A simple demonstration of shear-flow instability
aa r X i v : . [ phy s i c s . e d - ph ] J a n A simple demonstration of shear-flow instability.
Tom Howard ∗ and Ana Barbosa Aguiar Met Office, FitzRoy Road, Exeter, EX1 3PB, United Kingdom. (Dated: January 6, 2021)
Abstract
We describe a simple classroom demonstration of a fluid-dynamic instability. The demonstrationrequires only a bucket of water, a piece of string and some used tealeaves or coffee grounds.We argue that the mechanism for the instability, at least in its later stages, is two-dimensionalbarotropic (shear-flow) instability and we present evidence in support of this. We show results of anequivalent basic two-dimensional numerical non-linear model, which simulates behavior comparableto that observed in the bucket demonstration. Modified simulations show that the instability doesnot depend on the curvature of the domain, but rather on the velocity profile. IDEOS
The manuscript cites video clips which are available password-protected on the vimeovideo-sharing site. We strongly encourage readers to watch these video clips, in particularthe following two, as they add a lot to the description given by the text and figures.The experiment: VideoS1 available at https://vimeo.com/278481176 (password: Wel-come123)The numerical simulation (slow motion): VideoS2 available at https://vimeo.com/399593365 (password: Welcome123),This article may be downloaded for personal use only. Any other use requires prior permis-sion of the author and AIP Publishing. This article appeared in American Journal of Physics88, Issue 12, 1041 (2020) and may be found at https://doi.org/10.1119/10.0002438
I. INTRODUCTION
Shear-flow instability is a fundamental process in fluid dynamics, and is associated withthe destruction of parallel laminar flow. As such, it can be seen as fundamental to the onsetand existence of both two- and three-dimensional turbulence. Geophysicists usually use theterm “barotropic instability” in the context of the instability of a horizontally-sheared fluidon a rotating planet. The word “barotropic” distinguishes this from baroclinic instability,in which horizontal temperature gradients play a key role. Baroclinic instability is themajor large-scale weather-generating process of the mid-latitudes (the region between thetropics and the polar circles), whereas barotropic instability is important as an instabilitymechanism for jets and vortices. Both instabilities may be associated with the polygonalshapes often observed in the atmospheres of rotating planets.
The instructional value of laboratory demonstrations is well-recognized, both in the con-text of geophysical fluid dynamics in particular, and in the context of fluid dynamicsin general (Shakerin and references therein). A classroom demonstration (e.g. ) of baro-clinic instability can be prepared based on the “dishpan” experiments of the mid-twentiethcentury. To study barotropic instability in the laboratory, sophisticated experiments some-what analogous to the dishpan experiments can be constructed using differentially rotatingboundaries.
The shear is introduced due to the differential rotation of concentric sec-2ions of the bottom of the container of the working fluid. However, such an apparatus isbeyond the budget of most classroom teaching of geophysical fluid dynamics.Reynolds describes experiments with a horizontal glass tube containing carbon disul-phide overlain by water. Carbon disulphide is a highly toxic liquid with a density approxi-mately 25% higher than water at room temperature. By tilting the tube, Reynolds induceda counterflow in the two liquids and this led to instability at the shear interface when the ve-locity difference was large enough. A version of this apparatus, using a carefully-establishedsaline/fresh water step to create the density difference, is a well-loved teaching aid at theUniversity of Cambridge. However, the equipment is quite specialized and the set-up pro-cedure is painstaking and time-consuming. One simpler classroom option is to introduceshear by moving a solid boundary through a fluid — for example a cylinder with verticalaxis, partially submerged in water, may be towed sideways. Food dye applied to the cylinderbefore immersion, or mica powder mixed into the water, exposes the vortices which appearin the wake. Kelley and Ouellette describe a laboratory approach aimed at undergraduatestudents, who use an electromagnetic technique to drive Kolmogorov flow (a type of cyclicshear flow which exhibits shear-flow instability under some forcing parameters) in a thinfluid layer, and measure it quantitatively with a webcam. The authors estimate costs of theorder of $ present an apparatus based on a tilted gravity-drivensoap tunnel, with which a variety of fluid-dynamic phenomena, including shear instability,can be demonstrated. Their estimated costs were of the order of $ II. BRIEF REVIEW OF THE MECHANISM OF SHEAR-FLOW INSTABILITY
To introduce the mechanism we begin with a two-dimensional conceptual model consist-ing of two regions of incompressible fluid of constant uniform density (let’s call them theNorth and South parts) each in uniform flow in opposite directions like two counterflowingcarriageways of traffic. Nothing divides them; their interface is initially a straight verticalplane. For simplicity in this section we do not consider a curved shear layer as seen in the3ucket. We show in subsection IV C that the curvature is not an essential feature of ourdemonstration.The fundamental cause of shear flow instability is this: interaction of the two parts canexchange momentum between them and release kinetic energy, which is then available todrive a more complex (sometimes chaotic) behavior in the region of the interface.
