Wave and Vortex Dynamics on the Surface of a Sphere

Motivated by the observed potential vorticity structure of the stratospheric polar vortex, we study the dynamics of linear and nonlinear waves on a zonal vorticity interface in a two-dimensional barotropic flow on the surface of a sphere (interfacial Rossby waves). After reviewing the linear problem, we determine, with the help of an iterative scheme, the shapes of steadily propagating nonlinear waves; a stability analysis reveals that they are (nonlinearly) stable up to very large amplitude. We also consider multi-vortex equilibria on a sphere: we extend the results of Thompson (1883) and show that a (latitudinal) ring of point vortices is more unstable on the sphere than in the plane; notably, no more than three point vortices on the equator can be stable. We also determine the shapes of finite-area multi-vortex equilibria, and reveal additional modes of instability feeding off shape deformations which ultimately result in the complex merger of some or all of the vortices. We discuss two specific applications to geophysical flows: for conditions similar to those of the wintertime terrestrial stratosphere, we show that perturbations to a polar vortex with azimuthal wavenumber 3 are close to being stationary, and hence are likely to be resonant with the tropospheric wave forcing; this is often observed in high-resolution numerical simulations as well as in the ozone data. Secondly, we show that the linear dispersion relation for interfacial Rossby waves yields a good fit to the phase velocity of the waves observed on Saturn’s ‘ribbon’

Wave and Vortex Dynamics on the Surface of a Sphere

Motivated by the observed potential vorticity structure of the stratospheric polar vortex, we study the dynamics of linear and nonlinear waves on a zonal vorticity interface in a two-dimensional barotropic flow on the surface of a sphere (interfacial Rossby waves). After reviewing the linear problem, we determine, with the help of an iterative scheme, the shapes of steadily propagating nonlinear waves; a stability analysis reveals that they are (nonlinearly) stable up to very large amplitude. We also consider multi-vortex equilibria on a sphere: we extend the results of Thompson (1883) and show that a (latitudinal) ring of point vortices is more unstable on the sphere than in the plane; notably, no more than three point vortices on the equator can be stable. We also determine the shapes of finite-area multi-vortex equilibria, and reveal additional modes of instability feeding off shape deformations which ultimately result in the complex merger of some or all of the vortices. We discuss two specific applications to geophysical flows: for conditions similar to those of the wintertime terrestrial stratosphere, we show that perturbations to a polar vortex with azimuthal wavenumber 3 are close to being stationary, and hence are likely to be resonant with the tropospheric wave forcing; this is often observed in high-resolution numerical simulations as well as in the ozone data. Secondly, we show that the linear dispersion relation for interfacial Rossby waves yields a good fit to the phase velocity of the waves observed on Saturn’s ‘ribbon’

The Roll-Up of Vorticity Strips on the Surface of a Sphere

We derive the conditions for the stability of strips or filaments of vorticity on the surface of a sphere. We find that the spherical results are surprisingly different from the planar ones, owing to the nature of the spherical geometry. Strips of vorticity on the surface of a sphere show a greater tendency to roll-up into vortices than do strips on a planar surface. The results are obtained by performing a linear stability analysis of the simplest, piecewise-constant vorticity configuration, namely a zonal band of uniform vorticity located in equilibrium between two latitudes. The presence of polar vortices is also considered, this having the effect of introducing adverse shear, a known stabilizing mechanism for planar flows. In several representative examples, the fully developed stages of the instabilities are illustrated by direct numerical simulation. The implication for planetary atmospheres is that barotropic flows on the sphere have a more pronounced tendency to produce small, long-lived vortices, especially in equatorial and mid-latitude regions, than was previously anticipated from the theoretical results for planar flows. Essentially, the curvature of the sphere's surface weakens the interaction between different parts of the flow, enabling these parts to behave in relative isolation

