Vertical Line Source Research Articles

Vertical line time domain green function and its applications in numerical simulation of ship seakeeping performance

In present study, a vertical line source time domain Green function is proposed and the subsequent numerical evaluation methods are described in detail. For which, Taylor expansion and Chebyshev approximation algorithms are originally extended from point source to line source for small parameters, as well as asymptotic expansions are extended for large parameters. Meanwhile, polynomials are originally proposed to replace the asymptotic expansions to improve the computational efficiency. The accuracy and stability of the proposed method are validated through several cases and good agreement can be obtained.Furthermore, a hybrid Green function method based on the vertical line source Green function and Rankine source is developed for time domain simulation of seakeeping performances of ships. With respect to the hybrid method, the flow domain is divided into inner part and outer part by an artificial control surface. The first-order Taylor Expansion Boundary Element Method (TEBEM) based on the Rankine panel distribution is applied in the inner part, while three-dimensional panel method based on the vertical line source Green function distribution is adopted in the outer part. Under this circumstance, motions of the single-ship (Slender Wigley and Blunt Wigley) and two-ship (two side-by-side ships) in head waves are numerically investigated. The numerical results agree well with the experimental data, which proved the feasibility and the accuracy of present hybrid Green function method for motion prediction of both single-ship and two-ship.

Study on vertical line source Green's function for hydrodynamic calculations of ocean structures in water with ice cover

The numerical calculation of panel integrals associated with the Green's function satisfying water surface condition with ice cover is the basis for the boundary element method (BEM) to solve hydrodynamic problems of ocean structures in ice regions. In order to simplify the calculation process and reduce the time consuming, the vertical line source representing the analytical integral of ice-covered Green's function over vertical line segment is firstly derived in this paper. Then a semi-analytical quadrature, which accumulates a series of vertical line sources distributing along the horizontal direction, can be derived for the panel integral. In the numerical implementation, the asymptotic analysis method is employed to eliminate the singularity at zero point of the denominator, variable substitution is performed to treat the integral at infinite and approximate polynomials are introduced to evaluate the Bessel functions. Through the calculations of vertical line and panel integrals of Green's function and its derivatives, the analytical expression of vertical line source is proved to be accurate and more efficient than the normal numerical quadrature. Based on this, a BEM program is developed and applied in the hydrodynamic calculations of submerged structures with ice cover. The computed results are found to be in excellent agreement those of the validated BEM based on point source, which further verifies the reliability and practicability of utilization of vertical line source.

Study of vertical line source dispersion in a turbulent channel flow

Turbulent dispersion of passive concentration released from a vertical line source in a fully-developed channel flow is studied using direct numerical simulation (DNS). The investigation into the turbulent transport processes of the momentum and concentration is conducted in both physical and spectral spaces, based on a systematic analysis of the statistical moments of the velocity and concentration fields, probability density functions (PDFs) of the concentration fluctuations, and the characteristic wavelengths of energetic eddies that dominate turbulent mixing processes. It is observed that the plume development experiences turbulent convective- and diffusive-stages through a comparative study of the spatial and temporal scales of the velocity and concentration fields. It is also observed that the meandering ratio is small near the ground and within the turbulent diffusive stage, and PDFs of concentration fluctuations are sensitive to both the wall-normal distance and the downstream distance from the concentrated line source.

Numerical investigation of wetting front migration and soil water distribution under vertical line source irrigation with different influencing factors

AbstractVertical line source irrigation (VLSI) is an underground irrigation method suitable for deep-rooted plants. Understanding the characteristics of the soil wetting body of the VLSI was the key to designing this irrigation system. On the basis of experimental verification of the reliability of the HYDRUS simulation results of VLSI under the conditions of soil texture (ST), initial water content (θi), line source buried depth (B), line source diameter (D) and line source length (L), numerical studies of the migration law of the wetting front of VLSI and the distribution characteristics of soil moisture were performed. The wetting front migration (WFM) was mainly influenced by ST, θi, D and L (P < 0.05), while B had little effect on WFM (P > 0.05). The shape of the soil wetting body changed little under different influencing factors. The water content contour was approximately ‘ellipsoidal’ around the line source. The soil moisture near the line source was close to the saturated moisture content. The moisture content around the line source gradually decreased outward, and the contour lines gradually became dense. According to the simulation results, a prediction model of multiple factors influencing the migration process of the VLSI wetting front was established. The predicted value was in good agreement with the measured value. The results of this research could provide a theoretical basis for further optimizing the combination of VLSI and irrigation elements.

