Columnar Vortices Research Articles

Non-local energy transfers in rotating turbulence at intermediate Rossby number

AbstractTurbulent flows subject to solid-body rotation are known to generate steep energy spectra and two-dimensional columnar vortices. The localness of the dominant energy transfers responsible for the accumulation of the energy in the two-dimensional columnar vortices of large horizontal scale remains undetermined. Here, we investigate the scale-locality of the energy transfers directly contributing to the growth of the two-dimensional columnar structures observed in the intermediate Rossby number ($\mathit{Ro}$) regime. Our approach is to investigate the dynamics of the waves and vortices separately: we ensure that the two-dimensional columnar structures are not directly forced so that the vortices can result only from association with wave to vortical energy transfers. Detailed energy transfers between waves and vortices are computed as a function of scale, allowing the direct tracking of the role and scales of the wave–vortex nonlinear interactions in the accumulation of energy in the large two-dimensional columnar structures. It is shown that the dominant energy transfers responsible for the generation of a steep two-dimensional spectrum involve direct non-local energy transfers from small-frequency small-horizontal-scale three-dimensional waves to large-horizontal-scale two-dimensional columnar vortices. Sensitivity of the results to changes in resolution and forcing scales is investigated and the non-locality of the dominant energy transfers leading to the emergence of the columnar vortices is shown to be robust. The interpretation of the scaling law observed in rotating flows in the intermediate-$\mathit{Ro}$ regime is revisited in the light of this new finding of dominant non-locality.

Stratospheric Sudden Warmings as Self-Tuning Resonances. Part II: Vortex Displacement Events

Abstract Vortex displacement stratospheric sudden warmings (SSWs) are studied in an idealized model of a quasigeostrophic columnar vortex in an anelastic atmosphere. Motivated by the fact that observed events occur at a fixed orientation to the earth’s surface and have a strongly baroclinic vertical structure, vortex Rossby waves are forced by a stationary topographic forcing designed to minimize excursions of the vortex from its initial position. Variations in the background stratospheric “climate” are represented by means of an additional flow in solid body rotation. The vortex response is determined numerically as a function of the forcing strength M and the background flow strength Ω. At moderate M it is found that a large response, with many features resembling observed displacement SSWs, occurs only for a narrow range of Ω. Linear analysis reveals that for this range of Ω the first baroclinic azimuthal wave-1 Rossby wave mode is close to being resonantly excited. A forced nonlinear oscillator equation is proposed to describe the nonlinear behavior, and a method for determining the relevant coefficients numerically, using unforced calculations of steadily propagating vortex “V states,” is adopted. The nonlinear equation predicts some qualitative details of the variation in the response at finite M. However, it is concluded that strongly nonlinear processes, such as wave breaking and filament formation, are necessarily quantitatively important in determining the amplitude of the near-resonant response at finite M.

Conservation laws for a vortex in a compressible nonequilibrium medium

The problem of conservation of magnitudes is considered for a vortex in a relaxing compressible medium. Heat release due to the relaxation of a nonequilibrium medium leads to the propagation of compression waves, which remove material. Traditional integrals of motion are inapplicable in this case. We pro-pose the concept of integral quantity, which is conserved with an arbitrary degree of accuracy despite the fact that waves cross the boundary of the integration domain. Based on this concept, a broad class of conservation laws is derived for axisymmetric disturbances of columnar vortices, including conservation of the circulation and total angular momentum of the vortex. For nonaxisymmetric disturbances, it is shown that the total angular momentum and properly defined energy integral are conserved. Numerical verification of the derived conservation laws is performed and the perspectives for using these conservation laws in numerical simulations are discussed.

Weak-field dynamo emerging in a rotating spherical shell with stress-free top and no-slip bottom boundaries

