Great Circle Research Articles

Mitigating stochastic uncertainty from weather routing for ships with wind propulsion

Reducing the shipping sector's contribution to climate change requires urgent emission reductions this decade. Both weather routing and wind propulsion offer immediate solutions, where combining sails with efficient routing amplifies the performance of each technology. However, while large emission savings are theoretically available, the impact of stochastic uncertainty from wind forecasts is unknown. Here, we present a novel approach that exploits fuel consumption calculations from over a thousand departures across three routes to characterise stochastic uncertainty. We show that routes with ideal wind conditions and long voyage times are most sensitive to uncertain forecast inputs, reducing savings from Flettner rotors and weather routing by up to 44% when a priori weather routing strategies are used. This paper further shows how an adaptive weather routing strategy can be used as an accurate prediction tool to reduce uncertainty on all routes investigated, reliably amplifying carbon savings from Flettner rotors by between 1.16 and 2.48 times typical great circle route savings. Overall, this paper provides greater assurance around the previously estimated carbon savings that serves to strengthen confidence in a wind-assisted decarbonisation strategy and its potential to provide essential emission reductions this decade.

The p-widths of a surface

The p-widths of a closed Riemannian manifold are a nonlinear analogue of the spectrum of its Laplace–Beltrami operator, which corresponds to areas of a certain min-max sequence of possibly singular minimal submanifolds. We show that the p-widths of any closed Riemannian two-manifold correspond to a union of closed immersed geodesics, rather than simply geodesic nets.We then prove optimality of the sweepouts of the round two-sphere constructed from the zero set of homogeneous polynomials, showing that the p-widths of the round sphere are attained by lfloor sqrt{p}rfloor great circles. As a result, we find the universal constant in the Liokumovich–Marques–Neves–Weyl law for surfaces to be sqrt{pi }.En route to calculating the p-widths of the round two-sphere, we prove two additional new results: a bumpy metrics theorem for stationary geodesic nets with fixed edge lengths, and that, generically, stationary geodesic nets with bounded mass and bounded singular set have Lusternik–Schnirelmann category zero.

Identifying gravity waves launched by the Hunga Tonga–Hunga Ha′apai volcanic eruption in mesosphere/lower-thermosphere winds derived from CONDOR and the Nordic Meteor Radar Cluster

Abstract. The Hunga Tonga–Hunga Ha′apai volcano eruption was a unique event that caused many atmospheric phenomena around the globe. In this study, we investigate the atmospheric gravity waves in the mesosphere/lower-thermosphere (MLT) launched by the volcanic explosion in the Pacific, leveraging multistatic meteor radar observations from the Chilean Observation Network De Meteor Radars (CONDOR) and the Nordic Meteor Radar Cluster in Fennoscandia. MLT winds are computed using a recently developed 3DVAR+DIV algorithm. We found eastward- and westward-traveling gravity waves in the CONDOR zonal and meridional wind measurements, which arrived 12 and 48 h after the eruption, and we found one in the Nordic Meteor Radar Cluster that arrived 27.5 h after the volcanic detonation. We obtained observed phase speeds for the eastward great circle path at both locations of about 250 m s−1, and they were 170–150 m s−1 for the opposite propagation direction. The intrinsic phase speed was estimated to be 200–212 m s−1. Furthermore, we identified a potential lamb wave signature in the MLT winds using 5 min resolved 3DVAR+DIV retrievals.

