Generation, propagation and breaking of an internal wave...

Generation, propagation and breaking of an internal wave beam

Heather A. Clark and Bruce R. Sutherland⋆

⋆Corresponding Author’s address: Dept. of Physics, University of Alberta, Edmonton, Alberta

T6G 2G7, Canada

ABSTRACT

We report upon an experimental study of internal gravity waves generated by the large-amplitudevertical oscillations of a circular cylinder in uniformly stratified fluid. Quantitative measurementsare performed using a modified synthetic schlieren technique for strongly stratified solutions of NaClor NaI. The oscillatory forcing leads to the development of turbulence in the region bounding thecylinder. This turbulence is found to be the primary source of the observed quasi-monochromaticwave beams, whose characteristics at early times differ from theoretical predictions and experi-mental investigations of waves generated by small-amplitude cylinder oscillations. In particular,their wavelength is set by the Ozmidov scale rather than the size of the cylinder. The wave fre-quency is set by the buoyancy frequency, N , if the cylinder frequency is larger or much less thanN . Otherwise it is set by the cylinder oscillation frequency. Over long times the finite-amplitudewaves that have propagated away from their source are observed to break down and the processis examined quantitatively through conductivity probe measurements and qualitatively throughunprocessed synthetic schlieren images. From an analysis of the location of wave breakdown wedetermine that the likely mechanism for breakdown is through parametric subharmonic instabil-ity. This conclusion is supported by fully nonlinear numerical simulations of the evolution of atemporally, though not spatially, monochromatic internal wave beam.

I INTRODUCTION

Density stratified fluids support the propagation of inter-nal gravity waves that arise from buoyancy restoring forces.Energy and momentum are transported in geophysical flu-ids by the internal waves radiating from localized sources.Observations, modelling and experiments have been usedto study in detail several generation mechanisms and theproperties of the resulting waves. In particular, topo-graphic forcing by tidal flow over features of the oceanfloor is observed to be a major source of oceanic internalwaves1,2,3, which are subsequently responsible for signif-icant diapycnal mixing4. Similarly, flow over mountainsmay generate moderate- to large-amplitude atmosphericwaves5, the turbulent breakdown of which has been ob-served directly through in-flight measurements6. Frittsand Alexander7 reviewed the generation of atmosphericinternal waves by several primary sources, including to-pography, convection, shear, geostrophic adjustment andwave-wave interactions. In general, the existence of suchdisturbances can have a significant nonlocal effect on themean flow through the propagation and breaking of inter-nal waves.

While turbulent flows are involved to varying extent ingeophysical sources of internal waves, the turbulent gener-ation process is currently not well understood. It has beenobserved in experiments8,9,10 and numerical simulations11

that turbulent sources generate waves in a narrow fre-quency range relative to the background buoyancy fre-quency. In a recent numerical study, Taylor and Sarkar12

found that the internal waves generated by oceanic bottom

boundary layer turbulence propagated at angles between35 and 60 degrees from the vertical. It was shown that lin-ear differential viscous decay could produce the observedspectral peak and decay in wave amplitudes. However, theproposed model may not be an adequate explanation inthe case of larger amplitude waves12. A key difference be-tween their numerics and laboratory experiments is the useof large-eddy simulation for the numerical boundary layerand the resulting loss of resolution of the finescale turbu-lence. Although a viscous model was in agreement with thenumerical results, experiments show the immediate gener-ation of narrow-band internal waves on a timescale thatis less than the viscous timescale required for differentialdecay.

In this paper we present the results of a study of in-ternal waves generated by large-amplitude oscillations ofa circular cylinder in strongly stratified fluids. We clas-sify the forcing as large-amplitude because in all casesthe half peak-to-peak amplitude of the cylinder displace-ment was on the order of, but less than, the radius ofthe cylinder. Moderate-amplitude forcing in Boussinesqfluids has been investigated in laboratory experiments bySutherland et al.13 and Sutherland and Linden14, for cir-cular and elliptical cylinders, respectively. In general, goodqualitative agreement was found between the experimentsand the linear, viscous, Boussinesq theory of Hurley andKeady15. However, the beam width was consistently un-derpredicted because the theory neglects the formation ofa viscous boundary layer around the cylinder 13,14. Thelarge-amplitude forcing used in the current work results inboundary layer separation so that the internal waves arelaunched effectively by an oscillatory turbulent patch. In

this sense, the extension of previous experimental work toinclude large-amplitude effects also alters the generationprocess.

As well as standard sodium chloride (NaCl) solutions, anew experimental technique16 that is applied in this workis the use of sodium iodide (NaI) to produce stronger strat-ifications than those in conventional tank experiments.Whereas typical stratifications have a density change onthe order of 5% over the depth of the fluid, the density dif-ferences in this study are approximately 20% for solutionsof NaCl and 50% for NaI. The use in this work of signif-icant variations in the background density profile is moti-vated by the phenomenon of anelastic growth in amplitudefor upward-propagating atmospheric waves17. When thevertical distance traversed by a propagating wavepacketis a significant fraction of the local density scale height,non-Boussinesq effects result in an increase in wave am-plitude with height. As a result, nonlinear effects mayhave a more pronounced influence on the evolution of suchinternal waves. Previous work18 has shown that weaklynonlinear effects modify the amplitude growth of internalwavepackets, thus affecting the location and intensity ofwave breaking, should it occur. In the current experi-mental work the waves are generated at finite amplitudeand the vertical scale of propagation is such that non-Boussinesq growth may be measurable. The use of bothNaCl and NaI solutions allows for observation of the wavebehaviour over different fractions of a density scale height,since the vertical scale of propagation remains the samewhile the strength of the stratification is varied.

While the large-amplitude forcing acts to modify thesource region by causing boundary layer separation, it alsohas the effect of generating moderate- to large-amplitudewaves. Tabaei and Akylas19 showed through an asymp-totic analysis that nonlinear effects are relatively insignif-icant for an isolated beam with slow along-beam modu-lations in a uniform, Boussinesq stratification. Dispersiveand viscous effects were found to be the dominant factorsin determining the propagation of isolated beams. A sub-sequent paper20 investigated the role of nonlinearity in sit-uations where there exists a region of interaction, namelythe reflection of a wave beam from a slope or the collisionof two beams. In such cases it was found that nonlineareffects result in the generation of higher-harmonic beamsthat propagate out of the interaction region and into the farfield. In this study we find that nonlinear effects have a sig-nificant influence on the evolution of the finite-amplitudewaves, resulting in wave breaking for a single beam in theabsence of critical layers. Rather than the slow modula-tions considered by Tabaei and Akylas19, we observe thatthe wave breaks down due to parametric subharmonic in-stability (PSI). This mechanism for wave breakdown hasbeen studied analytically and in laboratory experiments forunbounded plane internal waves and wave modes in a rect-angular domain21,22,23,24. Ours is the first examination ofPSI occurring for an internal wavebeam in continuouslystratified fluid.

A brief summary of the analytic results of Hurley and

Keady15 for small-amplitude cylinder oscillations is givenin §II. In §III we describe the experimental apparatus andthe implementation of the synthetic schlieren techniquein strongly stratified solutions. We also explain in detailthe analysis of wave frequency, wavenumber and ampli-tude from the schlieren processed data. The final contentof §III is a description of the qualitative criterion for theonset of wave breakdown. In §IV we present the resultsof the quantitative analysis of the wave field for coherentbeam structures. A discussion of the characteristics of theobserved instabilities and subsequent wave breakdown isthe subject of §V, where we also present our findings fromnumerical simulations of a localized wave beam. We sum-marize the results of the study in §VI.

II THEORY

The experiments in this work use strong stratifications inwhich the background density varies by up to 50% overthe fluid depth. Therefore, we calculate the backgroundbuoyancy frequency, N , using the non-Boussinesq relation

N2(z) = −gρ̄

dρ̄

dz, (1)

where g is the acceleration due to gravity and ρ̄(z) is thebackground density profile as a function of height. Giventhe non-Boussinesq form of the equation as shown above,the construction of a near-exponential density profile in theexperiments yields a uniform stratification with an approx-imately constant background buoyancy frequency denotedby N0.

