The structure, stability and interaction of geophysical vortices
Lead Research Organisation:
University of St Andrews
Department Name: Mathematics and Statistics
Abstract
Vortical structures or swirling masses of fluid abound in the Earth's atmosphere and oceans, environments strongly influenced by both the stable vertical density stratification and the planetary rotation. These structures, or vortices, exist over a wide range of spatial scales and generally at huge Reynolds numbers. Vortices are conspicuous features of planetary atmospheres in general, and they are believed to play a central role in shaping planetary-scale circulations. Their interactions can be extraordinarily complex, and to this day we have virtually no understanding how these interactions contribute toward the collective motion of the atmosphere and the oceans as a whole. The main objectives of this proposal is to provide a complete description of vortex stability and vortex interactions in geophysical flows.The planetary rotation is important when its associated vorticity is comparable to or larger than the relative vertical vorticity. This can be expressed by saying that the Rossby number Ro, the ratio of relative to planetary vorticity, is small compared to 1. On the other hand, the stratification is important when the buoyancy frequency is larger than the horizontal vorticity. Likewise this can be expressed by saying that the Froude number Fr, the ratio of horizontal vorticity to buoyancy frequency, is small. When the Froude number is smaller than or comparable to the Rossby number, itself small, the system of governing equation can be greatly simplified. Then, the flow is well modelled by the `quasi-geostrophic' (QG) equations. The QG model has enabled a comprehensive exploration of basic vortex dynamics, from isolated vortex equilibria and stability to 3D QG turbulence. We now understand why vortices in the QG model tend to be robust, how they react to external shear and strain, what precipitates strong interactions between both like-signed and opposite-signed vortices (potential vorticity anomalies), as well as general properties of vortex populations in turbulence. The important question is: how well do these results apply to finite Ro and Fr? Or, how are the QG results altered when using the full equations of motion? We intend to answer these fundamental questions.When Ro and Fr are not small, two new features arise. The first is the appearance of Inertia-Gravity Waves (IGWs) at super-inertial frequencies, i.e. at frequencies larger than those associated with the vortical motion. The IGWs often induce weak motions compared to those induced by the vortices, and therefore IGWs tend to be of secondary importance in many circumstances. Another feature arising from finite Ro and Fr is the added contribution of ageostrophic motions, which are missing in the QG model. Ageostrophic motions differ from the high frequency IGWs in that they are directly associated with vortices: they are generated in response to the instantaneous potential vorticity (PV) distribution and have a direct advective effect on the PV evolution. The retention of ageostrophic motion thus has a significant impact on the flow and implies potentially large departures from QG dynamics. It is this important difference between the full equations and the QG approximation that underlies the proposed work. We aim to understand and quantify these effects on vortex motions and vortex stability properties.
Publications
Bersanelli M
(2016)
Models of interacting pairs of thin, quasi-geostrophic vortices: steady-state solutions and nonlinear stability
in Geophysical & Astrophysical Fluid Dynamics
Boatto S
(2016)
N-body dynamics on closed surfaces: the axioms of mechanics.
in Proceedings. Mathematical, physical, and engineering sciences
Dritschel D
(2015)
Effect of Prandtl's ratio on balance in geophysical turbulence
in Journal of Fluid Mechanics
Dritschel D
(2017)
Comparison of variational balance models for the rotating shallow water equations
in Journal of Fluid Mechanics
Dritschel D
(2015)
The motion of point vortices on closed surfaces
in Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Dritschel DG
(2020)
Equilibria and stability of four point vortices on a sphere.