Numerical experiments are performed in order to investigate an MHD dynamo in a rotating spherical shell with stress-free top and no-slip bottom boundaries. The Ekman number, the Prandtl number, and the ratio of inner to outer radii are fixed as 10 −3, 1, and 0.35, respectively. The magnetic Prandtl number is varied from 5 to 50, and the modified Rayleigh number is increased from 1.5 to 10 times the critical Rayleigh number. The initial imposed magnetic field is either a weak or strong magnetic field, where the magnetic energy of the initial field is approximately two orders of magnitude smaller or larger than the kinetic energy of the quasi-steady state of non-magnetic thermal convection. For cases involving a weak initial magnetic field, self-sustained dynamo solutions are established when the magnetic Prandtl number is larger than or equal to 5, and the modified Rayleigh number is larger than or equal to 5 times the critical Rayleigh number. The solutions are categorized as a weak field-dynamo, where the mean magnetic energy is one order of magnitude smaller than the mean kinetic energy. The dynamo solutions are characterized by a radially two-layer spatial structure. The upper layer is dominated by a strong prograde zonal mean zonal flow with large-scale prograde propagating spiral vortices having a longitudinal wavenumber of 3. Toroidal kinetic energy is converted to toroidal magnetic energy through stretching of the field lines by large-scale prograde propagating spiral vortices. On the other hand, the lower layer contains small-scale retrograde propagating columnar convective vortices having a longitudinal wavenumber of 8. The magnetic field lines are not concentrated in the vortices, but rather wind around the vortices in each layer. Poloidal kinetic energy is converted to poloidal magnetic energy through winding of the field lines around small-scale retrograde propagating columnar convective vortices. For cases involving a strong initial magnetic field, self-sustained dynamo solutions are established when the magnetic Prandtl number is greater than or equal to 4 and the modified Rayleigh number is greater than or equal to 3 times the critical Rayleigh number. In contrast with the cases involving a weak initial magnetic field, all of the dynamo solutions are strong-field solutions. Bistability is observed when the magnetic Prandtl number is 5, and the modified Rayleigh number is approximately equal to 10 times the critical Rayleigh number, where weak-field and strong-field dynamo solutions coexist. However, transition between the weak- and strong-field solutions does not occur in this case.

Three-dimensional vortex dynamics in oscillatory flow separation

The dynamics of coherent columnar vortices and their interactions in an oscillatory flow past an obstacle are examined experimentally. The main focus is on the low Keulegan–Carpenter number range (0.2 < KC < 2), where KC is the ratio between the fluid particle excursion during half an oscillation cycle and the obstacle size, and for moderate Reynolds numbers (700 < Rev < 7500). For this parameter range, a periodic unidirectional vortex pair ejection regime is observed, in which the direction of vortex propagation is set by the initial conditions of the oscillations. These vortex pairs provide a direct mechanism for the transfer of momentum and enstrophy to the outer region of rough oscillating boundary layers. Vortices are observed to be short-lived relative to the oscillation time scale, which limits their propagation distance from the boundary. The instability mechanisms leading to vortex decay are elucidated via flow visualizations and digital particle image velocimetry (DPIV). Dye visualizations reveal complex three-dimensional vortex interactions resulting in rapid vortex destruction. These visualizations suggest that one of the instabilities affecting the spanwise vortices is an elliptical instability of the strained vortex cores. This is supported by DPIV measurements which identify the spatial structure of the perturbations associated with the elliptical instability in the divergence field. We also identify regions in the periphery of the vortex cores which are unstable to the centrifugal instability. Vortex longevity is quantified via a vortex decay time scale, and the results indicate that vortex pair lifetimes are of the order of an oscillation period T.

Equatorially asymmetric convection inducing a hemispherical magnetic field in rotating spheres and implications for the past martian dynamo

Abstract The convective instability in a rapidly rotating, self-graviting sphere sets up in the form of equatorially symmetric, non-axisymmetric columnar vortices aligned with the rotation axis, carrying heat away in the cylindrical radial direction. In this study, we present numerical simulations of thermal convection and dynamo action driven by internal heating (intended to model a planetary core subject to uniform secular cooling) in a rotating sphere where, from the classical columnar convection regime, we find a spontaneous transition towards an unexpected and previously unobserved flow regime in which an equatorially antisymmetric, axisymmetric (EAA) mode strongly influences the flow. This EAA mode carries heat away along the rotation axis and is the nonlinear manifestation of the first linearly unstable axisymmetric mode. When the amplitude of the EAA mode reaches high enough values, we obtain hemispherical dynamos with one single hemisphere bearing more than 75% of the total magnetic energy at the surface of the rotating sphere. We perform the linear analysis of the involved convective modes and the nonlinear study of this hydrodynamic transition, with and without dynamo action, to obtain scaling laws for the regime boundaries. As secular cooling in a full sphere (i.e. without inner core) is a configuration which has probably been widespread in the early solar system in planetary cores, including the core of Mars, we discuss the possible implications of our results for the past martian dynamo.

