Levitation, oscillations, and wave propagation in a stratified fluid
LLevitation, oscillations, and wave propagation in astratified fluid
Marina Carpineti , Fabrizio Croccolo , and Alberto Vailati Dipartimento di Fisica Aldo Pontremoli, Universit`a degli Studi di Milano, I-20133Milano, Italy Universite de Pau et des Pays de l’Adour, E2S UPPA, CNRS, TOTAL, LFCRUMR5150, Anglet, France.E-mail: [email protected]
Abstract.
We present an engaging levitation experiment that students can perform at homeor in a simple laboratory using everyday objects. A cork, modified to be slightlydenser than water, is placed in a jug containing tap water and coarse kitchen saltdelivered at the bottom without stirring. The salt gradually diffuses and determinesa stable density stratification of water, the bottom layers being denser than the topones. During the dissolution of salt, the cork slowly rises at an increasing height,where at any instant its density is balanced by that of the surrounding water. If thecork is gently pushed off its temporary equilibrium position, it experiences a restoringforce and starts to oscillate. Students can perform many different measurements of thephenomena involved and tackle non-trivial physical issues related to the behaviour of amacroscopic body immersed in a stratified fluid. Despite its simplicity, this experimentallows to introduce various theoretical concepts of relevance for the physics of theatmosphere and stars and offers students the opportunity of getting acquainted witha simple system that can serve as a model to understand complex phenomena suchas oscillations at the Brunt-V¨ais¨al¨a frequency and the propagation of internal gravitywaves in a stratified medium. a r X i v : . [ phy s i c s . e d - ph ] F e b evitation, oscillations, and wave propagation in a stratified fluid
1. Introduction
The levitation of macroscopic objects has fascinated humankind for a very long time.Its history is intertwined with the research performed by several Nobel laureates,because beyond its mystical aspects levitation can be profitably used both as a toolin fundamental research and in technological applications [1, 2]. Beyond its historicalrole in science, the investigation of levitation is still full of puzzling results, such as theapparent reversal of the gravity acceleration recently reported for a solid body at thesurface of a levitating liquid [3, 4].The fascination determined by levitation comes from the fact that we rarelyexperience it, as objects close to the surface of the Earth do not float freely into theair, due to the strong gravitational attraction and the small buoyancy force exerted byair onto solid bodies. When the gravitational force is not balanced, the object is infree fall, and in the frame of reference of the falling object, the acceleration of gravityis not felt. This determines weightlessness levitation conditions that can be employedprofitably to perform experiments in the absence of gravity on platforms such as droptowers, parabolic flights, and artificial satellites like the International Space Station [5].On Earth, the gravitational force acting on a body can be also balanced by the buoyancyforce acting onto it when the body is immersed in a fluid medium. When the density ofthe body equals that of the surrounding fluid, the weight of the body is exactly balancedby the buoyancy force and the body is in equilibrium inside the fluid neither rising norsinking, in a condition similar to levitation. However, the phenomenon of levitationdoes not involve simply the balance of forces, as occurs in buoyancy phenomena, butthe mechanical stability of the levitating body [6]. The particularity of a levitationphenomenon is that restoring forces drive back the body when it is moved away fromits equilibrium position.A levitation condition can occur in a stable stratified medium, where the densitydecreases as a function of height. A body can therefore be in equilibrium only at a certainheight where its density is exactly matched and a displacement in a direction parallel tothe density gradient determines a restoring force that brings back the object to the layerof fluid with the same density. This force gives rise to damped harmonic oscillationsthat occur at a frequency, known as Brunt-V¨ais¨al¨a frequency [7], which is determined bythe local density gradient present in the stratified fluid. The oscillations gradually dampboth through viscous dissipation and through the emission of transverse internal gravitywaves propagating in the plane perpendicular to the direction of oscillation. Examplesof natural systems where a density stratification occurs include giant planets and stars[8], where the mass is large enough to determine the confinement of a fluid phase. Atypical example is represented by the atmosphere of a planet, where the stratificationof a gaseous phase is due to the variation of pressure with altitude, which determines agradual decrease of density.In this work, we discuss a simple experiment that can be realized by studentsusing kitchenware to investigate levitation, oscillations at the Brunt-V¨ais¨al¨a frequency, evitation, oscillations, and wave propagation in a stratified fluid
2. Levitation in science