A simple model for predicting soil infiltration rate for vertical line source irrigation

In this study, HYDRUS-2D was used to simulate the soil infiltration rate of vertical line source irrigation (VLSI) under different soil textures and initial water contents, as well as for different lengths, diameters and buried depths of the line source. The soil texture, diameter and length of the line source are the dominant influential factors of the soil infiltration rate of VLSI, whereas the initial water content and buried depth of the line source are less influential. The variation of soil infiltration rate is in accordance with the Philip model. The model parameters, i.e., sorptivity (S) and steady infiltration rate (A), increase as the diameter (D) and length (L) of line source increases. A power function between S and D and L, and one between A and D and L, were established. The exponential value of the power function was determined based on the HYDRUS simulation results, thereby obtaining a simplified estimation model of the soil infiltration rate of VLSI containing D and L. The effectiveness of the simplified model was verified based on a laboratory experiment and currently available research. The MAE (mean absolute error, MAE) is 0.056–0.348, RMSE (Root mean square error, RMSE) is 0.073–0.442, PBIAS (percentage bias, PBIAS) is 1.642 – 2.893, and NSE (Nash-Sutcliffe efficiency coefficient, NSE) is close to 1 (NSE⩾0.973), indicating that the model has a high accuracy and can therefore be used for estimating the soil infiltration rate of VLSI. The simplified estimation model established in this study has only two undetermined parameters that can be obtained based on only one set of experimental data, simplifying the experimental scheme and improving the work efficiency.

Excitation of the Zenneck Wave by a Tapered Line Source Above the Earth or Ocean

A tapered vertical line source over the surface of the earth or ocean is explored as a means of communication. Two types of tapers for the line source are explored. One is a “Zenneck-wave” taper and the other is an “optimum” taper. The Zenneck-wave taper is chosen to excite the Zenneck wave to the extent possible. The Zenneck wave is a “nonphysical wave” that propagates on a semiinfinite lossy earth. The wave is nonphysical in the sense that a simple source such as a dipole will not excite a field that resembles the Zenneck wave at large distances from the source. However, by using a tapered line source, a field that resembles the Zenneck wave can be produced along the surface of the earth or ocean out to some distance. The optimum taper is chosen to optimize the field at a given horizontal distance from the line source on the surface of the earth or ocean. The optimum line source can be used to produce a field along the surface of the earth or ocean that is nearly constant over a large range of distances, something that does not seem easily attainable with other types of sources.

Similarity solution for one-dimensional heat equation in spherical coordinates and its applications

A similarity type of general solution for the one-dimensional heat equation in spherical coordinates is developed, and the solution is expressed by the Kummer functions. After introducing some useful properties of the general solution, the solution is then applied to establish analytical solutions of finite line source problems and moving boundary problems. For the finite line source problems, a continuous point source problem with a power-type source intensity is solved first, and then analytical solutions for a vertical finite line source problem and an inclined finite line source problem are established using the mirror image method and the solution of the continuous point source problem. For the moving boundary problems, a one-phase Stefan problem with power-type latent heat in spherical coordinates and a nonlinear heat conduction problem induced by a continuous point source are investigated, and analytical solutions are developed using the general solution. Computational examples are presented. For the finite line source problems, computational error caused by neglecting vertical heat transfer under the situation of a time-varying source intensity is investigated. For the moving boundary problems, an illustrative example of the one-phase Stefan problem is presented, and the computational results can be used to calibrate numerical solutions of Stefan problems.

A Simplified Infiltration Model for Predicting Cumulative Infiltration during Vertical Line Source Irrigation

Vertical line source irrigation is a water-saving irrigation method for enhancing direct water and nutrient delivery to the root zone, reducing soil evaporation and improving water and nutrient use efficiency. To identify its influencing factors, we performed computer simulations using the HYDRUS-2D software. The results indicate that for a given soil, the line source seepage area, but not the initial soil water content and buried depth, has a significant effect on the cumulative infiltration. We thus proposed a simplified method, taking into account the seepage area for predicting the cumulative infiltration based on the Philip model. Finally, we evaluated the accuracy of the simplified method using experimental data and found the cumulative infiltrations predicted by the simplified method were in very good agreement with the observed values, showing a low mean average error of 0.028–0.480 L, a root mean square error of 0.043–0.908 L, a percentage bias of 0.321–0.900 and a large Nash-Sutcliffe coefficient close to 1.0 (NSE ≥ 0.995). The results indicate that this simplified infiltration model, for which the only emitter parameter required is the seepage area, could provide a valuable and practical tool for irrigation design.