Simultaneous two-scale 3D-PTV measurements in turbulence under the influence of system rotation

We present a technique capable of measuring in a Lagrangian manner the three components of the flow velocity simultaneously at two different spatial resolutions. We use two synchronized 3D Particle Tracking Velocimetry (3D-PTV) systems equipped with filters to separate the two types of seeding particles and measure in volumes of 55 × 55 × 50 mm3 and 15 × 15 × 15 mm3, respectively. In this way, we obtain (a) the velocity field in a large observation volume with focus on the large-scale flow features and (b) velocity derivatives in a small observation volume providing small-scale quantities associated with vorticity and strain. The small observation volume is located in the center of the large one, which allows the small-scale quantities to be related to the large-scale flow patterns. To illustrate the capability of the simultaneous two-scale 3D Particle Tracking Velocimetry (STS 3D-PTV) method, the behavior of vorticity and strain is analyzed in shear-free turbulence under the influence of system rotation. We find that in the quasi-2D region, vortex stretching is strongly suppressed only inside columnar vortex tubes.

Swirling Flow in a Fixed Container: Generation and Attenuation of a Vortex Column

The development of a columnar vortex and its attenuation using radial rods at the bottom boundary of a stationary container are experimentally studied. The fluid motion is achieved combining two independent flows: a global circulation around the cylinder axis and a meridian flow generated by recirculating fluid through a central nozzle located at the vessel bottom. The resulting velocity field is analyzed under two conditions: with and without the meridian or suction flow. It is shown that in the second condition a columnar vortex merges and that its intensity increases when the suction flow rate is increased. The key role played by the bottom boundary layer in the vortex formation is demonstrated. In the last part of the work, the attenuation of the vortex intensity produced by a set of rods located at the vessel bottom is investigated. It is found that obstacles with heights of the order of the boundary layer thickness are enough to produce the total annihilation of the vortex column.

Zigzag instability of vortex pairs in stratified and rotating fluids. Part 1. General stability equations.

In stratified and rotating fluids, pairs of columnar vertical vortices are subjected to three-dimensional bending instabilities known as the zigzag instability or as the tall-column instability in the quasi-geostrophic limit. This paper presents a general asymptotic theory for these instabilities. The equations governing the interactions between the strain and the slow bending waves of each vortex column in stratified and rotating fluids are derived for long vertical wavelength and when the two vortices are well separated, i.e. when the radii R of the vortex cores are small compared to the vortex separation distance b. These equations have the same form as those obtained for vortex filaments in homogeneous fluids except that the expressions of the mutual-induction and self-induction functions are different. A key difference is that the sign of the self-induction function is reversed compared to homogeneous fluids when the fluid is strongly stratified: |max| < N (where N is the Brunt–Väisälä frequency and max the maximum angular velocity of the vortex) for any vortex profile and magnitude of the planetary rotation. Physically, this means that slow bending waves of a vortex rotate in the same direction as the flow inside the vortex when the fluid is stratified-rotating in contrast to homogeneous fluids. When the stratification is weaker, i.e. |max| > N, the self-induction function is complex because the bending waves are damped by a viscous critical layer at the radial location where the angular velocity of the vortex is equal to the Brunt–Väisälä frequency.In contrast to previous theories, which apply only to strongly stratified non-rotating fluids, the present theory is valid for any planetary rotation rate and when the strain is smaller than the Brunt–Väisälä frequency: Γ/(2πb2) ≪ N, where Γ is the vortex circulation. Since the strain is small, this condition is met across a wide range of stratification: from weakly to strongly stratified fluids. The theory is further generalized formally to any basic flow made of an arbitrary number of vortices in stratified and rotating fluids. Viscous and diffusive effects are also taken into account at leading order in Reynolds number when there is no critical layer. In Part 2 (Billant et al., J. Fluid Mech., 2010, doi:10.1017/S002211201000282X), the stability of vortex pairs will be investigated using the present theory and the predictions will be shown to be in very good agreement with the results of direct numerical stability analyses. The existence of the zigzag instability and the distinctive stability properties of vortex pairs in stratified and rotating fluids compared to homogeneous fluids will be demonstrated to originate from the sign reversal of the self-induction function.

