Earth and Environmental Sciences (90)
flm;geophysical and geological flows;waves in rotating fluids
Journal of Fluid Mechanics (90)
Ryan Test (90)
A singular vorticity wave packet within a rapidly rotating vortex: spiralling versus oscillating motions
Philippe Caillol
Journal: Journal of Fluid Mechanics / Volume 873 / 25 August 2019
This paper considers a free vorticity wave packet propagating within a rapidly rotating vortex in the quasi-steady regime, a long time after the wave packet strongly and unsteadily interacted with the vortex. We study a singular, nonlinear, helical and asymmetric shear mode inside a linearly stable, columnar and axisymmetric vortex on the $f$ -plane. The amplitude-modulated mode enters resonance with the vortex at a certain radius $r_{c}$ , where the phase angular speed is equal to the rotation frequency. The singularity in the modal equation at $r_{c}$ strongly modifies the flow in the three-dimensional helical critical layer, the region around $r_{c}$ where the wave/vortex interaction occurs. This interaction generates a vertically sheared three-dimensional mean flow of higher amplitude than the wave packet. The chosen envelope regime assumes the formation of a mean radial velocity of the same order as the wave packet amplitude, leading to the streamlines exhibiting a spiral motion in the neighbourhood of the critical layer. Radar images frequently show such spiral bands in tropical cyclones or tornadoes. Through matched asymptotic expansions, we find an analytical solution of the leading-order equations inside the critical layer. The generalized Batchelor integral condition applied to the quasi-steady, three-dimensional motion inside the separatrices yields a leading-order, non-uniform three-dimensional vorticity. The critical-layer pattern, strongly deformed by the mean radial velocity, loses its symmetries with respect to the azimuthal and radial directions, which makes the leading-order mean radial wave fluxes non-zero. Finally, a stronger wave/vortex interaction occurs with respect to previous studies where a steady neutral vortical mode or an envelope of larger extent was involved.
Diffusion of inertia-gravity waves by geostrophic turbulence
Hossein A. Kafiabad, Miles A. C. Savva, Jacques Vanneste
Journal: Journal of Fluid Mechanics / Volume 869 / 25 June 2019
Published online by Cambridge University Press: 30 April 2019, R7
The scattering of inertia-gravity waves by large-scale geostrophic turbulence in a rapidly rotating, strongly stratified fluid leads to the diffusion of wave energy on the constant-frequency cone in wavenumber space. We derive the corresponding diffusion equation and relate its diffusivity to the wave characteristics and the energy spectrum of the turbulent flow. We check the predictions of this equation against numerical simulations of the three-dimensional Boussinesq equations in initial-value and forced scenarios with horizontally isotropic wave and flow fields. In the forced case, wavenumber diffusion results in a $k^{-2}$ wave energy spectrum consistent with as-yet-unexplained features of observed atmospheric and oceanic spectra.
On constant vorticity water flows in the $\unicode[STIX]{x1D6FD}$ -plane approximation
Calin Iulian Martin
Journal: Journal of Fluid Mechanics / Volume 865 / 25 April 2019
Print publication: 25 April 2019
We consider here three-dimensional water flows in the $\unicode[STIX]{x1D6FD}$ -plane approximation. In a quite general setting we show that the only flow exhibiting a constant vorticity vector is the stationary flow with vanishing velocity field and flat free surface.
High-speed standard magneto-rotational instability
Kengo Deguchi
The large Reynolds number asymptotic approximations of the neutral curve of Taylor–Couette flow subject to an axial uniform magnetic field are analysed. The flow has been extensively studied since the early 1990s as the magneto-rotational instability (MRI) occurring in the flow may explain the origin of the instability observed in some astrophysical objects. Elsewhere, the ideal approximation has been used to study high-speed flows, which sometimes produces paradoxical results. For example, ideal flows must be completely stabilised for a sufficiently strong applied magnetic field. On the other hand, the vanishing magnetic Prandtl number limit of the stability should be purely hydrodynamic, so instability must occur when Rayleigh's stability condition is violated. Our first discovery is that this apparent contradiction can be resolved by showing the abrupt appearance of the hydrodynamic instability at a certain critical value of the magnetic Prandtl number. This is found using the asymptotically large Reynolds number limit but with a sufficiently long wavelength to retain some diffusive effects. Our second finding concerns the so-called Velikhov–Chandrasekhar paradox, namely the mismatch of the zero external magnetic field limit of the Velikhov–Chandrasekhar stability criterion and Rayleigh's stability criterion. We show for fully wide-gap cases that the high Reynolds number asymptotic analysis of the MRI naturally yields the simple stability condition that describes smooth transition from Rayleigh to Velikhov–Chandrasekhar stability criteria with increasing Lundquist number.
