Standard Horseshoe Research Articles

Low-mass planets falling into gaps with cyclonic vortices

ABSTRACT We investigate the planetary migration of low-mass planets ($M_p\in [1,15]\, \mathrm{ M}_{\oplus }$, here $\mathrm{ M}_{\oplus }$ is the Earth mass) in a gaseous disc containing a previously formed gap. We perform high-resolution 3D simulations with the fargo3d code. To create the gap in the surface density of the disc, we use a radial viscosity profile with a bump, which is maintained during the entire simulation time. We find that when the gap is sufficiently deep, the spiral waves excited by the planet trigger the Rossby wave instability, forming cyclonic (underdense) vortices at the edges of the gap. When the planet approaches the gap, it interacts with the vortices, which produce a complex flow structure around the planet. Remarkably, we find a widening of the horseshoe region of the planet produced by the vortex at the outer edge of the gap, which depending on the mass of the planet differs by at least a factor of two with respect to the standard horseshoe width. This inevitably leads to an increase in the co-rotation torque on the planet and produces an efficient trap to halt its inward migration. In some cases, the planet becomes locked in co-rotation with the outer vortex. Under this scenario, our results could explain why low-mass planets do not fall towards the central star within the lifetime of the protoplanetary disc. Lastly, the development of these vortices produces an asymmetric temporal evolution of the gap, which could explain the structures observed in some protoplanetary discs.

Functional Horseshoe Priors for Subspace Shrinkage

We introduce a new shrinkage prior on function spaces, called the functional horseshoe (fHS) prior, that encourages shrinkage toward parametric classes of functions. Unlike other shrinkage priors for parametric models, the fHS shrinkage acts on the shape of the function rather than inducing sparsity on model parameters. We study the efficacy of the proposed approach by showing an adaptive posterior concentration property on the function. We also demonstrate consistency of the model selection procedure that thresholds the shrinkage parameter of the fHS prior. We apply the fHS prior to nonparametric additive models and compare its performance with procedures based on the standard horseshoe prior and several penalized likelihood approaches. We find that the new procedure achieves smaller estimation error and more accurate model selection than other procedures in several simulated and real examples. Supplementary materials for this article, which contain additional simulated and real data examples, MCMC diagnostics, and proofs of the theoretical results, are available online.

Modifying the Height of Horseshoes: Effects of Wedge Shoes, Studs, and Rocker Shoes on the Phalangeal Alignment, Pressure Distribution, and Hoof-Ground Contact During Motion

This study was designed as a comparative study with the intention to accumulate fundamental data on a wide variety of farriery methods. Twenty-five warmblood/crossbred horses, allocated into five groups (n = 5), underwent radiological and kinetic examination of the barefoot hoof, the shod hoof with a standard horseshoe, and finally a modified horseshoe. For radiographic measurements, a special podoblock with embedded reference points and changeable ground surfaces was used. Kinetic examinations were performed by placing one sensor between the shoe and the hoof and the second sensor between the shoe and the ground and then walking the horses on four different ground surfaces. When wedges were applied, the palmar angle increased by approximately 5° on all surfaces. However, this effect was only seen on firm surfaces combined with studs. Using a rocker shoe, the palmar angle increased on a firm surface (0.6 ± 0.3°) and decreased on a deformable surface (0.8 ± 0.3°). No consistent data were noted for the orientation of the proximal and middle phalanx in relation to the palmar angle. Pressure distribution showed wedges and studs to cause an increased pressure load on both the toe and the heels on a firm surface. Rocker shoes led to pressure peaks at the inner section of the toe, and high pressure was exerted on the quarters. In conclusion, all modified horseshoes showed unintended side effects and their influence on biomechanical parameters varied depending on the ground surface.

Modifying Horseshoes in the Mediolateral Plane: Effects of Side Wedge, Wide Branch, and Unilateral Roller Shoes on the Phalangeal Alignment, Pressure Forces, and the Footing Pattern

The underlying study to this article examined the effects of a broad variety of modified horseshoes on the phalangeal alignment, the pressure force distribution, and footing pattern on different grounds to establish a comparative database for future scientific and practical work. Twenty-five warmblood horses, allocated to five groups (n = 5), underwent radiographic and kinetic examinations for the barefoot hoof, a standard horseshoe, and a corresponding modified horseshoe in each group. A novel modified podoblock was used, and the kinetic examinations were carried out using two sensor foils simultaneously, while walking on four different types of ground. On firm ground, only side wedges caused significant changes in the orientation of the distal phalanx, whereas on penetrable ground, also, the wide branch shoe showed a tendential skewer orientation of the distal phalanx. The effects of a unilateral roller shoe did not differ significantly compared with a standard horseshoe. The orientation of the middle and proximal phalanx as well as the distal interphalangeal joint space symmetry showed no constant data. Moreover, pressure forces increased considerably beneath the supported regions of a wide branch shoe and side wedges. Side wedges showed an inconstant influence on the footing pattern. In conclusion, all examined modified horseshoes showed unwanted side effects which should be taken into account when planning an orthopedical treatment. Furthermore, different ground conditions can influence biomechanical parameters of horseshoes significantly.