Although our experiments use large density gradients,we may treat the fluid as Boussinesq over our region offocus for the generation of the beams because density vari-ations are relatively small over the vertical extent of thecylinder. Previous experimental work on internal wavegeneration by oscillating cylinders13,14 has been comparedto the analytic solution of Hurley and Keady15 for thewaves generated by small-amplitude oscillations of a cylin-der in a viscous, uniform, Boussinesq stratification. Forthe purposes of comparing their predictions to our large-amplitude experiments, here we present their final equa-tions 15 recast in a modified set of variables as they relateto our analyses.

For a cylinder oscillating vertically with angular fre-quency ωc in a uniformly stratified fluid with backgroundbuoyancy frequency N0 such that ωc < N0, four wavebeams emanate from the source region at a fixed angleto the vertical, given by

Θ = cos−1

(

ωcN0

)

. (2)

The coordinate transformation from the (x, z) plane tocross-beam and along-beam coordinates, (σ, r), of thebeam in the first quadrant is given by

σ = −x cosΘ + z sin Θ, r = x sin Θ + z cosΘ. (3)

2 Phys. Fluids Clark & Sutherland

The coordinate transformations for other quadrants maybe obtained using symmetry properties of the solution13.

Assuming time-periodic solutions and ignoring viscosity,the spatial dependence of the streamfunction describingthe beam in the first quadrant is given by

Ψ =

{

12AcωcRce

−iΘ[

− σRc

− i√

1 − (σ/Rc)2]

, |σ/Rc| < 1

0, |σ/Rc| > 1,

(4)in which Ac is the half peak-to-peak amplitude of verticaloscillations of the cylinder, and Rc is the cylinder radius.The streamfunction itself is given by the real part of ψ =Ψ(x, z)e−iωct. From this, the velocity fields and other fieldsof interest can be determined. In particular, ur = ∂ψ

∂σ is thealong-beam component of velocity, and pressure is given by

p = −(iωcρ0 tanΘ)ψ, (5)

where ρ0 is a characteristic density. Thus, the period-averaged power transmitted along the beam per along-cylinder distance is

P =

∫ Rc

−Rc

〈urp〉 dσ =π

8ρ0A

2cω

3cR

2c tanΘ. (6)

III EXPERIMENTAL SETUP AND

ANALYSIS METHODS

We have performed a series of experiments using an oscil-lating cylinder to generate waves in strong stratifications.As illustrated schematically in Figure 1, a cylinder of ra-dius Rc was forced by a variable-speed motor to producevertical oscillations with half peak-to-peak amplitude Acand angular frequency ωc in a rectangular acrylic tank.In order to obtain disturbances that were uniform in thespanwise (y) direction, the length of the cylinder was 2 mmless than the inner separation of the tank walls. The tankof dimensions Wt = 122.3 cm and Lt = 15.5 cm was filledto a depth of Ht ≃ 55 cm. For both types of stratifica-tion, experiments were performed with the cylinder cen-tred approximately 12 cm above the bottom of the tank orapproximately 8 cm below the fluid surface. This allowsfor a comparison between the characteristics of upward-and downward-propagating waves. For all experiments thecylinder was located approximately 80 cm (≃ 2Wt/3) fromthe left side of the tank. The spatial region for quantita-tive analysis was restricted to the left side of the cylinder,as we assume symmetry of the wave properties about thevertical axis. As in the case of small-amplitude forcing,four wave beams were observed emanating from the sourceregion. Hereafter we will refer to the “primary beam” andthe “reflected beam” as they are shown in Figure 1. Theterms are used similarly for the case with the cylinder nearthe top of the tank, but the beam reflection occurs off ofthe fluid surface rather than the bottom of the tank.

Approximately exponential density stratifications weremade from solutions of either NaCl or NaI, as described byClark and Sutherland16. For each subset of experiments

x

z

Ht

Wt

Rc

2Ac, ωc

primarybeam

reflectedbeam

(a) Front View

y

ztank imagescreen

camera

lights

Lp LpLt Ls

(b) Side View

Figure 1: (a) Schematic diagram of the suspended cylin-der oscillating with constant frequency and amplitude. Anapproximate resulting wave beam pattern is shown. (b) Aside view of the synthetic schlieren apparatus. The dashedline represents the path of a light ray from the screen tothe camera.

the density profile was measured with a conductivity probeand the data were fitted with an exponential function ofthe form

ρ̄(z) = ρ(z0) exp[−(z − z0)/H ], (7)

where H is the density scale height and z0 is the small-est measured vertical coordinate. H ≃ 270 cm for NaClstratifications, and H ≃ 130 cm for experiments usingNaI. These values yield background buoyancy (angular)frequencies of N0 ≃ 1.9 rad/s and N0 ≃ 2.7 rad/s for NaCland NaI respectively, according to (1).

A Synthetic schlieren technique

For the quantitative analysis of the internal wave field, weimplemented the synthetic schlieren technique as depictedin Figure 1(b). Due to the large variations in backgrounddensity over the tank depth for both NaCl and NaI strat-ifications, we have used a modified form of the syntheticschlieren equations for non-Boussinesq fluids16. Thus wetake into account the full vertical dependence of the den-sity profile and the corresponding index of refraction foreach type of stratification.

Phys. Fluids Clark & Sutherland 3

−40 −30 −20 −10x [cm]

0

10

20

30

z[c

m]

0−10

40(a) Raw

σ

−40 −30 −20 −10x [cm]

0

(b) N2t [s−3] -2.5 2.5

σ

−40 −30 −20 −10x [cm]

0

(c) 〈N2t 〉 [s−3] 0.0 2.8

Figure 2: Synthetic schlieren is applied to yield the instantaneous value of N2t , plotted in (b) at a time corresponding to

the raw image (a). The envelope,⟨

N2t

⟩

, over one wave period is shown in (c). In each case the coordinate origin is atthe equilibrium position of the cylinder. For the experiment shown, the parameters are N0 = 2.7 rad/s, ωc = 1.96 rad/s,Rc = 2.98 cm, Ac = 1.5 cm.

From measurements of the apparent vertical displace-ment, ∆z, of an image of horizontal black and white linesplaced behind the tank (see Figure 2 (a)), we may calcu-late the perturbation to the squared buoyancy frequency,∆N2, due to waves within the tank13. In this work wefocus on the time derivative of this field, denoted by N2

t ,given by

N2t ≃ −zt

1

γ

[

1

2L2t + Lt n̄

(

Lpnp

+Lsna

)]

−1

. (8)

The lengths Ls = 15.5 cm and Lp = 1.7 cm are shown inFigure 1(b). Here n̄(z) is the index of refraction profile forthe background stratification, while np = 1.49 and na =1.0 are the indices of refraction of the acrylic tank wallsand air, respectively. The parameter γ(z) is evaluated interms of the background density and index of refractionprofiles as16

γ =1

g

ρ̄

n̄[a1 + a2(ρ̄− ρ0)], (9)

where ρ0 = 0.99823 g/cm3 is the density of fresh water and

a1 = 0.2458 cm3/g, a2 = −0.1208 cm6/g2 for NaCla1 = 0.1894 cm3/g, a2 = −0.0086 cm6/g2 for NaI.

Raw video footage was captured by a Cohu CCD cameraand recorded to SVHS tapes at a rate of 1 frame per 1/30seconds. The images were digitized using the DigImage25

software package with a spatial resolution of approximately0.12 cm. For each experiment, vertical time series weremade from the digitized video footage at 24 equally spacedhorizontal locations over a total horizontal distance of ap-proximately 40 to 50 cm, each corresponding to a widthof 1 pixel. DigImage was used to extract the time se-ries images and further processing to obtain N2

t (z, t) was

performed with a separate computer program. As shownin Figure 2(b), we reconstruct a spatial snapshot at anydesired time from the set of vertical time series images.In this figure, vertical wavenumbers greater than 1.0 cm−1

have been filtered in order to remove small-scale spatialnoise. Interpolation is also performed between grid pointsin the horizontal direction to smooth the image that re-sults initially from the computational reconstruction of thesnapshot. We calculate the envelope,

⟨

N2t

⟩

, as shown in

2(c), from√

2 times the root mean square of 16 snapshotsevenly spaced in time over one wave period14 to obtaininformation about the average wave amplitude and beamstructure in space and time. The direction of the increas-ing positive cross-beam coordinate, σ, is superimposed onthe primary beam in (b) and (c). In Figure 2 and the spa-tial plots that follow in this paper, the coordinate originhas been shifted to correspond to the equilibrium positionof the cylinder centre.