On the Importance of Using Directional Information in the Search for Lower Mantle Reflectors

Abstract The SS precursor signals are a powerful tool for mapping topography of mantle discontinuities, which are sensitive to the thermal and compositional structure of the mantle. The depth of mantle discontinuities is usually estimated using the differential travel time between the main arrival and its precursor. However, this method ignores potential travel path deviations that influence the travel time of precursor signals. Here, we use an approach that considers directivity information as well as travel-time measurements. Applying seismic array techniques, we measure slowness, back azimuth, and travel time of the signals, and use this information to backproject to the point of reflection. In our test dataset, we observe deviations from the predicted values in slowness and back azimuth in the range of 0.1–2.3 s/° and 1–20°, respectively. These values lead to reflection locations that can differ considerably from theoretical reflection points calculated with great circle plane paths as well as depths different from the depth calculated for in-plane propagation, with differences up to ∼150 km. Our results indicate that the travel-path deviations should be considered to avoid misinterpretation of mantle discontinuities and potentially reduce previously observed scatter in discontinuity depth.

VLF and Ionospheric D‐Region Perturbations Associated With WWLLN‐Detected Lightning in the South Pacific Region

AbstractThe subionospheric early very low frequency (VLF) perturbations observed on NWC (19.8 kHz) navigational transmitter signal monitored at a low‐latitude station, Suva (18.1°S, 178.5°E), Fiji, during campaign periods of November 2011, 2012, and 2014 and December 2014, are presented. Early VLF events are associated with D‐region conductivity changes mainly produced by lightning‐generated transient luminous events (TLEs). Early VLF events occurred both during daytime and nighttime, with a considerably higher occurrence at nighttime. VLF perturbations caused by lightning strokes located up to 100 km off the transmitter‐receiver great circle path (TRGCP) are attributed to narrow‐angle scattering, while lightning strokes 100–500 km off the TRGCP are considered to cause early VLF events by wide‐angle scattering. Using the World Wide Lightning Location Network data, for the first time, we have studied the relationship between the energy of lightning strokes and the level of VLF perturbations. Greater is the energy of lightning, greater would be the strength of the VLF perturbation. However, the low‐energy lightning stroke can also produce a comparable level of perturbation to that of strong lightning. The modeling results of scattered amplitude (M) and echo phase (ϕE) of the unusually long recovery early/fast VLF event showed a better exponential fit (r ∼ 0.9) than the logarithmic fit. Long‐wavelength propagation capability (LWPC) code modeling of nighttime early VLF events considering causative TLE width of 50 km column indicated a decrease in the D‐region reference height (Hʹ) by up to 30 km and an increase in the sharpness factor (β) by 0.25 km−1.

Least squares spherical harmonics approximation on the Cubed Sphere

The Cubed Sphere grid is an important tool to approximate functions or data on the sphere. We introduce an approximation framework on this grid based on least squares and on a suitably chosen subspace of spherical harmonics. The main claim is that for the equiangular Cubed Sphere with step size π/(2N), a relevant spherical harmonics subspace is the one of all SH of degree less than 2N. This choice, which matches the Nyquist’s cutoff frequency along the equatorial great circle, provides an approximation problem both well-posed and well-conditioned. A series of theoretical and numerical results supporting this fact are presented.

Rotation of a Ferromanganese Nodule in the Penrhyn Basin, South Pacific, Tracked by the Earth's Magnetic Field

AbstractFerromanganese nodules are cm‐sized, authigenic, abyssal manganese‐iron‐hydroxide concretions. They grow very slowly at rates of a few mm per million years. Although their ages are older than millions of years, they are often found half buried on the modern sediment surface. The mechanisms for the nodules' persistence at the surface without complete burial could be linked to their occasional motion or agitation. Here, we report evidence for the rotation of a nodule from the Penrhyn Basin, South Pacific detected by paleomagnetism. The paleomagnetic inclinations of specimens from the nodules' surface are consistent with the recent geomagnetic fields. The paleomagnetic directions from the surface to the core show successive changes and form a great circle with a pole at (azimuth = 53.9°, dip = 32.1°). This suggests that the nodule rotated along its pole while successively recording magnetizations. As the nodule was found on a gentle slope at the foot of an abyssal hill, it may have moved downslope due to bottom current underwashing. Rock magnetic analyses of the nodule suggest the presence of magnetite in single domain and vortex states. Low temperature magnetometry revealed that magnetite grains were heavily oxidized to maghemite, especially close to the core of the nodule. The rotation may have exposed the rising part of the nodule to oxidative pore water. Oxygenated Antarctic Bottom Water might have caused remagnetization due to low temperature oxidation of magnetite. The rotation would also facilitate the omnidirectional growth of the nodules' mixed layer of diagenetic buserite and hydrogenetic vernadite.

