Vortex Core Research Articles

Numerical Study of the Dynamics of a Cyclone Modeled by a Vertical Vorticity Vector Considering a Radial Velocity as a Function of Time

In this article, the work carried out is a numerical study of the dynamics of a cyclone modeled by a vertical vorticity vector by considering a radial velocity as a function of time t. We made a short general study on cyclones by modeling a vertical cyclone starting from the Navier-Stokes equations. A study of the vorticity as a function of the radius and as a function of the radial abscissa at different times has been made. We also studied the radial and ortho radial components of the velocity as a function of the radius and as a function of the radial abscissa by varying the Ekman number and the Rossby number by looking at their impacts when they are increased or they are reduced. In the same dynamics, the influence of the decrease in the suction speed over time on the radius of the vortex core has been studied. In short, we have studied the effects of the variation of the radial velocity as a function of time. We ended our study by showing that the Ekman number is the most important parameter in the dynamics of the cyclone, by showing the relationship that exists between the Ekman number and the growth of the rotational motion of a cyclone.

On the statistics of coherent vortices in incoherent environments

Coherent vortices are important building blocks of complex flows on all scales, governing dynamics in various applications from geophysics over engineering to the coherent-structures’ paradigm of turbulence. As a commonly studied prototype of this configuration, we consider a large-scale, line vortex evolving in a background flow of small-scale, incoherent turbulence, which, nevertheless, naturally occurs in trailing vortices and tornadoes. The main unsteadiness of the large-scale vortex in this configuration is called meandering. Despite a common observation in experiments since the 1970s and a multitude of studies, its origin and physical mechanism remain puzzling as of this writing. Nevertheless, we do have considerable experimental evidence that, in the range of typically assumed parameters, vortex meandering shares four universal characteristics; namely, (i) fluctuations of the vortex-centre position obey a Gaussian distribution with (ii) monotonously growing standard deviation downstream. Besides, (iii) the fluctuation kinetic energy and enstrophy increase downstream, whereas the main contribution comes from a dipolar vorticity fluctuation pattern confined to the vortex core. This corresponds to the (iv) typical spectral signature of power spectra spanning all resolved scales, whereas the variance level increases monotonously towards the low frequencies. We present the first theoretical model to explain all four experimental cornerstones of vortex meandering in one theory. Starting from the definition of the vortex centre, we first show that the meandering motion is linearly proportional to the leading expansion coefficients of a Karhunen–Loève decomposition of the vorticity field. This has been conjectured before but has never been shown. The configuration strongly suggests scale separation between the response times of the vortex and the surrounding turbulence. On this assumption, we derive a Langevin equation for the slow, large scales driven by the fast, small scales, represented as a stochastic forcing. The particular phenomenon of vortex meandering therewith is found to belong to the large class of Gauss–Markov processes. That is, vortex meandering is an (abstract) Brownian motion. From this understanding, we infer immediately that the Gaussian statistics are a consequence of the central limit theorem and that the growing standard deviation follows from a competition between external forcing and intrinsic resistance. In the equilibrium limit, we derive the spectral signature of the large scales as a power law of the frequency, response time and forcing variance. Eventually, comparing our theoretical model with an experimental database gathered at the ONERA, we find good overall agreement.

Vortex-induced vibrations of a cantilevered blunt plate: POD of TR-PIV measurements and structural modal analysis

Vortex-induced vibrations sustained by immersed bodies may induce early structural fatigue and acoustic noise radiation. The present study consists of an experimental characterization of the vortex shedding and induced vibrations of a cantilevered blunt rectangular aluminium plate of chord to thickness ratio 16.7, immersed in a uniform water flow in the hydrodynamic tunnel of the French Naval Academy Research Institute. Experiences have been conducted at zero degrees incidence for Reynolds numbers Rec (based on chord length) ranging from 2.5×105 to 9.5×105, leading to turbulent wake conditions. The hydrodynamic properties of the wake have been evaluated by statistical analysis, Proper Orthogonal Decomposition (POD) and vortex core identification of Time Resolved Particle Image Velocimetry fields. The structural response of the plate has been examined by laser vibrometry through modal analysis. The fluid–structure interactions have been analysed for three different vibration regimes: out of resonance, lock-off resonance and lock-in resonance with the 1st twisting mode. The features of the first nine POD modes were analysed according to the different vibration regimes to understand the physics proper to the near wake. In the out of resonance regime, the strength of the primary Karman vortices is preferentially controlled by the Strouhal and Reynolds numbers. The thickness based Strouhal number is 0.195. A secondary hydrodynamic excitation source of characteristic Strouhal number 0.227 coexists with the primary Karman vortex shedding mode. At the lock-off resonance, this secondary excitation source is identified as a second Karman vortex shedding mode which is coupled with the natural frequency. A local maximum of vibration magnitude is attained for this regime. The coexistence of these two Karman modes is responsible for the early occurrence of resonance. The lock-in resonance occurs when the primary vortex shedding mode locks with the natural frequency. It is characterized by an enlargement of the wake and by a reinforcement of the primary Karman vortices strength.

