From Celsius to Chaos to Cyclones: Using Temperature-Conversion Equations to Introduce Advanced Mathematical Concepts in Earth Science Courses

The Fahrenheit-to-Celsius temperature-conversion equation is a basic component of many introductory earth science courses. Despite its simplicity, it presents a challenge to students and instructors alike because residents of the United States are unfamiliar with the Celsius scale. By solving for the point at which these two temperature scales are equal, it is possible to use the equations for temperature conversion as a springboard to more advanced topics. It is demonstrated that temperature-conversion equations and chaotic equations can be solved using identical numerical and graphical techniques. As a result, the fundamental concepts of chaos theory and numerical methods can be introduced to students in the context of the simplest equations in the earth sciences. These solution methods are applied to the quantitative theory of the extratropical cyclone as an example of the utility and broad scope of this educational approach.     

Improved Determination of the Location of the Temperature Maximum in the Corona

The most used method to calculate the coronal electron temperature [\(T_{\mathrm{e}} (r)\)] from a coronal density distribution [\(n_{\mathrm{e}} (r)\)] is the scale-height method (SHM). We introduce a novel method that is a generalization of a method introduced by Alfvén (Ark. Mat. Astron. Fys. 27, 1, 1941) to calculate \(T_{\mathrm{e}}(r)\) for a corona in hydrostatic equilibrium: the “HST” method. All of the methods discussed here require given electron-density distributions [\(n_{\mathrm{e}} (r)\)] which can be derived from white-light (WL) eclipse observations. The new “DYN” method determines the unique solution of \(T_{\mathrm{e}}(r)\) for which \(T_{\mathrm{e}}(r \rightarrow \infty) \rightarrow 0\) when the solar corona expands radially as realized in hydrodynamical solar-wind models. The applications of the SHM method and DYN method give comparable distributions for \(T_{\mathrm{e}}(r)\). Both have a maximum [\(T_{\max}\)] whose value ranges between 1?–?3 MK. However, the peak of temperature is located at a different altitude in both cases. Close to the Sun where the expansion velocity is subsonic (\(r < 1.3\,\mathrm{R}_{\odot}\)) the DYN method gives the same results as the HST method. The effects of the other free parameters on the DYN temperature distribution are presented in the last part of this study. Our DYN method is a new tool to evaluate the range of altitudes where the heating rate is maximum in the solar corona when the electron-density distribution is obtained from WL coronal observations.     

Processing of chondritic and planetary material in spiral density waves in the nebula

Abstract— A widely held view of nebular evolution is that during the ～0.5 Ma while interstellar material was collapsing onto the disk, the latter grew in mass to the point of gravitational instability. It responded to this by losing axial symmetry, growing spiral arms that had the capacity to tidally redistribute disk mass (inward) and angular momentum (outward) and prevent further increase in the disk/protosun mass ratio. The spiral arms (density waves) rotated differently than the substance of the nebula, and in some parts of the disk, nebular material may have encountered the arms at supersonic velocities. The disk gas, and solid particles entrained in it, would have been heated to some degree when they passed through shock fronts at the leading edges of the spiral arms. The present paper proposes this was the energetic nebular setting or environment that has long been sought, in which the material now in the planets and chondritic meteorites was thermally processed.     

X-radiation (E > 10keV), Hα and microwave emission during the impulsive phase of solar flares

A study has been made of the variation in hard (E 10 keV) X-radiation, H and microwave emission during the impulsive phase of solar flares. Analysis shows that the rise-time in the 20–30-keV X-ray spike depends on the electron hardness, i.e., t _rise exp (0.87 ). The impulsive phase is also marked by an abrupt, very intense increase in H emission in one or more knots of the flare. Properties of these H kernels include: (1) a luminosity several times greater than the surrounding flare, (2) an intensity rise starting about 20–30 s before, peaking about 20–25 s after, and lasting about twice as long as the hard spike, (3) an effective diameter of 3000–6000 km for class 1 flares, representing less than 1/8-1/2 of the main flare, (4) a location lower in the chromosphere than the remaining flare, (5) essentially no expansion prior to the hard spike, (6) a position within 6000 km of the boundary separating polarities, usually forming on both sides of the neutral line near both feet of the same tube of force, (7) a shape often resembling isogauss contours of the photospheric field indicated on magnetograms and (8) total radiated energy less than l/50 that of the hard electrons. Correspondingly, impulsive microwave events are characterized by: (1) the detection of a burst at 8800 MHz for every X-ray spike ifthe number of electrons above 100 keV is greater than 10³³, (2) great similarity in burst structure with 20–32 keV X-rays but only at f > 5000 MHz, (3) typical low frequency burst cutoff between 1400–3800 MHz, and (4) maximum emission at f > 7500 MHz. Finally the H, X-ray and microwave data are combined to present a picture of the impulsive phase consistent with the above observations.     