Markovian inhomogeneous closures for Rossby waves and turbulence over topography
Jorgen S. Frederiksen, Terence J. O'Kane
Journal: Journal of Fluid Mechanics / Volume 858 / 10 January 2019
Print publication: 10 January 2019
Manifestly Markovian closures for the interaction of two-dimensional inhomogeneous turbulent flows with Rossby waves and topography are formulated and compared with large ensembles of direct numerical simulations (DNS) on a generalized $\unicode[STIX]{x1D6FD}$ -plane. Three versions of the Markovian inhomogeneous closure (MIC) are established from the quasi-diagonal direct interaction approximation (QDIA) theory by modifying the response function to a Markovian form and employing respectively the current-time (quasi-stationary) fluctuation dissipation theorem (FDT), the prior-time (non-stationary) FDT and the correlation FDT. Markov equations for the triad relaxation functions are derived that carry similar information to the time-history integrals of the non-Markovian QDIA closure but become relatively more efficient for long integrations. Far from equilibrium processes are studied, where the impact of a westerly mean flow on a conical mountain generates large-amplitude Rossby waves in a turbulent environment, over a period of 10 days. Excellent agreement between the evolved mean streamfunction and mean and transient kinetic energy spectra are found for the three versions of the MIC and two variants of the non-Markovian QDIA compared with an ensemble of 1800 DNS. In all cases mean Rossby wavetrain pattern correlations between the closures and the DNS ensemble are greater than 0.9998.
Velocity and acceleration statistics in rapidly rotating Rayleigh–Bénard convection
Hadi Rajaei, Kim M. J. Alards, Rudie P. J. Kunnen, Herman J. H. Clercx
Journal: Journal of Fluid Mechanics / Volume 857 / 25 December 2018
Print publication: 25 December 2018
Background rotation causes different flow structures and heat transfer efficiencies in Rayleigh–Bénard convection. Three main regimes are known: rotation unaffected, rotation affected and rotation dominated. It has been shown that the transition between rotation-unaffected and rotation-affected regimes is driven by the boundary layers. However, the physics behind the transition between rotation-affected and rotation-dominated regimes are still unresolved. In this study, we employ the experimentally obtained Lagrangian velocity and acceleration statistics of neutrally buoyant immersed particles to study the rotation-affected and rotation-dominated regimes and the transition between them. We have found that the transition to the rotation-dominated regime coincides with three phenomena; suppressed vertical motions, strong penetration of vortical plumes deep into the bulk and reduced interaction of vortical plumes with their surroundings. The first two phenomena are used as confirmations for the available hypotheses on the transition to the rotation-dominated regime while the last phenomenon is a new argument to describe the regime transition. These findings allow us to better understand the rotation-dominated regime and the transition to this regime.
Scattering of internal tides by barotropic quasigeostrophic flows
Miles A. C. Savva, Jacques Vanneste
Oceanic internal tides and other inertia–gravity waves propagate in an energetic turbulent flow whose length scales are similar to the wavelengths. Advection and refraction by this flow cause the scattering of the waves, redistributing their energy in wavevector space. As a result, initially plane waves radiated from a source such as a topographic ridge become spatially incoherent away from the source. To examine this process, we derive a kinetic equation which describes the statistics of the scattering under the assumptions that the flow is quasigeostrophic, barotropic and well represented by a stationary homogeneous random field. Energy transfers are quantified by computing a scattering cross-section and shown to be restricted to waves with the same frequency and identical vertical structure, hence the same horizontal wavelength. For isotropic flows, scattering leads to an isotropic wave field. We estimate the characteristic time and length scales of this isotropisation, and study their dependence on parameters including the energy spectrum of the flow. Simulations of internal tides generated by a planar wavemaker carried out for the linearised shallow-water model confirm the pertinence of these scales. A comparison with the numerical solution of the kinetic equation demonstrates the validity of the latter and illustrates how the interplay between wave scattering and transport shapes the wave statistics.