Detailed numerical simulation of multi-scale interface-vortex interactions of liquid jet atomization in crossflow

In this paper, a detailed numerical simulation of liquid jet atomization in crossflow is performed based on the volume of fluid method coupled with the adaptive mesh refinement technique. The multi-scale phase interface evolution and the interface-vortex interaction are reported. It is found that the momentum flux ratio, Weber number, density ratio and viscosity ratio are key variables for the empirical correlation to accurately predict the atomization characteristics. Three breakup regimes, i.e., column bag breakup, surface breakup and ligament breakup, occur due to the great mass and momentum exchanges at the interface. Specially, the Kelvin-Helmholtz instability induces axial surface waves that eventually develop into column bag breakup while the Rayleigh-Taylor instability and surface thinning can induce surface breakup. Relatively, the column bag breakup generates larger liquid structures, presenting a bimodal feature. The produced ligaments either shrink to droplets or further breakup into droplets depending on their size and shape, leading to a log-normal distribution of droplet size. The interface-vortex interaction is distinctive compared to single-phase flows. A vortex core is observed to simultaneously form, grow and dissipate within each bag during the life circle of the bag, and three types of counter-rotating vortex pair exist around the column root, bag membrane and liquid droplets. Vortical structures tend to concentrate near the interface with large deformations, possessing a strong perpendicularity between the phase interface and the vortex. This work offers fundamental basis for better understanding and organization of atomization.

The magnetic field driven superconductor–metal transition in disordered hole-overdoped cuprates

By solving the Bogoliubov–de Gennes equations for a d-wave superconductor, we explore how the interplay between disorder and the orbital depairing of an external magnetic field influences the superconductor–metal transition of the hole-overdoped cuprates. For highly disordered systems, we find granular Cooper paring to persist above the critical field where the superfluid stiffness goes to zero. We also show that because the vortices are attracted to regions where the superconducting pairing is already weak, the Caroli–de Gennes–Matricon zero-bias peak in the local density of states at the vortex cores disappears already at moderate disorder.

Experimental study of axial flow evolution and area-varying waves in concentrated vortices

The dynamics of axial flow is investigated in initially two-dimensional (2-D) concentrated line vortices that are subsequently cut by a thin wall. The vortices are formed in the wake of an airfoil oscillating sinusoidally at small amplitude, and their circulation and peak vorticity are controlled by the choice of oscillation frequency. The flow field is quantified in terms of the spatio-temporal evolution of axial (spanwise) velocity using one-component molecular tagging velocimetry, and simultaneous measurements of spanwise vorticity and axial velocity at selected planes away from the cutting wall using stereoscopic molecular tagging velocimetry. Results show that the characteristics of the vortices cut by the wall are initially in good agreement with the baseline 2-D vortices. However, as the axial flow develops near the cutting wall and propagates away from it farther downstream, the structure of the initially 2-D Gaussian-shaped vortices quickly changes to a highly distorted region(s) of vorticity near the wall. Data over planes one and two core radii away from the wall reveal appreciable axial velocity both toward and away from the wall, whereas at locations farther away the axial velocity is primarily away from the wall. The data reveal the existence of small amplitude area-varying waves that propagate away from the wall along the vortex core. These disturbances are associated with local changes in the vortex axial velocity and peak vorticity. The propagation speed of these disturbances is found to agree with predictions using the analytical model of Lundgren & Ashurst (J. Fluid Mech., vol. 200, 1989, pp. 283–307).

