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.     

The interaction of topography and shear flow in a homogeneous rotating fluid

Abstract

Laboratory experiments are described in which a cylindrical obstacle is moved azimuthally through a rotating cylindrical tank of fluid in which a basic azimuthal flow inversely dependent upon the tank radius is generated by means of a source-sink arrangement. A technique is described whereby the flow can be adjusted until, relative to the obstacle, it is forward near the centre of the tank and reversed near the rim, with a monotonic variation between these extremes. The sense of this shear, relative to the obstacle, can be altered so that it either opposes or coincides with the sense of the basic rotation. Both cases are investigated in the experiments. The results of both qualitative and quantitative measurements are presented, and some comparison with related theoretical work is attempted.     

Experiments on instability of columnar vortex pairs in rotating fluid

We present results from a new series of experiments on the geophysically important issue of the instability of anticyclonic columnar vortices in a rotating fluid in circumstances such that the Rossby number exceeds unity. The vortex pair consisting of a cyclonic and an anticyclonic vortex is induced by a rotating flap in a fluid which is itself initially in a state of solid-body rotation. The anticyclonic vortex is then subject to either centrifugal or elliptical instability, depending on whether its initial ellipticity is small or large, while the cyclone always remains stable. The experimental results demonstrate that the perturbations due to centrifugal instability have a typical form of toroidal vortices of alternating sign (rib vortices). The perturbations due to elliptical instability are of the form of sinuous deformation of the vortex filament in the plane of maximal stretching which corresponds to the plane of symmetry for the vortex pair. The initial perturbations in both cases are characterized by a definite wave number in the vertical direction. The characteristics of the unstable anticyclone are determined by the main nondimensional parameter of the flow - the Rossby number. The appearance of both centrifugal and elliptical instabilities are in accord with the predictions of theoretical criteria for these cases.     

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”.     

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.     

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.     

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.     

Steady-state convection in a rotating long cavity

Abstract

Convection experiments in a rotating long cavity were conducted to investigate the changes introduced by Coriolis-force to the well understood convection in the (nonrotating) long cavity. In the far convective regime, the results indicated that large-scale features of the convection were dominated by geostrophically balanced confined primary currents. Close to the transition regime, where the intruding currents gain a thickness close to the vertical dimension of the tank, secondary features became predominant. In the transition regime, these secondary features were not present and the simple picture of two cavity filling counter currents with laterally tilted isotherms could be confirmed.     

A Formal Theory for Vortex Rossby Waves and Vortex Evolution

A formal theory is presented for the balanced evolution of a small-amplitude, small-scale wave field in the presence of an axisymmetric vortex initially in gradient-wind balance and the accompanying changes induced in the vortex by the azimuthally averaged wave fluxes. The theory is a multi-parameter, asymptotic perturbation expansion for the conservative, rotating, f-plane, shallow-water equations. It extends previous work on Rossby-wave dynamics in vortices and more generally provides a new perspective on wave/mean-flow interaction in finite Rossby-number regimes. Some illustrative solutions are presented for a perturbed vortex undergoing axisymmetrization.     

Thermal convection in differentially rotating systems

Abstract

The onset of convection in a cylindrical fluid annulus is analyzed in the case when the cylindrical walls are rotating differentially, a temperature gradient in the radial direction is applied, and the centrifugal force dominates over gravity. The small gap approximation is used and no-slip conditions on the cylindrical walls are assumed. It is found that over a considerable range of the parameter space either convection rolls aligned with the axis of rotation or rolls in the perpendicular (azimuthal) direction are preferred. It is shown that by a suitable redefinition of parameters, results for finite amplitude Taylor vortices and for convection rolls in the presence of shear can be applied to the present problem. Weakly nonlinear results for transverse rolls in a Couette flow indicate the possibility of subcritical bifurcation for Prandtl numbers P less than 0.82. Heat and momentum transports are derived as functions of P and the problem of interaction between transverse and longitudinal rolls is considered. The relevance of the analysis for problems of convection in planetary and stellar atmospheres is briefly discussed.     

Large-scale vortices in helical turbulence of incompressible fluid

Abstract

The interaction of a mean flow with a random fluctuation field is considered. This interaction is described by the averaged Navier-Stokes equation in which terms nonlinear in the fluctuation field are expressed in terms of the mean flow and the statistical properties of the fluctuation field, which is assumed to be homogeneous, isotropic, and helical. Averaged equations are derived using a functional technique. These equations are solved for a mean background flow that depends linearly on the position vector. The solutions show that large-scale vortices may arise in this system.     