Invariant manifolds, phase correlations of chaotic orbits and the spiral structure of galaxies

In the presence of a strong   m = 2  component in a rotating galaxy, the phase-space structure near corotation is shaped to a large extent by the invariant manifolds of the short-period family of unstable periodic orbits terminating at L ₁ or L ₂. The main effect of these manifolds is to create robust phase correlations among a number of chaotic orbits large enough to support a spiral density wave outside corotation. The phenomenon is described theoretically by soliton-like solutions of a Sine–Gordon equation. Numerical examples are given in an N -body simulation of a barred spiral galaxy. In these examples, we demonstrate how the projection of unstable manifolds in configuration space reproduces essentially the entire observed bar–spiral pattern.     

The contribution of sulphur dioxide from ablating micrometeorites to the atmospheres of Earth and Mars

Atmospheric composition is a key control on climate and the habitability of planetary surfaces. Ablation of infalling micrometeorites has been recognised as one way in which atmospheric chemistry can be changed, especially at times in solar system history when the infall rates of exogenous material were high. Despite its potential to influence climate and habitability, extraterrestrial sulphur dioxide is currently an unquantified contribution to the atmospheres of the terrestrial planets. We have used flash pyrolysis to simulate the atmospheric entry of micrometeorites and Fourier-transform infrared spectroscopy to identify and quantify the sulphur dioxide produced from the carbonaceous meteorites Orgueil (CI1), ALH 88045 (CM1), Cold Bokkeveld (CM2), Murchison (CM2) and Mokoia (CV3). We have used this approach to understand the introduction of sulphur dioxide to the atmospheres of Earth and Mars from infalling micrometeorites. Sulphates, present in carbonaceous chondrites at a few wt.%, are resistant to thermal decomposition, limiting the yields of sulphur dioxide from unmelted micrometeorites. Infalling micrometeorites are a minor source of present-day sulphur dioxide on Earth and Mars, calculated to be up to around 2400 tonnes and about 350 tonnes, respectively. During the Late Heavy Bombardment (LHB), the much greater infall rates of micrometeoritic dust are calculated to be associated with average production rates of sulphur dioxide of around 20 Mt yr⁻¹ for the early Earth and 0.5 Mt yr⁻¹ for early Mars, for a LHB of 100 Myr. These rates of delivery of sulphur dioxide at high altitudes would have reduced the solar energy reaching the surfaces of these planets, via scattering of sunlight by stratospheric sulphate aerosols, and may have had detrimental effects on developing biospheres by promoting cooler climates and reducing the probability of liquid water on planetary surfaces.     

Modelling the spatial and temporal distribution of rainfall: a case study in the wet and dry tropics of North East Australia

Rainfall regimes with strong spatial and temporal variation are characteristic of many coastal regions of north and eastern Australia. In coastal regions of north eastern Australia, regimes vary considerably over short distances. This occurs because of changes in local topography, including the height and orientation of mountain ranges and the direction of the coastline with respect to the prevailing moist south east air stream. Northern Australia experiences a tropical monsoon climate with rainfall occurring predominantly during the summer months. Areas with a closer proximity to the coast typically experience the heavier rainfalls. While networks of rainfall gauges have been established and continuous records are available for most of these stations from the 1890s, their low distribution density relative to the complexity of rainfall pattern they are required to represent means that there remains a poor understanding of the spatial and temporal distribution of rainfall in the wet tropics. An enhanced knowledge of rainfall distribution in both space and time has the potential to deliver significant economic and environmental benefits to managers of natural resources. This paper reports on the application of a technique for estimating mean annual and mean monthly rainfall across the Herbert River catchment of north east Australia's dry and wet tropics. The technique utilises thin plate smoothing splines to incorporate both location and elevation into estimates of rainfall distribution. We demonstrate that the method can be applied successfully at the meso scale and within the domain of routinely available data. As such, the method has broad relevance for decision making.