The details of the mechanism are most easily understood in terms of a derived propertyof fluid motion called vorticity. Mathematically, vorticity is the curl of the velocity field. Ina two-dimensional flow context, we can regard the vorticity as a scalar, because its directionis always the same (normal to the plane of the motion). Shapiro offers a striking physicalinterpretation of vorticity, which we paraphrase here: “The vorticity is a measure of themoment of momentum of a small fluid particle about its center of mass. Suppose that youhad some very complicated [two-dimensional] motion in a liquid and that it were possibleby magic suddenly to freeze a small sphere of that liquid into a solid. During the freezingthe moment of momentum would be conserved. The vorticity of the fluid before freezingis exactly twice the angular velocity of the solid sphere just after its birth.” This vividinterpretation may sound fanciful, but the mental image it conjures is not so different fromwhat we see in the development of shear-flow instability: vorticity (which is initially presentas a continuous shear at the interface) is rapidly converted by the instability into the rotationof a number of vortices along the line of the interface.The mechanism is illustrated in Fig. (1), which is adapted from Batchelor. The processbegins with undisturbed counterflow and a straight interface as described above, shown bythe dashed straight line (this line would have a monotonic curve in the case of the bucketdemonstration). We introduce a small sinusoidal disturbance as shown. It can be shown thatdisturbed vorticity south of the dashed line (for example at D) induces eastward movementin the vorticity north of the dashed line, and likewise vorticity north of the dashed line (forexample at B) induces westward movement in the vorticity south of the dashed line. Bothof these effects concentrate vorticity at points like A, giving a positive feedback, with energyprovided by the kinetic energy of the counterflowing parts. The arrows indicate the directionof the movement of the vorticity induced by the disturbance, and show both the accumulationof vorticity at points like A (purple arrows) and the general rotation about points like A (red4rrows) which together lead to growth of the disturbance. The accumulation of vorticity (atpoints like A) and depletion of vorticity (at points like C) is also indicated by the thicknessof the wavy line. A comprehensive discussion of this case is given by Batchelor. A B C D Undisturbed flowUndisturbed flow
N EW S
FIG. 1. Schematic diagram showing the positive feedback of a small sinusoidal disturbance to theinterface of two counterflowing regions of fluid. The compass rose is for ease of reference in themain text; other orientations are, of course, possible.
In this simplified configuration the sinusoidal disturbance grows around nodes (like pointA for example) with fixed spatial locations where distinct vortices will, ultimately, form.Extending the traffic analogy, a typical vortex centre is like a fixed point on the road, withtraffic sweeping past one way on one side, and the other way on the other side.In a more general case where the velocities are not equal and opposite, for example inthe demonstration which we describe next, the vortices move along the line of an interfaceat a speed intermediate between the faster-moving fluid on one side and the slower-movingfluid on the other side of the interface. 5
IG. 2. Snapshot of three vortices in a bucket, visualized using tealeaves.