On the underwater sound field from impact pile driving: Arrival structure, precursor arrivals, and energy streamlines.

Underwater noise from impact pile driving is studied through measurements using a vertical line array (VLA) placed at range 120 m from the pile source (water depth 7.5 m) over which bathymetry varied gradually increasing to depth 12.5 m at the VLA. The data were modeled assuming the pile impact produces a radial expansion that acts as sound source and propagates along the pile at supersonic speed. This leads to the conceptualization of the pile as a discrete, vertical line source for which frequency- and source-depth-dependent complex phasing is applied. Dominant features of the pressure time series versus measurement depth are reproduced in modeled counterparts that are linearly related. These observations include precursor arrivals for which arrival timing depends on hydrophone depth and influence of a sediment sound speed gradient on precursor amplitude. Spatial gradients of model results are taken to obtain estimates of acoustic particle velocity and vector intensity for which active intensity is studied in the time domain. Evaluation of energy streamlines based on time-integrated active intensity, and energy path lines based on instantaneous (or very-short-time integrated) active intensity reveal interesting structure in the acoustic field, including an inference as to the source depth of the precursor.

Calculation of displacements in a half-space from a linear array based on analytical solutions of transient wave

In this paper, we propose a theoretical calculation method for the displacements of transient wave excited by a linear array coupled to an elastic solid with a longitudinal wave couplant. Firstly, the vertical and horizontal displacements excited by a vertical line source are solved based on exact and analytical solutions of transient elastic waves induced by a line source normally acting on an elastic half-space. Secondly, in this case, the displacements from a linear array can be determined by the superposition of the solutions of a number of discrete line forces mentioned above. All the theoretical results are compared to the finite element simulation results. The theoretical results are consistent with finite element simulation data. Therefore, the calculation method can save a lot of computing time. This paper provides a fast and accurate calculation method of transient wave displacements in a half-space generated by a linear array. The calculation method can be used to explore the sound field characteristics of a linear array transducer in the future.

Low-temperature high magnetic field powder x-ray diffraction setup for field-induced structural phase transition studies from 2 to 300 K and at 0 to 8-T field.

A low-temperature and high magnetic field powder x-ray diffractometer (XRD) has been developed at UGC-DAE CSR (UGC: University Grant Commission, DAE: Department of Atomic Energy, and CSR: Consortium for scientific research), Indore, India. The setup has been developed around an 18 kW rotating anode x-ray source delivering Cu-Kα x-rays coming from a vertical line source. It works in a symmetric θ-2θ parallel beam geometry. It consists of a liquid helium cryostat with an 8 T split-pair Nb-Ti superconducting magnet comprising two x-ray windows each covering an angular range of 65°. This is mounted on a non-magnetic type heavy duty goniometer equipped with all necessary motions along with data collection accessories. The incident x-ray beam has been made parallel using a parabolic multilayer mirror. The scattered x-ray is detected using a NaI detector through a 0.1° acceptance solar collimator. To control the motions of the goniometer, a computer programme has been developed. The wide-angle scattering data can be collected in a range of 2°-115° of 2θ with a resolution of ∼0.1°. The whole setup is tightly shielded for the scattered x-rays using a lead hutch. The functioning of the goniometer and the artifacts arising possibly due to the effect of stray magnetic field on the goniometer motions, on the x-ray source, and on the detector have been characterized by collecting powder XRD data of a National Institute of Standards and Technology certified standard reference material LaB6 (SRM-660b) and Si powder in zero-field and in-field conditions. Occurrence of field induced structural-phase transitions has been demonstrated on various samples like Pr0.5Sr0.5MnO3, Nd0.49Sr0.51MnO3-δ and La0.175Pr0.45Ca0.375MnO3 by collecting data in zero field cool and field cool conditions.