Although it hardly occurs spontaneously in nature, levitation has been profitably usedin science to perform several challenging experiments. A notable example of its usage infundamental research is represented by the celebrated experiment by Robert Millikan forthe measurement of the charge of the electron, where a negatively charged droplet of oilwas levitated by using an electrostatic field [9]. Millikan was awarded the Nobel prize inPhysics in 1923 for this experimental demonstration of the discrete nature of the electriccharge [10]. Levitation can also be achieved by using a magnetic field, as it happens forexample by placing a superconducting disk above a strong magnet [11, 12, 1]. Magneticlevitation has important technological applications for the development of frictionlessrailways [13, 14]. A strong magnetic field can also be used for challenging applicationslike the levitation of a living being, as it has been done in a popular experiment by AndreGeim and Michael Berry. By using a strong magnetic field, they were able to levitatea living frog, and this result allowed them to win the Ig-Nobel prize in Physics in theyear 2000 [15]. A few years later, in 2010, Geim shared the Nobel prize in Physics withKonstantin Novoselov for the discovery of graphene [16]. Beyond the use of externalfields, levitation can be also achieved by using radiation [1]. The phenomenon of acousticlevitation was studied by Rayleigh. Currently, acoustic levitation can be even achievedby amateurs at home or in the classroom, by building affordable instrumentation based evitation, oscillations, and wave propagation in a stratified fluid . The levitation ofparticles using light can be achieved by using optical tweezers, developed by ArthurAshkin [18], an invention that allowed him to win the Nobel Prize in Physics in 2018[19]. Optical tweezers allow the non-invasive manipulation of nanoparticles by takingadvantage of a focused Laser beam that exerts a non-isotropic radiation pressure,resulting in the trapping of a particle in a potential well. The introduction of opticaltweezers determined a revolution in the fields of Biophysics and Soft Matter, wherethey allow the investigation of the behaviour of colloidal particles, molecular motors,and cells, to mention just a few representative applications [20, 21].Neutral buoyancy represents a condition similar to levitation, very common in thecase of animals populating the waters of the Earth. Meaningful strategies adopted bymarine animals to achieve neutral buoyancy are represented by the swim bladder ofteleost fish, by the squalene oil contained in large quantities into the liver of sharks,and by the large hollow chambers built by the nautilus into its shell [22]. In the case ofmarine animals, neutral buoyancy is achieved in a medium of nearly constant density,and this feature determines an indifferent equilibrium condition, where the weight ofthe animal can be balanced by buoyancy irrespectively of the position of the animal.The lack of a unique equilibrium position and of a restoring force indicates that neutralbuoyancy cannot properly be considered a levitation phenomenon.At variance, the equilibrium condition realized in the presence of a stable densitystratification can be well described as levitation. In particular, when the density gradientevolves in time, it is possible to observe that an object, initially at rest, suddenly startsto rise under the action of invisible forces. At any instant, its equilibrium is at a well-defined height where it is subjected to restoring forces that make it oscillate in case itis slightly displaced. Several macroscopic systems exhibit these conditions and sharethe common feature that the mechanical energy associated with the local oscillationcan be transferred at a distance by internal gravity waves generated by the oscillationitself. Although internal gravity waves in a fluid cannot be visualized, the effect of theenergy transferred by them at a distance can be appreciated from the effect that it hason macroscopic bodies embedded in the fluid. An important example is represented bythe Earth’s atmosphere, where mountain reliefs and thunderstorms can interact withgravity waves to give rise to complex turbulent phenomena [7]. Another notable exampleis represented by oceans, where the density stratification is both determined by changesin salinity and temperature. Internal gravity waves have been reported also at themesoscale in laboratory experiments on non-equilibrium fluids, where the displacementof parcels of fluids is determined by the thermal energy k B T [23].
3. The experiment
The idea of this experiment came during the lockdown period due to the COVID-19pandemic when forced to stay far from our laboratories we were stimulated to invent evitation, oscillations, and wave propagation in a stratified fluid
Levitation is a surprising, intriguing, and counter-intuitive theme that offers the chanceto discuss many physical phenomena. It is extremely difficult to observe it in practiceand all the experiments involving it require a great technological effort. In this work,we describe a very simple levitation experiment, where a weighed-down cork, initiallylocated at the bottom of a glass jug containing water, suddenly starts levitating insidethe fluid and gradually rises (Fig. 1).