in Proceedings. Mathematical, physical, and engineering sciences
McKiver W
(2016)
Balanced solutions for an ellipsoidal vortex in a rotating stratified flow
in Journal of Fluid Mechanics
Plotka H
(2012)
Quasi-geostrophic shallow-water vortex-patch equilibria and their stability
in Geophysical & Astrophysical Fluid Dynamics
Plotka H
(2013)
Quasi-geostrophic shallow-water doubly-connected vortex equilibria and their stability
in Journal of Fluid Mechanics
Reinaud J
(2018)
The merger of geophysical vortices at finite Rossby and Froude number
in Journal of Fluid Mechanics
| Description | The research aimed to study the effects of finite Froude and Rossby numbers on the stability of vortices and on their interactions. These are relevant to stably stratified fluid environments submitted to a background rotation such as the Earth's atmosphere and oceans or the giant gas planets. Vortices are extremely important in these flows. For example, recent estimates suggest that vortices drive, collectively, a large portion of the transport of mass in the oceans [1]. Understanding how these vortices behave and interact is therefore crucial to our understanding of the dynamical atmosphere and oceans. More importantly, such structures are not resolved by General Circulation Models but need to be parametrised. Our understanding of the dynamics of such vortices can be used by modellers. The objectives of our research project have been achieved. The first part of the research was dedicated to the study of the stability of isolated vortices. Prior to our research, the stability of three-dimensional vortices had only been investigated in the limit of vanishing Froude and Rossby number, a regime known as the Quasi-Geostrophic regime. In this regime, exact steady solutions for ellipsoidal vortices of uniform potential vorticity are known. Early investigations of such equilibria by Zhmur and Shchepetkin (1991) [2] and Meacham (1992) [3] have been extended by Dritschel, Scott and Reinaud (2005) [4]. It should be noticed, that in the quasi-geostrophic limit, there is no dynamical asymmetry between a cyclonic and an anticyclonic vortex. They behave in exactly the same way, simply rotating in opposite directions. The dynamical asymmetry between cyclonic and anticyclonic vortices arises from ageostrophic effects at finite Rossby and Froude number. Additionally, at finite Rossby and Froude number, vortices may radiate inertia-gravity waves. By doing so, vortices lose energy, and can consequently deform. Plotka and Dritschel (2014) [5] however found quasi-equilibria in the full Shallow Water model, where vortices only radiate very small amplitude waves, and nearly conserve their shape. We extended the concept to three-dimensional, continuously stratified, rotating flow. The first main output of the research is published in the paper "Ellipsoidal vortices in rotating stratified fluids: beyond the quasi-geostrophic approximation" by Tsang, Y.-K. (PDRA) and Dritschel, D.G. (Co-Inv) [6]. The research considers the evolution of the upright ellipsoid of uniform potential vorticity. The shape ellipsoid of the ellipsoid is characterised initially by two aspect ratios. In the limit of vanishing Rossby and Froude number, the ellipsoidal vortex is a steadily rotating solution. By carefully initialising the finite Rossby number vortex, one can minimise the amount of `imbalance' and obtain a quasi-equilibrium. Depending on the values of the two aspect ratios and of the vortex Rossby number, it was shown that the vortices may (i) be quasi-stable; (ii) change their shape and reach a quasi-equilibrium, (iii) shed some potential vorticity to become a binary system (2 vortices), (iv) oscillate in shape or (v) tumble. The authors have classified the vortex evolution into these five categories over a large parameter space. Key Findings: A) For a given shape and given potential vorticity (in absolute value), anticyclones rotate faster than cyclones. This is due to the first order ageostrophic correction. B) Anticyclonic vortices are less stable than their cyclonic counterparts. C) The difference in behaviour between cyclonic and anticyclonic vortices mainly comes from ageostrophic effects rather than the emission of inertia-gravity waves. The latter remains small. The second main output is published in the paper "Balanced solutions for an ellipsoidal vortex in a rotating stratified flow" by McKiver, W.J. (PDRA) and Dritschel, D.G. (Co-Inv) [7]. This research builds on the previous paper by finding, analytically, the first order ageostrophic correction for the motion of uniform potential vorticity ellipsoids, while filtering inertia-gravity