Initializing HYSPLIT with satellite observations of volcanic ash: A case study of the 2008 Kasatochi eruption

AbstractThe current work focuses on improving volcanic ash forecasts by integrating satellite observations of ash into the Lagrangian transport and dispersion model, HYSPLIT. The accuracy of HYSPLIT output is dependent on the accuracy of the initialization: the initial position, size distribution, and amount of ash as a function of time. Satellite observations from passive infrared, IR, sensors are used both to construct the initialization term and for verification. Space‐based lidar observations are used for further verification. We compare model output produced using different initializations for the 2008 eruption of Kasatochi in the Aleutian Islands. Simple source terms, such as a uniform vertical line or cylindrical source above the vent, are compared to initializations derived from satellite measurements of position, mass loading, effective radius, and height of the downwind ash cloud. Using satellite measurements of column mass loading of ash to constrain the source term produces better long‐term predictions than using an empirical equation relating mass eruption rate and plume height above the vent. Even though some quantities, such as the cloud thickness, must be estimated, initializations which release particles at the position of the observed ash cloud produce model output which is comparable to or better than the model output produced with source terms located above and around the vent. Space‐based lidar data, passive IR retrievals of ash cloud top height, and model output agree well with each other, and all suggest that the Kasatochi ash cloud evolved into a complex three‐dimensional structure.

Numerical modeling of borehole-surface electromagnetic responses with 3-D finite difference method and comparison with physical simulations

Borehole-surface electromagnetic method (BSEM) is a high-accuracy electromagnetic prospecting method that uses AC-powered vertical finite line source as the excitation source. The observed surface electromagnetic field is inverted for the subsurface conductivity structure. In this paper, the total field is separated into a primary part, due to a horizontally layered host medium, and a secondary part due to 3-D heterogeneities. After solving for the primary field with an analytical method, the frequency-domain second-order differential equation for the secondary field is discretized by finite differences. A sparse matrix storage scheme is employed and a BiCGSSTAB(m) method with a diagonal matrix preconditioner is used to obtain the secondary field as well as the 3-D BSEM response of the model. We compare the result of anomalous responses of three-layered medium derived by 3-D forward modeling with the result of semi-analytical solution. We also perform physical simulation and 3-D numerical forward modeling based on similarity criterion. As a result, the shape of both anomalous response curves are the same, which validates the 3-D numerical simulation method. The anomalous fields of 3-D numerical forward and physical simulation share similar anomalous feature of symmetric bimodal structure that is consistent with its harmonic response curve. It demonstrates that borehole-surface electromagnetic method can be used not only for prospection by employing a multi-frequency response, but also can provide multi-angle information about subsurface anomaly by varying relative depths of vertical finite line source in the borehole.

Design and Implementation of a Coal-Bed Methane Fracture Monitoring System based on Virtual Instrument Technology

Monitoring of fracture orientation is of significance in the exploration and development of coal-bed methane; for this purpose, a monitoring system, based on the principle of differential electric resistivity tomography, has been widely used. The resolution and monitoring scope determine the detection depth of the coal-bed methane fracturing layer. In this article, a coal-bed methane fracture monitoring system with a resolution of less than approximately 10 µV and 90 channels is reported using virtual instrument technology. Based on the three-dimensional geoelectric field of vertical finite line source by the finite-difference method, the surface potential difference between different depths and the scope of surface fracture anomaly was calculated, which demonstrated the feasibility of this system. The system exhibited satisfactory performance for a coal-bed methane well in Jixian, Shanxi Province. Compared with a microseismic monitoring system, the coal-bed methane fracture system was more convenient and achieved similar results of fracture orientations at a depth of 1263 m. Moreover, it avoided the uncertainty of a single method. Therefore, it is expected that this system may be widely applied to the fracture monitoring of coal-bed methane.

Detrainment from a turbulent plume produced by a vertical line source of buoyancy in a confined, ventilated space

AbstractNew experiments are presented which explore the dynamics of a turbulent buoyant plume produced by a vertically distributed linear source of buoyancy of strength $f$ per unit height. In a uniform environment, the plume volume flux increases with height from the base of the source, $z$, as $q(z) = {2^{-1/3}} {\pi }^{2/3} \alpha ^{4/3} f^{1/3} z^2$ where the entrainment coefficient, $\alpha = 0.09\pm 0.01$. In an enclosed space, with a net upward vertical ventilation flow $Q_V$, the buoyant plume generates a steady ambient stratification. The lowest part of the space, $z<h_i$, where $q(h_i)=Q_V$, is filled with fluid supplied by the ventilation flow and there is a net upflow in the ambient. Above this, $z>h_i$, the ambient fluid is linearly stratified with a reduced gravity gradient $f/Q_V$, and has no net vertical motion. Instead, for $z>h_i$, the time-averaged volume flux in the plume equals the ventilation flow. The intermittent entrainment of ambient fluid into the plume is now matched by intermittent detrainment from the plume, and the mean buoyancy in the plume relative to the ambient remains constant. The supply of fresh ventilation fluid to the ambient in the linearly stratified zone only occurs through the local detrainment and consequent horizontal intrusion of fluid from the plume. This has key implications for design of ventilation systems, in which there may be vertically distributed sources of buoyancy.