III. DEMONSTRATIONA. Description
A bucket (ours has an internal diameter of 180 mm) is filled with tap water to a depthof about 30 mm. A few used tealeaves or coffee grounds are added as a simple flow tracer.A piece of string is attached to the center of the bucket handle and the bucket lifted a littleso that it is suspended on the string. A vigorous angular impulse (a flick) is applied tothe handle using the fingers, and the bucket allowed to spin for a few turns. This sets thewater at and near the wall in rotation. The bucket is then set back down, arresting itsrotation (and attenuating the motion of the water nearest to the wall). Typically, within afew seconds, the axisymmetry of the resulting flow breaks and a number of smaller vortices(typically two, three or four) appear — see Fig. (2) and our first video clip VideoS1, availableat https://vimeo.com/278481176 (password: Welcome123). These smaller vortices, whichrotate in the same sense as the initial rotation of the bucket, will typically merge into a near-axisymmetric single vortex in a few more seconds, before the motion dies completely. Withcareful observation, the vortices can also be seen on the top surface, for example by sprinklingbuoyant glitter onto the water. However the tealeaves (which have a small negative buoyancyand thus illustrate the flow near the bottom boundary) have the well-recognized tendencyto converge in the vortices, and this provides an impactful visualization. Incidentally, we6 IG. 3. Schematic diagram showing cross-sectional views of alternative experimental setups whichall exhibit similar instability. (a) Initial setup. (b) As (a) but in a larger outer bucket facilitatingthe addition of a rigid lid (see main text). (c) Taller and narrower. In this setup the large outerbucket helps to reduce wobble. Bucket handles and string not shown. came upon our demonstration by accident whilst playing with a version of the well-known“Einstein’s Tea Leaves” demonstration, which is widely used in teaching the behavior ofair near the surface of atmospheric pressure systems. A snapshot of the demonstration is shown in Fig. (2). The initial flick was of the orderof four revolutions per second and the rotation of the bucket was then arrested by replacingit on the table after about two seconds. However, the behavior seems to be quite robust tovariations as long as the initial flick is strong enough. We have observed the instability ina taller, narrower configuration (120 mm diameter by 100 mm deep) and in a setup witha rigid lid, arranged by filling a transparent cylindrical plastic container (a cake box) withwater, placing a larger bucket upside down over the box so that the base of the bucket formsa lid to the box, and then inverting the whole. Atmospheric pressure stops the water fromdraining out of the box (rather like a very squat manometer). Similar instability behavioris seen in each case. These alternative configurations are shown schematically in Fig. (3),and video clips of their behavior are linked from section VI.
B. Curricular Context
In a teaching context, this demonstration could, for example, be included in a sessionintroducing the general behavior of the mid-latitude atmosphere via the concept of fluid-dynamic instability. The initial velocity profile (which is shown in Fig. (4)) can be interpretedas a jet, whose instability leads to the production of several vortices. Whilst this is a7oose and incomplete model of the mid-latitudes, it does provide an engaging, hands-onapproach. Given the low cost, it is easy to provide a group of students with several bucketsso that they can experiment and observe the behavior in small groups. In the early 20thcentury barotropic and baroclinic instability were rival hypotheses for the explanation ofthe observed mid-latitude eddies, and so having introduced the general concept of fluid-dynamic instability via the buckets, the session can then move on to a demonstration ofbaroclinic instability such as the one described by Marshall and Plumb. Alternatively,the activity could play a role in demystifying observations of polygonal patterns in theatmosphere of rotating planets.
C. Nature of the instability
When the bucket is arrested, the water in contact with the bucket is also arrested (in ac-cordance with the no-slip condition). So now the depth-averaged azimuthal velocity reachesa peak near the wall but falls to zero at the wall.As discussed by Rabaud and Couder, Coles observed patterns of “rollers” with theiraxes parallel to the rotation axis (Coles’ Fig. 22(o)) in a Couette geometry (concentricindependently-rotating cylinders), associated with sudden starts and stops of the rotationof the outer cylinder, which created an inflection in the velocity profile. This description isstrikingly similar to our experiment and observations.Our crudely-measured velocity profile, from before any visible onset of instability, is shownin Fig. (4). The velocity profile at this stage satisfies the Rayleigh-Kuo criterion, which isa necessary (though not sufficient) condition for shear-flow instability. In our context thiscriterion is that the velocity profile must include an inflection point, i.e. a point at whichthe curvature of the velocity profile changes sign.In initial discussion with correspondents (see acknowledgments), various candidate pro-cesses were suggested for the observed instability, including centrifugal instability generatingturbulence followed by an up-scale energy transfer; an instability of the Ekman layer; andshear instability. Cullen (pers. comm.) noted that the observed behavior appears to bequasi-two-dimensional, and that such flow would not be observed in the first place if it wasunstable to three-dimensional disturbances. In the following section we show that a two-8 IG. 4. Estimated azimuthal velocity ( u ) profile against radius ( r ) from a typical experiment,before any instability was observable by eye, is shown by the black dots connected by a gray line.Speeds were estimated by manually tracking particles through several frames of a stop-motionfilm. Uncertainties in this profile are difficult to estimate but an error of the order of 0.02 m s − is plausible. It was apparent, however, that all velocities were positive at this stage. The pointrepresenting zero velocity at the outside wall ( r = 0 .