waves. Compared with the full dynamics (which includes inertia-gravity waves), it proves highly accurate for an arbitrarily orientated ellipsoid at moderate Rossby number. This confirms that waves play little role in the evolution of the vortex. The main influence of the finite Rossby number can be felt from the first order ageostrophic correction. The third main output is published in the paper "The merger of geophysical vortices at finite Rossby and Froude number" by Reinaud, J.N. (PI) and Dritschel, D.G. (Co-Inv) [8]. The paper considers, for the first time, the interaction of two co-rotating vortices in quasi-equilibrium at finite Rossby number. Special attention is paid to the strong form of the interaction where vortices merge to form a larger main structure. As in the previous problem, the initial conditions stem from exact equilibria in the limit of vanishing Rossby number. The equilibria are then adapted to the targeted finite Rossby number by a procedure which minimises the amount of waves radiated by the vortices. Contrarily to the single ellipsoidal vortex problem, the equilibria are not trivial and must be first obtained numerically. We first determined the equilibria in the quasi-geostrophic regime at unprecedented resolution. We then investigated (i) the effect of the Rossby number on the interaction and (ii) the effect of a vertical offset between the vortices. The latter is important because vertical velocities are typically much smaller than the horizontal velocities. There is a fundamental anisotropy in the flow. Key Findings: D) Horizontally aligned cyclones are able to merger from further apart than cyclones. This is due to ageostrophic effects which make two cyclones move toward each other, while two anticyclones move apart. These trends are related to a transfer of energy from the leading-order QG-`balanced' part of the flow to ageostrophic energy. E) Interacting anticyclonic vortices are sensitive to vertical shear. This shear is enhanced by offsetting the vortices vertically. Hence, vertically offset anticyclones may deform more than their cyclonic counterparts. As a consequence, vertically offset anticyclones may merge from further away than cyclones do. We have also generated equilibrium states for opposite-signed vortices at unprecedented resolution in the limit of vanishing Rossby and Froude numbers. This will allows us to extend the research in the future, and consider the interaction of two opposite-signed vortices at finite Rossby and Froude number. Such structures are particularly relevant to the transport of quantities (energy, momentum, heat) in the oceans, since such structures can self-propagate over substantial distances (e.g. across an ocean basin). Several other outputs addressed related fundamental properties of vortex interactions, the role of gravity waves, and generalisations to other physical systems. In Plotka and Dritschel (2012,2013) [5,9], the equilibrium shapes of vortices together with their stability were studied in the context of the single-layer quasi-geostrophic model. A comprehensive analysis covering a wide parameter space was used to determine marginal stability and the nature of instabilities, such as re-equilibration near another known stable solution. Key Findings: F) The free-surface elasticity, as measured by the Rossby deformation length LD, plays a profound role. G) When LD is small compared to a typical vortex radius, evolution rates slow down dramatically and interactions become `elastic', i.e. with virtually no loss of material, in stark contrast to the much studied large LD limit. H) New persistently oscillating states appear. In Bersanelli, Dritschel, Lancellotti and Poje [10], the first ever study of equilibrium states for pairs of surface or near-surface buoyancy lenses was conducted. Key findings: I) The development of a reduced model describing the lenses as ellipses. J) Forms of the equilibrium states for two lenses in different layers, as well as a full linear stability analysis and investigation of the nonlinear evolution of unstable equilibria. In Dritschel and Boatto [11] and in Boatto and Dritschel [12], it is shown how to extend well-known concepts for two-dimensional flows on the plane to general curved surfaces. This is relevant to flows in planetary atmospheres, which are typically of oblate spheroidal shape. Key findings: K) First consistent formulation of the equations governing point vortices on a general closed surface. Similar results obtained for self-gravitating masses. L) Vortex motion on even slightly oblate surfaces (characteristic of all planets in the solar system) is fundamentally different than on