Figure 1.
Shots of the cork during its rise. The marked point and the reference axisused for tracking are shown. Coordinates are fixed so that the origin coincides withthe position of the tracked point at the starting time. For each figure, the time ofacquisition from the beginning of the measurement is shown.
Apparently, no source of energy is involved in the process, and this leaves thestudents with a first scientific puzzle: i) where does the mechanical work needed toraise the cork come from?
Another striking aspect of this experiment is that when thelevitating cork is slightly displaced vertically from its position, it undergoes dampedharmonic oscillations (Fig. 2). This aspect leaves the students with a second puzzlingquestion: ii) what is the physical origin of the restoring force that drives the cork towardsits equilibrium position during oscillations?
Figure 2.
Shots of the cork during oscillation induced by a slight initial push. Avideo is included as supplemental material (video1).
When two levitating corks are hosted inside the same container, pushing one of thecorks vertically determines, after a latency time, the onset of oscillations in the secondcork as well (Fig. 3). This feature raises a third puzzling question for the students: iii)what is the origin of the interaction between the two corks?evitation, oscillations, and wave propagation in a stratified fluid Figure 3.
Two corks in a tank. After some oscillations of the right cork, also the leftcork starts to move although with an oscillation amplitude much lower than the firstcork. A video is included as supplemental material (video2).
For the experiment we had in mind, we needed an easy-to-find object, with a densityslightly larger than water. The choice was to weigh-down the cork of a wine bottle. Thecork was cut in two halves digging a hole in each of them to host a weight, for whichwe chose another item frequently available at home: a steel marble with a diameter of10 mm borrowed from a toy. We soaked the sphere in vinyl glue, pasted back the twohalves of the cork (see Fig. 4), and used ethyl cyanoacrylate glue along the junction.Finally, we cut thin slices of the cork, until it had a density slightly larger than waterand was able to sink into it. Figure 4.
The picture on the left shows the two halves of the cork, with thehemispherical dimples carved into them, both filled with vinyl glue and one of themhosting the metal sphere. The picture on the right shows the assembled cork. In thisconfiguration, the cork density was still smaller than that of water and it was necessaryto cut thin slices of the cork to make it slightly denser than water.
The experiment is prepared by pouring 500 ml of tap water into a cylindrical glassjug, and adding a defined amount of course kitchen salt without applying any stirring,so that the salt grains rapidly sediment to the bottom of the jug. We decided to usecoarse rather than fine salt to avoid its partial dissolution while settling at the bottomof the jug, and therefore guarantee to start from better-defined initial conditions. Theexperiment is started by rapidly putting the cork at the bottom before the salt has evitation, oscillations, and wave propagation in a stratified fluid , g/cm , starting from an estimationof the critical amount of salt of 40 g dissolved in 1 kg of tap water, although the corkdensity can slightly change from one experiment to another due to its moisture content.The cork ascent lasts usually many hours and we recorded it using a Fujifilm XT20camera programmed to shoot pictures at regular time intervals, typically every fiveminutes. The framing is chosen so as to capture the entire height of the liquid. In orderto study the oscillation process, we apply a punctual acceleration to the cork and recordits oscillations at 24 frames/second by using the same camera. Should this experimentbe proposed to students, even a readily available smartphone could be used to makemany of the measurements and this was indeed the procedure followed at the beginning.However, when available, an automated system is definitely preferable. Figure 5.
Picture of the starting time of a typical measurement. The cork lies at thebottom of the jug where the coarse salt has settled.
Both cork’s rise and oscillations were characterized using the Open Source software”Tracker” [24], a Java-based application that allows to track the different positions of evitation, oscillations, and wave propagation in a stratified fluid . cm ). To track the different positions of the cork, a mark is then placedat a recognizable point of it, such as a corner or a small detail, and the origin of thereference system is fixed on it. Due to a large number of frames to analyse, the choiceof a well-identifiable and non-ambiguous point is extremely important to allow the useof the auto-tracking procedure, much faster than the manual one. At the end of theprocess, the different positions of the point as a function of time are obtained.
4. Theory review
In this section, we will provide the basic physical concepts needed for the understandingof the hydrostatic equilibrium of a fluid, which can give rise to the levitation of a bodyembedded into it and, consequently, to its oscillations when it is gently pushed off itsequilibrium position. The theory of internal gravity waves will be also briefly outlined.
