On the nonlinear evolution of columnar vortices in a rotating environment

We examine the three-dimensional, nonlinear evolution of columnar vortices in a rotating environment. As the initial vorticity distribution, a wavetrain of finite amplitude Kelvin-Helmholtz vortices in shear is employed. Through direct numerical simulation of the Navier-Stokes equations we seek to better understand the process of maturation of the various three-dimensional modes of instability to which such vortical flows are subject, especially those which exist as a consequence of the action of the Coriolis force. In the absence of rotational influence, we thereby demonstrate that the nonlinear evolution of columnar vortices is most strongly controlled by one or the other of two mechanisms. One mechanism of instability is identifiable as a so-called elliptical instability, which promotes the initial bending of vortex tubes in a sinusoidal fashion, while the other is a hyperbolic mode, which is responsible for the development of streamwise vortex streaks in the "braids" between adjacent vortex cores. In the rotating case, anticyclonic vortices are strongly destabilized by weak background rotation, while rapid rotation stabilizes both the cyclones and anticyclones. The strong anticyclones are subject to two distinct forms of instability, namely a Coriolis force modified elliptical instability and an inertial (centrifugal) instability. The former instability is very similar to the nonrotating form of the elliptical instability as it promotes bending of vortex tubes, while the latter instability grows on the edge of the vortex core and generates streaks of vorticity, which surround the vortex core itself. These results of direct numerical simulation fully verify the results of previous linear stability analyses. Taken together, they provide a simple explanation for the broken symmetry that is often observed to be characteristic of the von Karman vortex streets that develop in the atmospheric lee of oceanic islands.     

Vortex evolution due to straining: a mechanism for dominance of strong,interior anticyclones

In this article we address two questions: Why do freely evolving vortices weaken on average, even when the viscosity is very small? Why, in the fluid's interior, away from vertical boundaries and under the influence of Earth's rotation and stable density stratification, do anticyclonic vortices become dominant over cyclonic ones when the Rossby number and deformation radius are finite? The context for answering these questions is a rotating, conservative, Shallow-water model with Asymmetric and Gradient-wind Balance approximations. The controlling mechanisms are vortex weakening under straining deformation (with a weakening that is substantially greater for strong cyclones than strong anticyclones) followed by a partially compensating vortex strengthening during a relaxation phase dominated by Vortex Rossby Waves (VRWs) and their eddy–mean interaction with the vortex. The outcome is a net, strain-induced vortex weakening that is greater for cyclones than anticyclones when the deformation radius is not large compared to the vortex radius and the Rossby number is not small. Furthermore, when the exterior strain flow is sustained, the vortex changes also are sustained: for small Rossby number (i.e., the quasigeostrophic limit, QG), vortices continue to weaken at a relatively modest rate, but for larger Rossby number, cyclones weaken strongly and anticyclones actually strengthen systematically when the deformation radius is comparable to the vortex radius. The sustained vortex changes are associated with strain-induced VRWs on the periphery of the mean vortex. It therefore seems likely that, in a complex flow with many vortices, anticyclonic dominance develops over a sequence of transient mutual straining events due to the greater robustness of anticyclones (and occasionally their net strengthening).     

Cyclonic-anticyclonic asymmetry and a new soliton concept for rossby vortices in the laboratory,oceans and the atmospheres of giant planets

Abstract

It is demonstrated in laboratory experiments with rotating shallow water that large scale Rossby vortices, greater than the Rossby-Obukhov radius in size, have dispersive and non-linear properties that are fundamentally different for the two possible polarities. We call this “cyclonic-anticyclonic asymmetry”. This asymmetry manifests itself in the following way: first, anticylones, unlike cyclones, do not undergo the dispersive spreading inherent in a linear wave packet. and therefore, having a considerably longer natural lifetime, are obvious candidates for Rossby solitons; second, dipolar vortices are, because of the comparatively rapid decay of a cyclone, transformed into anticyclonic solitons; third, anticyclones are much more readily generated by zonal flows of the type existing in planetary atmospheres. The evident dominance of anticyclones amongst the long-lived vortices in the atmospheres of giant planets strongly suggests that the cyclonic-anticyclonic symmetry plays a decisive role in the atmospheric cyclogenesis of large planets.