perfectly spherical surfaces (as often considered for simplicity). In particular, the oblateness destabilises vortex configurations. In Dritschel, Gottwald and Oliver (2017) [13], a systematic study of a class of balanced models (models which filter gravity waves) was conducted. A long-standing problem in the field is to identify appropriate or optimal models for describing the dominant, relatively low-frequency dynamics of rotating stratified flows. These dynamics are thought to be largely controlled by the distribution of potential vorticity (PV), and the precise way PV controls the flow is through so-called `inversion relations'. These are the mathematical expressions of geostrophic and hydrostatic balance, but they are not unique, and herein lies the problem. There are as a consequence very many balance models which purport to describe the low frequency dynamics, but it is difficult to quantify which is optimal. In this paper, we first construct a class of balance models and show how to choose which one is optimal. Key findings: M) There is a most-accurate balance model among a class of carefully-constructed models, all of which possess the complete set of conservation laws of the original equations. N) It is essential to consider fields like the ageostrophic vorticity (or acceleration divergence) to understand why some models poorly represent all scales consistently. In particular, some models can be ill-posed in the sense that derivatives do not converge. Meanwhile Reinaud (PI) continued to work on vortex interactions for quasi-geostrophic vortices with international co-authors, publishing another 18 papers in the field between 2015 and 2019. Bibliography: [1] Z. Zhang, W. Wang, and B. Qiu. "Oceanic mass transport by mesoscale eddies", Science, vol. 345, 2014, pp. 322-324. [2] Zhmur, V.V. and Shchepektin A.F. ,"Evolution of an ellipsoidal vortex in a stratified ocean: survivability of the vortex in flow with vertical shear", Izv. Akad. Nauk SSSR Atmos. Ocean. Phys., vol. 27, 1991, pp. 492-503. [3] Meacham S.P, "Quasigeostrophic, ellipsoidal vortices in a stratified flow", Dyn. Atmos. Oceans, vol. 16, 1992, pp. 189-223. [4] Dritschel, D.G., Scott, R.K. & Reinaud, J.N. The stability of quasi-geostrophic ellipsoid vortices", J. Fluid Mech. 536, 2005, pp. 401-421. [5] Plotka, H. and Dritschel, D.G.: Quasi-geostrophic shallow-water doubly-connected vortex equilibria and their stability", J. Fluid Mech,, vol. 723, 2013, pp. 40-68. [6] Tsang, Y.-K. and Dritschel, D.G. "Ellipsoidal vortices in rotating stratified fluids: beyond the quasi-geostrophic approximation", J. Fluid Mech., vol. 762, 2015, pp. 196-231. [7] McKiver, W.J. and Dritschel, D.G. "Balanced solutions for an ellipsoidal vortex in a rotating stratied flow", J. Fluid Mech. 802, 2016, pp. 333-358. [8] Reinaud, J.N. and Dritschel, D.G. "The merger of geophysical vortices at finite Rossby and Froude number", J. Fluid Mech. 848, 2018, pp. 388-410. [9] Plotka, H. and Dritschel, D.G., "Quasi-geostrophic shallow-water vortex-patch equilibria and their stability", Geophys. Astrophys. Fluid Dyn., vol. 106(6), 2013, pp. 574-595. [10] Bersanelli, M., Dritschel, D.G., Lancellotti, C. and Poje, A.C., "Models of interacting pairs of thin, quasi-geostrophic vortices: steady-state solutions and nonlinear stability", Geophys. Astrophys. Fluid Dyn., vol. 110(6), 2016, pp. 491-517. [11] Dritschel, D.G. and Boatto, S., "The motion of point vortices on closed surfaces", Proc. Roy. Soc. A vol. 471, 2015, 20140890. [12] Boatto, S. and Dritschel, D.G., "N-body dynamics on closed surfaces: the axioms of mechanics", Proc. Roy. Soc. A vol. 472, 2015, 20160020. [13] Dritschel, D.G., Gottwald, G.A. and Oliver, M. "Comparison of variational balance models for the rotating shallow water equations", J. Fluid Mech. vol. 822, 2017, pp. 689-716. |
| Exploitation Route | The research projet was theoretical by nature. Its outcome is however of interest for meteorologists and oceanographers, and in particular those involved in parametrisation. This is therefore of interest for governmental bodies such as the Met Office and commercial companies offering numerical weather prediction. Operational numerical weather prediction models and general circulation models cannot solve small scale features in the atmosphere and the oceans. The overall effects of these unresolved small scale on the large scales which can be resolved have to be parametrised. To be accurately parametrised, modellers need to have a physical insight on how these feature behave. Our research helps buid a database of knowledge on how the unresolved scale act in the flow. |
| Sectors | Environment,Other |