Let us consider a very general configuration suitable to describe fluids such as theatmosphere and oceans: a layer of fluid of density ρ , distributed in a spherical shell ofthickness L , located ar a distance R from the centre of mass of a distribution of totalmass M.A parcel of fluid of volume V is under the action of the gravity force F = ρV g determined by the Law of Universal Gravitation F = G mMR , where the acceleration ofgravity is given by g ( R ) = G MR . At a height z , the weight of the column of fluid aboveit determines a hydrostatic pressure p inside the fluid, governed by the equation: ∇ p = ρ g (1)In the case of an incompressible fluid like a liquid, the density is constant and thehydrostatic pressure is governed by Stevino’s equation p ( z ) = p − ρgz (2)which is a particular solution of Eqn. 1, where p is the pressure at the surface ofthe fluid z = 0.A consequence of the hydrostatic pressure of Eqn. 1 is that the surface S of a body ofvolume V completely immersed in the fluid is subjected to forces acting perpendicularlyto its surface. As the horizontal components are balanced, the body is under the actionof a buoyancy force F A directed vertically, which can be obtained by integrating thehydrostatic pressure provided by Eqn. 2 across the surface S of the body: F A = − (cid:12)(cid:12)(cid:12)(cid:12)(cid:73) S p ( z ) d S (cid:12)(cid:12)(cid:12)(cid:12) = (cid:90) V dpdz d V = ρ g V (3) evitation, oscillations, and wave propagation in a stratified fluid S is an element of the surface with outgoing normal, and we have appliedthe divergence theorem to convert a surface integral to a volume integral.The total force acting on the body can be obtained by adding to the buoyancy forcethe weight of the body ρ V g with the proper sign: F T = ( ρ − ρ ) g V (4)where ρ is the density of the body. An immediate consequence of Eqn. 4 is thatwhen the density of the body is matched with that of the fluid, F T = 0. Under theseconditions, the body is in an indifferent equilibrium condition called neutral buoyancy,i.e the equilibrium condition is achieved at each point of the fluid. Such conditionis achieved by several aquatic organisms, which have devised a series of strategies tocompensate for the variation of pressure as a function of depth [22]. A major limitationof neutral buoyancy is that it requires a very close matching between the density of thefluid and that of the body. Even a small difference between these quantities would giverise to a condition where the body is under the action of a constant acceleration, whichbrings it progressively farther from its initial position.A stable equilibrium condition can be achieved by immersing the body in a stratifiedcompressible fluid, like for example a gas stratified by the gravity force. Under thiscondition, the body, after a transient, migrates to the layer of fluid matching its density.This condition is difficult to obtain in practice because the average density of mostsolid bodies is much larger than that of a gas. Matching the density of the solid wouldrequire the use of a liquid, which is intrinsically not compressible and, therefore, cannotbe stratified by the gravity force.Notwithstanding that, a gravitationally stable density stratification can be obtainedin a liquid under non equilibrium conditions. In fact, if the liquid is under the action ofa vertical temperature gradient ∇ T its thermal expansion determines a density gradient ∇ ρ = ∂ρ∂T ∇ T . Moreover, if the liquid is a mixture of two components, a variation ofthe concentration c with height determines a density gradient ∇ ρ = ∂ρ∂c ∇ c . In general,combining these two terms, the overall vertical density gradient is given by: ∇ ρ = ρ ( α ∇ T + β ∇ c ) (5)where α is the thermal expansion coefficient, and β is the solutal expansioncoefficient.Under the condition that the overall density gradient points downwards, the densityprofile is gravitationally stable, and a body of density ρ such as ρ min < ρ < ρ max canlevitate in a layer of a stratified fluid. Let us now consider the case where a body of density ρ and volume V is immersed ina layer, with the same density, of a gravitationally stable stratified fluid. If the volume evitation, oscillations, and wave propagation in a stratified fluid V undergoes a small displacement z in the vertical direction, the density gradient canbe assumed to be locally constant so that the density changes linearly with height: ρ ( z ) = ρ + ∂ρ∂z z (6)Combining Eqns. 1, 3, and 6, one can calculate the buoyancy force acting on volume V : F B = − (cid:73) S p ( z ) d S = (cid:90) V dpdz d V = ρ g V + ∂ρ∂z g V z (7)and recalling Eqn. 4 the total force acting on V is: F T = ∂ρ∂z