Kinetic energy budget analysis of spiraling columnar critical convection in a rapidly rotating spherical shell

The kinetic energy budget analysis of spiraling columnar critical convection emerging in a rapidly rotating spherical shell is performed. In the cylindrical-radially elongated spiraling convection mode appearing as a critical mode at small Prandtl numbers, the kinetic energy generated in the inner region of the spherical shell is transported in a cylindrically radial manner and is dissipated near the outer boundary around the equator. However, when the Prandtl number is increased, the dynamical energy flux turns in the axial direction rather than in the cylindrically radial direction. The kinetic energy generated inside the shell is transported in the direction of the rotating axis and is dissipated near the outer boundary at the same cylindrically radial location. The existence of the axial component of the dynamical energy flux is attributed to the ageostrophic flows in the columnar vortices induced by viscous damping and buoyancy force. In spite of the existence of the axial component of dynamical energy flux, the obtained geometry of the axially integrated kinetic energy budget is consistent with the results using a two-dimensional rotating annulus model. Therefore, the interpretation of spiraling structure with the radial propagation properties of topographic Rossby waves is applicable to the three-dimensional spiraling columnar convection emerging in a rotating spherical shell. Flow patterns calculated from the dispersion relation of two-dimensional topographic Rossby waves in a rotating spherical shell well explain the structure of the three-dimensional spiraling columnar convection.

Baroclinic multipole formation from heton interaction

In a two-layer quasi-geostrophic model, the interaction between two opposite-signed hetons (baroclinic vortex pairs) is studied analytically and numerically, for singular and finite-area vortices.For point vortices, using trilinear coordinates, it is shown that the possible evolutions depend on the deformation radius Rd: for large Rd, the layers decouple, vortices pair in each layer and their trajectories are open; for medium Rd, the exchange of opposite-sign partners between layers becomes possible; for small Rd, two other regimes appear: one where hetons remain unaltered during their evolution but follow open trajectories, and one where hetons occupy only a bounded subdomain of space at all times. Conditions for invariant co-rotation of the heton pair are derived and analyzed.Then, the nonlinear evolutions of finite-area heton pairs, with piecewise-constant vorticity, are computed with contour dynamics. When the central cyclonic vortex is initially aligned vertically, a transition occurs between three nonlinear regimes as layer coupling increases: for weak coupling, the vortices pair horizontally and drift away in opposite directions; for moderate layer coupling, the core vortex splits into two parts, one of which remains as a tilted columnar vortex at the center; for stronger layer coupling, each anticyclone pairs with part of the cyclone in each layer, thus forming an L-shaped dipole, a new coherent structure of two-layer flows. When the initial distance between the central and satellite vortices is increased, the velocity shear at the center decreases and the central vortex remains vertically aligned, thus forming a Z-shaped tripole, also a newly observed vortex compound. Such tripoles also compete with oscillating states, in which the core vortex periodically aligns and tilts, a regime observed when layer coupling is moderate and as vortices become closer in each layer. This Z-shaped tripole forms for various values of stratification and of initial distances between vortices, and is therefore a robust vortex compound in two-layer quasi-geostrophic flows.

Instability of a vertical columnar vortex in a stratified fluid

In this article, we study experimentally the evolution of a vertical columnar vortex in a stratified fluid. Three different measurement techniques are used. Particle image velocimetry allows us to monitor the time evolution of the characteristics of the vortex (Froude and Reynolds numbers). Dye visualizations reveal the existence of an instability for Froude numbers smaller than one, which creates an undulation of the vortex centerline. Synthetic schlieren visualization shows that the density structure of the unstable mode is very similar to the structure found recently numerically for the radiative instability of a Lamb–Oseen vortex (Riedinger et al. in J Fluid Mech, 2010). The experimental stability diagram and unstable wavelengths are compared with these numerical results. A secondary instability associated with the presence of critical layers is also observed for Froude numbers larger than one.

Three-dimensional vortex visualization in stratified spin-up

We present the results of three-dimensional time-dependent numerical simulations of incremental spin-up of a thermally stratified fluid. The fluid inside a vertical cylindrical container of radius R and height 2H is water characterized by the kinematic viscosity ν and thermal diffusivity κ. Initially, its density (temperature) varies linearly with height and is characterized by a constant buoyancy frequency N, which is proportional to the density gradient. The system undergoes an abrupt change in the rotation rate from its initial value Ωi, when the fluid is in a solid-body rotation state, to the final value Ωf. The aim of this contribution is to show the formation of columnar vortices in a high Rossby number spin-up flow.

Waves on a columnar vortex in a strongly stratified fluid

This paper investigates the discrete bounded waves sustained by a vertical columnar Rankine vortex in a strongly stratified fluid. We show that these waves are very different from their well-known counterpart in homogeneous fluid (Kelvin vortex waves); they exist only for nonzero azimuthal wavenumber m, their frequency lies in the interval [0,mΩ] (Ω is the angular velocity in the vortex core) and they are unstable because of an outward radiation from the vortex. The instability mechanism is explained in terms of an over-reflection phenomenon by means of a Wentzel–Kramers–Brillouin–Jeffreys analysis for large axial wavenumber.