| Description | Modelling vortex motion in the Gulf of Mexico |
| Organisation | City University of New York (CUNY) |
| Country | United States |
| Sector | Academic/University |
| PI Contribution | Our modelling approaches were extended to study near surface vortices and mixing of tracers in the Gulf of Mexico, through a new collaboration with Professor Andrew Poje at CUNY, New York. Collaboration was started to examine the transport of tracers in and around cyclonic vortices in the Gulf of Mexico. Using an idealised model developed in St Andrews, and related to that used in the present grant, we were able to analyse general properties of the mixing, both horizontally and vertically, due to vortex motion, deformation and interaction. This is an ongoing collaboration which will extend well beyond the current grant. |
| Start Year | 2011 |
| Description | Systematic initialization for the semigeostrophic scaling |
| Organisation | University of Sydney |
| Country | Australia |
| Sector | Academic/University |
| PI Contribution | This research is a long-running fruitful collaborative effort involving Professors Georg Gottwald (Syndey University) and Marcel Oliver (Jacobs University, Bremen). This research takes a novel approach to deriving reduced "balanced" models useful for approximating the complex dynamics of shallow-water flows in geophysical fluid dynamics. See also "in progress outputs". Professor Gottwald is an international expert on dynamical systems, with application to geophysical flows. He has provided valuable knowledge and understanding to the project, and has independently collaborated extensively with the other collaborator, Professor Marcel Oliver. |
| Start Year | 2004 |
| Description | Banded planetary circulation patterns: Observations, simulations and theory |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | Yes |
| Primary Audience | |
| Results and Impact | Popular talk given to physics student, researchers and senior academics discussing research on geophysical fluid flows, part of which was carried out during the present grant. A general popular review of research carried out on the dynamics of geophysical flows. This was aimed at an audience with little or no background in the subject, and was intended to inspire students and early career researchers to consider a career in fluid dynamics. |
| Year(s) Of Engagement Activity | 2011 |
| Description | Dundee Science Festival (Family fun days) |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | Yes |
| Primary Audience | |
| Results and Impact | Running an exhibit on fluid dynamics for the public at "Sensations" in Dundee. A series of experiments and videos were presented to the general public to show them the fascination of fluid dynamics |
| Year(s) Of Engagement Activity | 2011 |
| Description | Fife Schools Science Fair |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | Yes |
| Primary Audience | |
| Results and Impact | This was a one-day exhibition targeting local science students, showing them the kind of research on fluid dynamics we do in St Andrews. A series of presentations were given to groups of students and their teachers from a variety of local schools. The students were also allowed to participate in various fluid dynamical experiments. Written materials were provided for them to take away. |
| Year(s) Of Engagement Activity | 2012 |
| Description | Fife Science Festival (Science Discovery day) 2012 |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | Yes |
| Primary Audience | |
| Results and Impact | This was a one-day science fair open to the general public. The vortex dynamics research group led by Prof D Dritschel ran a series of fluid dynamical experiments, showed videos and explained the science behind fluid behavior, in particular as applied to the Earth's atmosphere and oceans. Over 2000 people attended throughout the day. Public exhibition discussing fluid dynamical research to a large public group |
| Year(s) Of Engagement Activity | 2012 |
| Description | Fife Science Festival (Science Discovery day) 2013 |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | Yes |
| Primary Audience | |
| Results and Impact | This was a one-day science fair open to the general public. The vortex dynamics research group led by Prof D Dritschel ran a series of fluid dynamical experiments, showed videos and explained the science behind fluid behavior, in particular as applied to the Earth's atmosphere and oceans. Over 2000 people attended throughout the day. This was a public exhibition discussing our research in fluid dynamics to a large audience |
| Year(s) Of Engagement Activity | 2012 |