According to our concept, the Rossby soliton is a “real” vortex; unlike a wave, it retains some fluid particles within it throughout its lifetime. Two similar solitons can merge by mutual collisions. This picture of a “vortical” soliton differs in an essential way from the earlier idea due to Maxworthy and Redekopp (1976) of purely “wave-like” Rossby solitons that can freely pass through one another.

Laboratory experiments were performed by us to simulate the new Rossby solition, with special reference to naturally-occurring vortices of the same general type as Jupiter's great red spot. The experimental data presented contradict the idea of “pure wave solitons” but confirm our concept of “vortical solutions”.     

An experimental study of global instabilities due to the tidal (elliptical) distortion of a rotating elastic cylinder

Abstract

Broad band secondary instability of elliptical vortex motion has been proposed as a principal source of shear-flow turbulence. Here experiments on such instability in an elliptical flow with no shear boundary layer are described. This is made possible by the mechanical distortion in the laboratory frame of a rotating fluid-filled elastic cylinder. One percent ellipticity of a 10 cm diameter cylinder rotating once each second can give rise to an exponentially-growing mode stationary in the laboratory frame. In first order this mode is a sub-harmonic parametric Faraday instability. The finite-amplitude equations represent angular momentum transfer on an inertial time scale due to Reynolds stresses. The growth of this mode is not limited by boundary friction but by detuning and centrifugal stabilization. On average, a generalized Richardson number achieves a marginal value through much of the evolved flow. However, the characteristic flow is intermittent with the cycle: rapid growth, stabilizing momentum transfer from the mean flow, interior re-spin up, and then again. Data is presented in which, at large Reynolds numbers, seven percent ellipticity causes a fifty percent reduction in the kinetic energy of the rotating fluid. In the geophysical setting, this tidal instability in the earth's interior could be inhibited by sub-adiabatic temperature gradients. A near adiabatic region greater than 10 km in height would permit the growth of tidally destabilized modes and the release of energy to three-dimensional disturbances. Such disturbances might play a central role in the geodynamo and add significantly to overall tidal dissipation.     

Slow quasigeostrophic unstable modes of a lens vortex in a continuously stratified flow

This work addresses the linear dynamics underlying the formation of density interfaces at the periphery of energetic vortices, well outside the vortex core, both in the radial and axial directions. We compute numerically the unstable modes of an anticyclonic Gaussian vortex lens in a continuously stratified rotating fluid. The most unstable mode is a slow mode, associated with a critical layer instability located at the vortex periphery. Although the most unstable disturbance has a characteristic vertical scale which is comparable to the vortex height, interestingly, the critical levels of the successively fastest growing modes are closely spaced at intervals along the axial direction that are much smaller than the vortex height.     

Laboratory experiments on fronts

Abstract

Supercritically unstable density fronts near a vertical wall in a rotating, two-layer fluid were created on a laboratory turntable by withdrawing the outer wall of an annulus with a narrow gap, and allowing buoyant fluid from within the annulus to collapse toward a state of quasi-geostrophic balance. The resulting “coastal” current has a nearly uniform potential vorticity and is bounded by a front on which ageostrophic, wave-like disturbances grow. If the current width is comparable to the Rossby radius of deformation, the dominant length scale of disturbances is proportional to the width of the current. On the other hand, if the upper layer is much wider than the Rossby radius, then the observed length scale is a constant multiple of the Rossby radius. If the vertical boundary is omitted in the experiments, so that we are left with a circular anticyclonic vortex, the observed length scales and large-amplitude behaviour of disturbances are identical to those for the boundary currents, indicating that the wall has no significant influence on the flow.

