Robust propagation of internal coastal Kelvin waves in complex domains
Chenyang Ren, Xianping Fan, Yiling Xia, Tiancheng Chen, Liu Yang, Jin-Qiang Zhong, H. P. Zhang
RRobust propagation of internal coastal Kelvin waves in complexdomains
Chenyang Ren, Xianping Fan, Yiling Xia, TianchengChen, Liu Yang, Jin-Qiang Zhong, ∗ and H. P. Zhang
1, 5, † School of Physics and Astronomy and Institute of Natural Sciences,Shanghai Jiao Tong University, Shanghai 200240, China School of Physics Science and Engineering,Tongji University, Shanghai 200092, China Ocean University of China, Qingdao 266100, China School of Physics and Astronomy, University of Manchester,Manchester M139PL, United Kingdom Collaborative Innovation Center of Advanced Microstructures, Nanjing, China (Dated: February 9, 2021) a r X i v : . [ phy s i c s . f l u - dyn ] F e b bstract We experimentally investigate internal coastal Kelvin waves in a two-layer fluid system on arotating table. Waves in our system propagate in the prograde direction and are exponentiallylocalized near the boundary. Our experiments verify the theoretical dispersion relation of the waveand show that the wave amplitude decays exponentially along the propagation direction. We furtherdemonstrate that the waves can robustly propagate along boundaries of complex geometries withoutbeing scattered and that adding obstacles to the wave propagation path does not cause additionalattenuation.
Over recent decades, tools from topology have shed new light on a wide range of physicalphenomena. One of such phenomena is the emergence of robust boundary states undertopological protection. For example, Hall current in two-dimensional semiconductors canrobustly and unidirectionally propagate along the boundaries which may have defects andcomplex shapes; its persistence is guaranteed by topological arguments [1–5]. Topologicallyprotected states have also been found in classical systems, including two-dimensional chiralmaterials [6–9], isostatic lattices in two dimensions [10], and photonic systems [11].Recently, fluid flow phenomena have been examined with topological methods [12–17];examples include oceanic Kelvin waves [12, 15] and atmospheric Lamb waves [17]. Robustboundary states, behaving like the topologically protected states, have also been found inhighly nonlinear systems, such as the wall states in the rotating Rayleigh–Bénard system[18]. Analysis of the shallow-water model [12] shows that the earth’s rotation breaks thetime-reversal symmetry and leads to a gap between the low and high frequency wavebands. Eastward-propagating Kelvin waves with frequencies in the gap can be considered astopologically protected boundary states at the equator, where the earth’s rotation changessigns [12]. Coastal Kelvin waves trapped near a sharp boundary have also been analyzedfrom a topological point of view [16], because the robust propagation of these waves alongirregular coasts has been well documented [19, 20]. Current theoretical analysis has notbeen able to rigorously establish coastal Kelvin waves as topologically protected boundarystates and topological origin of these boundary waves remains to be explored [16].As an important geophysical phenomenon, coastal Kelvin waves have been extensively ∗ [email protected] † [email protected] IG. 1. Schematic diagram and photography of the experimental setup. The various componentsare explained in the text. studied by theoreticians to explore the effects of boundary conditions, stratification, andbottom topography [21–24]. In addition, experimental studies of coastal Kelvin waves,which have some technical challenges, have also been carried out to investigate modal wavestructures, evolution of nonlinear waves, and reflection of solitary waves [25–27]. In thiswork, we focus on linear wave propagation phenomenon and possible interactions betweenlinear waves and obstacles. To that end, we set up a two-layer fluid system on a rotatingtable and use a computer-controlled wave-maker to excite coastal Kelvin waves. Wavepropagation along the fluid interface is quantified by particle image velocimetry (PIV). Ourmeasurements show that the wave in our system propagates in the prograde direction whiledecaying exponentially along the path and that measured dispersion relation is in goodagreement with inviscid theory prediction for low-frequency waves. We further demonstratethat internal coastal Kelvin waves robustly propagate along various irregular boundarieswithout being scattered into the bulk and that the complex geometry of the boundariesdoesn’t lead to additional wave dissipation.A schematic drawing and a photograph of the experimental apparatus are shown in Fig.1. It consists of a rotary table, a fluid tank, a wave-maker and a