08 m) is assumed, based on the no-slipcondition, rather than estimated from observations. The analytical profile used in the modellingexperiments (see section IV) is shown by the continuous black line. The large circle on the left-handside of the plot represents the bucket seen from above, to clarify the context of the profile. dimensional model (which admits no variation in the vertical direction) exhibits behaviorsimilar to that observed when the model is initialized with a profile based on the measuredprofile of Fig. (4).
IV. NUMERICAL MODELA. Description
We describe a non-linear two-dimensional numerical model of the instability seen in thebucket. The stream function, ψ , and vorticity, ζ , are convenient tools for such a model.For our purpose, each is a scalar function of space and time. Mathematically, we define thestream function as a field, the 2-D Laplacian of which is the vorticity field. Physically, this9ives the useful properties that the instantaneous contours of the stream function (which maybe familiar to some readers as “streamlines”) are everywhere tangent to the instantaneousvelocity vector, and that regions of rapid movement can be identified by closely-packedstreamlines. The streamlines indicate the paths that particles in the flow would take if theflow did not evolve over time. In our case the flow does evolve over time, so the stream linesare not identical to particle paths (see VideoS3).The values of stream function and vorticity do not depend on the coordinate systemin which they are evaluated. For our purpose it is convenient to use a polar coordinatesystem: we use a polar grid with 256 azimuthal grid points and 112 radial grid points. Theintegration procedure is: • Given a vorticity field • Invert the Laplacian to obtain the stream function, ψ , given in polar coordinates by u = − ∂ψ/∂r , v = ∂ψ/r∂θ , where u is azimuthal velocity component, v is meridional(i.e. radial) velocity component, r is radius and θ is the azimuthal angle (in radians,defined positive in the direction of increasing u ) • Evaluate the components of velocity ( u, v ) from the gradient of the stream function • Advect the vorticity field for one time-step. “Advection” refers to movement by theflow of the fluid; thus in this context “advect” the vorticity means move the vorticity,using the velocity that we evaluated. We use a simple upwind explicit scheme with asmall time-step to accomplish this. • RepeatInversion of the Laplacian is a linear problem that can be addressed by a Fourier de-composition in the azimuthal direction. Owing to the linearity, the modes can be treatedindependently. This requires one tri-diagonal matrix inversion for each mode.We initialized our model with the smooth velocity profile shown by the black line inFig. (4). This velocity is a heuristic analytical function of the radius, as follows: u ( r ) = u α exp( − α γ ), where α = ( R − r ) /w , R is the outer radius, and u , w, γ are tunable parame-ters, w being a scale width for the velocity jet. The fit was chosen by eye such that the peakazimuthal speed and the radius at which it occurred matched well with the measured profile.10e used w = 0 .
008 m, u = 1 .
05 m s − , γ = 1 .
3. This gives a profile which is simple to workwith, whilst maintaining consistency with the measurements to within their uncertainties.A more complicated alternative would be to use piecewise linear interpolation (as shownby the gray lines) between each of the measurements, but this seems unjustified given theuncertainties. The radial gradient of vorticity changes sign at about r = 0 . r = 0 .
055 m and r = 0 .