Investigations of string cavitation and air back suction during injection duration in the scaled-up diesel fuel injection nozzle

The flow inside the diesel fuel injector nozzle is directly associated with the subsequent jet atomization and fuel–air mixture performance, and finally affects the combustion efficiency and emission. In this study, high-speed microscope imaging technology is applied to capture the complex multiphase flow and near-field spray during the process of fuel injection. Besides normal optical experiments for nozzle internal flow and spray, the submerged jet experiment for comparison is also carried out to investigate the phenomenon of air back suction. Combined with the simulation of a three-component multiphase model, the difference between air and cavitation vapor is analyzed. Results show that the submerged jet experiment will not change the internal flow field but can eliminate the influence of air. The shaded area in the hole obtained by the shadow method may be the sucked air or vapor generated by string-type cavitation phase change. Both can coexist in the hole and are closely related to the swirling intensity and the spray structure, respectively. When the needle lift is low, the internal flow field has larger swirling intensity resulting in a kind of hollow spray. The low-pressure region exists in the vortex core, nearly at the axis of the nozzle orifice. The ambient air will be sucked into the orifice to form an air column under the action of pressure difference, which is also easy to occur under low injection pressure. When the needle lift is high, the weak swirl intensity leads to hole-to-hole string cavitation which has weak cavitation intensity and obvious periodicity. At this time, the results of non-submerged and submerged jet experiments are basically the same. In addition, the cavitation bubble near the orifice outlet was observed, for the first time, to move upstream under the action of the pressure difference inside and outside the orifice in the submerged jet experiment. It can be predicted that in the non-submerged jet experiment, the ambient air will be sucked into the orifice to form an air column.

Glitches due to quasineutron-vortex scattering in the superfluid inner crust of a pulsar

We revisit the mechanism of vortex unpinning caused by the neutron-vortex scattering \cite{prad1} in the inner crust of a pulsar. The strain energy released by the crustquake is assumed to be absorbed in some part of the inner crust and causes pair-breaking quasi-neutron excitations from the existing free neutron superfluid in the bulk of the inner crust. The scattering of these quasi-neutrons with the vortex core normal neutrons unpins a large number of vortices from the thermally affected regions and results in pulsar glitches. We consider the geometry of a cylindrical shell of the affected pinning region to study the implications of the vortex unpinning in the context of pulsar glitches. We find that a pulsar can release about $\sim 10^{11} - 10^{13}$ vortices by this mechanism. These numbers are equivalent to the glitch size of orders $\sim 10^{-11} - 10^{-9}$ for Vela-like pulsars with the characteristic age $\tau \simeq 10^4$ years. We also suggest a possibility of a vortex avalanche triggered by the movement of the unpinned vortices. A rough estimate of the glitch size caused by an avalanche shows an encouraging result.

Amplitude structure of optical vortices determines annihilation dynamics.

We show that annihilation dynamics between oppositely charged optical vortex pairs can be manipulated by the initial size of the vortex cores, consistent with hydrodynamics. When sufficiently close together, vortices with strongly overlapped cores annihilate more quickly than vortices with smaller cores that must wait for diffraction to cause meaningful core overlap. Numerical simulations and experimental measurements for vortices with hyperbolic tangent cores of various initial sizes show that hydrodynamics governs their motion, and reveal distinct phases of vortex recombination; decreasing the core size of an annihilating pair can prevent the annihilation event.

Evolution characteristics of streamwise vortex of crossing shock wave/turbulent boundary layer interaction

The streamwise vortex generated by the crossing shock wave/turbulent boundary layer interaction seriously affects the performance of the hypersonic inlet. The shock structure, vortex core, and vortex intensity in the flowfields of weak, moderate, and strong strength interaction are experimentally and numerically investigated at Mach 2.95. The evolution characteristics of the streamwise vortex are explored. The reflection and intersection of shock waves in the interaction flowfield divide the evolution process of the streamwise vortex into four stages. The typical conical vortex evolutes into two vortexes. The vortex close to the symmetric plane further evolutes into two vortexes in the weak and strong strength flowfields. A separated flow from the focus on the flat plate produces a new streamwise vortex in the strong strength flowfield. The streamwise vortex core trace is almost parallel to the symmetric plane after the pair of streamwise vortexes collide at the symmetric plane. The vorticity intensity of the streamwise vortex increases and then decreases along the flow direction, which is greatly influenced by the interaction strength.