Efficient Computation of the Coupling Between a Vertical Line Source and a Slot

A novel spectral formulation is proposed here for the coupling integral between a cylindrical wave with arbitrary radial wavenumber and azimuthal dependence, and a slot placed on a plane orthogonal to the wavefront. Such a kind of integral is massively encountered in electromagnetic modeling of stratified structures comprising slots and vertical pins or vias. The twofold spatial integral defining the coupling is transformed into an equivalent onefold spectral integral, the integration path of which is selected to obtain Gaussian decay of the integrand. Both propagating and evanescent cylindrical modes are considered. An arbitrary stratification orthogonal to the slot plane can be considered, as well as an arbitrary current on the slot. Numerical comparisons against the standard spatial approach are shown to validate the new formulation, and its advantage in terms of computation cost is investigated in depth.

A semi-analytical model for multi-stage fractured horizontal wells

Multi-stage fractured horizontal wells are widely applied when developing tight reservoirs and shale gas reservoirs. As such, testing and evaluating fractured horizontal wells’ productivity become necessary for further improving the wells’ performance. Analyzing post-fracturing transient pressure data provides estimation of some key parameters that affect the productivity, including effective fracture lengths, fracture conductivities, fracture skin factors and average formation permeability. This paper presents a semi-analytical model based on Green’s functions and the source/sink method to facilitate the transient pressure analysis for a multi-stage fractured horizontal well in a closed box-shaped reservoir. Four kinds of fluid flow in the multi-stage fractured horizontal well system, including fluid from the reservoir to the fractures and from the reservoir to the horizontal wellbore, fluid inside the fractures as well as fluid flow inside the horizontal wellbore, are all taken into consideration. Compared with previous models, this study considers the pressure drops caused by pipe flow inside the wellbore. For the model’s accuracy and robustness, the fractures and the horizontal wellbore are discretized into vertical plane segments and horizontal line segments, respectively. Correspondingly, the fluid flow from the reservoir to the fracture as well as the fluid flow directly from the reservoir to the horizontal wellbore at each segment are modeled by analytical solutions of vertical plane source and horizontal line source, respectively. In addition, the fluid flow inside the fractures is modeled as 1-D linear flow. The fluid flow inside the horizontal wellbore is described with Penmatcha and Aziz’s model (1999). Finally, interface flux- and pressure-continuity conditions are used to couple the equations of aforementioned four kinds of fluid flow. The effects of fluid flow directly from the reservoir to the horizontal wellbore, fracture spacing, fracture lengths, fracture conductivities, and fracture skin factors on the transient pressure behavior are studied and type curves are generated. The results suggest that, in a tight or shale-gas reservoir, fracture stage, fracture lengths, conductivities and skin factors have significant influence on the transient pressure behavior during a testing period while the fluid flow directly from the reservoir into the horizontal wellbore reduces the pressure drops slightly. Then, a field case is analyzed and reliable results are obtained. This model can be further applied to optimize the fracture spacing and fracture lengths for a multi-stage fractured horizontal well.

Static Deformation of a Uniform Half-Space with Rigid Boundary Due To a Vertical Dip-Slip Line Source