Infinite horseshoes and complex dynamics in physical systems

Horseshoe maps have played an important role in the study of nonlinear dynamical systems. Here we study maps associated to two simple physical systems: a four-hill potential and an open billiard. They turn out to be very different from the standard horseshoe maps: one iteration stretches and folds a square in phase space an infinite number of times before placing it across itself. In this paper we explore in further depth these infinite horseshoe maps. We show that the infinite folding action requires that the maps are not defined in part of the rectangle. Our exploration also shows that infinite horseshoes provide valuable information on the complexity of the system. In particular, they provide a visual way to code the wide variety of complex orbits existing in the dynamical systems considered.

No-choke flow in USBR standard horseshoe channel cross-section

Choking problems occur in transitions that have a bed rise higher than a maximum allowable value. Due to the complexity of the United States Bureau of Reclamation (USBR) horseshoe cross-section, a time-consuming trial-and-error procedure is inevitable as a way to determine the maximum allowable bed rise. This paper presents straightforward solutions to calculate a choke-free maximum bed rise for practical changes in bed elevations. The maximum relative errors associated with the proposed equations are less than 1%. A graphical solution is presented to obtain the upstream operating ranges based on non-dimensional specific energy curves. In addition, a direct relationship is developed to determine the critical depth in the transition zone which bears less than 1% relative error. Finally, a choking solution is discussed by considering the energy head loss occurred in the channel transition zone.

Iterative Formulas and Estimation Formulas for Computing Normal Depth of Horseshoe Cross-Section Tunnel

This note presents iterative formulas that can be used to calculate the normal depth for all types of horseshoe cross sections. Based on the principle of gradual optimization fitting, general estimation formulas were developed for direct computation of the normal depth for all types of the standard horseshoe cross sections. The estimation formulas have high accuracy for hydraulic designers to use in comparison with the accurate normal depths determined from iterative formulas, and the maximum relative error is less than 0.5%.

Technical Note: Formulas for Computing Geometry and Critical Depth of General Horseshoe Tunnels

This technical note presents general equations for the geometric elements of all types of horseshoe cross-sections used in water conveyance systems such as for irrigation, drainage, and water supply projects. We derived iterative formulas for calculating critical depth of general horseshoe cross-sections, and these formulas can be used for computer program development to find accurate solutions of critical depth. Based on the principle of gradual optimization fitting, general estimation formulas were also developed for direct computation of critical depth for all types of standard horseshoe cross-sections. The estimation formulas have high accuracy for hydraulic designers to use in comparison to accurate critical depth determined from the iterative formulas, and the maximum relative error is less than 1%.

ON THE HORSESHOE DRAG OF A LOW-MASS PLANET. I. MIGRATION IN ISOTHERMAL DISKS

We investigate the unsaturated horseshoe drag exerted on a low-mass planet by an isothermal gaseous disk. In the globally isothermal case, we use a formal- ism, based on the use of a Bernoulli invariant, that takes into account pressure effects, and that extends the torque estimate to a region wider than the horse- shoe region. We find a result that is strictly identical to the standard horseshoe drag. This shows that the horseshoe drag accounts for the torque of the whole corotation region, and not only of the horseshoe region, thereby deserving to be called corotation torque. We find that evanescent waves launched downstream of the horseshoe U-turns by the perturbations of vortensity exert a feed-back on the upstream region, that render the horseshoe region asymmetric. This asymmetry scales with the vortensity gradient and with the disk's aspect ratio. It does not depend on the planetary mass, and it does not have any impact on the horseshoe drag. Since the horseshoe drag has a steep dependence on the width of the horseshoe region, we provide an adequate definition of the width that needs to be used in horseshoe drag estimates. We then consider the case of locally isothermal disks, in which the tempera- ture is constant in time but depends on the distance to the star. The horseshoe drag appears to be different from the case of a globally isothermal disk. The difference, which is due to the driving of vortensity in the vicinity of the planet, is intimately linked to the topology of the flow. We provide a descriptive inter- pretation of these effects, as well as a crude estimate of the dependency of the excess on the temperature gradient.

OPTIMIZATION AND THE MIRANDA APPROACH IN DETECTING HORSESHOE-TYPE CHAOS BY COMPUTER

We report on experiences with an adaptive subdivision method supported by interval arithmetic that enables us to prove subset relations of the form [Formula: see text], and thus check certain sufficient conditions for chaotic behavior of dynamical systems in a rigorous way. Our proof of the underlying abstract theorem avoids referring to any results of applied algebraic topology and relies only on the Brouwer fixed point theorem. The second novelty is that the process of gaining the subset relations to be checked is, to a large extent, also automatized. The promising subset relations come from solving a constrained optimization problem via the penalty function approach. Abstract results and computational methods are demonstrated by finding embedded copies of the standard horseshoe dynamics in iterates of the classical Hénon mapping.