B Measurements of wave frequency

In the majority of cases, the wave frequencies were mea-sured using time series of N2

t for t ∈ [5Tc, 15Tc], where Tcis the oscillatory period of the cylinder. For several ex-periments with low forcing frequencies, the experimentaldata spans fewer than 15 periods, so the time window forfrequency measurements was reduced. The standard timeinterval was chosen so that the waves were well-developedand reasonably steady. Figure 3 shows an example of theanalysis process for one experiment in a NaI stratificationwith N0 ≃ 2.7 rad/s, Rc = 2.98 cm, Ac = 2.0 cm, andωc = 1.53 rad/s. We constructed the horizontal time se-ries of the data, as plotted in Figure 3(a), at a verticaldistance of 5Rc from the equilibrium vertical position of


−40 −30 −20 −10x [cm]

25

35

45

55

t [s]

0

(a)N2t [s−3] -4.0 4.0

−3

−2

−1

0

1

2

3

N2 t

[s−

3]

25 35 45 55t [s]

−4

4(b)

0.25

0.50

0.75

1.00

1.25

arbit

rary

unit

s

1 2 3 4ωigw [s−1]

(c)

N0

ωc0 5

0

Figure 3: (a) A horizontal time series of the N2t field is shown with vertical lines bounding the region of spatial averaging.

A profile through the contour plot at the location of the leftmost solid dark line is shown in (b), with the correspondingpower spectrum of the Fourier transform in time plotted in (c). The forcing and buoyancy frequencies are marked onthe horizontal axis.

the cylinder. Eleven equally spaced profiles over the re-gion x ∈ [−6Rc,−5Rc], which is between the thick verticallines in (a), were Fourier transformed in time to obtainpower spectra in frequency space. The spatial region waschosen to be close enough to the cylinder for the waves tobe indicative of the properties at generation, but outside ofthe turbulent patch so that meaningful synthetic schlierenmeasurements were possible. This interval was found tobe the most appropriate choice to apply over all experi-ments. The field for one representative profile is plotted in(b), with the corresponding spectrum shown in (c). Theaverage of the 11 spectra was computed before finding thelocation of the peak in frequency, which we denote by ωigw.For an improved measurement, ωigw was obtained from aparabolic fit to the three discrete points that defined thepeak of the spectrum. The uncertainty, δωigw, in the mea-surement was taken to be the half-width of the paraboliccurve. In general, the wave and cylinder frequencies neednot be equal because of the turbulent generation process.A comparison between measured wave and cylinder fre-quencies is the topic of §A.

C Measurements of wavenumber

Beginning with a snapshot of the N2t field, the image was

reflected about x = 0 and rotated such that the lines ofconstant phase of the waves appeared horizontal. Throughthis process the horizontal and vertical coordinates of theimage become r and σ respectively, as plotted in Fig-ure 4(a) for a typical experiment with N0 = 2.7rad/s,ωc = 1.96rad/s, Rc = 2.98 cm and Ac = 2.0 cm. Thesnapshot shown corresponds to a time t ∈ [4Tigw, 5Tigw].Note that the r axis as shown does not start from zero. Forsmall values of the radial coordinate the schlieren processeddata is unreliable due to the presence of the cylinder andthe surrounding three-dimensional turbulence. We haverestricted our analysis to σ ≥ 0 because of potential inter-

ference between the primary beam and the reflected beamin the lower flank, for which σ < 0. Although interferenceis not predicted by theory, the turbulent generation pro-cess results in more diffuse boundaries for the experimentalbeams. The heavy dashed lines in 4(a) correspond to theedges of the original schlieren image that has undergonereflection and rotation. The amplitude is not necessarilyzero in the upper corners of (a), but the region for dataacquisition did not include the areas outside of the dashedcurves. In some cases it was evident that a clockwise rota-tion by the angle π/2−Θ = π/2−cos−1(ωigw/N0) resultedin beams that were not horizontal despite time series im-ages confirming that the waves had reached quasi-steadystate. Although we have been unable to determine thecause of this observation, in these cases a correction of upto ∼ 0.1 rad was applied to the rotation angle so that avertical profile through the image would be close to per-pendicular to the phase lines. A spatially averaged profile,plotted as the solid curve in Figure 4(b), was computedfrom 11 evenly spaced profiles over r ∈ [5Rc, 6Rc], whichare represented by the dashed vertical lines in (a). Thedashed curves in (b) are the envelope of the r-averaged N2

t

profiles over one wave period, with the lower curve beingthe reflection of the positive amplitudes obtained from therms. The profile shown at a particular phase moderatelyovershoots the envelope because the experimental signalincludes noise and so is not perfectly sinusoidal in time.We then calculated the power spectrum, shown in (c), ofthe Fourier transformed data.

In all cases we observed a distribution of power overa range of wavenumbers, which is expected for a beamof internal waves. However, here we focus on the mag-nitude of the wavenumber with the maximum associatedpower in the Fourier spectrum, and denote it as k∗σ. Foreach spectrum as shown in Figure 4(c), the peak value wasfound from a quadratic fit to the three points with themaximum amplitudes. The resulting peak values from 16


10 15 20 25 30r [cm]

5

10

15

20

25

σ[c

m]

5 350

30(a)N2

t [s−3] -2.7 2.7

−2

−1

0

1

2

N2 t[s−

3]

5 10 15σ [cm]

0−3

3

〈N2t 〉

(b)

0.05

0.10

0.15

0.20

0.25

0.30

arbit

rary

unit

s

1 2 3 4|kσ| [cm−1]

(c)

0 50.00

Figure 4: Contours of N2t are shown in (a) after reflection and rotation about the position of the cylinder. The heavy

dashed line demarcates the regions in the upper corners of the image where no schlieren information is available. Ther-averaged profile is given by the solid curve in (b), with the envelope over one wave period given by the dashed curve.The power spectrum resulting from the Fourier transform of the solid curve is plotted in (c) as a function of cross-beamwavenumber.

evenly spaced snapshots were then averaged in time fort ∈ [4Tigw, 5Tigw]. The wave period was calculated fromthe measured wave frequency as Tigw = 2π/ωigw. Thisinterval in time was chosen because the beams were well-developed and significant distortions due to wave instabili-ties were not yet present. The uncertainty in the measure-ment of the cross-beam wavenumber, δk∗σ, was taken tobe the standard deviation determined from the averagingprocess. This procedure provides a characteristic value foran analysis of the wave lengthscale and is used in furthercalculations that involve the polarization relations. We ac-knowledge that this treatment is a simplication and in us-ing this value we do not intend to imply that the wave fieldis entirely monochromatic in space. However, the charac-teristics of the Fourier wavenumber spectra are consistentwith the selection of the dominant wavenumber to describethe dynamics.

D Measurements of wave amplitude

For the analysis of wave amplitudes, we have focused onthe N2

t field for times t ∈ [4Tigw, 5Tigw] and we com-pute the envelope as described in §A. In order to capturethe properties of the beam, we have computed a profilein the σ direction, averaged over a radial coordinate ofr ∈ [10 cm, 20 cm]. Although we work with the envelope,⟨

N2t

⟩

, for this analysis, some “patchiness” is still evidentin the final image that we use for further analysis. Av-eraging in the r-direction reduces some of the variabilityin the signal that is an artifact of processing rather thanan indication of the wave structure. The lower bound ofthe spatial interval was chosen such that the measurementswould be outside of the turbulent boundary layer aroundthe cylinder. The characteristic amplitude, denoted byAN2

t

, was taken as the maximum value extracted from ther-averaged profile, with its uncertainty given by the stan-dard deviation.

−12 −10 −8x [cm]

25

27

29

31

z[c

m]

−1423

33(a)

−12 −10 −8x [cm]

−14

(b)

−12 −10 −8x [cm]

−14

(c)

Figure 5: Close-up view of synthetic schlieren backgroundshowing evolution of image characteristics. (a) Typical dis-tortions of the image due to waves. (b) The image whensatisfying the criterion for breakdown. (c) Loss of resolu-tion of the lines.