The Airy stress function for a vertical dip-slip line source buried in a homogeneous, isotropic, perfectly elastic half-space with rigid boundary is obtained. This Airy stress function is used to derive closed- form analytical expressions for the stresses and displacements at an arbitrary point of the half-space caused by vertical dip-slip line source. The variation of the displacements and stress fields with distance from the fault and depth from the fault is studied numerically. Keywords - Dip-slip faulting, Half-space, Rigid boundary, Static deformation I. Introduction Strains and stresses within the Earth constitute important precursors of earthquakes. Therefore, the determination of the static deformation of an Earth model around surface faults is important for any scheme for prediction of earthquakes. Static dislocation models are used to analyze the static deformation of the medium caused by earthquake faults. Steketee (1958a, b) applied the elasticity theory of dislocations in the field of seismology. For the sake of simplicity, Steketee ignored the curvature of the Earth, its gravity, anisotropy and non-homogeneity and dealt with a semi-infinite, non-gravitating, isotropic and homogeneous medium. Homogeneity means that the medium is uniform throughout, whereas isotropy specifies that the elastic properties of the medium are independent of direction. Maruyama (1964) calculated all the sets of Green's functions required for the displacement and stress fields around faults in a half-space. Jungels and Frazier (1973) described a finite element variational method applied to plain strain analysis. This technique presents a suitable tool for the analysis of permanent displacements, tilts and strains caused by seismic events. The accuracy of technique was demonstrated by comparing the numerical results for the static field due to long dislocation in a homogeneous half-space from closed form analytical solution with those obtained from the finite element method. Sato (1971) and Sato and Yamashita (1975) derived the expressions for the static surface deformations due to two-dimensional strike slip and dip-slip faults located along the dipping boundary between the two different media. Freund and Barnett (1976) gave a two-dimensional analysis of surface deformation due to dip- slip faulting in a uniform half-space, using the theory of analytic functions of a complex variable. Singh and Garg (1986) obtained the integral expressions for the Airy stress function in an unbounded medium due to various two- dimensional seismic sources. Singh et al. (1992) followed a similar procedure to obtain closed-form analytical expression for the displacements and stresses at any point of either of two homogeneous, isotropic, perfectly elastic half-spaces in welded contact due to two-dimensional sources. Singh and Rani (1991) obtained closed-form analytical expressions for the displacements and stresses at any point of a two-phase medium consisting of a homogeneous, isotropic, perfectly elastic half-space in welded contact with a homogeneous, orthotropic, perfectly elastic half-space caused by two-dimensional seismic sources located in the isotropic half-space. Bonafede and Rivalta (1999a) obtained analytical solutions for the elementary tensile dislocation problem in a layered elastic medium composed of two welded, semi-infinite half-spaces. A plain - strain configuration was considered and different rigidities and Poisson ratios were assumed for the two half- spaces. The elementary dislocation problem refers to a dislocation surface over which a jump discontinuity with constant amplitude (Burgers vector) is prescribed for the displacement field. Similar dislocation models in homogeneous half-spaces (e.g. Okada, 1992) are often employed to model dyke injection within the crust (e.g. Bonaccorso and Davis 1993), although a constant-displacement discontinuity, in general, is not the most realistic description of dyke opening. Bonafede and Rivalta (1999b) obtained the solutions for the displacement and stress fields produced by a vertical tensile crack, opening under the effect of an assigned overpressure within it, in the proximity of the welded boundary between two media characterized by different elastic parameters. Singh et al.(2011) obtained analytical expressions for stresses at an arbitrary point of homogeneous, isotropic, perfectly elastic half-space with rigid boundary caused by a long tensile fault of finite width.

Deformation of a Uniform Half-Space with Rigid Boundary Due to a Strike-Slip Line Source

Closed-form analytical expressions for the displacement and stresses at an arbitrary point caused by strike-slip line source buried in a homogeneous, isotropic, perfectly elastic half-space with rigid boundary are obtained. These expressions are used, further, to find the expressions for the displacement and stresses caused by vertical as well as horizontal strike-slip line source. The variation of the displacement and stress fields due to vertical strike-slip line source and horizontal strike-slip line source with distance from the fault and depth from the fault is studied numerically. Keywords - Half-space, rigid boundary, static deformation, strike-slip faulting

Dynamics of a viscous layer flowing radially over an inviscid ocean

AbstractWe present a theoretical and experimental study of a viscous fluid layer spreading over a deep layer of denser, inviscid fluid. Specifically, we study an axisymmetric flow produced by a vertical line source. Close to the source, the flow is controlled viscously, with a balance between radial compressive stresses and hoop stresses. Further out, the flow is driven by gradients in the buoyancy force and is resisted by viscous extensional and hoop stresses. An understanding of these different fluid-mechanical relationships is developed by asymptotic analyses for early times and for the near and far fields at late times. Confirmation of the late-time, far-field behaviour is obtained from a series of laboratory experiments in which golden syrup was injected into denser solutions of potassium carbonate. We use our mathematical solutions to discuss a physical mechanism by which horizontal viscous stresses in a spreading ice shelf, such as those in West Antarctica, can buttress the grounded ice sheet that supplies it.