Standard horseshoe cross section geometry

The standard horseshoe cross section is found in many tunnels for open-channel and pressurized conduit water conveyance. The exact geometric relationships for this cross section are needed for hydraulic analysis and mathematical modeling purposes because look-up tables of numerically approximated values are typically as slow, or slower, when implemented in computer programs. Implicit solutions of the unsteady hydraulic equations require exact, analytical partial derivatives of some of the relationships, both for accuracy and numerical stability. The equations for cross-sectional area, wetted perimeter, top width, and depth to area centroid are presented as functions of the depth of water in standard horseshoe sections, in addition to the partial derivatives of wetted perimeter with respect to water depth, and depth to area centroid with respect to water depth. Sample unsteady-flow simulation results for a horseshoe cross section are compared to those of a comparably sized circular cross section.

BASIN EROSION IN THE TWIN-WELL DUFFING OSCILLATOR: TWO DISTINCT BIFURCATION SCENARIOS

A study is made of the control-phase portraits encountered in the twin-well Duffing oscillator, concentrating on the loss of stability of attractors confined to a single well of the potential. This loss of stability may be understood by examining the bifurcations which precede a single-well attractor touching its basin boundary. Two distinct bifurcation scenarios are described in which the basin boundary develops a fractal structure. This fractal structure accompanies the development of a homoclinic tangency between the inset and outset of the saddle whose inset determines the separatrix action. In the first scenario described, this fractal structure, or chaotic saddle, grows due to a sequence of fractal–fractal basin implosions which are caused by the completion of Smale cycles or heteroclinic chains; the subharmonics involved occur in decreasing order, after the main homoclinic tangency. The second scenario focuses on the bifurcational events which necessarily prepare the creation of a chaotic saddle; the subharmonics involved appear in increasing order, before the main homoclinic event. Both basin erosion scenarios are consistent with a geometric model of the twin-well Duffing oscillator based on a three layer template; in the first scenario, the full three-layer template participates; while in the second scenario, only two layers, a standard horseshoe, play a role.

Structure of laminar juncture flows

A computational study of both steady and periodic laminar horseshoe vortex flows generated upstream of a cylinder/flat plate juncture is presented. The flowfields are simulated using the full three-dimensional unsteady Navier-Stokes equations and a time-accurate implicit algorithm. A new type of laminar horseshoe vortex topology is identified. For the case of a single primary vortex, this new topology is found to be independent of the computational grid and is also supported by recent experimental flow visualizations. The flat plate skin-friction portraits corresponding to the new and to the standard horseshoe vortex topologies are equivalent, pointing out the nonunique relation between the wall limiting streamline pattern and the three-dimensional flow above the plate. For the new topology, the foremost line of coalescense is an attachment rather than a separation line. This unusual feature illustrates the fact that convergence of skin-friction lines is a necessary but not sufficient condition for separation. As the Reynolds number increases, the flow topology evolves from a single to multiple primary horseshoe vortices, in agreement with experimental observations. At least two different types of triple horseshoe vortex systems are shown to be possible. Above a certain value of the Reynolds number, the juncture flow becomes unsteady and periodic at a frequency that increases with Reynolds number. The unsteady horseshoe vortex process upstream of the cylinder is found in qualitative agreement with experiment. Horseshoe vortices are periodically generated and convected toward the juncture. Vorticity intensification by vortex stretching, and the eruption of vorticity from the plate surface are observed.

Water Surface Profile for Horseshoe Tunnel

A method of calculating water surface profile for USBR standard horseshoe tunnel is developed. The method uses direct-integration process on energy principle. Equations describing geometry of this particular cross section are listed. Numerical values of hydraulic exponents are presented in both table and curves. Application examples show the easiness of profile calculation and comparisons with step-by-step procedures.

Hydraulic Elements for USBR Standard Horseshoe Tunnel

The hydraulic elements for the USBR standard horseshoe tunnel are identified and computed. These elements are described in relation to the varying value of the central flow depth.

Transition from Circular to Rectangular Section

Two methods are widely used for the design of transitions from circular to rectangular box sections. One method is based on a straight line variation of area between the circular section and rectangular box section with the provision of circular corners. This method results in a surface profile that is difficult to form, varying from a parabolic curve to a straight line; if approximations are made in the surface profile, an uneven surface results. The second method is based on the provision of oblique circular quarter cones at the corners, with radius of the circular corners varying in a straight line from zero at the rectangular box section to radius R at the circular section. With this method, a straight line variation in area is not achieved; a very serious hydraulics defect is. The method described in this paper is a modification of previous methods to achieve an almost straight line variation in area through the transition, with a minimum cost in form work. This method is based on the travel of a point on a circular section through the length of transition to its corresponding point on a rectangular box section. The circular section gradually takes the shape of the rectangular box section within the transition, with no concentration of flow at any section of the transition, due to geometric change of shape. The proposed method is also applicable to any other type of transition, e.g., from circular to standard horseshoe shape.