David G. Dritschel

Contour dynamics and contour surgery: Numerical algorithms for extended, high-resolution modelling of vortex dynamics in two-dimensional, inviscid, incompressible flows

Abstract The complex flow situations that regularly develop in a two-dimensional vortical flow h ave tradionally, indeed almost exclusively, been studied using Eulerian numerical methods, particularly spectral methods. These Eulerian methods have done remarkably well at modelling low to moderate Reynolds number flows.However, at the very high Reynolds numbers typical of geophysical flows, Eulerain methods run into difficulties, not the least of which is sufficient spatial resolutions. On the other hand, Lagrangian methods are and contour dynamics methods, are inherently inviscid. It would appear, therefore, that Lagrangian mehtods ideally suited for the modelling of flows at very high Reynolds numbers. Yet in practice, Lagrangian methods have themselves been limited by the frequent, extraordinary increase in the spatial complexity of inviscid flows. As a consequence, Lagrangian methods have been restricted to relatively simple flows which remain simple. Recently, an extension of contour dynamics, “contour surgery”, has enabled the modelling of complex inviscid flows in wholly Lagrangian terms, This extension overcomes the buildup of small-scale structure by truncating, in physical space, the modelled range of scales. The results of this truncation, or “surgery”, is to make feasible the computation of flows having a range of scales spanning four to five orders of magnitude, or one to two orders of magnitude greater than ever considered by Eulerian-Lagrangian methods. This paper discusses the history of contour dynamis which led to contour surgery, gives details of the contour surgery algorithm for planar, cylindrical, spherical, and quasi-geostrophic flow, presents new results obtained with high-resolutions calculations, including the first every comparison between contour surgery and a traditional pseudo- spectral method, and outlines some outstanding problems facing dynamics/ surgery.

Contour surgery: a topological reconnection scheme for extended integrations using contour dynamics

Abstract A numerical algorithm is described which, it is believed, can accurately model the dynamics of a two-dimensional, inviscid, incompressible fluid with unparalled spatial resolution. The fluid is assumed, however, to be divided into regions of uniform vorticity, conservation of vorticity ensuring that this remains true for all time. Like contour dynamics, the algorithm is concerned with following the evolution of the boundaries of vorticity discontinuity (contours). Unlike contour dynamics, the algorithm automatically removes vorticity features smaller than a predefined scale. For example, two contours enclosing the same uniform vorticity merge into one if they are close enough together. Also, the curvature along a contour is not allowed to exceed the inverse of the cutoff scale. At present, calculations with contour surgery resolve fluid motions extending over four to five orders of magnitude of scales (13 to 20 octaves). Such high-resolution pictures of two-dimensional vortex dynamics have been facilitated by and indeed depend critically upon a nonlocal adaptive node adjustment scheme, and a variety of tests quantify the accuracy of the technique.

Multiple Jets as PV Staircases: The Phillips Effect and the Resilience of Eddy-Transport Barriers

Abstract A review is given that focuses on why the sideways mixing of potential vorticity (PV) across its background gradient tends to be inhomogeneous, arguably a reason why persistent jets are commonplace in planetary atmospheres and oceans, and why such jets tend to sharpen themselves when disturbed. PV mixing often produces a sideways layering or banding of the PV distribution and therefore a corresponding number of jets, as dictated by PV inversion. There is a positive feedback in which mixing weakens the “Rossby wave elasticity” associated with the sideways PV gradients, facilitating further mixing. A partial analogy is drawn with the Phillips effect, the spontaneous layering of a stably stratified fluid, in which vertically homogeneous stirring produces vertically inhomogeneous mixing of the background buoyancy gradient. The Phillips effect has been extensively studied and has been clearly demonstrated in laboratory experiments. However, the “eddy-transport barriers” and sharp jets characteristic o...

Quantification of the inelastic interaction of unequal vortices in two‐dimensional vortex dynamics

The interaction of two isolated vortices having uniform vorticity is examined in detailed contour dynamics calculations, and quantified using a diagnostic that measures the coherence of the final state. The two vortices have identical vorticity, leaving two basic parameters that determine the evolution: the radius ratio and separation distance. It is found that the term ‘‘vortex merger’’ inadequately describes the general interaction that takes place. Five regimes are found: (1) elastic interaction, (2) partial straining‐out, (3) complete straining‐out, (4) partial merger, and (5) complete merger. Regime 5 is what used to be called ‘‘merger,’’ but occurs in less than one‐quarter of the parameter space. Contrary to popular belief, inelastic vortex interactions (IVI’s) do not always lead to vortex growth. In fact, in over half of the parameter space, smaller vortices are produced. These results bring into question commonly accepted ideas about nearly inviscid two‐dimensional turbulence.