At very large amplitude the growing waves lead to the formation of cyclone-anticyclone vortex pairs. For very wide currents, both the mean flow and the disturbances are first confined to a region within a few Rossby radii of the front. However, both the mean flow and the turbulent eddy motions slowly propagate into the previously stationary upper layer until, eventually, the whole of the upper layer is turbulent.     

Problems of simulation of large,long-lived vortices in the atmospheres of the giant planets (jupiter,saturn, neptune)

Large, long-lived vortices are abundant in the atmospheres of the giant planets. Some of them survive a few orders of magnitude longer than the dispersive linear Rossby wave packets, e.g. the Great Red Spot (GRS), Little Red Spot (LRS) and White Ovals (WO) of Jupiter, Big Bertha, Brown Spot and Anne's Spot of Saturn, the Great Dark Spot (GDS) of Neptune, etc. Nonlinear effects which prevent their dispersion spreading are the main subject of our consideration. Particular emphasis is placed on determining the dynamical processes which may explain the remarkable properties of observed vortices such as anticyclonic rotation in preference to cyclonic one and the uniqueness of the GRS, the largest coherent vortex, along the perimeter of Jupiter at corresponding latitude.We review recent experimental and theoretical studies of steadily translating solitary Rossby vortices (anticyclones) in a rotating shallow fluid. Two-dimensional monopolar solitary vortices trap fluid which is transported westward. These dualistic structures appear to be vortices, on the one hand, and solitary waves, on the other hand. Owing to the presence of the trapped fluid, such solitary structures collide inelastically and have a memory of the initial disturbance which is responsible for the formation of the structure. As a consequence, they have no definite relationship between the amplitude and characteristic size. Their vortical properties are connected with geostrophic advection of local vorticity. Their solitary properties (nonspreading and stationary translation) are due to a balance between Rossby wave dispersion and nonlinear effects which allow the anticyclones, with an elevation of a free surface, to propagate faster than the linear waves, without a resonance with linear waves, i.e. without wave radiation. On the other hand, cyclones, with a depression of a free surface, are dispersive and nonstationary features. This asymmetry in dispersion-nonlinear properties of cyclones and anticyclones is thought to be one of the essential reasons for the observed predominance of anticyclones among the long-lived vortices in the atmospheres of the giant planets and also among the intrathermocline oceanic eddies.The effects of shear flows and differences between the properties of monopolar vortices in planetary flows and various laboratory experiments are discussed. General geostrophic (GG) theory of Rossby vortices is presented. It differs essentially from the traditional quasi-geostrophic (QG) and intermediate-geostrophic (IG) approximations by the account of (i) all scales between the deformation radius and the planetary scale and (ii) the arbitrary amplitudes of vortices. It is shown that, unlike QG- and IG-models, the GG-model allows for explaining the mentioned cyclonic-anticyclonic asymmetry not only in planetary flows, but also in laboratory modeling with vessels of near paraboloidal form.     

Convection driven by centrifugal bouyancy in a rotating annulus

Abstract

Drift rates and amplitudes of convection columns driven by centrifugal bouyancy in a cylindrical fluid annulus rotating about a vertical axis have been measured by thermistor probes. Conical top and bottom boundaries of the annular fluid region are responsible for the prograde Rossby wave like dynamics of the convection columns. A constant positive temperature difference between the outer and the inner cylindrical boundaries is generated by the circulation of thermostatically controled water. Mercury and water have been used as converting fluids. The measurements extend the earlier visual observations of Busse and Carrigan (1974) and provide quantitative data for an eventual comparison with nonlinear theories of thermal Rossby waves. The measured drift frequencies are in general agreement with linear theory. Of particular interest is the decline of the amplitude of convection with increasing Rayleigh number in a region beyond the onset of convection.     