The role of boundary conditions in the simulation of rotating,stratified turbulence

Abstract

In this paper we use the CASL method to explore the role of boundary conditions in determining the long-time behaviour of rotating, stratified, quasi-geostrophic turbulence. We find that initially two-dimensional (sufficiently tall) columns of potential vorticity (PV) break down through three-dimensional instability to give a fully three-dimensional flow consisting of ellipsoidal structures. This is the case both for rigid-lid (isothermal) vertical boundary conditions and for vertically periodic boundaries. However, the rigid boundary case gives rise to semi-ellipsoids at both the top and bottom boundaries, and, for sufficient domain depths, preferred depths for the formation of ellipsoids in the interior. By contrast, in the vertically periodic case, the distribution of ellipsoids is homogeneous in depth.

The role of the horizontal boundaries is indirect, but still significant. In all cases doubly periodic horizontal boundary conditions are imposed. We consider a range of initial conditions where in each case equal numbers of two-dimensional columns of positive and negative vorticity are used, taking up a fixed, but relatively small fraction of the domain (approximately 5%). Thus when there is only a small number of vortices, they have larger radius. When the initial number of vortices is small enough (i.e., when the radius is not small compared with the horizontal domain width), at long time there is a two-dimensionalisation giving rise to a single column of positive PV and a single column of negative PV, as has been observed in some previous simulations. We find the same phenomenon for both vertically periodic and rigid lid boundary conditions, but it occurs over a broader range of initial conditions in the vertically periodic case. However, in all cases fully three-dimensional final states are regained when the number of vortices is increased while keeping the fraction of the domain occupied by vortices fixed, i.e., when the vortex radius/domain width ratio is sufficiently small.     

A note on the geostrophic flow past a crater in a rotating fluid

Abstract

Laboratory experiments are described on the flow past a solid obstacle in a rotating, homogeneous fluid. Specifically, the obstacle has the form of a walled crater specially constructed so that the volume of the depression is identically equal to the volume of the walls. The results show that closed streamlines occur rather more easily above such topography than above other obstacle types of the same scale but that the conditions for closure are determined essentially by the detailed geometry of the crater, the value of the Rossby number, and the depth of the fluid. The observed flow patterns are analysed and classified and attempts to quantify the most common flow type are made.     

Barotropic instability of a boundary jet on a sloping bottom

Abstract

The stability of a shear flow on a sloping bottom in a homogeneous, rotating system was investigated by means of a laboratory experiment.

The basic flow was driven near a vertical wall of a circular container by a ring-shaped plate that contacted with a free surface of the working fluid and rotated relative to the fluid container. The velocity profile was asymmetric in the radial direction and had only one inflection point. The velocity profile was well expressed by a linear theory for the vertical shear layer.

The effect of the circular geometry was checked by comparing experimental results obtained in two fluid systems in which only the sign of the curvature was opposite and it was confirmed that circular geometry was not essential for the shear flow on the sloping bottom in this experiment.

It was found that the sloping bottom stabilizes the basic flow only when the drift direction of the topographic Rossby wave is opposite to that of the basic flow. The viscous dissipation in both the Ekman layer and the interior region was also important in determining the critical Rossby number.

The eddy fields caused by the instability can be classified into two types: One is the stationary eddy field in which a row of eddies moves along the basic flow without changing form. The other is the flow pattern in which eddies have finite life times and their configuration is not well organized. When the sloping bottom does not stabilize the basic flow, the former flow pattern is realized, otherwise the latter flow pattern appears.

The wave numbers of the eddies in the regular flow pattern were observed as a function of the Rossby number. The relation did not fit to linear preferred modes predicted by an eigenvalue problem.     

Spontaneous Emission of Gravity Waves by Interacting Vortex Dipoles in a Stratified Fluid: Laboratory Experiments

Results from a new series of experiments on the geophysically important issue of spontaneous emission of internal gravity waves during unsteady interactions of vortical structures are presented. Vortex dipoles are a common element of a quasi-two-dimensional turbulent flow. Vortex dipoles perform translational motion and can collide with other vortices. During collision events the flow is unsteady and unbalanced and a further adjustment process associated with these events can therefore result in the spontaneous emission of gravity waves. Our laboratory experiments demonstrate that gravity waves are emitted when two translating vortex dipoles interact (collide) in a layered fluid, in accord with the current theoretical results. The emission was evident both in a two-layer system and in a fluid with a linear distribution of density with depth. The waves were generated during the period of deceleration of the secondary dipoles which constitute a vortex quadrupole emerging immediately after the collision of the primary dipoles.     