The stability and energetics of corotating uniform vortices

Equilibrium shapes of two-dimensional rotating configurations of uniform vortices are numerically calculated for two to eight corotating vortices. Additionally, a perturbation series is developed which approximately describes the vortex shapes. The equilibrium configurations are subjected to a linear stability analysis. This analysis both confirms existing results regarding point vortices and shows that finite vortices may destabilize via a new form of instability derived from boundary deformations. Finally, we examine the energetics of the equilibrium configurations. We introduce a new energy quantity called ‘excess energy’, which is particularly useful in understanding the constraints on the evolution of unstable near-equilibrium configurations. This theory offers a first glance a t nonlinear stability. As an example, the theory explains some features of the merger of two vortices.

A general theory for two-dimensional vortex interactions

A general theory for two-dimensional vortex interactions is developed from the observation that, under slowly changing external influences, an individual vortex evolves through a series of equilibrium states until such a state proves unstable. Once an unstable equilibrium state is reached, a relatively fast unsteady evolution ensues, typically involving another nearby vortex. During this fast unsteady evolution, a fraction of the original coherent circulation is lost to filamentary debris, and, remarkably, the flow reorganizes into a set of quasi-steady stable vortices. The simplifying feature of the proposed theory is its use of adiabatic steadiness and marginal stability to determine the shapes and separation distance of vortices on the brink of an inelastic interaction. As a result, the parameter space for the inelastic interaction of nearby vortices is greatly reduced. In the case of two vortex patches, which is the focus of the present work, inelastic interactions depend only on a single parameter : the area ratio of the two vortices (taking the vorticity magnitude inside each to be equal). Without invoking adiabatic steadiness and marginal stability, one would have to contend with the additional parameters of vortex separation and shape, and the latter is actually an infinitude of parameters.

Vortex stripping and the erosion of coherent structures in two‐dimensional flows

This paper studies the erosion of a monotonically distributed vortex by the joint action of inviscid stripping, induced by an externally imposed adverse shear, and viscous diffusion, either in the form of Newtonian viscosity or hyperviscosity. It is shown that vortex erosion is greatly amplified by the presence of diffusion; abrupt vortex breakup or gradual quasi‐equilibrium evolution depend crucially on the strain to peak vorticity ratio and on the Reynolds number. Peculiar, unexpected effects are observed when hyperviscosity is used in place of Newtonian viscosity.

The nonlinear evolution of rotating configurations of uniform vorticity

The nonlinear evolution of perturbed equilibrium configurations of constant-vorticity vortices is calculated. To illustrate a variety of nonlinear behaviour, we consider the following relatively simple configurations: the corotating configurations of N vortices whose linear stability has been treated in a previous study; the elliptical vortex; and the annular vortex. Our calculations test for non linear stability as well as categorize the possible forms of stability and instability. The energy ideas announced in the previous study are found to greatly constrain vortex evolution. In particular, we show that two vortices and an elliptical vortex may evolve into each other, and that an annular vortex may break cleanly into five co-rotating vortices.

The stability of a two-dimensional vorticity filament under uniform strain

The quantitative effects of uniform strain and background rotation on the stability of a strip of constant vorticity (a simple shear layer) are examined. The thickness of the strip decreases in time under the strain, so it is necessary to formulate the linear stability analysis for a time-dependent basic flow. The results show that even a strain rate γ (scaled with the vorticity of the strip) as small as 0.25 suppresses the conventional Rayleigh shear instability mechanism, in the sense that the r.m.s. wave steepness cannot amplify by more than a certain factor, and must eventually decay. For γ < 0.25 the amplification factor increases as γ decreases; however, it is only 3 when γ e 0.065. Numerical simulations confirm the predictions of linear theory at small steepness and predict a threshold value necessary for the formation of coherent vortices. The results help to explain the impression from numerous simulations of two-dimensional turbulence reported in the literature that filaments of vorticity infrequently roll up into vortices. The stabilization effect may be expected to extend to two- and three-dimensional quasi-geostrophic flows.

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 highresolution 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’.