g V z (8)By applying the second principle of dynamics ρ V ¨ z = F T , one obtains the equationof motion of a harmonic oscillator (notice that ∂ρ∂z is negative) :¨ z = gρ ∂ρ∂z z (9)where oscillations occur at the Brunt-V¨ais¨al¨a frequency [7]: N = (cid:115) − gρ ∂ρ∂z (10)Oscillations at the Brunt-V¨ais¨al¨a frequency also occur when a parcel of fluid isdisplaced vertically with respect to the surrounding fluid. This process occurs in naturalsystems such as the atmosphere and the oceans and in astrophysical systems such asstars. Interestingly, the same physical mechanism drives oscillations at mesoscopiclength scales during a thermal diffusion process occurring in a mixture of fluids [23]. Inthat case, the density stratification stems from temperature and concentration gradient,thus resulting in an additional contribution related to the temperature gradient.Anyway, the equation can be written in the same form of Eq.10 of the present paper ifone considers the dependence from the density gradient. When the volume V oscillates, its mechanical energy is gradually transferred to thesurrounding fluid. Part of the energy undergoes viscous dissipation, while another partgets transferred at a distance by the layered fluid in the form of an internal gravitywave. Internal gravity waves are qualitatively similar to the surface waves in a liquid.However, their density gradient is, of course, much smaller than that of surface waves. Inthe following we will describe the emission of transverse waves by a body that undergoesvertical harmonic oscillations at the Brunt-V¨ais¨al¨a frequency in a stratified fluid . Fora more rigorous and general treatment of internal gravity waves see Tritton [26] andNappo [7]. evitation, oscillations, and wave propagation in a stratified fluid u , pressure p (cid:48) and density ρ (cid:48) are: ∇ · u = 0 (11) ρ (cid:48) ∂ u ∂t + ρ (cid:48) u · ∇ u = −∇ p (cid:48) + ρ (cid:48) g (12) ∂ρ (cid:48) ∂t + u · ∇ ρ (cid:48) = 0 (13)Equation 11 is the continuity equation and expresses the conservation of mass, whileEqn. 12 is the second law of dynamics and Eqn. 13 expresses the fact that the fluid isnot compressible.We assume that the variables for the fluid in motion with velocity u can be writtenas a superposition of an hydrostatic contribution and a small perturbation: p (cid:48) ( x, y, z, t ) = p ( z ) + δp ( x, y, z, t ) (14) ρ (cid:48) ( x, y, z, t ) = ρ ( z ) + δρ ( x, y, z, t ) (15)where p and ρ obey the hydrostatic Eqns. ∇ p = ρg and ∂ρ/∂t = 0, respectively.By inserting Eqns. 14 and 15 into Eqns 11-13, we obtain: ∇ · u = 0 (16) ρ ∂ u ∂t + δρ ∂ u ∂t + ρ u · ∇ u + δρ u · ∇ u = −∇ δp + δρ g (17) ∂δρ∂t + u · ∇ ρ + u · ∇ δρ = 0 (18)We now assume that the perturbations of density and pressure are small withrespect to the hydrostatic quantities, | δp/p | (cid:28) | δρ/ρ | (cid:28)
1, and linearize theequations in u , δp , and δρ : ∇ · u = 0 (19) ρ ∂ u ∂t = −∇ δp + δρ g (20) ∂δρ∂t + u · ∇ ρ = 0 (21)We look for solutions of the equations in the form of propagating harmonic wavesof wave vector k and frequency ω : u = U exp [ i ( k · x − ωt )] (22) δρ = R exp [ i ( k · x − ωt )] (23) δp = P exp [ i ( k · x − ωt )] (24) evitation, oscillations, and wave propagation in a stratified fluid x (horizontal), and z (vertical). We assume that the parcel of fluid is located in the originof the coordinate system at time t = 0, and undergoes harmonic motion in the verticaldirection. This implies that the amplitude U of the oscillation in Eqn. 22 must bedirected in the vertical direction, that is U = (0 ; U ).By imposing that Eqn. 22 must be a solution of the continuity Eqn. 19 we obtainthat the velocity must be perpendicular to the wave vector: U · k = 0 ⇒ k z = 0 (25)Therefore, the velocity wave defined by Eqn. 22 is transverse, and its wave vectoris directed horizontally, k = ( k ; 0).By imposing that Eqns. 23 and 24 are solutions of Eqns. 20 and 21 we get: k P = 0 ⇒ P = 0 (26) i ω ρ U = g R (27) i ω R = d ρ d z U (28)Equation 26 implies that the pressure does not propagate and the dissipation ofenergy does not occur through the emission of a sound wave. Multiplication of Eqn. 27and 28 allows to determine the frequency of the internal gravity waves, which coincideswith the Brunt-V¨ais¨al¨a frequency of the oscillating parcel of fluid: ω = (cid:115) − gρ ∂ρ∂z = N (29)Notice that, although this theoretical model confirms that the harmonic motion ofa parcel of fluid in the vertical direction gives rise to transverse internal gravity wavespropagating horizontally, the wave number and the velocity of the wave are not predictedby it, and in principle, there are no constraints on the values that k and the velocitycan assume.