E Wave breakdown: qualitative observations

In order to gain insight into the mechanism for the ob-served instabilities, a qualitative examination was per-formed of the raw video footage for a subset of experi-ments. The collection of experiments included both NaCland NaI stratifications, as well as the full range of cylin-der radius, amplitude, and oscillation frequency. A con-sistent, although qualitative, criterion was chosen to markthe onset of significant disturbances to the beam struc-ture; we will refer to this phenomenon as wave breakdown.The horizontal black and white lines of the schlieren im-age were monitored visually for the first occurrence of thelines appearing to be oriented vertically at a location out-side of the turbulent region bounding the cylinder. Anexample of the visual characteristics of the schlieren imageis shown in Figure 5 for an experiment in a NaCl stratifi-cation with N0 = 1.9 rad/s, ωc = 1.53 rad/s, Rc = 2.98 cm,and Ac = 2.0 cm. The spatial region is the same for eachframe, with time increasing from (a) to (c). A typical im-


age resulting from a coherent wave beam is shown in (a),which includes visible deflections of the lines from theirundisturbed orientation. The region is shown in (b) 4 sec-onds (≃ 1Tigw) later with clear qualitative changes occur-ring in the image. Our criterion for wave breakdown issatisfied at (x, z) ≃ (−11 cm, 28 cm), where the lines be-come vertical and appear to be “overturning.” The imageshown corresponds to a time of approximately 60 seconds(≃ 15Tigw) from the start of the oscillations of the cylin-der. The image in (c), taken 7 seconds (≃ 1.7Tigw) afterthe time of (b), shows blurring and the inability of thecamera to resolve each separate line due to the develop-ment of fully three-dimensional structures in the flow. Weinterpret this evolution of the raw image as evidence ofthe evolution of the beam instability, but these featuresalone do not provide significant insight into the instabilitymechanism. We provide a discussion of potential causesfor wave breakdown in §B.

For each experiment, several frames separated by 1 sec-ond were captured from the video recording, starting atapproximately the time of breakdown. In each case, thestill images were used to estimate the time and location atwhich breakdown occurred, which could then be comparedfor different forcing parameters of the cylinder. We empha-size that this analysis focuses on the first occurrence of theimage “overturning” shown in Figure 5. Such overturningoften was subsequently observed at different locations inthe tank for later times in the experiment.

IV WAVE STRUCTURE AND TRANS-

PORT

A Wave frequencies

In Figure 6 we compare the normalized wave frequency,ωigw/N0, to the normalized forcing frequency, ωc/N0. In(a) the average power spectrum for each experiment isshown with an offset on the horizontal axis correspond-ing to the value of ωc/N0. The spectra have been rescaledsuch that the maximum is the same arbitrary value forall experiments. Note that each spectrum displays similarcontent to that shown in Figure 3, although the spectra forFigure 6 are the average results and the ωigw axis has beennormalized by the buoyancy frequency. We include thepresentation of the frequency spectra in this form to pro-vide additional information about the shape of the spectrafor varying forcing frequency. For example, an increase inthe width of the spectral peaks is evident for cylinder fre-quencies that are approaching or above the buoyancy fre-quency. In Figure 6(b) the value of ωigw with peak poweris plotted with a solid circle. The open circles correspondto secondary, lower amplitude peaks in the frequency spec-trum. The dashed line with a slope of 1 corresponds to theprediction of linear theory that the wave frequency is equalto the forcing frequency. Two additional dashed lines cor-responding to frequency superharmonics are also shown in(b). In the low forcing frequency experiments, the major-ity of the power in the measured spectrum was at twice

0.2

0.4

0.6

0.8

1.0

ωig

w/N

0

0.2 0.4 0.6 0.8 1.0 1.2ωc/N0

0.00.0

(a)

0.2

0.4

0.6

0.8

1.0

ωig

w/N

0

0.2 0.4 0.6 0.8 1.0 1.2ωc/N0

0.00.0

slope=1

slope=2

slope=3

(b)

Figure 6: (a) Profiles of the normalized power spectrumas a function of normalized wave frequency ωigw/N0. Eachprofile corresponds to a different experiment in which thecylinder frequency is ωc. Each power spectrum is horizon-tally offset by ωc/N0 as indicated by the horizontal axis.The spectra are normalized by the maximum power andscaled for visual clarity. The peaks from each of thesecurves are plotted in (b) with solid circles. Open circlesshow the locations of secondary peaks in the spectra forthe lowest frequency cases. The dashed line correspondsto the linear theory prediction that the wave frequencyshould match the forcing frequency if the latter is belowN0. A typical horizontal error bar is shown in the upperleft-hand corner.

the forcing value with a peak also occurring at the forcingfrequency. For the lowest forcing frequency, another small-amplitude peak can be seen in the spectrum at three timesthe forcing value, but frequency tripling is not permitted inother experiments due to the upper limit of the buoyancyfrequency. Also according to linear theory, no waves can begenerated directly by the cylinder for ωc aboveN0, which isindicated on the plot by the dotted vertical line. Althoughthe spectra are broader for forcing frequencies above thebuoyancy frequency, we nonetheless observed propagating


0.5

1.0

1.5

2.0

2.5

3.0

(k∗ σ)−

1[c

m]

1.0 2.0 3.0 4.0 5.0 6.0Rc [cm]

0.00.0

(a)

0.5 1.0 1.5 2.0LO [cm]

0.0 2.5

(b)

ωc/N0 ∈ (0.5, 0.6)

ωc/N0 ∈ (0.6, 0.7)

ωc/N0 ∈ (0.7, 0.8)

ωc/N0 ∈ (0.7, 0.8)

Figure 7: The inverse of the cross-beam wavenumber as a function of (a) Rc and (b) the Ozmidov scale, LO. A typicalhorizontal error bar is shown in the upper left-hand corner of (b). In the legend, the direction of the arrows correspondsto the direction of propagation of the primary beam. Symbols are designated according to the relative forcing frequencyfor each experiment.

internal waves with ωigw ≃ 0.5N0 in these experimentsdue to generation by the oscillatory turbulent patch. Theobserved relative frequency corresponds to propagation atapproximately 60◦ from the vertical. This result is near thelargest angle of the ranges for turbulently generated wavesfound numerically by Taylor and Sarkar12 (35◦– 60◦) andexperimentally by Dohan and Sutherland9 (42◦– 55◦).

Agreement with the theoretical prediction is closest forthe range ωc/N0 ∈ [0.5, 0.8], where we see that the exper-imental data lie on the theoretical curve within the errorbars in all cases. In the following sections, we focus onexperiments in this frequency range because the waves ex-hibit the most coherent beam structure, thereby facilitat-ing our analysis and interpretation of the data.

B Cross-beam wavelengths

In the linear regime the length scale of the waves is com-pletely determined by the size of the cylinder15. In thecurrent experiments, for which Ac is of the same order asRc, we anticipate that the large-amplitude forcing may in-fluence the wavenumber. In Figure 7(a) the inverse of thecharacteristic cross-beam wavenumber, 1/k∗σ, is plotted asa function of the cylinder radius. From the plot we ob-serve that the points corresponding to the smallest valuesof oscillation frequency are separated from the remainingexperiments. We conclude that Rc alone is not an ade-quate predictor of the wavenumber, particularly for smalloscillation frequencies.

Due to the turbulent nature of the wave generation pro-cess, we propose that the Ozmidov scale, LO, should setthe value of 1/k∗σ. The Ozmidov scale is a measure of thevertical extent of the largest turbulent eddies that havesufficient kinetic energy to overturn in a given stratifica-

tion. In terms of the turbulent dissipation rate ε and thebuoyancy frequency26,

LO = ε1/2N−3/2. (10)

To estimate ε, we assume that the energy of the trans-mitted waves is small in comparison with the total energyinput by the cylinder, so that the turbulent dissipation ratemay be approximated by the rate of energy input per unitmass.

From observations of the unprocessed experimentaldata, we find that when the cylinder begins to oscillate,fluid in the bounding region above or below the cylinderis displaced and flows around the cylinder to the oppositeside. This appears to be the primary process occurring inthe generation of the turbulent patch, and it is also con-sistent with our observation that in quasi-steady state theoscillatory turbulence is out of phase with the cylinder.We consider the area in the x − z plane, A, of fluid thatis displaced by the motion of the cylinder with amplitudeAc during a quarter cycle. A direct calculation of this areathrough integration yields

A = AcRc

√

1 − A2c

4R2c

+R2c

[

π − 2 sin−1

(√

1 − A2c

4R2c

)]

.