Radiative instability in stratified vortices

This paper investigates the stability of a columnar vortex in an inviscid stratified fluid. By means of a WKBJ analysis for large axial wave number, we demonstrate that the normal modes can be stable or unstable owing to the emission of internal waves from the vortex. This phenomenon is shown to be analogous to the radioactive decay of nuclei in quantum mechanics. The destabilized character of the wave emission is shown to be associated with the presence of a critical point in the radial structure of the normal mode. The theoretical predictions for the frequency and growth rate of the normal modes are shown to be in good agreement with numerical results for two examples.

Calculation of unsteady flow past a cube on the wall of a narrow channel using URANS and the Spalart–Allmaras turbulence model

UDC 532.517.4 The results of numerical simulation of fully developed turbulent flow in a channel with the cube on the lower wall (for the characteristic Reynolds number Re = 40,000) within the framework of the traditional approach to the solution of unsteady Reynolds-averaged Navier-Stokes equations (URANS) in combination with the semiempirical Spalart-Allmaras turbulence model (with a correction for rotation) have been presented. A de- tailed comparative analysis of the results of numerical simulation of local and integral flow characteristics and the Martinuzzi experimental data has shown that the self-oscillating regime of flow past the cube is a su- perposition of oscillations of the arms of a horseshoe vortex and the rear arched and detached vortex struc- tures. Using fast Fourier transformation, it has been found that the oscillations are of a bimodal character in the longitudinal and vertical directions, whereas in the transverse direction, they are of a unimodal character. Introduction. The problem on unsteady flow past a cube in a narrow channel with the fully developed turbu- lent flow being formed at the inlet is of particular interest for testing today's calculation algorithms of computational hydrodynamics. In this case channel flow is of a self-oscillating character and is distinguished by a complex vortex structure (Fig. 1). Flow separation begins ahead of the cube and further develops from its frontal face side and on the lateral walls. A horseshoe vortex that interacts with the near wake of the cube results. We recognize the following basic large-scale flow elements for their subsequent analysis and comparison to experimental data: 1) a horseshoe vor- tex; 2) a vortex ensemble consisting of a vortex attached to the face side of the cube and two columnar vortices near the lateral walls; 3) a rear detached vortex; 4) an attached vortex behind the end wall of the cube, which is not iden- tified in the Martinuzzi experiment, as should be noted, but is visualized in many calculations; 5) an arched vortex with two oppositely rotating bends, which is formed in the near wake. Experimental data to which we compare results of numerical investigations were obtained by Martinuzzi (1) and have been introduced into the ERCOFTAC (European Research Community on Flow, Turbulence and Combustion) databank (2). About ten large-scale numerical investigations based on solution of steady and unsteady Reynolds-aver- aged Navier-Stokes equations (RANS ⁄ URANS) and on application of the large-eddy model (see, e.g., (3-5)) have been carried out over a period of 15 years. It is noteworthy that, except for the results obtained by Durbin with the V2F turbulence model (6), none of the works gives satisfactory agreement between the results of calculations within the framework of the RANS ⁄ URANS methodology and the experimental data. This work is primarily aimed at substantiating the applicability of the URANS approach to the prediction of turbulent separated flows. The commercial FLUENT v6.3 package (7) is used. Preliminary verification and approval of the package with a catalog of semiempirical models realized in it, which were carried out on two-dimensional test problems such as flow past a circular cylinder (8) and circulation flow in a square cavity with a moving upper bound- ary (9, 10), have shown satisfactory correlation with experimental data, when the two-parametric model of shear-stress transfer MSST and the Spalart-Allmaras (SA) vortex-viscosity model were selected. It has been shown in (8-10) that the SA model in its traditional formulation substantially overstates turbulent-viscosity values, which renders it unsuit- able for calculations of turbulent separated flows. For further testing, we have selected the SA model modified with allowance for rotation.