5. Results and discussion
We performed various experiments using different amounts of salt, ranging between 10and 46 g/l. In each measurement, we tracked the cork’s rise until its position wasapproximately at half of the water height. Then, we took movies of the oscillations inorder to measure the Brunt-V¨ais¨al¨a frequency.In Figure 1 we show a sequence of images taken when the cork had a low density.Thanks to the relatively large speed of the process – it lasted approximately three hours– it was possible to follow the rise until the cork reached the water surface. Both themarked point and the reference system used with Tracker are shown. In Figure 6 thecorresponding plot of the position of the cork versus time is shown. In the beginning,shots were taken every five minutes but, due to the high rise speed, the time intervals evitation, oscillations, and wave propagation in a stratified fluid
Figure 6.
Cork’s rise obtained by tracking the position of the marked point shown inFig. 1.
It is also very difficult to guarantee that the salt fills uniformly the bottom of thejug. Moreover, the finite size of the cork is not negligible with respect to the size of thejug. Therefore, the presence of the cork itself perturbs salt dissolution and therefore thetypical times of the process. Finally, the temperature of the water is not controlled atall, therefore a change in the transport properties of the mixture is to be expected fromone measure to the other.It is interesting to focus on the energy balance of the system in order to understandthe relationship between the different mechanisms acting on the fluid and the cork. Ifone writes down the total mechanical energy of the cork as the sum of its gravitationalpotential energy plus its kinetic energy, when the cork is lifted from the bottom ofthe jug, the conservation of mechanical energy is obviously not satisfied, since the corkpotential energy increases without any reason. In order to solve this apparent energymystery, one should seek the origin of the force lifting the cork. The cork is subjectedto the weight force, which is constant in time, and to the buoyancy determined by thesurrounding liquid, which changes in time. The work needed to raise the cork is providedby the gradual increase of the density of the liquid surrounding the cork, determined by evitation, oscillations, and wave propagation in a stratified fluid k B T . Theenergy is, in the end, taken from the thermal energy of the system, so that an accuratemeasurement of the fluid temperature should reveal a decrease of the temperature duringthe rise of the salt and the cork. This would, of course, be possible only in an idealadiabatic system not exchanging energy with other systems. In the experiment describedhere, the jug and the water are actually in thermal equilibrium with the surroundingenvironment, and the temperature increase is then not measurable since the environmentacts as the ultimate thermal reservoir. Nevertheless, it is interesting to observe that inthe initial condition, with the salt and the cork at the bottom of the jug, the potentialenergy of the system is at a minimum, while after some time, both the salt and the corkhave been lifted increasing their potential energy at the expense of the enthalpy of thesalt solution and ultimately to thermal energy. This is one of the cases of conversion ofthermal energy to mechanical energy.At any time during its rise, the cork floats in a stratified medium and, if set inmotion with a slight tap, it starts to oscillate at the Brunt-V¨ais¨al¨a frequency that isdirectly related to the local density gradient and gravity as shown in Eqn. 10. In Fig.2, we show a sequence of shots of the cork that oscillates after a slight tap, given whenit had reached approximately half height within the liquid. At a visual inspection, itwas evident that the regular oscillations of the cork were periodically slowed down untilthe cork almost stopped, to start again to oscillate after a while. A video is includedas supplemental material (video1). In Fig. 7 we show a sequence of tracking of theoscillations taken at different times during the cork rise for a slow process where weused the cork with its final density. The periodical regular slowing down of the corkthat can be observed in almost all the figures follows a complex dynamics, which cannotbe trivially related to that of a damped harmonic oscillator. However, by fitting thefirst oscillations with the formula for a damped harmonic oscillator y ( t ) = A ∗ cos ( ω ∗ t ) ∗ exp ( − ( t/τ )) + y (30)one gets a typical oscillation frequency ω in the range from 2.67 to 3.45 rad/s anda damping constant τ from 2.38 to 4.87. The values of the frequency resulting from thissimple fit are reported in Fig. 12.A more accurate scrutiny of Fig. 7 reveals that the dynamics of the oscillations issimilar to that observed for a beating phenomenon that occurs when two waves of slightlydifferent frequencies interfere. As discussed at the end of Section 4.3, the oscillationsdue to the gradient of density are expected to damp emitting transverse internal gravitywaves that propagate perpendicularly to the direction of oscillation. Due to the finitesize of the container, we expect that these waves are reflected back by the jug walls andfinally interfere with the cork oscillation. evitation, oscillations, and wave propagation in a stratified fluid Figure 7.