On the spatial segregation of helicity by inertial waves in dynamo simulations and planetary cores
P. A. Davidson, A. Ranjan
Journal: Journal of Fluid Mechanics / Volume 851 / 25 September 2018
Print publication: 25 September 2018
The distribution of kinetic helicity in a dipolar planetary dynamo is central to the success of that dynamo. Motivated by the helicity distributions observed in numerical simulations of the Earth's dynamo, we consider the relationship between the kinetic helicity, $h=\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\times \boldsymbol{u}$ , and the buoyancy field that acts as a source of helicity, where $\boldsymbol{u}$ is velocity. We show that, in the absence of a magnetic field, helicity evolves in accordance with the equation $\unicode[STIX]{x2202}h/\unicode[STIX]{x2202}t=-\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{F}+S_{h}$ , where the flux, $\boldsymbol{F}$ , represents the transport of helicity by inertial waves, and the helicity source, $S_{h}$ , involves the product of the buoyancy and the velocity fields. In the numerical simulations it is observed that the helicity outside the tangent cylinder is predominantly negative in the north and positive in the south, a feature which the authors had previously attributed to the transport of helicity by waves (Davidson & Ranjan, Geophys. J. Intl, vol. 202, 2015, pp. 1646–1662). It is also observed that there is a strong spatial correlation between the distribution of $h$ and of $S_{h}$ , with $S_{h}$ also predominantly negative in the north and positive in the south. This correlation tentatively suggests that it is the in situ generation of helicity by buoyancy that establishes the distribution of $h$ outside the tangent cylinder, rather than the dispersal of helicity by waves, as had been previously argued by the authors. However, although $h$ and $S_{h}$ are strongly correlated, there is no such correlation between $\unicode[STIX]{x2202}h/\unicode[STIX]{x2202}t$ and $S_{h}$ , as might be expected if the distribution of $h$ were established by an in situ generation mechanism. We explain these various observations by showing that inertial waves interact with the buoyancy field in such a way as to induce a source $S_{h}$ which has the same sign as the helicity in the local wave flux, and that the sign of $h$ is simply determined by the direction of that flux. We conclude that the observed distributions of $h$ and $S_{h}$ outside the tangent cylinder are consistent with the transport of helicity by waves.
Fast and slow resonant triads in the two-layer rotating shallow water equations
Alex Owen, Roger Grimshaw, Beth Wingate
Published online by Cambridge University Press: 02 July 2018, pp. 18-45
In this paper, we examine triad resonances in a rotating shallow water system when there are two free interfaces. This allows for an examination in a relatively simple model of the interplay between baroclinic and barotropic dynamics in a context where there is also a geostrophic mode. In contrast to the much-studied one-layer rotating shallow water system, we find that as well as the usual slow geostrophic mode, there are now two fast waves, a barotropic mode and a baroclinic mode. This feature permits triad resonances to occur between three fast waves, with a mixture of barotropic and baroclinic modes, an aspect that cannot occur in the one-layer system. There are now also two branches of the slow geostrophic mode, with a repeated branch of the dispersion relation. The consequences are explored in a derivation of the full set of triad interaction equations, using a multiscale asymptotic expansion based on a small-amplitude parameter. The derived nonlinear interaction coefficients are confirmed using energy and enstrophy conservation. These triad interaction equations are explored, with an emphasis on the parameter regime with small Rossby and Froude numbers.