Effect of the wind speed on aerodynamic behaviours during the acceleration of a high-speed train under crosswinds

A high-speed train that accelerates under crosswinds can potentially encounter dramatic aerodynamic effects and safety risks. The goal of this study is to reveal the effect of wind speed on the instantaneous vortex structures, pressure distribution, aerodynamic loads and safety indicators of a high-speed train that is accelerating under crosswinds. The three-dimensional and incompressible IDDES method combined with moving overset grid technology is applied to obtain reliable results. The main results are summarized as follows: two typical inclined vortices on the leeward side are observed, and the amplitude and area of the negative pressure in the vortex core region increase with increasing wind speed. The leeward side and the roof exhibit severe pressure fluctuations, and the sensitivity of the pressure from wind speed on the head car is stronger than that on other carriages. For aerodynamic force/moment coefficients, local maximum/minimum values appear during the acceleration process; a reasonable inference for this phenomenon is that a certain range of yaw angles causes the force/moment coefficient to reach the local maximum/minimum values. Wind speed has a significant increasing effect on the maximum values of the lateral force coefficient and rolling moment coefficient, especially for the head car. Although the overturning coefficient and derailment coefficient are less than the prescribed safety thresholds, the overturning risk needs more attention than the derailment risk considering that the maximum value of the overturning coefficient is twice that of the derailment coefficient. The obtained numerical results can provide a preliminary reference for train accelerations under crosswind.

Experimental and numerical studies of the effects of the contraction ratios on the swirling flow characteristics of the model combustor outlet in lean gas turbines

Large Eddy Simulation (LES) is utilized in this work to study isothermal turbulent swirling flow characteristics for a lean partially premixed gas turbine model combustor. The goal is to investigate the influence of contraction ratio (CR) variation on the vortex breakdown phenomena features and the turbulence structure in these combustors. Particle image velocimetry (2D-PIV) measurements were carried out on a single-case model combustor using (air-air) phase to perform the mean flow behavior and the validation process under atmospheric conditions with CR equal to 0 (largest diameter, Dinlet = Doutlet), the equivalence ratio (φ) of 0.3, swirl number of (Sn) = 0.66 and Reynolds number of (Re) = 42,000. While five test cases with various CRs were applied in LES under the same conditions using (air-CH4) phase to predict the vortical structure and the precession vortex core (PVC). The CRs ranged as follows: 0, 50 %, 60 %, 70 % and 80 % (smallest diameter). Experimental and LES results show that there was a reasonable agreement between the PIV and LES results with maximum deviations of 2.8 % and 4.65 % for the normalized mean axial and radial velocities, respectively. As the CR was increased, the pressure drop and the mean axial velocity increased linearly at the contraction part. In addition, the downstream stagnation point and the size of the central toroidal recirculation zone (CTRZ) stabilized when CR = 60 %, 70 %, and 80 %. The swirl level was further enhanced by 40 % when CR = 60 % and 80 %. The presence of CR successfully suppressed the pressure and axial velocity fluctuations generated in the shear layer zones by 35 %. LES data was analyzed by Fast Fourier Transformation (FFT) and Proper Orthogonal Decomposition (POD) to investigate the PVC dynamics. The FFT showed four dominant frequencies as a result of the appearance of the PVC and vortex breakdown oscillation. The frequency of PVC (fpvc) decreased linearly with an increase in the CR where the fpvc was 20.8 Hz when CR = 0 and became 12.5 Hz at CR = 80 %. In contrast, the other frequencies were still constant even with increased CR at 187 Hz, 562 Hz, and 979 Hz. The POD analysis showed the first mode was more energetic and had 40 % of the total pressure fluctuations.

Topological data analysis of vortices in the magnetically-induced current density in LiH molecule.

A novel strategy for extracting axial (AV) and toroidal (TV) vortices in the magnetically-induced current density (MICD) in molecular systems is introduced, and its pilot application to LiH molecule is demonstrated. It exploits differences in the topologies of AV and TV cores and involves two key steps: selecting a scalar function that can describe vortex cores in MICD and its subsequent topological analysis. The scalar function of choice is ΩBα based on the velocity-gradient Ω method known in research on classical flows. The Topological Data Analysis (TDA) is then used to analyze the ΩBα scalar field. In particular, TDA robustly assigns distinct topological features of this field to different vortex types in LiH: AV to saddle-maximum separatrices which connect maxima to 2-saddles located on the domain's boundary, TV to a 1-cycle of the super-level sets of the input data. Both are extracted as the most persistent features of the topologically-simplified ΩBα scalar field.

