations of detrital pyrite and uraninite in sedimentary rocks can quantitatively constrain Earth surface redox conditions, we constructed a model of grain erosion from chemical weathering and physical abrasion to place an upper limit on ancient environmental O2 concentrations. Even conservative model calculations for deltaic depositional systems with suffi cient transport distances (approximately hundreds of kilometers) show that redox-sensitive detrital grains are remarkably sensitive to environmental O2 concentrations, and they constrain the Archean and early Paleoproterozoic atmosphere to have <3.2 × 10 –5 atm of molecular O2. These levels are lower than previously hypothesized for redox-sensitive detrital grains, but they are consistent with estimates made from other redox proxy data, including the anomalous fractionation of sulfur isotopes. The binary loss of detrital pyrite and uraninite from the sedimentary record coincident with the rise of oxygen indicates that atmospheric O2 concentrations rose substantially at this time and were never again suffi ciently low (<0.01 atm) to enable survival and preservation of these grains in short transport systems.