Spherical vortices in rotating fluids
M. M. Scase, H. L. Terry
Journal: Journal of Fluid Mechanics / Volume 846 / 10 July 2018
Published online by Cambridge University Press: 08 May 2018, R4
Print publication: 10 July 2018
A popular model for a generic fat-cored vortex ring or eddy is Hill's spherical vortex (Phil. Trans. R. Soc. A, vol. 185, 1894, pp. 213–245). This well-known solution of the Euler equations may be considered a special case of the doubly infinite family of swirling spherical vortices identified by Moffatt (J. Fluid Mech., vol. 35 (1), 1969, pp. 117–129). Here we find exact solutions for such spherical vortices propagating steadily along the axis of a rotating ideal fluid. The boundary of the spherical vortex swirls in such a way as to exactly cancel out the background rotation of the system. The flow external to the spherical vortex exhibits fully nonlinear inertial wave motion. We show that above a critical rotation rate, closed streamlines may form in this outer fluid region and hence carry fluid along with the spherical vortex. As the rotation rate is further increased, further concentric 'sibling' vortex rings are formed.
Triadic resonances in the wide-gap spherical Couette system
A. Barik, S. A. Triana, M. Hoff, J. Wicht
Journal: Journal of Fluid Mechanics / Volume 843 / 25 May 2018
Print publication: 25 May 2018
The spherical Couette system, consisting of a viscous fluid between two differentially rotating concentric spheres, is studied using numerical simulations and compared with experiments performed at BTU Cottbus-Senftenberg, Germany. We concentrate on the case where the outer boundary rotates fast enough for the Coriolis force to play an important role in the force balance, and the inner boundary rotates slower or in the opposite direction as compared to the outer boundary. As the magnitude of differential rotation is increased, the system is found to transition through three distinct hydrodynamic regimes. The first regime consists of the emergence of the first non-axisymmetric instability. Thereafter one finds the onset of 'fast' equatorially antisymmetric inertial modes, with pairs of inertial modes forming triadic resonances with the first instability. A further increase in the magnitude of differential rotation leads to the flow transitioning to turbulence. Using an artificial excitation, we study how the background flow modifies the inertial mode frequency and structure, thereby causing departures from the eigenmodes of a full sphere and a spherical shell. We investigate triadic resonances of pairs of inertial modes with the fundamental instability. We explore possible onset mechanisms through numerical experiments.
Librational forcing of a rapidly rotating fluid-filled cube
Ke Wu, Bruno D. Welfert, Juan M. Lopez
The flow response of a rapidly rotating fluid-filled cube to low-amplitude librational forcing is investigated numerically. Librational forcing is the harmonic modulation of the mean rotation rate. The rotating cube supports inertial waves which may be excited by libration frequencies less than twice the rotation frequency. The response is comprised of two main components: resonant excitation of the inviscid inertial eigenmodes of the cube, and internal shear layers whose orientation is governed by the inviscid dispersion relation. The internal shear layers are driven by the fluxes in the forced boundary layers on walls orthogonal to the rotation axis and originate at the edges where these walls meet the walls parallel to the rotation axis, and are hence called edge beams. The relative contributions to the response from these components is obscured if the mean rotation period is not small enough compared to the viscous dissipation time, i.e. if the Ekman number is too large. We conduct simulations of the Navier–Stokes equations with no-slip boundary conditions using parameter values corresponding to a recent set of laboratory experiments, and reproduce the experimental observations and measurements. Then, we reduce the Ekman number by one and a half orders of magnitude, allowing for a better identification and quantification of the contributions to the response from the eigenmodes and the edge beams.
