Introduction

This special issue includes contributions from many of the 34 participants of the “International Conference on Geophysical and Astrophysical Vortex Interactions”, which took place from 11 to 14 June 2019 in St Andrews, UK. The contributions exhibit a diverse range of research in this specialised area, which has grown significantly over the past two decades due to applications to both geophysical flows (e.g. atmospheres and oceans) and astrophysical flows (e.g. magneto-hydrodynamics).

Such flows are typically characterised by enormous Reynolds numbers, i.e. exceedingly weak viscous dissipation. This favours turbulence with activity over a wide range of scales, and also the formation and persistence of a multitude of coherent structures, such as vortices, fronts and jets. These structures have been the subject of research for decades, going back to the early work of Helmholtz, Kelvin and Thomson in the nineteenth century.

Remarkably, a precise definition of such structures is generally lacking, and instead pragmatic solutions must be found. If a vortex contains vorticity or “potential vorticity” (the relevant tracer for a rotating stratified flow), then what criterion should mark its boundary? There are many proposals, but no universally adopted definition. Despite this, our understanding of vortex dynamics and interactions has progressed unabated. A precise definition is often not needed –only one that is “sensible” for the context under study, one that suits the quantitative analysis carried out to achieve the goal of the investigation.

Another remarkable feature of geophysical and astrophysical flows is the prevalence of structure having diverse forms – not just vortices but fronts and jets. This feature arises from fundamental anisotropies associated with background gradients of density, of potential vorticity and of magnetic fields. Such gradients support waves of various types, and these waves have the propensity to steepen and break in the nonlinear dynamics, leading to mixing and mean flow modifications. Surprisingly, this mixing underlies the emergence, and maintenance, of coherent structures in complex nonlinear flows. One may follow the latest in the article by M.E. McIntyre and references therein. Striking examples of jet emergence in turbulent quasi-geostrophic shallow-water flows are provided in the article by B.H. Burgess, who furthermore develops a scaling theory for the late-time dynamics in the inviscid limit.

The concept of “balance”, particularly in geophysical fluid dynamics, also stems from the anisotropies naturally present in these flows. In a balanced flow, certain waves are “slaved” to lower-frequency motions associated with the advection of nearly materially conserved dynamical tracers like potential vorticity. These “balanced” motions are considered devoid of higher-frequency waves. Theoretically, however, this is impossible except in cases where the low-frequency flow is exactly steady. Recent developments in the theory of balance may be found in the article by G.T. Masur and M. Oliver. They show just how close one can reach the theoretical ideal of balance.

Balance in not just a theoretical concept, it is immensely useful practically. Balance enables one to simplify the governing equations of motion (often greatly) by filtering faster waves in some manner (here there are many choices leading to many models). The first successful weather forecast by Jule Charney in the 1950s employed “quasi-geostrophic” balance after recognising that earlier attempts by Lewis Richardson in the 1920s failed due to the difficulties of accurately modelling high-frequency wave motions. Since then, much research has been conducted to develop and employ balance models which more accurately capture the dynamics of more complete, wave-permitting models. The article by W.J. McKiver discusses an application of balance models to accurately model the dynamics of an isolated vortex in a three-dimensional rotating stratified flow, while the article by A. Orozco Estrada et al. exhibits internal gravity wave radiation (partial loss of balance or adjustment to a new balanced state) occurring in the merger of anti-cyclonic lens-shaped vortices in the laboratory.

Basic studies of vortex dynamics in a variety of contexts can be found throughout this special issue. For example the article by J.N. Reinaud and X. Carton considers the basic process of vertical alignment of three-dimensional vortices in a rotating, stratified (quasi-geostrophic) flow. This has been extensively studied in simpler models (e.g. two layers), but never comprehensively in three dimensions. Vertical alignment is like vortex merger except the vortices are confined to distinct, non-overlapping height ranges. The article by M.M. Jalali and D.G. Dritschel examines the many diverse ways two oppositely signed vortices may interact in a quasi-geostrophic shallow-water flow. The problem is complicated by the fact that it depends on the vortex size and strength ratio, the Rossby deformation length and the proximity of the two vortices. When they are close enough, they typically destabilise and break down into more complex structures.

Several articles address the important problem of vortex formation, primarily in an oceanic context. We understand that vortices may emerge from a turbulent background by merging with others of like sign, but in realistic contexts one must also consider the impact of the β-effect, large-scale flows, topography, coastlines, etc. Many of these influences are examined in the observational/numerical study of M. Morvan et al. following the life cycle of meso-scale vortices in the Gulf of Aden. This is complemented by a pair of theoretical studies, one by C. de Marez et al. examining how deep sub-meso-scale vortices may be generated by interactions with a coast (idealised as a vertical wall) on the β-plane, and another by M. Morvan et al. examining how a descending density current can destabilise and form meso-scale and sub-meso-scale eddies. Finally, the article by D. Stepanov et al. discusses the way that an unsteady oceanic meso-scale eddy field organises surface tracers into tight clusters and ribbons, as commonly observed after oil spills.

The intrinsically nonlinear nature of vortex interactions, together with the multitude of factors influencing them in geophysical and astrophysical flows, makes their study challenging. This can be regarded as an opportunity to improve our understanding of our physical environment and, potentially, to create positive change. Advances in computer power have greatly aided this research by enabling increasingly realistic simulations. But this special issue emphasises that theory still plays a critical role in this research. It is important to break down, reduce, distil, experiment, create new models and thereby learn.

We gratefully acknowledge support from the (EPSRC) UK Fluids Network special interest group on multi-scale processes in geophysical fluid dynamics.