Particle Image Velocimetry(PIV) system for flow-field measurements. The rotary table is fixed securely on the floor.Both its rotating rate Ω (clockwise) and rotational acceleration a can be adjusted precisely.3e use a rectangular fluid tank that has inner dimensions of × × (mm ). Itsbottom plate, made of oxidized aluminum, is mounted on the rotary table. Its sidewalls,as well as the top lid, are made of 10 mm thick transparent Plexiglas plates. The tank hasfour rounded corners in a radius of 150 mm, designed to eliminate the corner vortices duringthe spin-up of the fluid and reduce fluid mixing. The tank is filled with two fluid layers,both are 100 mm in thickness. The upper fluid layer is fresh water and the lower one issaltwater. Their densities are ρ =0 . g/cm and ρ =1 . g/cm , respectively. In orderto prepare the two fluid layers with sharp density interface, we first fill the lower half of thetank with the salty water. A low-density sponge in size of × × (mm ) is gentlyplaced on the salty water surface. Fresh water at a very slow flow rate of . × − m/sis injected on top of the sponge. It permeates through the sponge, and then flows steadilyonto the fluid tank. During this filling process [28], we observe no noticeable mixing betweenthe top and bottom fluid layer, and two fluid layers with an interfacial thickness δ ≈ mmis prepared. After the tank is filled, the rotary table spins up to a rotation rate Ω witha steady acceleration a . For all experiments we choose Ω =1.88 rad/s and a =2 . × − rad/s . The acceleration a is low enough to prevent mixing between the two fluid layers.The fluid system then reaches a state of solid-body rotation after the time-scale of spin-up, τ = L/ ( ν Ω) / , which is approximately 20 minutes in the present experiment. Here L is thelong dimension of the tank and ν is the fluid viscosity.The internal waves are excited by a wave-maker made of a rigid sphere. The spherehas a radius R =15 mm, connected through an electric guide rail to a step motor. Duringthe experiment, the sphere is driven to oscillate vertically in the form of a cosine function h ( t )= h + H cos ( ωt ) . The equilibrium position h of the sphere is selected so that the spherecenter locates right at the interface of the two layers. For all measurements, the center of thesphere is 18 mm away from the long lateral wall of tank. We use an Arduino micro-controllerto control the oscillation frequency ω and amplitude H of the wave-maker.We take velocity measurements using a PIV system installed on the rotary table [29].A thin vertical light-sheet (1 mm in thickness) powered by a continuous solid-state laserilluminates the seeding particles over various locations in the fluid tank (insets in Figs. 2-5). Hollow glass beads, with a diameter of 50 µ m and an effective density of 1.030 g/cm ,are used as the seeding particles. Since these particles sediment three times faster in freshwater than in salty water, they accumulate at the interface during the spin-up process and4 IG. 2. (a) A snapshot of seeding particles in a vertical plane f (see inset). (b) The correspondinginstantaneous velocity field extracted from our PIV system. The background coloration correspondsto the vertical velocity component v z ( x, z ) . The central dark region denotes the position of thewave-maker. We define the positive horizontal direction of x to be leftward, i.e., the propagationdirection of the waves. (c) The vertical velocity component v z ( x ) measured at various fluid heights.Inset: a schematic drawing of top view of tank. The circle M denotes the position of wave-maker.The dashed line f is the position of the laser light-sheet in PIV measurements. The view angle ofcamera is indicated by black arrow. The red arrows present the propagation path of waves. form a bright region in raw particle images. From the bright region, we can track the fluidinterface η ( x ) , see red line in Fig. 2(a). Images of the particles are captured at a frame rateof 10 fps by a high-resolution Basler acA2040-90um camera ( × pixels) mountedin the co-rotating frame. Two-dimensional velocity maps are obtained by cross-correlatingtwo consecutive images taken at a time interval according to the flow speeds. Each velocityvector is calculated from an interrogation windows ( × pixels), with overlap ofneighboring sub-windows [30]. We choose the measurement area as a rectangular region of5 × (mm ) crossing the interface of the two fluid layer, reaching a spatial resolution of1.33 mm in the velocity fields.We first examine the property of unidirectional propagation of the Kelvin wave [31]. Forthis purpose, we install the wave-maker at the middle of the long side of the tank (positionM in inset of Fig. 2(c)). The fluid velocity over a vertical