(11)Assuming that Ac . Rc, which is the case for our experi-ments, this expression may be approximated by A ∼ AcRc.To estimate the characteristic speed of the displaced fluidwe assume that it travels a vertical distance of 2Rc in atimescale of ω−1

c , so that v ∼ 2ωcRc. From these calcula-tions, we find that the rate of kinetic energy input per unitlength along the cylinder for the displaced fluid is approx-imately ρ0(AcRc)(2ωcRc)

2ωc. To obtain the input rate of


energy per unit mass, we divide this quantity by the fluiddensity and the area of the region of energy input, whichto leading order is πR2

c . Thus, we estimate

ε ∼ (AcRc)(2ωcRc)2ωc

πR2c

∼ AcRcω3c , (12)

and so from (10)

LO ∼√

AcRc

(ωcN

)3/2

. (13)

In Figure 7(b) (k∗σ)−1 is plotted as a function of LO.

We do not expect perfect collapse of the data because ofthe complicated features of the experiments. However, Fig-ure 7(b) demonstrates that a clearer trend emerges througha comparison between the wave lengthscale and a length-scale of turbulence. In particular, the separation of thedata according to frequency, as mentioned above for (a), isreduced in (b). Here we note that the values of LO that wecalculate based on (13) are between approximately 1 cmand 2 cm. Although LO is crudely estimated, the relation-ship between (k∗σ)

−1 and LO is of order 1, which makes adirect correspondence between these quantities more phys-ically plausible.

The estimated values of LO are consistent with the ver-tical extent of turbulent patches surrounding the cylinder,as shown in Figure 8. This shows an unprocessed syn-thetic schlieren image at a time approximately four waveperiods from the start of the cylinder oscillation, whichcorresponds to the lower bound of the time interval forthe wavenumber analysis, as described in §C. Fully threedimensional disturbances in a stratified fluid scatter lightpassing through the tank from the background image ofblack and white lines. We identify the resulting regionwhere the image is blurred as being turbulent. Figure 8shows that the turbulent boundary layer above the cylin-der is of comparable extent to the Ozmidov scale estimatedusing (13). This result is typical of all the large-amplitudecylinder oscillation experiments we have performed. Out-side the turbulent boundary layer, the image of black andwhite lines is distorted but not blurred, indicating the pres-ence of spanwise-coherent internal waves emanating fromthe oscillatory turbulent patch.

Thus it is reasonable to suppose that LO gives a measureof the largest lengthscales of the turbulent patch that actsas a source for the internal waves. Some of the scatter inthe data shown in Figure 7 may be attributed to the valueof k∗σ. Since we have retained only the peak wavenumberin our analysis, the results for each experiment may beaffected differently by this simplification depending on thetrue wavenumber distribution.

C Wave amplitudes

Although we measured the amplitude AN2

t

directly, we mayuse the Boussinesq polarization relations to find the am-plitude of the vertical displacement of the waves, given by

Aξ =AN2

t

N30kx sinΘ

=AN2

t

N30k

∗

σ cosΘ sin Θ. (14)

−12 −9 −6 −3x [cm]

−5

0

5

z[c

m]

LO

t ≃ 4Tigw

0−10

10

Figure 8: Snapshot of the unprocessed synthetic schlierenimage in the near-cylinder region for an experiment withN0 = 1.9 rad/s, Rc = 2.98 cm, Ac = 2.0 cm and ωc =1.53 rad/s. The coordinate origin corresponds to the cen-tre of the cylinder. Fully three dimensional turbulencesurrounding the cylinder blurs the image of black andwhite lines behind the tank. The estimated Ozmidov scale,LO ≃ 1.76 cm, is shown in relation to the turbulent layerat the top of the cylinder.

0.02

0.04

0.06

0.08

0.10

Aξ/λ

x

0.3 0.6 0.9 1.2 1.5 1.8 2.1Ac/LO

NaCl

NaCl

NaI

NaI

0.0 2.40.00

Figure 9: Vertical displacement amplitude, Aξ, normalizedby horizontal wavelength, λx, shown as a function of nor-malized cylinder amplitude. Upward(downward)-pointingarrows denote measurements of upward(downward)-propagating waves.


We normalize Aξ by the horizontal wavelength, λx =2π/kx, to provide a more physically meaningful interpre-tation of the data. The final values shown in Figure 9 werecalculated as

Aξλx

=AN2

t

2πN30 sinΘ

, (15)

which have been plotted as a function of Ac/LO. As shownin Figure 9, there is a clear separation in the ratio Aξ/λxfor NaCl and NaI experiments, with an approximately con-stant ratio of ∼ 2.5% for all NaI experiments. In previouswork9,10 with NaCl stratifications it was observed acrossexperiments that Aξ/λx collapsed to a value in the range of2-4%, regardless of the forcing amplitude. We do not havea clear explanation for the observed increase in the ratiofor NaCl experiments. The effect of the smaller value of N0

for NaCl stratifications is magnified by the cubic power ofN0 in (14). However, there is no obvious physical reason toanticipate that this difference between the amplitude ratiosshould be based on stratification alone. We have estimatedthe Reynolds number based on the cylinder diameter, oscil-lation amplitude, frequency and a characteristic kinematicviscosity of the fluid. Viscosity measurements were per-formed using an Anton Paar DMA 500 density meter forsolutions of NaI with densities between fresh water and ap-proximately 1.6 g/cm3. For the experiments of focus here,we estimate Reynolds numbers of approximately 1500-2000for both types of stratification. Thus viscosity differencesbetween solutions of NaI and NaCl do not account for theresults shown in Figure 9. The separation of the resultsfor NaCl and NaI may indicate that the wave behaviouris outside the applicability of the polarization relations,which were derived under the assumptions of linear the-ory. There also exists the possibility that a quantity otherthan Aξ/λx, with a different dependence on N0, remainsrelatively constant despite changes in the forcing ampli-tude. We have presented the results as shown to allow forcomparison within the context of previous studies.

For all calculations in the analysis of the wave gener-ation, we have assumed that the fluid can be treated asBoussinesq, because the extent of vertical propagation issmall in comparison with the density scale height. Asshown in Figure 9, the vertical error bars are sufficientlylarge as to produce a region of overlap between the datafor upward- and downward-propagating waves. If trendsin the data are significant, the slightly reduced amplitudesfor downward-propagating cases may indicate that the re-gion of observation is at the threshold of non-Boussinesqasymmetry between the directions of vertical propagation.However, we cannot conclude that the data shown containsevidence of non-Boussinesq wave behaviour.

D Wave power

We use the measurements of the characteristic wavenumberand amplitude, as described in the previous subsections, tocalculate the average energy flux of the primary beam, andhence the wave power. The present analysis is restricted toexperiments for which δk∗σ/k

∗

σ ≤ 0.1 and δAN2

t

/AN2

t

< 0.2

200

400

600

800

1000

Pexpt

[erg

/s]

1000 2000 3000Pthy [erg/s]

0 40000

ωc/N0 ∈ (0.5, 0.6)

ωc/N0 ∈ (0.6, 0.7)

ωc/N0 ∈ (0.7, 0.8)

ωc/N0 ∈ (0.7, 0.8)

Figure 10: Experimentally measured power versus theo-retically predicted power of the primary beam. 1 erg =1 g cm2 s−2. Note that the horizontal and vertical scalesdiffer significantly.

simultaneously, so that the values used for the calculationof power are as unambiguous as possible.

From the Boussinesq polarization relations we obtain thefollowing expression for the time-averaged vertical energyflux of a monochromatic plane wave with wavenumber kσ:

〈FE〉 =1

2ρ0

A2N2

t

N30 cosΘ sinΘk3

σ

. (16)

The use of this expression in our analysis requires thatmodes other than kσ do not contribute significantly to theN2t field. Based on the breadth of the wavenumber spectra,

we have modified the above expression to account for thecontributions from all cross-beam modes, kn, with non-negligible power. We replace the characteristic amplitudeand wavenumber with a sum over modes, i.e.