Classification of positive elliptic-elliptic rotopulsators on Clifford tori

We prove that positive elliptic-elliptic rotopulsator solutions of the n-body problem in spaces of constant Gaussian curvature that move on Clifford tori of nonconstant size either lie on great circles, or project onto regular polygons. We additionally prove for the case that the configurations project onto regular polygons that all masses are equal and show that all these different types of positive elliptic-elliptic rotopulsator exist.

Segregated solutions for a critical elliptic system with a small interspecies repulsive force

We consider the elliptic system−Δui=ui3+∑j=1j≠iq+1βijuiuj2inR4,i=1,…,q+1, when α:=βij and β:=βi(q+1)=β(q+1)j for any i,j=1,…,q. If β<0 and |β| is small enough we build solutions such that each component u1,…,uq blows-up at the vertices of q polygons placed in different great circles which are linked to each other, and the last component uq+1 looks like the radial positive solution of the single equation.

Gauss maps of harmonic and minimal great circle fibrations

We investigate Gauss maps associated to great circle fibrations of the euclidean unit 3-sphere mathbb {S}^3. We show that the associated Gauss map to such a fibration is harmonic, respectively minimal, if and only if the unit vector field generating the great circle foliation is harmonic, respectively minimal. These results can be viewed as analogues of the classical theorem of Ruh and Vilms about the harmonicity of the Gauss map of a minimal submanifold in the euclidean space. Moreover, we prove that a harmonic or minimal unit vector field on mathbb {S}^3, whose integral curves are great circles, is a Hopf vector field.

Contact Path Method of Generating Spherical Conjugated Curves as Bevel Gearing Tooth Profiles

Abstract This paper proposes a general method to generate spherical conjugated curves based on their contact path. The main objective of generating conjugated curves is to use them as gearing tooth profiles. The derivation is based on the prescribed spherical contact path, the equation of meshing, and the rigid body motion between two bodies. This paper demonstrates the method by assigning great circle and small circle contact paths in circular pitch pair transmission to generate conjugated curves. With the great circle contact path, the conjugated curves are exact spherical involute, and with the small circle contact path, the conjugated curves are spherical cycloid. The contact properties of the conjugated curves, such as induced curvature and the sliding ratio, are also provided with mathematical equations. Based on the results, the procedures and the geometries between the planar and spherical contact path methods have corresponding relationships. In contrast to the Camus theorem, the exact spherical involute could be generated from a similar procedure of planar involute cases. The ultimate goal of this research topic is to propose a general approach to generating conjugated profiles in planar, spherical, and spatial cases. The successful derivation of the spherical cases in this paper paves the way for spatial cases in the future.

Optimization of Waypoints on the Great Circle Route Based on Genetic Algorithm and Fuzzy Logic

Determining the appropriate number and position of waypoints on a great circle route (GCR) helps to shorten the sailing distance, reduce the number of course changes, and well-approximate the GCR through a small number of rhumb line (RL) legs. In this study, a genetic algorithm-based method (i.e., the GA method) is proposed to optimize the positions of waypoints on the GCR when the number of waypoints is given. Furthermore, a fuzzy logic-based evaluation method for the number of waypoints (i.e., the FL method) is proposed to judge whether to add a new waypoint or stop the process by using the non-fixed values while considering both the number of waypoints and the remaining benefit of the GCR. According to the example demonstration results, the two methods proposed in this study can well-determine the number and position of waypoints and provide effective support for ocean route planning.