plane f is measured. Figure 2(a)shows an example of the particle density field captured through our PIV system with theoscillation amplitude H =20 mm and frequency ω =1 . rad/s. The interface of the two fluidlayers η ( x ) is marked by the red curve, that is determined by the average height of the brightarea for each horizontal position x . It shows oscillations on the left side of the wave-maker,but a parabolic profile η ( x )= η ( x )= bx on the right, caused by the centrifugal force (SeeSupplemental Material [32]). We choose the vertical interfacial position at x =0 as the zeroof the fluid height, i.e., η ( x =0)=0 . These are direct observations that the internal waveindeed propagates leftward unidirectionally. (See Supplementary Movie for the propagationof the Kelvin wave).Figure 2(b) presents the instantaneous velocity field v ( x, z ) extracted from the PIVsystem. A prominent feature shown in the velocity field is the apparent periodic oscillationsof both the velocity components v x and v z on the left for x>R . The magnitude of fluidvelocity on the right for x< − R , however, is nearly zero. In the region of | x | ≤ R , thefluid flow is complex due to the large-amplitude disturbances of the oscillation source.Along the interface z = η ( x ) for x>R , we find that the maxima and nodes of v z appearalternatively, while v x remains close to zero. In Fig. 2(c) we plot the vertical fluidvelocities v z against the horizontal position x for various fluid heights. Velocity datafor the oscillation-source region ( | x | ≤ R ) are excluded. For x>R the velocity profiles v z ( x ) can be best described by an oscillation function with its amplitude decaying exponentially v z ( x )= v z ( z ) exp ( − α v x ) cos ( kx − ωt ) . Here v z is the oscillation amplitude that depends on z and is maximum at z =0. k is the wave number and λ v d = α − v is the decay length of v z . For each fluid height z the crest and trough of v z appears periodically. For x< − R ,no significant velocity component is observed ( v z ≈ ). We stress that such a dynamicalfeature of unidirectional propagation has been observed in a wide range of the experimentalparameters ( ω , H ).To further illustrate the propagation dynamics of internal coastal Kelvin wave, we movethe wave-maker to the right side of tank (position M in inset of Fig. 3(b)), leaving a large6 IG. 3. (a) The interfacial profiles ζ ( x ) of Kelvin waves propagating along a straight fluid boundary.Results for H =15 mm, ω =1.40 rad/s, and 1.90 rad/s. Black dots: the raw data of ζ ( x ) . Black solidlines: fitted curves of ζ ( x ) according to formula (1). Red solid lines: the amplitude A as a functionof x , A = A exp ( − αx ) . (b) The measured damping cofficient α (black dots for H =15 mm and redtriangles for H =20 mm) and wave number k (purple) as functions of the oscillation frequency ω .Red solid line: dispersion relation expressed by formula (2). Inset: a schematic drawing of top viewof tank. distance for the leftward propagation of the Kelvin waves. We perform a set of measurementsof the interfacial position ζ ( x ) , choosing the range of the wave frequency as 1.30 rad/s ≤ ω ≤ ζ ( x )= η ( x ) − η ( x ) for various frequencies.Here η ( x )= bx is the parabolic interface measured when the wave-maker is turned off. Ourexperimental data of ζ ( x ) , which are shown in the black dots, agree well with the followingfunction: ζ ( x, t ) = A exp( − αx ) cos ( kx − ωt + φ ) , (1)as shown by the fitted black curves. α is the damping cofficient and k is wavenumber. Thewave amplitude decays exponentially along the prorogation direction x as implied by the redlines; for progressive linear interfacial waves, dissipation mainly occurs at the sidewalls andinterfacial boundary layer [33]. Figure 3(b) shows the experimental data of k , α as functionsof ω . With increasing ω , both α and k increase. In the range of ω ≤ . rad/s, we find7hat the wavenumber k is about one order in magnitude larger than the damping coefficient α [33–36], meaning that viscous effects are weak for these waves. The dispersion relation forcoastal Kelvin wave can be solved in a inviscid two-layer model [37] (also see supplementarymaterials [32]): ω = gk ( ρ − ρ ) ρ coth ( kD ) + ρ coth ( kD ) , (2)where D , D is the depth of two-layer fluids respectively. The red line in Fig. 3(b) showsthe theoretical results, without free parameters, from Eq. (2), which is in close agreementwith the experimental data in the low-frequency range when ω ≤ . rad/s. Therefore,the inviscid dispersion relation is confirmed in low-frequency waves. We also explore theinfluence of forcing amplitude H on α ; as shown in Fig. 3(b), the α parameters measuredwith two amplitudes are nearly identical. The observed amplitude-independence of thedamping coefficient suggests that the Kelvin waves in our experiments propagate in thelinear regime, which is consistent with the dispersion relation for linear waves measured inFig. 