〈FE〉 =1

2ρ0

1

N30 cosΘ sinΘ

∑

n

A2n

k3n

. (17)

The images used in the present analysis were processedusing the same procedure as for the wavenumber analy-sis, as described in §C. Signal attenuation with increasingσ and a geometric effect caused by the image rotation re-sulted in a region of zero amplitude for the largest values ofσ. Based on the properties of the FFT algorithm that wehave employed, we account for this in our analysis by scal-ing the amplitude of a mode according to the ratio Lσ/L,where Lσ is a measure of the beam width and L is thelength of the spatial domain for the Fourier transform. Inthis case the amplitude, An, that one would obtain fromthe transform is related to the amplitude of the real signal,A, through

An =

(

LσL

)

A. (18)

We find that the squared amplitude is given in terms ofthe power by

A2n =

(

2

Lσ/L

)2

Pn, (19)


in which Pn represents the squared magnitude of theFourier coefficient of the n’th mode of the N2

t field.To obtain the total power of the primary beam, we mul-

tiply the vertical energy flux by the area of a horizontalcross-section through the beam, LcLx = LcLσ/ cosΘ, inwhich Lc is the length of the cylinder. Through this stepand the substitution of (19) into (17) we arrive at the ex-pression for the total measured power of the experimentalbeam:

Pexpt =2ρ0LcL

2

N30 cos2 Θ sinΘLσ

∑

n

Pnk3n

. (20)

In order to use the above expression we also require a quan-titative method of determining the beam width, Lσ. Weexpress Lσ in terms of a multiple of a characteristic wave-length, λ∗, as

Lσ = αλ∗ =2πα

k∗σ. (21)

Substituting this expression into (18) with An → A∗ cor-responding to the amplitude of the waves with kn → k∗σ,we obtain

α =A∗

A

(

k∗σL

π

)

=1

α

√P∗

A

(

k∗σL

π

)2

. (22)

Thus we find

α =P1/4∗√A

(

k∗σL

π

)

. (23)

For our estimate of α, we take the power of the mode forwhich kn is closest to k∗σ, and we use the characteristicamplitude, AN2

t

, as described in §D. This yields valuesof α ∈ [1.6, 2.2] across all experiments. Comparison withthe structure of the original N2

t profiles in the σ directionshows that this range of α is reasonable when we considerthat α characterizes the number of wavelengths containedin the primary beam.

In order to gain insight into the effects of the large-amplitude forcing on the resulting energy transport of thewaves, we compare (20) with the theoretically predictedtime averaged power of the primary beam using (6). De-noting the prediction as Pthy and recasting the expression(6) in terms of our experimental parameters, we obtain

Pthy =π

8ρ0

(

N20

ω2igw

− 1

)1/2

ωigwR2c(Acωc)

2Lc. (24)

The use of both ωigw and ωc in the calculation of Pthy

arises from the conversion of different parameters in (6)to the variables in our notation. The product of Acωc isthe magnitude of the maximum velocity of the oscillatingcylinder, whereas the single power of ωigw is a result ofour distinction between the properties of the waves andthe cylinder. We have expressed Pthy in terms of the wavefrequency rather than the angle from the vertical becausefrequency was the directly measured quantity. For calcu-lations of both Pexpt and Pthy we have used characteristicdensities of ρ0 = 1.25 g/cm3 for NaI stratifications andρ0 = 1.1 g/cm3 for NaCl stratifications. These values were

estimated from the experimental measurements of the den-sity at the vertical level of our analysis of wavenumber andamplitude.

Figure 10 shows a plot of the experimentally measuredpower versus the theoretical prediction. The large verti-cal error bars on the data points are a result of addingsignificant contributions from several of the variables in(20). Namely, the largest contributions to the final errorwere due to the uncertainties in Θ, Lσ, and Pn, the lastof which was estimated from the standard deviation of thetime-averaged spectrum.

The experimental measurements and theoretical predic-tions are similar for small values of Pthy, but the two val-ues deviate more with increasing forcing intensity. In gen-eral, we expect the experimental values to be less than thetheoretically predicted power, because the coupling of thecylinder to the internal waves is affected by the develop-ment of the turbulent boundary layer. This behaviour isobserved for all experiments, but the low values of Pexpt

for large Pthy also suggest that we may be observing a sat-uration of the wave field. Where the theory predicts anincreasing rate of energy transport by waves, we hypoth-esize that much of the forcing energy is lost to turbulentkinetic energy in the bounding region of the cylinder. Herewe may use the expression for ε, (12), to estimate the totalturbulent dissipation rate. The product of ρ0ε with thecharacteristic volume of displaced fluid, ∼ AcRcLc, yieldsan approximate dissipation rate of ρ0(AcRc)

2Lcω3c . Us-

ing characteristic experimental values, we obtain an esti-mate of ∼ 2000 erg/s. Therefore, it is reasonable to observesmaller wave powers than predicted by theory. In partic-ular, the estimated dissipation rate is comparable to thediscrepancy between the values of Pthy and Pexpt as theforcing intensity is increased.

V WAVE INSTABILITIES AND

BREAKING

A In-situ probe measurements

Synthetic schlieren provides a means of measuring quanti-tatively the structure and amplitude of internal waves inspace and time. In the previous sections we have demon-strated the application of synthetic schlieren to experi-ments in strongly stratified fluids. We have used a sep-arate quantitative technique for one characteristic experi-ment, in which measurements were made of the waves ata fixed location as they evolved in time. This providesan independent means through which we may observe theestablishment of the wave field and its subsequent break-down.

A conductivity probe was used to perform a verticaltraverse of the background stratification in the region ofinterest, with approximately 42 measurements of voltage,V , per vertical centimeter. The probe was then placed atfixed locations in space for a series of three experimentsin which a cylinder with Rc = 4.43 cm and Ac = 2.0 cmwas oscillating with frequency ωc = 1.96 s−1 approximately


−1.0

−0.5

0.0

0.5

1.0

ξ[c

m]

(a)

−1.5

1.5

−1.0

−0.5

0.0

0.5

1.0

ξ[c

m]

(b)

−1.5

1.5

−1.0

−0.5

0.0

0.5

1.0

ξ[c

m]

30 60 90 120 150t [s]

5(c)

0 180−1.5

1.5

Figure 11: Time series of the vertical displacement mea-sured at z = 45, 40, 35 cm in (a), (b), and (c) respectively,with the same cylinder and forcing parameters in each case.

10 cm above the bottom of the tank. The probe providedmeasurements of the voltage with a resolution in time of∆t = 0.05 s. The horizontal location of the probe was ap-proximately 30 cm from the center of the cylinder whilethe vertical coordinate was set at 45, 40, and 35 cm suc-cessively above the bottom of the tank.

The time series measurements of voltage were translatedinto densities using the function ρ̄(V ) obtained from a lin-ear fit to four discrete measurements of voltage for densi-ties in the range of [0.998, 1.30] g/cm3. With waves in thefluid, the perturbation density at each vertical level wascomputed by subtracting the initial background density.The wave vertical displacement, ξ, was then calculated ac-cording to

ρ = −dρ̄dzξ, (25)

where ρ is the perturbation density and the backgrounddensity gradient at each of the three vertical levels wasfound through

dρ̄

dz=dV

dt

dρ̄

dV

dt

dz. (26)

Time series of the vertical displacement are shown inFigure 11 for each vertical level. The motor driving theoscillations of the cylinder was turned on at t = 30 s for(a) and t = 20 s for (b), (c), and was turned off at t ≃ 150 sfor each experiment. In all cases, we observe at first a reg-ular, periodic signal in time, with growth in amplitude asthe initial transient reaches and passes the location of the

probe. Based on the background stratification and theforcing frequency, we expect close agreement between thecylinder and wave frequencies because ωc/N0 ∈ [0.5, 0.8],as discussed in §A. A closer examination of the earlytimes in Figure 11 yields a wave period that is similarto the cylinder period of Tc = 2π/ωc = 3.2 s. For eachplot the vertical axis is the same to allow for direct com-parison of the wave amplitudes at varying location in thevertical. For an experiment with the same forcing param-eters as the experiment under discussion, the estimatedvertical displacement amplitude (as described in §C) wasAξ = 0.5 cm. Although the spatial location of the mea-surements differ, the conditions should be the most similarto those at z = 35 cm in the present analysis. The ap-propriate time for comparison of the wave amplitudes isapproximately 4 wave periods after the oscillations of thecylinder began, which is t ≃ 33 s. While we do not ex-pect exact agreement between the measurements becauseof the different techniques of instantaneous measurementand averaging over space and time, the magnitude of thevertical displacement is similar to the value of 0.5 cm ob-tained using synthetic schlieren. As shown in Figure 11,after a finite time the signal becomes irregular and thewave amplitude varies erratically. We consider this changein the regularity of the signal the signature of the onsetof instability. The occurrence of large wave amplitudes orirregularity in the time series correlates well with the ob-servation of significant distortions in the schlieren image ofhorizontal lines behind the tank, as described in §E.