Webgis Application For Searching Nu Mosque In East Java Using The Haversine Method

Indonesia is the country with the largest number of mosques in the world. In East Java, there are many Islamicorganizations, one of which is Nahdhotul Ulama (NU). Nahdhotul Ulama (NU) is one of the socio-religiousorganizations in Indonesia which was founded on 16 Rajab 1344 or to coincide with January 31, 1926 in Surabaya onthe initiative of two leading traditional scholars at that time, K.H Hasyim Asy'ari and K.H Abdul Wahab Hasbullah.Geographic Information System is a computer system that is used to collect, check, integrate, and analyze informationrelated to the earth's surface. Basically, the term geographic information system is a combination of three mainelements, namely system, information, and geography. Thus, an understanding of these three main elements will bevery helpful in understanding GIS. The haversine is an important equation in navigation, giving the great circle thedistance between any two points on the sphere of longitude and latitude. It is a special case of the more commonformula in spherical trigonometry, the law of haversines, relating the sides and angles of a spherical triangle (SphericalTrigonometry).From the calculation results of the haversine algorithm the first trial formula is at the user's location at 1. Jl. RayaSurabaya - Malang, Sawahan, Buduran, Buduran District, Sidoarjo Regency, East Java 61252 found the closest mosquebased on the heversine formula is Darussalam Mosque, jl. Kedemang Singomenggolo no 98 Buduran with a distanceof 1,396 Km. From the calculation results of the haversine algorithm, the first trial formula at the user's location on Jl.Raya Suko No.55, RT.01/RW.01, Ngemplak, Cemeng Kalang, Sidoarjo District, Sidoarjo Regency, East Java 61224found the closest mosque based on the heversine formula is Miftahul Jannah Mosque, jl. Banjar poh village BanjarBendo with a distance of 1,878 Km

Image-based approximation of derivatives of traditional differential metrics of angular distortion in map projections

ABSTRACT Map projections are imaging procedures used to depict geographic features. We adopt the traditional differential metric and exploit the intrinsic image properties of map projections to establish an image-based differential metric for evaluating distortions in map projections, obtaining an effective, practical, and relatively accurate metric. We use bivariate polynomial functions to approximate the forward and inverse formulae of map projections. Thereafter, the proposed metric is conveniently calculated using the partial derivatives of the approximate forward functions based on polynomial functions, while complicated differential calculations are avoided. Moreover, multiple sampling and image filters mitigate the influence of imaging noise and achieve a high computation precision. Experiments were conducted using the NASA G.Projector mapping software to generate images from more than 200 map projections. Explicit equations of map projections were not required owing to the use of the mapping software. These images were then evaluated using the proposed metric through an implementation in the Julia programming language. The corresponding results confirmed that the proposed metric avoided the drawbacks of the great circle arc metric and provided considerably low errors (1.12° on average) and high consistency (0.999 on average) with respect to the traditional differential metric. Although there were errors, experimental results indicated that feasibility and high usability were achieved by the image-based method for evaluating distortions in small-scale map projections.

Data-Driven Probabilistic Methodology for Aircraft Conflict Detection Under Wind Uncertainty

Assuming the availability of a reliable aircraft trajectory planner, this paper presents a probabilistic methodology to detect conflicts between aircraft in the cruise phase of flight in the presence of wind velocity forecasting uncertainty. This uncertainty is quantified by ensemble weather forecasts, the members of which are regarded as realizations of correlated random processes and used to derive the eastward and northward components of the wind velocity. First, the Karhunen-Loève expansion is used to obtain a series expansion of the components of the wind velocity in terms of a set of uncorrelated random variables and deterministic coefficients. Then, the uncertainty generated by these uncorrelated random variables in the outputs of the aircraft trajectory planner is quantified using the arbitrary polynomial chaos technique. Finally, the probability density function of the great circle distance between each pair of aircraft is derived from the polynomial expansions using a Gaussian kernel density estimator and used to estimate the probability of conflict. The arbitrary polynomial chaos technique allows the effects of uncertainty in complex nonlinear dynamical systems, such as those underlying aircraft trajectory planners, to be quantified with high computational efficiency, only requiring the existence of a finite number of statistical moments of the random variables of the Karhunen-Loève expansion while avoiding any assumptions on their probability distributions. To demonstrate the effectiveness of the proposed conflict detection method, numerical experiments are conducted via an optimal control-based aircraft trajectory planner for a given wind velocity forecast represented by an ensemble prediction system.