The evolution of sub-mesoscale,coherent vortices on the β-plane

Abstract

An investigation is made of the evolution of small-scale, axisymmetric vortices in a stratified fluid with spatially variable Coriolis parameter. The criteria for smadness are a horizontal scale less than or equal to the first internal radius of deformation and a vertical scale less than or equal to that of the ambient stratification. These circumstances match those of Sub-mesoscale, Coherent Vortices frequently observed in the oceans. The dynamical model is the balance equations, which include various effects of finite Rossby number. The principal topics are the regime of nearly uniform propagation, the development of an equilibrium ratio of vertical and horizontal scales (i.e., Burger number selection), and the occurrence of various types of instability for vortices with extremes in amplitude or shape.     

Radiating Vortices in Geophysical Fluid Dynamics

The dynamics of a single vortex on a beta-plane is discussed in this paper. A barotropic, an equivalent barotropic, one-and-a half and two-layer models are considered. The momentum and energy balances are used to describe the evolution of a vortex. A quasi-stationary balance of the Rossby, Zhukovsky-Kutta forces and the force induced by Rossby-wave radiation, describes the dynamics of the barotropic vortex. A net Coriolis force occurs if the fluid is stratified. The difference between the dynamics of cyclones and anticyclones results directly from the Coriolis force acting on a single vortex in a stratified fluid.All vortices radiate Rossby waves in the quasigeostrophic approximation but intense anticyclones propagate steadily in a one-and-a half layer model. A critical amplitude that bounds radiating and steadily propagating anticyclones is found. Steady propagation of anticyclones in general is impossible in a two-layer fluid due to the radiation of a barotropic Rossby-wave. Some solutions of solitary wave type which are known for a two-layer model, survive owing to wave interference.A single vortex can extract energy from a Rossby wave if synchronism conditions are satisfied. The wave interference again plays a crucial role in this case. The wave interference also determines the energy exchange of vortices located at larger distances. If the distance between the vortices is shorter than the length of the radiated waves, modon may be formed due to a small energy loss.The unbounded monotonic variation of the planetary vorticity is a characteristic feature of a beta-plane approximation. As a result, a single vortex propagates up to a 'rest latitude' where it disappears. The evolution of a single barotropic vortex over bottom topography provides another example of a background vorticity distribution with a local extremum above hills (valleys) or ridges (troughs). Physics of its movement differs from a beta-plane case, but if a vortex lies over broad topography, equations are similar and the evolution of a vortex manifests the same typical features. Particularly, a cyclonic vortex tends to drift to the top of a hill or a ridge. An anticyclonic vortex, on the contrary, slides to the bottom of a valley or a trough.An interaction of a barotropic vortex with a broad mean flow is tractable qualitatively on the basis of previous results. Numerical examples illustrating absorption of a small vortex by a larger one and a vortex movement across the flow, are direct analogies of the vortex evolution over a hill and a ridge, respectively. At the same time, strong influence of strain drastically changes the vortex structure.     

A mechanism for angular momentum mixing

Abstract

If a contained homogeneous, rotating fluid is forced near to a resonance for elastoid-inertia waves, strong vortices are observed to form. Numerical experiments reported here lend support to the explanation that these are due to a redistribution of the angular momentum by the waves. If the waves grow until the Angular momentum gradient is overturned somewhere, turbulent mixing there make the redistribution irreversible, resulting in a vortex. The process is analogous to the formation of steps in a stratified fluid by breaking internal waves.     