On triadic resonance as an instability mechanism in precessing cylinder flow
Thomas Albrecht, Hugh M. Blackburn, Juan M. Lopez, Richard Manasseh, Patrice Meunier
Published online by Cambridge University Press: 27 February 2018, R3
Contained rotating flows subject to precessional forcing are well known to exhibit rapid and energetic transitions to disorder. Triadic resonance of inertial modes has been previously proposed as an instability mechanism in such flows, and that idea was developed into a successful model for predicting instability in a cylindrical container when departures from solid-body rotation are sufficiently small. Using direct numerical simulation and dynamic mode decomposition, we analyse instabilities of precessing cylinder flows whose three-dimensional basic states, steady in the gimbal frame of reference, may depart substantially from solid-body rotation. In the gimbal frame, the instability can be interpreted as resulting from a supercritical Hopf bifurcation that results in a limit-cycle flow. In the cylinder frame of reference, the basic state is a rotating wave with azimuthal wavenumber $m=1$ , and the instability satisfies triadic-resonance conditions with the instability mode maintaining a fixed orientation with respect to the basic state. Thus, we are able to demonstrate the existence of two alternative but congruent explanations for the instability. Additionally, we show that basic states may depart substantially from solid-body rotation even with modest cylinder tilt angles, and growth rates for instabilities may be sufficiently large that nonlinear saturation to disordered states can occur within approximately ten cylinder revolutions, in agreement with experimental observations.
Rapidly rotating precessing cylinder flows: forced triadic resonances
Juan M. Lopez, Francisco Marques
Journal: Journal of Fluid Mechanics / Volume 839 / 25 March 2018
Print publication: 25 March 2018
Rapidly rotating cylinder flows subjected to low-amplitude precessional forcing are studied numerically over a range of cylinder and precessional rotation rates. For sufficiently small rotation rates, viscous effects lead to a forced overturning flow that is steady in the precession (table) frame of reference. Increasing the rotation rates, this forced flow loses stability in a Hopf bifurcation, which can be either supercritical or subcritical, and may preserve or break the symmetry of the system, depending on the parameter regime studied. Regardless of these details of the Hopf bifurcation, it is found that the Hopf instability is associated with a slightly detuned triadic resonance between the forced overturning flow and two free Kelvin modes (inviscid eigenmodes of the rotating cylinder). Further increases in rotation rates lead to a sequence of secondary instabilities which also follow a generic pattern irrespective of the parameter regime investigated. The relationship between this sequence of instabilities and the resultant nonlinear dynamics with the experimentally observed phenomenon of resonant collapse is discussed.
Poleward propagation of near-inertial waves induced by fluctuating winds over a baroclinically unstable zonal jet
Marine Tort, Kraig B. Winters
We investigate the excitation and radiation of near-inertial internal gravity waves continuously excited by a latitudinally confined temporally fluctuating wind in a numerical model of a stratified ocean on a $\unicode[STIX]{x1D6FD}$ -plane at mid-latitude. The surface wind forcing contains both high- and low-frequency components which excite propagating waves and a baroclinically unstable zonal jet respectively. Wentzel–Kramers–Brillouin (WKB) ray theory implies that near-inertial waves propagate strictly towards the equator. We seek to refine this view here by (i) adding the non-traditional Coriolis force (accounting for the horizontal component of the Earth's rotation) into the equations of motion, in order to allow poleward sub-inertial propagation to occur, and (ii) relaxing the conceptual constraint of no zonal variability, to allow the zonal jet to undergo instability, to meander and to sustain an active field of mesoscale eddies, potentially impacting the excitation of near-inertial waves. The key results are that, while (i) permits weakly stratified waveguides with sub-inertial poleward wave propagation to develop in accord with theory, the sub-inertial energy flux observed is very small compared with the equatorward flux. Thus, in terms of energy radiated from the storm track, non-traditional effects are small for wind-driven near-inertial waves. The consequences of (ii) are much more pronounced. Refinement (ii) produces a radiating wave field that is bidirectional, i.e. with both poleward and equatorward components. We show that the presence of regions of significant background vorticity with horizontal scales significantly smaller than the width of the storm track provides the scale selection mechanism to excite waves with sufficiently super-inertial frequencies to propagate poleward distances of the order of 1000 km.