Thermo-hydraulic performance of a cryogenic printed circuit heat exchanger for liquid air energy storage

• The nature of local flow and heat transfer was analyzed using vortex identification. • The reverse flow suppresses local heat transfer and increases the flow resistance. • The air side has larger thermal resistance (∼ 58%) in the absence of phase change. • The best comprehensive performance can be achieved at an inclined angle of 45°. • The pressure drop and heat transfer correlations for the best angle were developed. The liquid air energy storage assisted by liquefied natural gas is a promising large-scale storage method, but its development is limited by the lack of thermo-hydraulic data on the cryogenic printed circuit heat exchanger. A simplified model for conjugate heat transfer was built and solved by a commercial code FLUENT 14.5. Model validation was conducted by comparing results with experimental data. The nature of flow and heat transfer was revealed by quantifying the vortex strength and visualizing the vortex core. The influence of the inclined angle was investigated, and the Nusselt number and friction factor correlations for the best angle were developed. The minimum internal temperature approach for optimum system operation is approximately 6 K, located in the middle of the heat exchanger. Results indicate that the evolution of Dean vortices plays a crucial part in the augmentation of heat transfer. The secondary flow caused by elbows forces the fluid cores to strike the windward side periodically, which disturbs the boundary layer and enhances the fluid mixing. Entropy production analysis shows that the thermal entropy generation has a dominant share of total irreversibility. The best performance is achieved at an inclined angle of 45° since the irreversibility is reduced by augmenting the heat transfer.

Vortex Lens: Interactive Vortex Core Line Extraction using Observed Line Integral Convolution.

This paper describes a novel method for detecting and visualizing vortex structures in unsteady 2D fluid flows. The method is based on an interactive local reference frame estimation that minimizes the observed time derivative of the input flow field v(x, t). A locally optimal reference frame w(x, t) assists the user in the identification of physically observable vortex structures in Observed Line Integral Convolution (LIC) visualizations. The observed LIC visualizations are interactively computed and displayed in a user-steered vortex lens region, embedded in the context of a conventional LIC visualization outside the lens. The locally optimal reference frame is then used to detect observed critical points, where v=w, which are used to seed vortex core lines. Each vortex core line is computed as a solution of the ordinary differential equation (ODE) · w(t)=w(w(t), t), with an observed critical point as initial condition (w(t0), t0). During integration, we enforce a strict error bound on the difference between the extracted core line and the integration of a path line of the input vector field, i.e., a solution to the ODE · v(t)=v(v(t), t). We experimentally verify that this error depends on the step size of the core line integration. This ensures that our method extracts Lagrangian vortex core lines that are the simultaneous solution of both ODEs with a numerical error that is controllable by the integration step size. We show the usability of our method in the context of an interactive system using a lens metaphor, and evaluate the results in comparison to state-of-the-art vortex core line extraction methods.

End boundary effects on wakes dynamics of inclined circular cylinders

We perform direct numerical simulations to characterize the three-dimensional wake dynamics of long inclined circular cylinders with inhomogeneous end boundary conditions. Three Reynolds numbers, Re=100, 200 and 300, corresponding to the regimes of laminar flow, mode A* wake, and mode B wake, respectively, are considered to reveal the roles of the intrinsic secondary instabilities and the extrinsic end boundary effects in shaping the three-dimensional flows. At Re=100, the end boundary effects are felt over the entire cylinder span by inducing oblique vortex shedding, which is associated with stronger spanwise flow in the wake than a parallel shedding. The Strouhal number of the oblique shedding is related to that of the parallel shedding of straight cylinder by the cosine law, considering the combined inclination angle and oblique angle. At Re=200, the intrinsic secondary instability results in large-scale vortex dislocation, precluding the propagation of the end boundary effects towards further span. Nevertheless, for the inclined cases, oblique shedding is still observed within limited span from the upstream end boundary. The oblique vortices feature intact and straight vortex cores, and are related to the two-dimensional flow at Re=200 (that are void of vortex dislocations) from the viewpoint of cosine law. Further along the span, the oblique vortices destabilizes with the formation of small-scale vortices, and the flow transitions to the typical mode A* wake. At Re=300, the highly three-dimensional flow near the end boundary creates disturbances that travel along the cylinder span, creating vortex dislocations for cases with low inclination angles. For high inclination angle, oblique vortex shedding is again observed over the cylinder span, and is not disrupted by vortex dislocations of either intrinsic or extrinsic causes. The present study offers renewed insights into the three-dimensional wake dynamics of inclined cylinders, and lays the foundation for the designs of long flexible engineering structures and related flow control techniques.