Steady Flow Analysis of Francis Turbine

The flow features are presented on different openings in Francis turbine by unsteady numerical simulation. The unsteady flow results are as follows:when the opening is 12 mm, pressure fluctuation at runner outlet near hub is not obvious and pressure fluctuation amplitudes near midstream and shroud are bigger. Helical vortex is situated on conical tube on opening 12 mm. It will disappear at the elbow tube. Pressure fluctuation of low frequency in draft tube is mainly located at conical and elbow tube. The frequency of helical vortex is low and is about 0.233 times that of runner frequency. And pressure fluctuation amplitudes in conical tube is higher. Frequencies of two points on the same measurement section are almost equivalent. But when the opening is 24 mm, there is no back flow region at runner outlet near hub and pressure fluctuation amplitude is smaller. And the ones near midstream and shroud are greatly higher. The frequency is several times that of product of runner frequency and blade number. The columnar vortex formed in conical tube will become small helical vortex in the elbow tube. The result is that the long vortex rope forms the cutting effect like drilling rig on the empennage. It will have serious effect on the tubine efficiency and stability. The length of vortex rope provides the theoretical basis for studying the depth of Francis turbine draft tube and optimization design.

On the formation of cyclones and anticyclones in a rotating fluid

It is commonly observed that the columnar vortices that dominate the large scales in homogeneous, rapidly rotating turbulence are predominantly cyclonic. This has prompted us to ask how this asymmetry arises. To provide a partial answer to this we look at the process of columnar vortex formation in a rotating fluid and, in particular, we examine how a localized region of swirl (an eddy) can convert itself into a columnar structure by inertial wave propagation. We show that, when the Rossby number (Ro) is small, the vortices evolve into columnar eddies through the radiation of linear inertial waves. When the Rossby number is large, on the other hand, no such column is formed. Rather, the eddy bursts radially outward under the action of the centrifugal force. There is no asymmetry between cyclonic and anticyclonic eddies for these two regimes. However, cyclones and anticyclones behave differently in the intermediate regime of Ro∼1. Here we find that the transition from columnar vortex formation to radial bursting occurs at lower values of Ro for anticyclones, with the transition for anticyclones occurring at Ro∼0.5, and that for cyclones at Ro∼2. Thus, in a homogeneous turbulence experiment conducted at, say, Ro=1, we would expect to see more cyclones than anticyclones. The reason for this asymmetry at Ro∼1 is explained.

Instability and breakdown of a vertical vortex pair in a strongly stratified fluid

The dynamics of a counter-rotating pair of columnar vortices aligned parallel to a stable density gradient are investigated. By means of numerical simulation, we extend the linear analyses and laboratory experiments of Billant & Chomaz (J. Fluid Mech. vol. 418, p. 167; vol. 419, pp. 29, 65 (2000)) to the fully nonlinear, large-Reynolds-number regime. A range of stratifications and vertical length scales is considered, withFrh< 0.2 and 0.1 <Frz< 10. HereFrh≡U/(NR) andFrz≡Ukz/Nare the horizontal and vertical Froude numbers,UandRare the horizontal velocity and length scales of the vortices,Nis the Brunt–Väisälä frequency, and 2π/kzis the vertical wavelength of a small initial perturbation. At early times withFrz< 1, linear predictions for the zigzag instability are reproduced. Short-wavelength perturbations withFrz> 1 are found to be unstable as well, with growth rates only slightly less than those of the zigzag instability but with very different structure. At later times, the large-Reynolds-number evolution diverges profoundly from the moderate-Reynolds-number laboratory experiments as the instabilities transition to turbulence. For the zigzag instability, this transition occurs when density perturbations generated by the vortex bending become gravitationally unstable. The resulting turbulence rapidly destroys the vortex pair. We derive the criterion η/R≈ 0.2/Frzfor the onset of gravitational instability, where η is the maximum horizontal displacement of the bent vortices, and refine it to account for a finite twisting disturbance. Our simulations agree for the fastest growing wavelengths 0.3 <Frz< 0.8. Short perturbations withFrz> 1 saturate at low amplitude, preserving the columnar structure of the vortices well after the generation of turbulence. Viscosity is shown to suppress the transition to turbulence for Reynolds numberRe≲ 80/Frh, yielding laminar dynamics and, under certain conditions, pancake vortices like those observed in the laboratory.

A model for vortical plumes in rotating convection

In turbulent rotating convection a typical flow structuring in columnar vortices is observed. In the internal structure of these vortices several symmetries are approximately satisfied. A model for these columnar vortices is derived by prescribing these symmetries. The symmetry constraints are applied to the Navier–Stokes equations with rotation in the Boussinesq approximation. It is found that the application of the symmetries results in a set of linearized equations. An investigation of the linearized equations leads to a model for the columnar vortices and a prediction for the heat flux (Nusselt number) that is very appropriate compared to the results from direct numerical simulations of the full governing equations.