072 m for the measuredprofile.We add a field of small random vorticity ‘noise’ to initialize any instability. Typicalmagnitudes (root-mean-square) of the noise are set to be of the order of 0.1% of typicalvalues of the initial vorticity profile. To sidestep the difficulty with the singularity at thepole we truncate the grid at a small but non-zero radius and approximate the behavior ofthe fluid within this radius as a blob of material which is allowed to revolve in rigid rotationaround the pole but is otherwise identical to the rest of the fluid. (We suggest that this is onlya minor constraint on the motion, and we note that a version of the laboratory demonstrationnot shown here including a comparable fixed rigid central cylinder of ∼
20 mm diameterexhibits very similar instability behavior.) This leads naturally to a Neumann boundarycondition on ψ for the k = 0 mode (which represents the average) of the stream function atthe inner boundary: ∂ψ k ∂r = − ζ k r , inner boundary , ( k = 0) , (1)since ζ k r ( k = 0) is the azimuthal velocity of the rigid circular blob. Konijnenberg etal. use a similar boundary condition (their equation 5.8). For the other modes, we applya Dirichlet boundary condition at the inner boundary: ψ k = 0 , inner boundary , ( k = 0) , (2)which represents the constraint that no fluid can pass through the circular boundary of therigid blob. At the outer boundary we apply a Dirichlet boundary condition to all modes(including k = 0): ψ k = 0 , outer boundary, all k. (3)Again, for the k = 0 mode, this represents the constraint that no fluid can pass through theouter circular boundary of the domain (the wall of the bucket), and in the case of the k = 011ode, this effectively sets the arbitrary zero for the stream function so that the solution isunique.The inner boundary conditions for the vorticity are ∂ζ k ∂r = 0 ( k = 0) , and ζ k = 0 ( k = 0) . Treating the circular blob as rigid implies that its vorticity has a single value at any time.Our implementation of Eq. (1) ensures that this vorticity is the average of the vorticity ofthe fluid just outside the blob (since the k = 0 mode represents the average).We do not include any explicit model of diffusion. Our first-order upwind scheme isknown to be diffusive, and one might optimistically argue that this property of the two-dimensional numerical model mimics the damping influence of the bottom boundary in thebucket. However we have not made any quantitative comparison of these damping factors. B. Numerical modelling results and discussion
Summary: the modelled evolving flow exhibits instability which is visually similar to thatobserved in the bucket, when initialized with a velocity profile similar to that measured. Theemergent modelled preferred wavenumber (which is three in this instance) is the same asthat observed, and the modelled growth rate is consistent with that observed.A common visualization tool for water flows is to add coloured dye to some parts of thewater. The dye acts as a passive “tracer”: showing the movement without influencing themovement. In our physical demonstration the tea leaves perform a similar function, exceptthat they tend to congregate in the vortices due to bottom boundary effects which are notsimulated in our numerical model. It is relatively simple to add a simulation of a numericalpassive tracer to the numerical model, and the result gives an intuitive sense of the behavior,because the tracer looks like phenomena which can be seen in the real world — for examplethe patterns seen in creamer added to coffee.A video clip of the simulation, VideoS2, available at https://vimeo.com/399593365 (password: Welcome123), shows a slow motion (one-fifth speed) animation of some aspectsof the simulation, concentrating on the most interesting period of flow evolution. The topleft-hand panel shows our tracer. The tracer starts out as a sinusoidal pattern of 4 radialcycles and 7 azimuthal cycles We chose 7 cycles to ensure that we did not prejudice the12isual emergence of three vortices. Green shows positive values of tracer and purple showsnegative values of tracer. The tracer diffuses, so we reset it at about 4.5 seconds to capturethe most interesting part of the evolution.A alternative intuitive visualization is shown in the top right-hand panel: a cloud of dotsmoving with the flow. Physically these are analogous to paper dots floating on the surfaceof the water in the bucket. The initial positions of the dots are chosen randomly. A short“tail” shows the recent track of each dot. The passive tracer of the top left panel is shownagain faintly in gray for comparison.Neither the tracer nor the dots are subject to the process which tends to concentrate thetea leaves in the vortices in the physical demonstration, and the formation of three separatevortices is seen more clearly in an animation of the stream function, shown by color-filledcontours in the bottom left panel. You can see that three sets