Tracking of the cork oscillation taken at different times of the same measurewith 30 g/l NaCl.
To test this hypothesis we performed a further experiment using a rectangularplexiglass tank as a container. We prepared another cork similar to the first one andimmersed both of them in water and salt. As soon as the corks had reached a properheight, we slightly pushed one of them and verified that after a while also the other onestarted to oscillate. This is a piece of evidence that a signal propagated horizontallyfrom the first cork to the second, as expected for internal gravity waves. In Figure 3an image of the two corks is shown, while a video is included as supplemental material(video2). Gravity waves in a transparent fluid are almost invisible, and this featuremakes them hard to investigate, but some of their features can be determined from theireffect on bodies immersed in the fluid, such as the two corks that we used. A remarkableexample is represented by the velocity of the wave, which can be obtained from the ratio d c t m = v = 6 , ± , cm/s , where d c = 8 , ± , cm is the distance between the two corks,measured from the images, and t m (cid:39) , ± , s is the time interval between the push evitation, oscillations, and wave propagation in a stratified fluid v w ≈ , ± , cm/s . Figure 8.
Sequence of images taken with a frame rate of 8 img/s at different times ofa portion of the sheet of paper placed behind the tank between the two corks. Timeincreases from left to right and from top to bottom. The horizontal line is deformeddue to the variation of the refractive index produced by the transit of the internalgravity wave. A video is available as supplemental material (video3)
We have tracked three different peaks spanning the space between the two corksobtaining an average value for v w , v av = 6 , ± , cm/s .We also measured the frequency of the internal gravity wave by using theline deformations along the vertical direction, induced by the index of refractioninhomogeneities shown in Fig. 8. Using the Tracker program, fixed points of the linewere chosen and followed in time during the passage of the wave to measure theirdisplacement along the y-axis. A typical result is shown in Fig. 10.The continuous line is the best fit of the experimental points with Eq. 30. Byaveraging the oscillation frequencies obtained by tracking different points of the line,we obtained an average value for ω = 4 . ± . rad/s that corresponds to a frequency f = 0 , ± . Hz for the internal gravity wave. This value is close to the oscillationfrequency of the cork pushed in motion, ω c = 4 . rad/s , as measured by tracking the evitation, oscillations, and wave propagation in a stratified fluid Figure 9.
Peak position of the wave detectable as a deformation of the line shownin Fig. 8. The solid line is the best linear fit of the experimental points and its slopegives an estimate of the wave speed that, for this curve is 6 , ± , cm/s . Figure 10.