3(b). We note that nonlinear phenomena, such as solitary waves and triads interactions,do not occur in our experiments in the linear regime.For high frequencies ω> . rad/s, the damping coefficient α increases sharply withincreasing ω . In this case, the theoretically predicted dispersion relation is not accuratebecause of the strong frictional damping, and the wavelength λ L = 2 π/k and decay length λ d = α − become comparable. This is shown in Supplementary Movie S2 [32]: high-frequencywave decays rapidly along the propagation path. Hence, in our following experiments, wemainly use the wave with frequency ω ≤ . rad/s to further investigate the robustness ofthe coastal Kelvin wave propagation in complex fluid domains [16].In our experimental settings, we have installed obstacles (approximately 5 times largerthan the Rossby radius of deformation in size) of different geometric shapes in the fluid tankalong the propagation path of the Kelvin wave. These obstacles change the geometry ofthe fluid boundary as well as the propagation path, but do not alter the level of topologicalcomplexity of the system [1–5].In a series of measurements, we first install a semi-cylindrical obstacle O near the wall,as shown in the inset of Fig. 4(a). The relative interfacial position ζ for the waves of threefrequencies ω is measured in the vertical plane f at two times separated by half an oscillation8 IG. 4. The interfacial profiles ζ ( s ) of Kelvin waves in plane f when it propagates surroundingan semi-cylindrical obstacle. The experiment is performed with H =15 mm, ω =1.57 rad/s (a), 1.90rad/s (b) and 2.26 rad/s (c), respectively. Dots: the raw data of ζ ( s ) . Solid lines: fitted curvesof ζ ( s ) using formula (1). The s -coordinate presents the travel distance of the wave. Black circle:position of the wave-maker. Gray rectangular: position of the semi-cylindrical obstacle. Inset: aschematic drawing of top view of tank. The diameter of the semi-cylindrical obstacle O is mm. period. We see that with low frequencies ω =1.57 rad/s and ω =1.90 rad/s, the Kelvin wavecan travel leftward around the obstacle and reach the wall to the left of obstacle, ratherthan being reflected by the obstacle. The Kelvin wave with a high frequency, ω =2.26 rad/s,however, decays to almost zero, when it arrived to the left of obstacle, because of the strongenergy dissipation under large frequency, as discussed in Fig. 3.For further demonstration of the robustness of near-shore propagation of Kelvin wave,we install a barrier board O that is right-angle to the long lateral wall of the tank (see insetof Fig. 5(d)). The fluid interface is measured along all sections of the wave propagationpaths, i.e., in the vertical planes of f , f and f . In Fig. 5(a)-(c) we present the interfacialpositions ζ ( s ) captured at two oscillation phases that are half-period apart in planes f , f and f respectively. The experiment is performed with ω =1.73 rad/s, H =15 mm. Figure5(a) shows within a short travel distance s< mm that the energetic Kelvin wave is9 IG. 5. (a-c) The interfacial profiles ζ ( s ) of Kelvin waves propagating surrounding a thinrectangular obstacle (see inset). A mm thick obstacle board O with a length L c =210 mmis installed , perpendicular to the lateral wall of the tank (inset). Dots: the raw data of ζ ( s ) . Solidlines: fitted curves of ζ ( s ) using formula (1). The red arrows and circles indicate the directionof wave propagation. Black circle: position of the wave-maker. Gray rectangular: position of theobstacle board. (d) The wave oscillation amplitudes A ( s ) as a function of the travel distance s .Symbols: data measured with an obstacle board in length L c =110 mm. The solid lines presentexponential function A ( s )= A exp( − αs ) fitted to the straight-boundary data (open symbols). Inset:a schematic drawing of top view of tank. The dashed lines f , f , f are the positions of the laserlight-sheet in PIV measurements, captured by the camera with corresponding number at differentangles (black arrows). propagating leftward. As the wave comes across the barrier board O at s =200 mm, figures5(a) and 5(b) indicate that rather than being reflected backward, the wave makes a sharpright turn of 90 degrees and then travels in the direction perpendicular to the long lateralwall of the tank along board O. Another striking behavior is observed when the wave arrivesat the end of board O at 410 mm ≤ s ≤