Transport and Mixing of Chemical Air Masses in Idealized Baroclinic Life Cycles

The transport, mixing, and three-dimensional evolution of chemically distinct air masses within growing baroclinic waves are studied in idealized, high-resolution, life cycle experiments using suitably initialized passive tracers, contrasting the two well-known life cycle paradigms, distinguished by predominantly anticyclonic (LC1) or cyclonic (LC2) flow at upper levels. Stratosphere-troposphere exchange differs significantly between the two life cycles. Specifically, transport from the stratosphere into the troposphere is significantly larger for LC2 (typically by 50%), due to the presence of large and deep cyclonic vortices that create a wider surf zone than for LC1. In contrast, the transport of tropospheric air into the stratosphere is nearly identical between the two life cycles. The mass of boundary layer air uplifted into the free troposphere is similar for both life cycles, but much more is directly injected into the stratosphere in the case of LC1 (fourfold, approximately). However, the total mixing of boundary layer with stratospheric air is larger for LC2, owing to the presence of the deep cyclonic vortices that entrain and mix both boundary layer air from the surface and stratospheric air from the upper levels. For LC1, boundary layer and stratospheric air are brought together by smaller cyclonic structures that develop on the poleward side of the jet in the lower part of the middleworld, resulting in correspondingly weaker mixing. As both the El Niño-Southern Oscillation and the North Atlantic Oscillation are correlated with the relative frequency of life cycle behaviors, corresponding changes in chemical transport and mixing are to be expected

Generalized Kirchhoff Vortices

A family of exact solutions of the Euler equations is presented: they are generalizations of the Kirchhoff vortex to N confocal ellipses. Special attention is given to the case N=2, for which the stability is analyzed with a method similar to the one used by Love [Proc. London Math. Soc. 1, XXV 18 (1893)] for the Kirchhoff vortex. The results are compared with those for the corresponding circular problem

TAVI-in-homograft (TiH): open transcatheter aortic valve replacement in calcified aortic homograft case reports

Background Redo surgery in patient who underwent aortic valve replacement with an aortic homograft can result technically challenging because of the massive calcification of the conduit. Case presentation We present a case of a patient who underwent open surgery on cardiopulmonary bypass assistance to implant a standard transcatheter aortic bioprosthesis through aortotomy in an off-label procedure and we discuss its safety and feasibility. Conclusions The combination of open cardiac surgery and open trans-aortic implant of a transcatheter prosthesis may reduce the surgical risk shrinking the technical difficulties that the implantation of a standard surgical prosthesis would have given

The Antarctic Stratospheric Sudden Warming of 2002: A Self-Tuned Resonance?

The extraordinary Antarctic stratospheric warming event of 2002 was characterized by a remarkable vertical structure, with the vortex observed to divide at upper levels in the stratosphere but not at lower levels: such ‘partially’ split vortex events are relatively rare. A simple, yet fully three-dimensional, model is constructed to investigate the dynamics of this unique event. Planetary waves are excited on the model vortex edge by a lower boundary forcing characterized by two parameters: an amplitude hF and a frequency ωF, measured relative to a stationary frame. For realistic forcing amplitudes, a partial vortex split resembling that observed during the 2002 event is found only within a specific, narrow band of forcing frequencies. Exploiting the relative simplicity of our model, these frequencies are shown to be those causing a ‘self-tuning’ resonant excitation of the gravest linear mode, during which nonlinear feedback causes an initially off-resonant forcing to approach resonance

Filamentation of Unstable Vortex Structures via Separatrix Crossing: A Quantitative Estimate of Onset Time

The onset of filamentation for compact vortex structures in two-dimensional incompressible flows is elucidated. An estimate is presented for the filamentation time of an unstably perturbed Kirchhoff ellipse, obtained from a linear analysis of the geometry of the instantaneous corotating streamfunction

Two-Layer Geostrophic Vortex Dynamics. Part 1. Upper-Layer V-States and Merger

We generalize the methods of two-dimensional contour dynamics to study a two-layer rotating fluid that obeys the quasi-geostrophic equations. We consider here only the case of a constant-potential-vorticity lower layer. We derive equilibrium solutions for monopolar (rotating) and dipolar (translating) geostrophic vortices in the upper layer, and compare them with the Euler case. We show that the equivalent barotropic (infinite lower layer) case is a singular limit of the two-layer system. We also investigate the effect of a finite lower layer on the merger of two regions of equal-sign potential vorticity in the upper layer. We discuss our results in the light of the recent laboratory experiments of Griffiths and Hopfinger (1986). The process of filamentation is found to be greatly suppressed for equivalent barotropic dynamics on scales larger than the radius of deformation. We show that the variation of the critical initial distance for merger as a function of the radius of deformation and the ratio of the layers at rest is closely related to the existence of vortex-pair equilibria and their geometrical properties

Filamentation of Unstable Vortex Structures via Separatrix Crossing: A Quantitative Estimate of Onset Time

The onset of filamentation for compact vortex structures in two-dimensional incompressible flows is elucidated. An estimate is presented for the filamentation time of an unstably perturbed Kirchhoff ellipse, obtained from a linear analysis of the geometry of the instantaneous corotating streamfunction