We consider the full three‐dimensional Ginzburg–Landau model of superconductivity with applied magnetic field, in the regime where the intensity of the applied field is close to the ‘first critical field’ at which vortex filaments appear, and in the asymptotics of a small inverse Ginzburg–Landau parameter . This onset of vorticity is directly related to an ‘isoflux problem’ on curves (finding a curve that maximizes the ratio of a magnetic flux by its length), whose study was initiated in [22] and which we continue here. By assuming a nondegeneracy condition for this isoflux problem, which we show holds at least for instance in the case of a ball, we prove that if the intensity of the applied field remains below , the total vorticity remains bounded independently of , with vortex lines concentrating near the maximizer of the isoflux problem, thus extending to the three‐dimensional setting a two‐dimensional result of [28]. We finish by showing an improved estimate on the value of in some specific simple geometries.
more » « less Award ID(s):
 2000205
 NSFPAR ID:
 10444102
 Publisher / Repository:
 Oxford University Press (OUP)
 Date Published:
 Journal Name:
 Proceedings of the London Mathematical Society
 Volume:
 126
 Issue:
 3
 ISSN:
 00246115
 Page Range / eLocation ID:
 p. 10151062
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
More Like this

Inviscid spatial Landau damping is studied experimentally for the case of oscillatory motion of a twodimensional vortex about its elliptical equilibrium in the presence of an applied strain flow. The experiments are performed using electron plasmas in a Penning–Malmberg trap. They exploit the isomorphism between the twodimensional Euler equations for an ideal fluid and the driftPoisson equations for the plasma, where plasma density is the analog of vorticity. Perturbed elliptical vortex states are created using [Formula: see text] strain flows, which are generated by applying voltages to electrodes surrounding the plasma. Measurements of spatial Landau damping (also called criticallayer damping) are in agreement with previous studies in the absence of an applied strain, where the damping is due to a resonance between the local fluid motion and the vortex oscillations. Interestingly, the damping rate does not change significantly over a wide range of applied strain rates. This can be accurately predicted from the initial vorticity profile, even though the resonant frequency is reduced substantially due to the applied strain. For higher amplitude perturbations, nonlinear trapping oscillations also exhibit behavior similar to the strainfree case. In principle, higherorder effects of the applied strain, such as separatrix crossing of peripheral vorticity and interactions with harmonics of the fundamental resonance, are expected to change the damping rate. However, this occurs only for conditions that are not realized in the experiments described here. Vortexincell simulations are used to investigate the possible roles of these effects.more » « less

Abstract Tropical cyclone formation and evolution at, or near, the Equator is explored using idealized three‐dimensional model simulations, starting from a prescribed, initial, weak counterclockwise rotating vortex in an otherwise quiescent,
nonrotating environment. Three simulations are carried out in which the maximum tangential wind speed (5 m s) is specified at an initial radius of 50, 100, or 150 km. After a period of gestation lasting between 30 and 60 hr, the vortices intensify rapidly, the evolution being similar to that for vortices away from the Equator. In particular, the larger the initial vortex size, the longer the gestation period, the larger the maximum intensity attained, and the longer the vortex lifetime. Beyond a few days, the vortices decay as the cyclonic vorticity source provided by the initial vortex is depleted and negative vorticity surrounding the vortex core is drawn inwards by the convectively driven overturning circulation. In these negative vorticity regions, the flow is inertially/centrifugally unstable. The vortex evolution during the mature and decay phases differs from that in simulations away from the Equator, where inertially unstable regions are much more limited in area. Vortex decay in the simulations appears to be related intimately to the development of inertial instability, which is accompanied by an outward‐propagating band of deep convection. The degree to which this band of deep convection is realistic is unknown. 
We simulate the motion of a commensurate vortex lattice in a periodic lattice of artificial circular pinning sites having different diameters, pinning strengths, and spacings using the timedependent GinzburgLandau formalism. Above some critical DC current density Jc, the vortices depin, and the resulting steadystate motion then induces an oscillatory electric field E (t) with a defect "hopping" frequency f0, which depends on the applied current density and the pinning landscape characteristics. The frequency generated can be locked to an applied AC current density over some range of frequencies, which depends on the amplitude of the DC as well as the AC current densities. Both synchronous and asynchronous collective hopping behaviors are studied as a function of the supercell size of the simulated system and the (asymptotic) synchronization threshold current densities determined.more » « less

Abstract Polygonal eyewall asymmetries of Hurricane Michael (2018) during rapid intensification (RI) are analyzed from ground‐based single Doppler radar. Here, we present the first observational evidence of the evolving wind field of a polygonal eyewall during RI to Category 5 intensity by deducing the axisymmetric and asymmetric winds at 5‐min intervals. Spectral time decomposition of the retrieved tangential wind structure shows quantitative evidence of low (1–4) azimuthal wavenumbers with propagation speeds that are consistent with linear wave theory on a radial vorticity gradient, suggesting the presence of rapidly evolving vortex Rossby waves. Dual‐Doppler winds from the NOAA P‐3 Hurricane Hunter airborne radar provide further evidence of the three‐dimensional vortex structure that supports growth of asymmetries during RI. Both reflectivity and tangential wind fields show polygonal structure and propagate at similar speeds, suggesting a close coupling of the dynamics and the convective organization during the intensification.

In this paper a model for viscous boundary and shear layers in three dimensions is introduced and termed a vortexentrainment sheet. The vorticity in the layer is accounted for by a conventional vortex sheet. The mass and momentum in the layer are represented by a twodimensional surface having its own internal tangential flow. Namely, the sheet has a mass density perunitarea making it dynamically distinct from the surrounding outer fluid and allowing the sheet to support a pressure jump. The mechanism of entrainment is represented by a discontinuity in the normal component of the velocity across the sheet. The velocity field induced by the vortexentrainment sheet is given by a generalized Birkhoff–Rott equation with a complex sheet strength. The model was applied to the case of separation at a sharp edge. No supplementary Kutta condition in the form of a singularity removal is required as the flow remains bounded through an appropriate balance of normal momentum with the pressure jump across the sheet. A pressure jump at the edge results in the generation of new vorticity. The shedding angle is dictated by the normal impulse of the intrinsic flow inside the bound sheets as they merge to form the free sheet. When there is zero entrainment everywhere the model reduces to the conventional vortex sheet with no mass. Consequently, the pressure jump must be zero and the shedding angle must be tangential so that the sheet simply convects off the wedge face. Lastly, the vortexentrainment sheet model is demonstrated on several example problems.more » « less