440 mm. Figures 5(b) and 5(c) illustrate that at thislocation the wave changes again sharply its propagation direction and turns around the edgeof board O clockwisely, with its propagation path nested inside the vicinal region next to theobstacle surface. This robust near-shore flow continuously returns to the tank wall at s =650mm and then propagates leftward, although its amplitude has significantly decayed shown inFig. 5(a). This robust and unidirectional propagation of the coastal Kelvin wave shows that10ts dynamics is insensitive to the details of the boundary geometry [14, 16]. In addition, wemeasure the profile of internal Kelvin waves in the direction perpendicular to the boundary(see supplementary materials for Figure S1 [32]). Our measurements show that the internalKelvin wave is trapped exponentially near the boundary [31]. For a wave of frequency ω = 1 . rad/s, Fig. S1 shows a decaying length (in the y-direction) of . ± . mm.This is consistent with a direct estimation of the Rossby radius of deformation Λ = ω/ k Ω with the wave frequency and vector data from Fig. 3(b) [38].The general topological property of the Kelvin waves is further investigated by measuringthe wave amplitudes along various traveling paths surrounding different obstacles. Aspointed out in Fig. 3, the amplitude of the Kelvin wave decays exponentially with increasingtravel distance due to the viscous damping. Here, we examine whether the decay length λ d is independent of the path of wave propagation. In Fig. 5(d) we show the wave amplitude A as a function of the propagation distance s measured under various settings of boundaryconditions and different oscillation frequencies ω . The open symbols represent the results forKelvin waves propagating along a straight lateral wall, while the filled symbols are the resultsobtained when the Kelvin waves bypass along obstacles. We find that the decay length λ d decreases with increasing ω . For a fixed ω , the two sets of data collapse approximately into asingle curve A ( s )= A exp ( − αs ) . These results imply that the decrease of the wave amplitude A along a propagation path is determined by the traveling distance s , but independent onthe geometry of the path. Previous theoretical studies predicted that the Kelvin waves withfrequencies of ω < have no additional attenuation of amplitude in the process of diffractionat different barriers, by means of hydrodynamic theories [19, 39, 40].We have experimentally investigated internal coastal Kelvin waves in a two-layer fluidsystem on a rotating table and focused on slow waves with frequencies lower than thetable rotating frequency. Our experiments have shown that these waves, localized nearthe tank boundary, propagate in the same direction as the table rotation and decayexponentially along the propagation path. Wave dispersion relation from the inviscid theoryhas been experimentally verified with low-frequency waves when their propagations is nearlyunaffected by dissipation. Protruding objects, including a half-cylinder and a perpendicularplate, are used as obstacles to modulate the wave propagation. Our experiments showthat low-frequency waves can robustly propagate along the complex boundary (containingprotruding objects) without being scattered and that the protruding objects do not cause11dditional wave dissipation; similar observations have been made in other topologicallyprotected boundary states [7, 8, 12, 14, 15]. However, as stressed in [16], the topologicalorigin of coastal Kelvin wave remains to be theoretically established with the correspondingtopological invariant. Our experimental results and previous theoretical analysis suggest thatfluid dynamical systems may provide a fertile platform to further extend the applications oftopological methods and ideas. ACKNOWLEDGMENTS
We acknowledge financial support from National Natural Science Foundation of China(Grants No. 11774222, No. 12074243, and No. 11422427) and from the Program forProfessor of Special Appointment at Shanghai Institutions of Higher Learning (Grant No.GZ2016004). The experimental studies at Tongji University were supported by the NationalNatural Science Foundation of China (Grant No. 11772235) and a NSFC/RGC JointResearch (Grant No. 11561161004). We thank Mingji Huang and Siyuan Yang for usefuldiscussions. [1] C. L. Kane and E. J. Mele, Z topological order and the quantum spin Hall effect, Phys. Rev.Lett. , 146802 (2005).[2] C. L. Kane and E. J. Mele, Quantum spin Hall effect in graphene, Phys. Rev. Lett. , 226801(2005).[3] M. Z. Hasan and C. L. Kane, Colloquium: topological insulators, Rev. Mod. Phys. , 3045(2010).[4] B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Quantum spin Hall effect and topologicalphase transition in HgTe quantum wells, Science , 1757 (2006).[5] B. I. 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