Tracking of the position of a point on the line shown in Fig. 8 during thepassage of the gravitational wave. The continous line is the fit obtained using Eqn.30. cork position. This result confirms that the internal gravity wave has the same frequencyas the Brunt-V¨ais¨al¨a wave that generates it, as shown in Eqn. 29, although it must bestressed that the image quality and the strong damping of the waves make it difficultto obtain a precise estimation of the frequencies.To have better quality measurements of Brunt-V¨ais¨al¨a frequencies in differentconditions, we finally fitted the individual cork’s oscillations shown in Figs. 2 and7. In order to fit the entire curves and not only the first oscillations, we used aphenomenological fitting function more refined with respect to Eq. 30 that takes intoaccount both the oscillation damping and the beatings between the cork oscillation and evitation, oscillations, and wave propagation in a stratified fluid y ( t ) = A ∗ cos ( ω ∗ t + φ ) ∗ cos (∆ ω ∗ t + φ ) ∗ exp ( − ( t/τ ) γ ) + y (31)where ω , ∆ ω , φ , φ , τ , γ and y are the free parameters. Eq. 31 is the productof two cosine functions which typically describes beatings that occur when two waves ofsimilar frequencies interfere. In beating phenomena, ω is the average of the frequenciesof the interfering waves, while ∆ ω is their difference. The stretched exponential term exp ( − ( t/τ ) γ ) with γ = 2 turned out to be the best function to account for the severedamping of the two oscillating terms that have different time constants. In Fig. 11 weshow the remarkable result of the fit. It can be noticed that while Eqn. 30 is able to fitonly the first oscillations, Eqn 31 is able to account for all the oscillations features. Figure 11.
Fit of the oscillation tracking for a measurement performed with 30 g/lNaCl. Data are taken 4 hours after the beginning of data acquisition, when the corkwas almost 1,5 cm from the bottom of the jug.
As shown in Fig. 12, for all the data shown in Fig. 7, ∆ ω is much smaller than ω and this result confirms that the two beating waves have very close frequencies. As aconsequence, ω gives a good estimation of the Brunt-V¨ais¨al¨a frequency and turns outto be in very good agreement with the frequency ω that is obtained by fitting the datawith the much simpler Eqn. 30. In Fig. 12, it can be appreciated that the frequency ofoscillation decreases in time as expected because the density gradient diminishes whilethe concentration approaches the final equilibrium value.From the Brunt-V¨ais¨al¨a frequency, it is possible to calculate the value of the densitygradient ∂ρ∂z from Eqn. 10. Having fitted all the oscillations measured with all thedifferent salt concentrations, we found for the frequency N = ω values between 1 , , ∂ρ∂z ranges from 236 kg/m to 2140 kg/m corresponding to concentration variations ∂c∂z = ∂ρ∂z / ∂ρ∂c ranging from 0 . m − to 3 . m − , where for a solution of NaCl in water atsmall concentration ∂ρ∂c = 720 kg/m [27]. evitation, oscillations, and wave propagation in a stratified fluid Figure 12.
The plot shows the values of ω (blue squares) and ∆ ω (red squares), asobtained by fitting the data in Fig. 7 with Eqn. 31. Open squares are the values of ω obtained by fitting the first oscillations of the same data with Eqn. 30. Both ω and ω give a good estimation of the Brunt-V¨ais¨al¨a frequency.
6. Conclusions
We have discussed an engaging, inexpensive, and simple experiment that allowsintroducing undergraduate students to many arguments on hydrostatic, buoyancy, andlevitation in a stratified fluid. The general scientific framework of levitation, and thetheoretical description of the phenomena that are experimentally accessible, are alsopresented in this work. Brunt-V¨ais¨al¨a oscillations are observed and the measurementof their frequency may be used to estimate the concentration gradient inside the fluid.Moreover, the effect of gravity waves can be observed and measured. We propose aphenomenological function to fit the data, able to keep into account both the dampingof the oscillations and the beatings with the internal gravity wave they generate. Themotion of the cork determines a partial remixing of the stratified fluid, thus reducingthe concentration gradient that drives the oscillations. In this respect, the experimentrepresents a remarkable example of a classical system where performing a measurementsignificantly affects the state of the system. This feature makes a long series ofrepeated measurements not useful, in contrast with the usual physical systems usuallyencountered by students during laboratory activities. The experiment is exploitable bystudents of any age as the level of deepening can be tuned as a function of the students’knowledge. However, it is particularly suitable for undergraduate students that canappreciate all its implications.
Acknowledgements
Work partially supported by the European Space Agency, CORA-MAP TechNESContract No. 4000128933/19/NL/PG. This research was carried under the frameworkof the E2S UPPA Hub Newpores and Industrial Chair CO2ES supported by the evitation, oscillations, and wave propagation in a stratified fluid
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