Shear flow driven magnetized planetary wave structures in the ionosphere

The generation and further linear and nonlinear dynamics of planetary ultra-low-frequency (ULF) waves are investigated in the rotating dissipative ionosphere in the presence of inhomogeneous zonal wind (shear flow). Planetary ULF magnetized Rossby type waves appear as a result of interaction of the medium with the spatially inhomogeneous geomagnetic field. An effective linear mechanism responsible for the intensification and mutual transformation of large scale magnetized Rossby type and small scale inertial waves is found. For shear flows, the operators of the linear problem are not self-conjugate, and therefore the eigenfunctions of the problem may not be orthogonal and can hardly be studied by the canonical modal approach. Hence, it becomes necessary to use the so-called nonmodal mathematical analysis. The nonmodal analysis shows that the transformation of wave disturbances in shear flows is due to the non-orthogonality of eigenfunctions of the problem in the conditions of linear dynamics. Using numerical modeling, the peculiar features of the interaction of waves with the background flow as well as the mutual transformation of wave disturbances are illustrated in the ionosphere. It has been shown that the shear flow driven wave perturbations effectively extract an energy of the shear flow increasing the own energy and amplitude. These perturbations undergo self-organization in the form of the nonlinear solitary vortex structures due to nonlinear twisting of the perturbation’s front. Depending on the features of the velocity profiles of the shear flows the nonlinear vortex structures can be either monopole vortices or vortex streets and vortex chains.     

Linear spin-up in a sliced cylinder

Abstract

spin-up and spin-down in a circular tank with a uniformly sloping bottom are studied experimentally and numerically for small values of the relative change in the angular velocity of the tank. Generally, the initial single-cell flow evolves into a number of smaller vortices. The evolution is compared with an analytical model based on an expansion of the flow field in linear Rossby waves (Pedlosky and Greenspan, 1967). Although it is possible to tune the experimental parameters in such a way that agreement with the theory is found, in most cases the experiments show shedding of vortices in the initial stage of the spin-up or spin-down, a phenomenon not described by the analytical model. Nonetheless, in such cases the analytical model still accounts for other observations: the alternating generation of cyclonic and anticyclonic vortices in the eastern part of the tank and their subsequent westward motion.     

Some interactions between small numbers of baroclinic,geostrophic vortices

Abstract

Various interactions between small numbers (two and four) of baroclinic, geostrophic point vortices in a two-layer system are studied with attention to the qualitative changes in behavior which occur as size of the deformation radius is varied.

A particularly interesting interaction, which illustrates the richness of baroclinic vortex dynamics, is a collision between two hetons. (A heton is a vortex pair in which the constituent vortices have opposite signs and are in opposite layers. The “breadth” of a heton is the distance between its constituent vortices. A translating heton transports heat.) When two hetons, which initially have different breadths, collide, the result is either an exchange of partners, or a “slip-through” collision in which the initial structures are preserved. It is shown here that the outcome is always an exchange, provided the deformation radius is sufficiently small. This strongly contrasts with a collision between pairs of classical, one-layer vortices in which no exchange occurs if the initial ratio of the breadths is sufficiently extreme.

Finally the transport of passive fluid by a translating baroclinic pair is investigated. A pair of vortices in the top layer transports no lower layer fluid if the distance between the vortices is less than 1.72 deformation radii. By contrast, the size of the region trapped by a heton increases without bound as the spacing between the vortices increases.     

Barotropic instability of coastal flows as a boundary-value problem: linear and non-linear theory

The barotropic instability is traditionally viewed as an initial-value problem wherein wave perturbations of a laterally sheared flow in a homogeneous uniformly rotating fluid that temporally grows into vortices. The vortices are capable of mixing fluid on the continental shelf with fluid above the continental slope and adjacent deep-sea region. However, the instability can also be viewed as a boundary-value problem. For example, a laterally sheared coastal flow is perturbed at some location, creating perturbations that grow spatially downstream. This process leads to a time periodic flow that exhibits instability in space. This article first examines the linear barotropic instability problem with real frequency and complex wavenumber. It is shown that there exists a frequency band within which a spatially growing wave is present. It is then postulated that far downstream the spatially unstable flow emerges into a chain of identically axisymmetric vortices. Conservation of mass, momentum and energy fluxes are applied to determine the diameter, spacing and the speed of translation of the vortices.     