B Qualitative analysis of synthetic schlieren

As discussed in previous sections, we observed the break-down of propagating beams through both quantitative andqualitative techniques. Here we provide the results of fur-ther investigations of the wave breakdown and the associ-ated instability mechanism. First we note that the mea-sured normalized vertical displacement amplitudes, Aξ/λx,as presented in §C, were well below the magnitude requiredfor overturning instability. Non-Boussinesq growth of thewave amplitudes to overturning values does not account forthe observations because the density scale height was largein comparison with the distance of vertical propagation.

The qualitative data, obtained from unprocessedschlieren images as described in §E, yielded several sig-nificant results. For some experiments, the breakdowncriterion was not satisfied at any time that the cylinderwas oscillating. It was common to these experiments thatthe forcing frequency was very near to or above the back-ground buoyancy frequency, or the amplitude of oscillationwas the smallest of our parameter range. This result forhigh frequency forcing is consistent with the quantitativesynthetic schlieren measurements, in which the wave sig-nal was weaker and less coherent than for mid-range forcingfrequencies. Thus, we should not necessarily expect signifi-cant growth and transition to instability for the waves gen-erated by high-frequency forcing. The occurrence of wavebreakdown for both upward- and downward-propagating


(a) (b) (c)

Figure 12: Schematic illustration of potential causes for wave breakdown: (a) beam superposition due to surface reflection,(b) beam-beam interference, and (c) breakdown of a freely propagating beam due to instability.

beams confirms that non-Boussinesq growth of upward-propagating waves is not responsible for the behaviour.Another trend in the observations is that the time of wavebreakdown varied significantly across experiments. For thesame cylinder radius and amplitude, a change in the forc-ing frequency yielded an opposite change in the observedtime of breakdown, i.e. a decrease (increase) in frequencyresulted in a later (earlier) breakdown. This is a physi-cally reasonable consequence of the change in the timescalefor beam development. The effects of cylinder radius andamplitude on the breakdown time and location appear tobe dominated by the forcing frequency in this qualitativeanalysis.

There are several possible scenarios, which are repre-sented schematically in Figure 12, that could lead to thebreakdown of waves as observed. We expect that nonlin-ear effects are the most significant in regions of beam self-interaction, such as when the primary beam reflects offof the surface, or in a beam-beam interaction that couldarise through multiple reflections off of the tank walls andthe fluid surface. These situations are depicted in (a) and(b), respectively. We also consider the possibility of thebreakdown of a freely propagating, non-interfering primarybeam, as shown in (c). Breakdown in a region of beamreflection is the most straightforward to identify from ob-servations because of the proximity to the surface or thebottom of the tank. In the case of beam-beam interactionsat mid-depth, we expect that the location of the breakdownwould vary significantly according to the wave frequencyand the corresponding angle of propagation. Through ge-ometrical considerations, the region of interference wouldmove farther from the position of the cylinder with de-creasing forcing frequency.

The greatest insight into the underlying mechanism forthe instability has been obtained from a comparison of thebreakdown location across experiments, for which the dataare plotted in Figure 13. The experiments have been sep-arated into upward- and downward-propagating beams in(a) and (b) respectively, with further distinctions made ac-cording to the forcing frequency relative to the backgroundbuoyancy frequency, as shown by the legend. For reference,we also include lines with slopes predicted by the value ofωc/N0 according to (2), by which we can place approximatebounds on the expected location of the primary beam ata given frequency. Note that in this qualitative analysis ofvideo footage, the experiments were not restricted to the

frequency range that was used for quantitative analyses. InFigure 13, a marker indicates the location of wave break-down for each experiment. We observe that as a group,the markers are displaced somewhat upward in (a) anddownward in (b) relative to a line through the centre ofthe cylinder. This is consistent with our observations fromquantitative synthetic schlieren that the primary beam wasnot centred about the equilibrium position of the cylinder.Such an effect is evident in Figure 4(a), in which the ro-tated beam is displaced from a line through the centreof the cylinder. Therefore, the locations of the markersagree well with our expectation that the path of the pri-mary beam emanates from a turbulent patch above andbelow the cylinder. The horizontal coordinates of break-down were clustered within approximately 20 cm from thecentre of the cylinder, and there were no apparent trends inthis location based on forcing parameters. The initial wavebreakdown occurred at a vertical coordinate that does notcorrespond to a region of surface or bottom wave reflec-tion, nor does the horizontal distance change significantlywith frequency as it would in the case of interacting beamsillustrated in Figure 12. So, having ruled out beam-beaminteractions, we conclude that the waves contained in a sin-gle beam undergo a transition to instability independentlyof interactions with boundaries or other beams. A can-didate mechanism for the breakdown of an isolated beamis parametric subharmonic instability (PSI), whereby en-ergy is transfered to waves of lower frequency and higherwavenumber than the primary disturbance27,28,24. Thishypothesis provides the motivation for the numerical sim-ulations described in the following subsection.

C Numerical Simulations

A two-dimensional, fully nonlinear, Boussinesq code18 wasused to simulate the evolution of an internal wave beam.Previous work18,29 has focused on plane waves and spa-tially localized wavepackets, often with uniform structurein the along-stream direction. The simulations presentedhere are not an attempt to model accurately all of thecharacteristics of the oscillating cylinder experiments. Ourobjective in performing simulations was to investigate po-tential instabilities in an established beam of finite width.For this study the spatial domain was horizontally and ver-tically periodic with a resolution of 128 by 512 points inthe x and z directions respectively. The code uses finite


x [cm]

z[c

m]

0-10-20-30-40-10

0

10

20

30

40

ωc/N0 = 0.4

0.5

0.6

0.7

0.8 0.9

(a)

x [cm]

0-10-20-30-40

10

0

-10

-20

-30

-40

ωc/N0 = 0.6

0.7

0.8

(b)

ωc/N0 ∈ (0.3, 0.4)

ωc/N0 ∈ (0.4, 0.5)

ωc/N0 ∈ (0.5, 0.6)

, ωc/N0 ∈ (0.6, 0.7) ,

, ,ωc/N0 ∈ (0.7, 0.8)

ωc/N0 ∈ (0.8, 0.9)

Figure 13: Schematic of experimental apparatus (to scale) with a marker denoting the location of wave breakdown foreach experiment as indicated in the legend. Results for upward- and downward-propagating primary beams are shownin (a) and (b), respectively.

differencing in the vertical with periodic upper and lowerboundary conditions and was run with 64 spectral modes inthe horizontal direction. While a horizontally plane wavestructure can be resolved with far fewer modes, the finitebeam width in the current study means that increased hori-zontal resolution was required for adequate sampling acrossthe signal.

We have initialized the simulations with a perturbationin the form of plane wave structure in the cross-beam (σ)direction with a Gaussian envelope to determine the beamwidth. In order to satisfy the doubly-periodic boundaryconditions, the full disturbance consisted of a superpositionof three identical beams separated by a fixed distance. Thebeams decayed sufficiently rapidly with σ to prevent anincrease in amplitude of the neighbouring beams due tosuperposition. Given a domain x ∈ [0, L], z ∈ [0, H ], thebeam separation was given by

σs =LH√L2 +H2

. (27)

This distance guaranteed the periodicity of the structureby positioning the centre line of the two secondary beamsat the appropriate corners of the domain. The centre line ofthe primary beam was from corner to corner of the domain,regardless of the dimensions L and H . These parametersdetermined the angle of the beam to the vertical direction,and hence the frequency ωigw, through the relations

Θ = tan−1

(

L

H

)

= cos−1

(

ωigw

N0

)

. (28)

The maximum amplitude of the perturbation, cross-beam wavenumber, and standard deviation of the Gaussianenvelope, denoted by a0, kσ, and σ0 respectively, are freeparameters. The initial structure of the streamfunction is

then given by

ψ(σ, t = 0) = a0

{

exp

[−σ2

2σ20

]

cos(kσσ) + exp

[−(σ − σs)2

2σ20

]

cos [

+ exp

[−(σ + σs)2

2σ20

]

cos [kσ(σ + σs)]

}

.