Tilted incompressible Coriolis modes in spheroids
D. J. Ivers
The incompressible flow of a uniform fluid, which fills a rigid spheroid rotating about an arbitrary axis fixed in an inertial frame, is dominated at small Rossby and Ekman numbers by the rotation through the Coriolis force. The effects of rotation on the flow can be found by treating the Coriolis force modified by a pressure gradient as a skew-symmetric bounded linear operator $\boldsymbol{{\mathcal{C}}}$ acting on smooth inviscid incompressible flows in the spheroid. It is shown that the space of incompressible polynomial flows of degree $N$ or less in the spheroid is invariant under $\boldsymbol{{\mathcal{C}}}$ for any $N$ . The skew symmetry of $\boldsymbol{{\mathcal{C}}}$ implies the Coriolis operator $\boldsymbol{{\mathcal{C}}}$ is non-defective for such flows with an orthogonal set of eigenmodes (inertial and geostrophic modes) which form a basis for the finite-dimensional space of spheroidal polynomial flows. The eigenmodes are tilted if the rotation axis is not aligned with the symmetry axis of the spheroid. The non-defective property of $\boldsymbol{{\mathcal{C}}}$ enables enumeration of the modes and proof of their completeness using the Weierstrass polynomial approximation theorem. The fundamental tool, which is required to establish invariance of spheroidal polynomial flows under $\boldsymbol{{\mathcal{C}}}$ and completeness of the Coriolis modes, is that the solution of the polynomial Poisson–Neumann problem, i.e. Poisson's equation with Neumann boundary condition and polynomial data, in a spheroid is a polynomial. The Coriolis modes of degree one and all geostrophic modes are explicitly constructed. Only the modes of degree one have non-zero angular momentum in the boundary frame.
Green's functions for Rossby waves
R. C. Kloosterziel, L. R. M. Maas
Journal: Journal of Fluid Mechanics / Volume 830 / 10 November 2017
Compact solutions are presented for planetary, non-divergent, barotropic Rossby waves generated by (i) an impulsive point source and (ii) a sustained point source of curl of wind stress. Previously, only cumbersome integral expressions were known, rendering them practically useless. Our simple expressions allow for immediate numerical visualization/animation and further mathematical analysis.
Damping of magnetohydrodynamic waves in a rotating fluid
Binod Sreenivasan, Ghanesh Narasimhan
Journal: Journal of Fluid Mechanics / Volume 828 / 10 October 2017
Published online by Cambridge University Press: 12 September 2017, pp. 867-905
Print publication: 10 October 2017
The long-time evolution of a flow structure subject to background rotation and a coaxial uniform magnetic field is investigated in this paper. The conditions of magnetic Reynolds number $Rm\ll 1$ and Rossby number $Ro\ll 1$ apply, while the condition of magnetic interaction parameter $N\gg 1$ ensures that nonlinear inertial forces are small in the system. Cylindrical polar coordinates $(s,\unicode[STIX]{x1D719},z)$ are used, where the velocity and the induced magnetic field are axisymmetric. Two regimes are analysed in the inviscid limit, that of strong rotation, where the inertial wave frequency is much higher than the Alfvén wave frequency, and that of weak rotation, where the Alfvén wave frequency is dominant. In either regime, the evolution consists of a damped wave-dominated phase followed by a diffusion-dominated phase. For strong rotation, the laws of energy decay in the damped wave phase are obtained by considering the decay of the fast and slow magneto-Coriolis (MC) waves individually. The diffusion-dominated phase obeys the decay laws in the well-known quasistatic approximation. The wave–diffusion transition time scale indicates that the wave phase of decay is very long, so that small-scale turbulence is characterized by damped wave motions. The ratio of kinetic to magnetic energies of the slow MC wave in the early stages of evolution is $O(Le^{2})$ , where $Le$ is the initial ratio of the inertial wave to Alfvén wave time scales. The induced magnetic field is hence far more efficient than the velocity in supporting slow MC waves for $Le\ll 1$ . In the regime of weak rotation, the fast and slow MC wave solutions merge and tend to the classical damped Alfvén wave solution. Here, the decay laws in non-rotating magnetohydrodynamic turbulence (Moffatt, J. Fluid Mech., vol. 28, 1967, pp. 571–592) are recovered. Computations of the general solution for the long-time decay of an isolated vortex confirm the theoretical energy scalings as well as the wave–diffusion transition time scale of the kinetic energy. It is shown that a magnetically damped system that initially generates Alfvén waves because of relatively weak rotation can subsequently give rise to MC waves. Small-scale motions of $Rm\sim 1$ in the Earth's core probably generate slow magnetostrophic waves only for $Le>0.1$ , which suggests that a mean intensity of ${\sim}10~\text{mT}$ or higher is plausible for the toroidal magnetic field within the core.