An experimental study of baroclinic wave interactions in a two-layer system

Abstract

Baroclinic waves in a rotating two-layer shear flow are observed by measuring fluctuations in the height of the interface using a technique which does not disturb the flow. Decomposition of the shape of the interface into azimuthal Fourier components shows that the wave spectrum is dominated by a single wave number, but other components have significant amplitudes. Nonlinear interactions between these components are isolated by comparing the results with the predictions of a linear stability theory. The observed interactions are reminiscent of those found by Hide, Mason and Plumb (1977) in a different system, thus demonstrating that they are a fundamental property of baroclinic waves.     

A numerical study of detached shear layers in a rotating sliced cylinder

Abstract

The low Rossby number flow in a rotating cylinder with an inclined bottom, of small slope, is examined when part of the lid of the container is rotating at a slightly different rate. The resulting flow is calculated numerically by solving the governing equations for the two-dimensional geostrophic motion which approximates the flow in most of the fluid including the inertially-modified E ^¼ -layers. The presence of ageostrophic regions, on the container walls and beneath the velocity discontinuity on the lid, is accounted for in the governing equations and their boundary conditions. This study supplements previous work on this configuration, in which the zero Rossby number flow was calculated and experimental results were presented, by enabling a direct comparison to be made between the results of the low Rossby number theory and the experiments. The numerical results for a range of Rossby and Ekman numbers compare well with those from the experiments despite a severe limitation on the size of the Rossby number arising from the analysis in the ageostrophic part of the detached shear layer.     

On the Structures Observed in Thin Rotating Layers of a Conductive Fluid and the Anomalies of the Geomagnetic Field

The results of the laboratory and numerical experiments in circular rotating trays with thin layers of a conductive fluid under the MHD generation of small-scale velocity fields are presented. The configurations of constant magnets for MHD generation were determined based on the numerical calculations with shallow water equations. Both the laboratory and numerical experiments with rotating trays demonstrate the emergence of nonaxisymmetric structures and large-scale near-circular vortices caused by the energy transfer from the system of the externally generated small-scale vortices to the large-scale velocity fields under the action of the Coriolis force. The near-circular vortex has areas with differential rotation when the angular velocity of rotation decreases with the radius. The single large-scale vortices and wide jet flows arise in the regimes of subrotation and superrotation relative to the external rotation depending on its angular velocity. The emergence of the flow structures with the azimuthal wave number m = 2 is demonstrated, and their probable relation to the anomalies of the geomagnetic field observed on the Earth’s surface is considered.     

Point vortex dynamics for coupled surface/interior QG and propagating heton clusters in models for ocean convection

Abstract

The dynamic behavior of baroclinic point vortices in two-layer quasi-geostrophic flow provides a compact model for studying the transport of heat in a variety of geophysical flows including recent heton models for open ocean convection as a response to spatially localized intense surface cooling. In such heton models, the exchange of heat with the region external to the compact cooling region reaches a statistical equilibrium through the propagation of tilted heton clusters. Such tilted heton clusters are aggregates of cyclonic vortices in the upper layer and anti-cyclonic vortices in the lower layer which collectively propagate almost as an elementary tilted heton pair even though the individual vortices undergo shifts in their relative locations. One main result in this paper is a mathematical theorem demonstrating the existence of large families of long-lived propagating heton clusters for the two-layer model in a fashion compatible to a remarkable degree with the earlier numerical simulations. Two-layer quasi-geostrophic flow is an idealization of coupled surface/interior quasi-geostrophic flow. The second family of results in this paper involves the systematic development of Hamiltonian point vortex dynamics for coupled surface/interior QG with an emphasis on propagating solutions that transport heat. These are novel vortex systems of mixed species where surface heat particles interact with quasi-geostrophic point vortices. The variety of elementary two-vortex exact solutions that transport heat include two surface heat particles of opposite strength, tilted pairs of a surface heat particle coupled to an interior vortex of opposite strength and two interior tilted vortices of opposite strength at different depths. The propagation speeds of the tilted elementary hetons in the coupled surface/interior QG model are compared and contrasted with those in the simpler two-layer heton models. Finally, mathematical theorems are presented for the existence of large families of propagating long-lived tilted heton clusters for point vortex solutions in coupled surface/interior QG flow.     

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.