Impact of wearing a head-mounted display on localization accuracy of real sound sources

For augmented reality experiences, users wear head-mounted displays (HMD) while listening to real and virtual sound sources. This paper assesses the impact of wearing an HMD on localization accuracy of real sources. Eighteen blindfolded participants completed a localization task on 32 loudspeakers while wearing either no HMD, a bulky visor HMD, or a glass visor HMD. Results demonstrate that the HMDs had a significantly impact on participants’ localization performance, increasing local great circle angle error by 0.9°, and that the glass visor HMD demonstrably increased the rate of up–down confusions in the responses by 0.9–1.1%. These results suggest that wearing an HMD has a sufficiently small impact on real source localization that it can safely be considered as an HMD-free condition in most but the most demanding AR auditory localization studies.

Accurate TCP Position and Orientation Trajectory Generation in 6DOF Robotic Manipulators and CNC Machine Tools using FIR Filtering and Haversine Synchronisation

The use of robotics in manufacturing is rapidly growing. One promising application in particular is the use of robots as machining platforms. Robotic machining offers many advantages such as large operating envelopes, flexibility in operations and low installation and running costs. Robotic machining platforms much like computer numerically controlled (CNC) machine tools are commanded by numerical controllers (NCs) using the well established G-code part programming language (ISO-6983). Toolpaths are defined in the work piece coordinate system (WCS) as a series of discrete tool centre point (TCP) positions and tool orientations. The role of the NC is to generate smooth feed drive or robot joint commands such that the tool or end effector travels along the commanded toolpath whilst satisfying TCP position and orientation tolerances and robot/machine tool kinematic constraints. Interpolating tool orientations directly in the WCS requires complex real-time spherical interpolation computations. Overcoming this challenge, this paper introduces a C4 continuous tool path trajectory generation method for 6DOF motion. The tool centre point position is smoothed in the WCS, however, the tool orientation is smoothed directly in the rotary coordinate system without the requirement for spherical interpolation. The interpolated tool orientation trajectory follows the path of the great circle arc using the modified Haversine method eliminating non-linear interpolation errors present in linear interpolation of rotary axis positions (machine coordinate system interpolation). The proposed method is validated by simulation of a 6DOF robotic manipulator.

MORGen: an Algorithm to Compute Spreading Centre and Transform Geometries from Simple Initial Plate Boundaries and Euler Rotations

The age structure of the global ocean floor is a key feature in paleogeographic reconstructions, which in turn forms the quantitative basis for Earth System Science. However, much of the ocean floor in paleogeographic reconstructions has been lost to subduction. The age structure of such lost ocean floor is constructed from the reconstructions of adjacent continents, using the relative rotations, around Euler poles to predict the geometry of spreading centres and transform faults. Building such mid-ocean ridge features in paleogeographic reconstructions is laborious, as it requires redrawing of ridge-transform systems upon every Euler pole shift in the model. In this paper, we present the Mid-Ocean Ridge Generator (MORGen) algorithm, based on pyGPlates. MORGen reduces the laborious task by automating the drawing of mid-ocean ridge geometries from geometrically simple plate boundary input assuming ridge-perpendicular spreading and adjusts ridge geometries in a simplest-scenario fashion by gradually adjusting ridge orientation and transform fault length upon Euler pole shifts, inspired by observations from the modern sea floor. The code takes as input curved line features, representing approximate divergent plate boundaries, and a set of Euler poles. These are then converted into spreading centre-transform geometries. Upon Euler pole shifts, the geometries are adjusted to fit the set of small circles and great circles dictated by the new Euler pole. For studies of paleo-environment and paleo-oceanography MORGen can be used in combination with other algorithms for full reconstructions of ocean floors, including their age, bathymetry, and roughness. For in-situ preserved ocean floor, the paleo-age distribution can be reconstructed directly in high resolution from geophysical and geological data from the modern ocean floor and MORGen would not normally be the option of choice. In cases where models contain ocean floor that has now been subducted, MORGen straightforwardly facilitates mid-ocean ridge geometry reconstruction. To illustrate how well the MORGen algorithm reproduces real ocean floor age structure, we show a synthetic ridge evolution for the South Atlantic and Southern Oceans and compare this to geophysically constrained ocean floor geometry. In addition, we show examples of use cases where direct (re)construction of mid ocean ridges is not possible: now-subducted ocean basins in the Mediterranean region and an ocean in a future supercontinent scenario.