Internal shear layers from librating objects
Stéphane Le Dizès, Michael Le Bars
In this work, we analyse the internal shear layer structures generated by the libration of an axisymmetric object in an unbounded fluid rotating at a rotation rate $\unicode[STIX]{x1D6FA}^{\ast }$ using direct numerical simulation and small Ekman number asymptotic analysis. We consider weak libration amplitude and libration frequency $\unicode[STIX]{x1D714}^{\ast }$ within the inertial wave interval $(0,2\unicode[STIX]{x1D6FA}^{\ast })$ such that the fluid dynamics is mainly described by a linear axisymmetric harmonic solution. The internal shear layer structures appear along the characteristic cones of angle $\unicode[STIX]{x1D703}_{c}=\text{acos}(\unicode[STIX]{x1D714}^{\ast }/(2\unicode[STIX]{x1D6FA}^{\ast }))$ which are tangent to the librating object at so-called critical latitudes. These layers correspond to thin viscous regions where the singularities of the inviscid solution are smoothed. We assume that the velocity field in these layers is described by the class of similarity solutions introduced by Moore & Saffman (Phil. Trans. R. Soc. Lond. A, vol. 264, 1969, pp. 597–634). These solutions are characterized by two parameters only: a real parameter $m$ , which measures the strength of the underlying singularity, and a complex amplitude coefficient $C_{0}$ . We first analyse the case of a disk for which a general asymptotic solution for small Ekman numbers is known when the disk is in a solid plane. We demonstrate that the numerical solutions obtained for a free disk and for a disk in a solid plane are both well described by the asymptotic solution and by its similarity form within the internal shear layers. For the disk, we obtain a parameter $m=1$ corresponding to a Dirac source at the edge of the disk and a coefficient $C_{0}\propto E^{1/6}$ where $E$ is the Ekman number. The case of a smoothed librating object such as a spheroid is found to be different. By asymptotically matching the boundary layer solution to similarity solutions close to a critical latitude on the surface, we show that the adequate parameter $m$ for the similarity solution is $m=5/4$ , leading to a coefficient $C_{0}\propto E^{1/12}$ , that is larger than for the case of a disk for small Ekman numbers. A simple general expression for $C_{0}$ valid for any axisymmetric object is obtained as a function of the local curvature radius at the critical latitude in agreement with this change of scaling. This result is tested and validated against direct numerical simulations.
On the instabilities of a potential vortex with a free surface
J. Mougel, D. Fabre, L. Lacaze, T. Bohr
In this paper, we address the linear stability analysis of a confined potential vortex with a free surface. This particular flow has been recently used by Tophøj et al. (Phys. Rev. Lett., vol. 110(19), 2013, article 194502) as a model for the swirling flow of fluid in an open cylindrical container, driven by rotating the bottom plate (the rotating bottom experiment) to explain the so-called rotating polygons instability (Vatistas J. Fluid Mech., vol. 217, 1990, pp. 241–248; Jansson et al., Phys. Rev. Lett., vol. 96, 2006, article 174502) in terms of surface wave interactions leading to resonance. Global linear stability results are complemented by a Wentzel–Kramers–Brillouin–Jeffreys (WKBJ) analysis in the shallow-water limit as well as new experimental observations. It is found that global stability results predict additional resonances that cannot be captured by the simple wave coupling model presented in Tophøj et al. (2013). Both the main resonances (thought to be at the root of the rotating polygons) and these secondary resonances are interpreted in terms of over-reflection phenomena by the WKBJ analysis. Finally, we provide experimental evidence for a secondary resonance supporting the numerical and theoretical analysis presented. These different methods and observations allow to support the unstable wave coupling mechanism as the physical process at the origin of the polygonal patterns observed in free-surface rotating flows.