of closed streamlines appear,corresponding to three vortices. Thus the three areas of closed streamlines correspond tothe three areas in which the tea leaves tend to congregate in the physical experiment. Thebottom right panel shows the dots with their “tails” again, this time overlain on contours ofthe stream function (streamlines). By watching this panel closely (and, if possible, stoppingthe animation after the three sets of closed streamlines have appeared and stepping throughone frame at a time) you can see for yourself that the instantaneous movement of theparticles is always along the streamlines, even though the particle paths are not identical tostreamlines (because the streamlines evolve).Video clip VideoS3, available at https://vimeo.com/279323035 (password: Welcome123),shows the evolution of the stream function in real time over the entire 30-second-long sim-ulation. The sense of the imposed initial rotation was clockwise in both the physicaldemonstration and the simulation.Figure (5) shows snapshots of the evolving simulated stream function after the instabilityhas begun to grow. To investigate the growth of the instability during a particular simulationwe focus on the radial velocity at approximately the mid radius (about 45 mm). A Hovmollerdiagram is a common way of plotting meteorological data. The axes are typically longitudeor latitude (x-axis) and time (y-axis) with the value of some field represented through coloror shading. Figure (6(a)) is a Hovmoller diagram of the radial velocity at mid radius, whichillustrates the inception, growth, and advection (movement by the flow) of the vorticesarising from the instability. 13
IG. 5. Streamlines (contours of the stream function) of the numerical simulation at 6.75, 8.25and 9.75 seconds (from left to right). Contour interval 0.0005 m s − . The dashed line shows theboundary of the center blob (see section IV A). The whole tripolar pattern rotates clockwise: thevortex at about “11 o’clock” in the left-hand panel is at about “10 o’clock” in the center panel andabout “6 o’clock” in the right-hand panel (see video clips VideoS2 [URL will be inserted by AIP]and VideoS3 [URL will be inserted by AIP]). To estimate the growth rate of the modelled instability we use the maximum radialvelocity at the mid radius as a simple metric of the strength of the instability. The evolutionof this metric is shown in Fig. (6(b)). Choosing a small section of the steepest part of thecurve (delineated by the straight lines on the plot) facilitates estimation of the growth rate.We find an e-folding time of approximately 0.6 seconds. Growth rates in the experiment andmodel can be visually compared by comparing the video clips; they appear similar. We donot have a precise measurement of the growth rate in the experiment, but the three vorticesare well-established by time t=8.2 seconds on the video clip, unidentifiable at 6.2 seconds,and arguably detectable at around 7.2 seconds. These observations are not inconsistent withthe e-folding time of 0.6 seconds obtained from the model.The model is quite basic, perhaps even crude by contemporary standards. However, a two-dimensional linear stability analysis yields very similar results in terms of wavenumber andgrowth rate of the instability (see appendix B). The cross-corroboration of the three experi-ments (laboratory, numerical and analytical) gives confidence in the numerical model results,in spite of its simplicity, and supports our suggestion that the instability is fundamentallytwo-dimensional. The strong similarity between the behavior seen in the two-dimensionalmodel and the observed behavior suggests that, although a centrifugal instability may be14 (a) radial speed at half−radius (m s ) - azimuthal angle ti m e ( s ) p p − 0 .0 50 .0 00 .0 5 e − e − e − e − (b) tim e ( s ) m a x r a d i a l s p ee d ( m s - ) FIG. 6. (a) Hovmoller diagram illustrating growth of radial velocity at half-radius. (b) Time seriesof maximum radial speed at mid-radius, on a logarithmic scale. E-folding time of the most rapidgrowth is approximately 0.6 seconds. involved in the creation of the profile we measured, a two-dimensional process with negli-gible variation in the vertical direction — we suggest the barotropic (shear-flow) instabilityassociated with the change in sign of the radial gradient of vorticity — can account for theevolution of the flow forwards in time from our axisymmetric measured profile.
C. Modified numerical model: cyclic straight channel