Not wholly belonging: British planning’s uncertain European connections

ABSTRACT This article takes a long view of British planning’s connections with continental Europe, locating Brexit within historic uncertainties about the country’s international outlook, interests and position. In 1948, Churchill portrayed Britain at the intersection of three ‘great circles’: the British Empire, the wider English-speaking world (principally the USA) and Europe. This notion is drawn on to show how the strong earlier European links of British planning were seriously disrupted or severed by twentieth-century wars. These drew both country and planning approach closer to its ‘distant friends’ within the other ‘great circles’. As former imperial ties faded and the USA relationship became less special, Britain looked again to Europe but without shedding these habitual links. Even after Britain joined the European Communities in 1973, its strongest international planning connections remained with the USA and its former Empire and Dominions. In the 1990s, the EU promoted spatial planning but Britain remained largely aloof until the ‘New Labour’ governments of 1997–2010. Yet growing Euroscepticism saw this relative enthusiasm fade, with Brexit reviving uncertainties, now about whether EU approaches should be jettisoned and a more deregulated planning system created. The article predicts (or at least hopes) that current anti-Europe thinking will itself fade.

Atmospheric and ionospheric waves induced by the Hunga eruption on 15 January 2022; Doppler sounding and infrasound

SUMMARY The massive explosive eruption of the Hunga volcano on 15 January 2022 generated atmospheric waves that were recorded around the globe and affected the ionosphere. The paper focuses on observations of atmospheric waves in the troposphere and ionosphere in Europe, however, a comparison with observations in East Asia, South Africa and South America is also provided. Unlike most recent studies of waves in the ionosphere based on the detection of changes in the total electron content, this study builds on detection of ionospheric motions at specific altitudes using continuous Doppler sounding. In addition, much attention is paid to long-period infrasound (periods longer than ∼50 s), which in Europe is observed simultaneously in the troposphere and ionosphere about an hour after the arrival of the first horizontally propagating pressure pulse (Lamb wave). It is shown that the long-period infrasound propagated approximately along the shorter great circle path, similar to the previously detected pressure pulse in the troposphere. It is suggested that the infrasound propagated in the ionosphere probably due to imperfect refraction in the lower thermosphere. The observation of infrasound in the ionosphere at such large distances from the source (over 16 000 km) is rare and differs from ionospheric infrasound detected at large distances from the epicenters of strong earthquakes, because in the latter case the infrasound is generated locally by seismic waves. An unusually large traveling ionospheric disturbance (TID) observed in Europe and associated with the pressure pulse from the Hunga eruption is also discussed. Doppler sounders in East Asia, South Africa and South America did not record such a significant TID. However, TIDs were observed in East Asia around times when Lamb waves passed the magnetically conjugate points. A probable observation of wave in the mesopause region in Europe approximately 25 min after the arrival of pressure pulse in the troposphere using a 23.4 kHz signal from a transmitter 557 km away and a coincident pulse in electric field data are also discussed.