Irreversible switching of vortex core in Pac-man nanodisks induced by rotating magnetic fields

Vortex has attracted great attention recently due to its potential applications in information storage, nano-oscillators and logic devices. The control of vortex polarity is key to the devices based on vortex. In this paper, an irreversible vortex core (VC) switching in Pac-man disks has been investigated by micromagnetic simulations. The switching process is closely related to the evolution of energy densities and there is a sharp peak in the local exchange energy density when VC switching occurs. Irreversible switching of VC can be realized due to the weak interaction between reversed vortex and rotating magnetic field. In the process of VC reversal, the positive and negative magnetization regions of the gyrofield are separated. Moreover, the switching time of VC decreases monotonously with frequency under rotating magnetic fields. When one direction of the rotating magnetic field is turned off, the difference in switching time is attributed to the different motion behaviors of vortex. This work benefits the potential storage applications in vortex-based spintronic devices.

Majorana zero modes in a magnetic and superconducting hybrid vortex

We propose and investigate a new platform for the realization of Majorana zero modes in a thin-film heterostructure composed of an easy-plane ferromagnet and a superconductor with spin-orbit coupling. The system can support an energetically favorable bound state comprising a magnetic vortex and a superconducting vortex. We show that a hybrid vortex thus created can host a robust zero-energy Majorana bound state at its core over a wide range of parameters, with its partner zero mode located at the boundary of a disk-shaped topological region. We identify a novel mechanism underlying the formation of the topological phase that, remarkably, relies on the orbital effect of the magnetization field and not on the usual Zeeman effect. The in-plane components of magnetization couple to electrons as a gauge potential with nonzero curl, thus creating an emergent magnetic field responsible for the gapped topologically nontrivial region surrounding the vortex core. Our construction allows the mobility of magnetic vortices to be imposed on the Majorana zero mode at the core of the superconducting vortex. In addition, the system shows a rich interplay between magnetism and superconductivity which might aid in developing future devices and technologies.

Experimental investigation on wake characteristics of wind turbine and a new two-dimensional wake model

Wind tunnel experiments are performed to investigate the wake characteristics of a model wind turbine using Particle Image Velocimetry (PIV) and Hot-wire velocimetry. Results show that the velocity deficit at the hub height is the largest, and the stratification of the velocity shear layer at the blade tip is obvious. The instantaneous turbulence kinetic energy (TKE) level is much larger than the mean TKE, especially downstream of the blade tip. In the x/dT=4 position of the wake region, the mean TKE increased by two times due to the blade tip disturbance. The Reynolds stresses also increase and the instantaneous value is the highest along the blade’s tip. A thin vortex band appears at the root and tip of the blade, the vortex core expands and diffuses with the development of the wake along the downstream direction. The two-dimensional distributions of velocity spectra fΦu reveal that the velocity fluctuation at x/dT=2 location is small and that at x/dT=4 position is the largest. Based on Jensen’s wake model, a new theoretical wake model is proposed to predict the velocity distribution in the wind turbine wake. The calculated results of the wake model are in good agreement with the experimental data.

Baroclinic effects on the distribution of tropical cyclone eye subsidence

Solutions of the secondary (transverse) circulation equation for an axisymmetric, gradient balanced vortex are used to better understand the distribution of subsidence in the eye of a tropical cyclone. This secondary circulation equation is derived using both the physical radius coordinate r and the potential radius coordinate R. In the R-coordinate version, baroclinic effects are implicit in the coordinate transformation and are recovered in the final step of transforming the solution for the streamfunction Ψ back from R-space to r-space. Two types of elliptic problems for Ψ are formulated: 1) the full secondary circulation problem, which is formulated on 0 ≤ R < ∞, with the diabatic forcing due to eyewall convection appearing on the right-hand side of the elliptic equation; 2) the restricted secondary circulation problem, which is formulated on 0 ≤ R ≤ Rew, where the constant Rew is the potential radius of the inside edge of the eyewall, with no diabatic forcing but with the streamfunction specified along R = Rew. The restricted secondary circulation problem can be solved semi-analytically for the case of vertically sheared, Rankine vortex cores. The solutions identify the conditions under which large values of radial and vertical advection of θ are located in the lower troposphere at the outer edge of the eye, thereby producing a warm-ring thermal structure.