The instability is also simulated when the curvature is artificially removed, modelling ahypothetical straight channel with cyclic lateral boundaries.A well-recognized advantage of numerical modelling is the ease of modifying the experi-ment. To show that the existence of an instability depends on the velocity profile, and noton the curvature of the domain, we modified the numerical simulation. The modified simula-tion consists of a straight two-dimensional domain with cyclic boundaries in the primary ( x )direction and walls bounding the other ( y ) direction. Thus the y direction replaces the radial( r ) direction of the previous simulation, and we choose the width (in the y direction) of thedomain to match the radius of the bucket. The velocity profile that was used to initialize15he previous simulation is here the initial x -direction velocity, and we choose the length (inthe x direction) of the domain to be approximately the circumference of the bucket (about0.5 meters). We use a rectilinear grid in place of the polar grid of the previous simulation,and remove any of the terms describing effects associated with the curvature of the domain.In order to study the effect of only the removal of the curvature, we retain the cyclic lateralboundary conditions.This simulation exhibits a similar instability to that seen in the curved simulation (seeFig. (7)).The cyclic lateral boundaries create a periodic domain and this quantizes the allowablewavenumbers. We also performed a simulation with cyclic lateral boundaries and a doubledlength (in the x direction). The initial velocity profile remained unchanged. The double-length simulation exhibited an instability with six nodes, compared to the three nodes ofthe standard length straight channel. Since the wavelength is the channel length divided bythe number of nodes, this finding (that the wavelength is the same in spite of doubling thechannel length) is consistent with the expectation that the selected azimuthal wavelengthfor the shear instability scales with the shear layer width, rather than the available length.In other words, the number of vortices seen in the bucket demonstration (typically three)depends on the width of the shear layer. By experimenting with different initial flicks anddifferent periods of suspension before the bucket is set back down, different numbers ofvortices may sometimes be produced. Three or four was our most frequent outcome.Our straight-channel results suggest that the essential requirement for the instability isthe velocity profile: a curved domain is not a requirement for the instability. V. SUMMARY AND CONCLUSIONS
The highlight of this paper is that it presents a very simple, very low-budget classroomdemonstration of dynamic instability in a rotating fluid. The demonstration can be usedto introduce the concept of fluid-dynamic instability in a classroom context, or, given itssimplicity, it can be ported to outreach activities. Thanks to the low cost, multiple sets of theequipment can readily be provided to create a participatory activity. The activity can playa role in demystifying observations of polygonal patterns in geophysical flows. The breakingsymmetry of an initially axially-symmetric flow in water is visualized using tealeaves or16offee grounds. Consistency of the observed phenomenon with two-dimensional toy modelssuggests that the instability is essentially two-dimensional in nature and furthermore theexistence of the instability does not depend on the curvature of the domain, but on theshape of the initial velocity profile.We close with a speculation. Given a sufficiently strong initial flick, if the flow is left toevolve, the multiple vortices ultimately merge into a single axisymmetric vortex centered onthe middle of the bucket. This is reminiscent of the upscale cascade, which is a feature oftwo-dimensional flows. Given the similarity between the instability in the early stages of ourexperiment and our numerical and analytical models, which are two-dimensional, could thismerging be a manifestation of the upscale cascade? Further work would be needed beforea more robust assertion about this could be made, but we note that a similar merging isexhibited in the late stages of flow in our numerical model.
VI. SUPPORTING INFORMATION
The text is supported by the following supplementary information:Video clip of demonstration: VideoS1: https://vimeo.com/278481176 (password: Wel-come123)Video clip of numerical simulation (slow motion): VideoS2: https://vimeo.com/399593365 (password: Welcome123),Video clip of numerical simulation (real time): VideoS3: https://vimeo.com/279323035 (password: Welcome123)Video clip of taller, narrower configuration (120 mm diameter by 100 mm deep): VideoS4: https://vimeo.com/323816117 (password: Welcome123)Video clip of configuration with rigid lid: VideoS5: https://vimeo.com/323821255 (password: Welcome123) 17 ppendix A: Stream function of the simulated straight cyclic channel
FIG. 7. Top: Stream function of the simulated straight cyclic channel flow at 6.75 seconds. Contourinterval 0.0005 m s − . Bottom: as top, but data from the curved simulation reprojected onto astraight plot for ease of comparison. For the given velocity profile, these two plots illustrate theresult that although curvature modifies the behavior (the two plots are not identical), it is not arequirement for instability (both configurations exhibit instability). Appendix B: Linear stability analysis