The interaction of two co-rotating quasi-geostrophic vortices in the vicinity of a surface buoyancy filament

In this paper, we investigate the interaction between two like-signed quasi-geostrophic uniform potential vorticity internal vortices in the vicinity of a surface buoyancy anomaly filament in a three dimensional, stably stratified and rapidly rotating fluid. The surface buoyancy distribution locally modifies the pressure fields and generates a shear flow. We start the study by first considering the effects of a uniform linear horizontal shear on the binary vortex interaction. We confirm that a cooperative shear facilitates the merger of a pair of vortices while an adverse shear has the opposite effect. We next investigate the binary vortex interaction in the vicinity of the surface buoyancy filament explicitly. Here, not only the filament generates a shear flow, but it also responds dynamically to the forcing by the vortex pair. The filament destabilises and forms buoyancy billows at the surface. These billows interact with the internal vortices. In particular, a surface billow may pair with one of the internal vortices. In such cases, the like-signed internal vortex pair may separate if they are initially moderately distant from each other.     

A steady nonlinear and singular vortex Rossby wave within a rapidly rotating vortex. Part II: Application to geophysical vortices

In a previous paper, Caillol [Geophys. Astrophys. Fluid Dyn., 2014, 108] investigated the steady nonlinear vortical structure of a singular vortex Rossby mode that has survived to a strong critical-layer-like interaction with a linearly stable, columnar, axisymmetric and dry vortex. We presented a general theory for this wave/mean flow interaction through the nonlinear critical layer theory and calculated the mean azimuthal and axial winds induced at the critical radius at the end of this interaction in the final stage. We here apply that theory to rapidly rotating geophysical vortices: tropical cyclones, cold-air mesocyclones and tornadoes. We find that the numerous assumptions invoked in that paper agree well with the reality of those intense vortices. We also find that in spite of a lack of moist-convection modelling, this dry vortex is fairly well accelerated at the critical radius by such a shear wave with a magnitude of order the square root of the damped-wave amplitude. The intensification level strongly depends on the aspect ratio, height of the system: rapid vortex and parent vortex, over core radius. The thinner the vortex is, the sharper the intensification is. This result is in sharp contrast to the numerous numerical simulations on VR wave/vortex interactions that yield a much smaller intensification of order the square of the wave amplitude. This weakly nonlinear approach nevertheless fails to model small vertical wavelength VR wave/vortex interactions for their related asymptotic expansions are divergent and for they yield strongly nonlinear VR waves coupled with evolving critical layers whose extent can no longer be considered as thin.     

Shear flow instability of rotating hydromagnetic fluids

Abstract

The effect of an axial magnetic field on the linear stability of shear flows in rotating systems is examined by extending Busse's analysis of the nonmagnetic case to fluids of high magnetic diffusivity in the presence of a magnetic field. The shear is caused by differential rotation which creates slight deviations from a state of rigid rotation, corresponding to a small Rossby number. It is found that the Rossby number for the onset of instability is larger when a magnetic field is present than when it is absent.     

A note on the free-surface effect on the topographically induced vorticity field in a homogeneous rotating flow

Abstract

The vorticity field induced by a topography in a homogeneous rotating fluid bounded by a free-surface is investigated. It is found that the integrated strength of the topographically bound vortex is zero and that the volume of the free-surface perturbations is equal to the volume of the topography.     

Experiments in ocean circulation modelling

Abstract

In a laboratory model ocean, fluid in a rotating tank of varying depth is subjected to “wind-stress”, For a certain range of the parameters, Ekman number E and Rossby number R, a homogeneous fluid displays steady, westward intensified flow. For the same range of E and R, a two-layer fluid can have baroclinic instabilities. The parameter range for the various kinds of instabilities is mapped in a regime diagram. The northward transport in the western boundary current is measured as it varies with Rossby number for both homogeneous and two-layer fluid.