Our stability analysis follows quite closely that described by Barbosa Aguiar et al. In ouranalysis, we consider only the barotropic mode; we ignore friction; and we assume constantdepth, implying a beta parameter of zero. Thus we have a two-dimensional problem withvorticity conserved following the flow. We write all the dependent variables as the sum of afixed base state (indicated by an overline) and a small perturbation (indicated by a prime),e.g. u = u + u ′ . We linearize about the base state with base-state azimuthal speed u = u ( r )(we use the fitted analytical profile shown in Fig. 4 of the main text) to give ∂ζ ′ ∂t + ur ∂ζ ′ ∂θ + v ∂ζ∂r = 0 , (B1)where r is radius, θ is the azimuthal angle (in radians), t is time, ζ is the base-state vorticity,and ζ ′ is the perturbation vorticity. v is the radial component of velocity, which is of18erturbation order, but we can omit the prime because v = 0. It is convenient to define θ as increasing in the direction of positive u . We introduce a stream function ψ such that u = − ∂ψ/∂r , v = ∂ψ/r∂θ and ζ = ∇ ψ .The solution to Eq. (B1) can be expressed in terms of the sum of Fourier modes of theform ψ ′ = Re ˜ ψ ( r, k ) exp { i ( σt − kθ ) } , (B2)where ˜ ψ ( r, k ) is independent of θ , iσ is a (possibly complex) growth rate and k is wavenumber( k = 1 , , ... ). Since the problem is linear, the modes do not interact and we can considerthem separately.We seek to identify the fastest-growing modes. We discretize in the r direction andeventually arrive at σ DDD φ − k (cid:20) ur (cid:21) DDD φ − k (cid:20) dζrdr (cid:21) φ = 0 (B3)where DDD is a matrix representing the discrete form of the operator d dr − k r + 1 r ddr and φ is a vector representing the discrete form of ˜ ψ ( r, k ). Quantities in square bracketsare diagonal matrices of known discrete values. Eq. (B3) can be further manipulated to givea set of eigenvalue problems σk φ = MMM φ, k = 1 , , ... (B4)where MMM = MMM ( k ) = DDD − (cid:18)(cid:20) ur (cid:21) DDD + (cid:20) dζrdr (cid:21)(cid:19) A careful treatment of the boundary conditions is important in the non-linear model (seemain text) but we have found that the results of our linear model are not sensitive to theuse of Dirichlet or Neumann boundary conditions. The growth rate of the fastest-growingradial eigenfunction for each wavenumber up to nine is shown in Fig. (8). The e-folding time(which is the reciprocal of the growth rate) for wavenumbers 3 or 4 is about 0.6 seconds (asin the numerical model: see section IV B of the main text).19
IG. 8. Growth rate, iσ vs wavenumber, k , for the profile shown by the black line in Fig. (4) ofthe main text ACKNOWLEDGMENTS
This work was supported by the Met Office Hadley Centre Climate Programme fundedby BEIS and Defra.Thanks to Grae Worster and an anonymous referee of an earlier draft of this paper forpointing out the centrifugal instability of the boundary layer mentioned in section III C. Weacknowledge the contribution of the anonymous referees who helped to improve earlier draftsof this article. Thanks to Simon Hammett for the stop-motion filming, which facilitated anestimation of the velocity profile. Thanks to Mike Cullen, Paul Billant, Michael McIntyre,Mike Bell, Richard Wood, Adam Scaife, Eddy Carrol, Andy White, Nigel Wood, DavidThomson, Philip Brohan and Matt Palmer for various helpful discussions, emails, adviceand encouragement. © British Crown Copyright 2020, Met OfficeThis article may be downloaded for personal use only. Any other use requires prior permis-sion of the author and AIP Publishing. This article appeared in American Journal of Physics88, Issue 12, 1041 (2020) and may be found at https://doi.org/10.1119/10.0002438 ∗ tom.howard@metoffice.gov.uk O. M. Phillips, “Shear-flow turbulence,” Annual Review of Fluid Mechanics, (1), 245–264(1969). J. P. Kossin and W. H. Schubert, “Mesovortices in hurricane Isabel,” Bull. Amer. Meteor. Soc.,pp. 151–153 (2004), doi:10.1175/BAMS-85-2-151. A. Adriani, A. Mura, G. Orton, C. Hansen, F. Altieri, M. L. Moriconi, J. Rogers, G. Eichst¨adt,T. Momary, A. P. Ingersoll, G. Filacchione, G. Sindoni, F. Tabataba-Vakili, B. M. Dinelli,F. Fabiano, S. J. Bolton, J. E. P. Connerney, S. K. Atreya, J. I. Lunine, F. Tosi, A. Migliorini,D. Grassi, G. Piccioni, R. Noschese, A. Cicchetti, C. Plainaki, A. Olivieri, M. E. O’Neill,D. Turrini, S. Stefani, R. Sordini, and M. Amoroso, “Cluster of cyclones encircling Jupiter’spoles,” Nature, , 216–219 (2018), doi:10.1038/nature25491. L. Illari, J. Marshall, P. Bannon, J. Botella, R. Clark, T. Haine, A. Kumar, S. Lee, K. J. Mackin,G. A. McKinley, M. Morgan, R. Najjar, T. Sikora, and A. Tandon, “ ”Weather in a Tank”:Exploiting laboratory experiments in the teaching of meteorology, oceanography, and climate,”Bull. Amer. Meteor. Soc., , 1619–1632 (2009), doi:/10.1175/2009BAMS2658.1. J. Marshall and R. A. Plumb,
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