For each spatial coordinate pair (x, z) in the domain,a transformation to the σ coordinate was performed using(3). The value of ψ was calculated for the resulting value ofσ according to (29), and was then assigned at the originalgrid point. Small-amplitude randomly generated noise wasalso superimposed on the field over the entire domain toseed any physical instabilities evenly.

For a beam in the first quadrant, the vertical compo-nent of the group velocity is positive. Therefore, in orderto obtain the correct signs of the horizontal and verticalwavenumbers kx and kz, they were calculated as

kx = |kσ| cosΘ, kz = −|kσ| sin Θ, (30)

where kσ < 0.The parameters that determine the flow were chosen

to model the experimental conditions. In all cases, thebackground velocity was zero and N0 was the value deter-mined from the density profile, as described in §III. For agiven experiment, the characteristic cross-beam wavenum-ber, k∗σ, frequency, ωigw, and vertical displacement ampli-tude, Aξ, were known. The initial streamfunction ampli-tude was determined through polarization relations as

a0 =ωigw

kxAξ =

ωigw

|k∗σ| cosΘAξ, (31)

such that a0 > 0. The value of Aξ was determined sim-ilarly using a polarization relation, so the final value ofa0 should be considered an estimate due to the propaga-tion of uncertainties in our experimental measurements.


We also have an approximate measure of the beam widthfrom experiments in terms of the parameter α, given by(23). For the numerical beam initialization, we have at-tempted to obtain approximately 2 wavelengths across thewidth of the beam for consistency with observations andthe measurements of α. Contours of the vorticity field, ζ,are shown at initialization in Figure 14(a). Note that theextrema of the contour range are the same for each panel.

For all simulations that were initialized using parame-ters comparable to experimental conditions, an instabilitydeveloped along the central beam after an initial period ofregular propagation of phase lines through the beam at aconstant angle. The onset of the instability occurred alongthe centre line of the beam, where the initial amplitude waslargest, and the transition appeared visually to occur alongthe entire length of the beam simultaneously, as shown inFigure 14(b). Therefore, we have confidence that the in-stability is physical and is not caused by boundary effectsin the numerical formulation. Within the beam structure,waves began to develop at a larger angle to the verticaldirection, and hence a lower frequency, than the initial dis-turbance. A cascade of energy to smaller scales was alsoobserved. As in other studies of PSI for internal waves24,we find that the disturbance which grows fastest from thebackground noise is that with frequency half that of thewavebeam, as illustrated in Figure 14(c). The phase linesof the waves including the developed instability align wellwith the expected direction for the subharmonic, therebysupporting the conclusion that PSI was the primary mech-anism for the breakdown of the wave beams in the numer-ical context. In general, the instability grew in amplitudeuntil overturning began to occur, after which the simu-lations broke down rapidly. For waves with smaller initialamplitudes, the simulations ran for the full time of the cor-responding experiment. However, the development of PSIwas responsible for a complete loss of the coherent beamstructure. This effect may explain our observations in rawexperimental footage of the sudden spread at late times oflarge disturbances in the tank that did not correspond withthe expected location of beams. A similar effect was ob-served in experiments by McEwan30, who noted that den-sity microstructure became evident surprisingly rapidly inregions outside of breaking due to PSI.

The use of experimentally realistic parameters to initial-ize the simulations facilitates a comparison between theobserved time of PSI onset in the simulations and the ex-perimentally observed time of significant visual distortionsof the schlieren image, as described in §B. Although we cannot specify what physical effect was occurring at the timeof observation, the hypothesis that PSI arose in the exper-iments may be supported or refuted through an order-of-magnitude comparison with the results of the simulations.Runs were performed with parameters modelling five char-acteristic experiments in two different stratifications witha range of forcing frequencies. We have found that the es-timated time of the onset of PSI in the simulations differsfrom the experimental breakdown time by a maximum ofa factor of 2. This is reasonable agreement if one considers

that the simulations serve to model approximately some ofthe characteristics of the experimentally measured waves.Using the numerics, we have verified that instabilities be-came evident in a physically reasonable timescale givenphysically realistic input parameters. There were cases inwhich the experimental time was approximately 30% lessthan the time from simulations and vice versa, so no sys-tematic pattern emerged for these particular runs. Thesimilarity of the experimental and numerical timescales forthe development of instabilities serves as support for thehypothesis that PSI was the cause for breakdown of thebeams in the experiments.

VI CONCLUSIONS

We have studied the generation, propagation, and even-tual breakdown of internal wave beams generated by thelarge-amplitude vertical oscillations of a cylinder in strongstratifications of NaCl or NaI. Quantitative measurementsof wave frequency, wavenumber, and amplitude were madeusing a generalized form of synthetic schlieren that takesinto account the full vertical profile of the density and in-dex of refraction for fluids with significant density varia-tions with height. The large-amplitude forcing produced aturbulent boundary layer around the cylinder that modi-fied the generation region and the resulting characteristicsof the wave beams.

It was found that the wave frequency was equal to theforcing frequency only for the interval ωc/N0 ≈ 0.5 − 0.8.For small forcing frequencies, beams were generated athigher harmonics and with larger associated power thanthe primary beam. With forcing frequencies above N0, thewaves generated by the localized turbulent patch had fre-quencies of approximately 0.5N0. In all cases, the waveswith the maximum associated power were observed in therange ωigw/N0 ≈ 0.5 − 0.8 regardless of the forcing fre-quency. These results indicate a preferred frequency rangethat is in agreement with previous work on the turbu-lent generation of internal waves9,10,12. Wave frequencyselection in the typical range occurred despite a domi-nant frequency component, due to forcing by the cylinder,of the turbulent source. Coherent quasi-monochromaticbeam structures were observed emanating directly from thesource without an intermediate spatial region of waves witha broad frequency distribution. Thus, differential viscousdecay does not account for the observations of frequencyselection in this study.

The Ozmidov scale, which characterizes the verticalscale of the eddies in stratified turbulence, was found tobe more predictive of the lengthscale of the waves than thecylinder radius alone. Also as a consequence of the tur-bulent generation mechanism, the wave amplitudes werefound to be an approximately constant fraction of the hor-izontal wavelength, which has been noted in previous ex-perimental studies10. However, the magnitude of this ra-tio differed for experiments in NaCl and NaI stratifications.This observation is currently unexplained and requires fur-ther investigation.


10 20 30kxx

10

20

30

kxz

00

-1.5 1.5(a)Nt = 0

10 20 30kxx

0

-1.5 1.5(b)Nt = 250

π2− Θ′

π2− Θ

10 20 30kxx

0

-1.5 1.5(c)Nt = 300

Figure 14: Contours of the magnitude of the vorticity, ζ [s−1], at (a) initialization, (b) approximately the onset timeof the instability, and (c) a late time in the evolution of the wave field. The angles π

2− Θ = π

2− cos−1(ωigw/N0) and

π2− Θ′ = π

2− cos−1(ωigw/2N0) are shown in (c).

Qualitative observations from unprocessed videos wereused to characterize the time and location of wave break-down in the experiments. With the motivation of examin-ing potential instabilities for a temporally monochromaticwave beam, fully nonlinear numerical simulations wereperformed with experimentally realistic input parameters.The results of the simulations showed the development ofparametric subharmonic instability at times comparable tothe observed values for the experiments. Based on this out-come, we conclude that PSI of the isolated primary beamwas responsible for the breakdown of the waves in the ex-periments at relatively late times.

ACKNOWLEDGMENTS

The laboratory experiments were performed in the lab-oratory of Paul F. Linden, Department of Mechanical andAerospace Engineering, UCSD, with their analysis con-ducted at the University of Alberta. The authors wish tothank Linden for his comments on the manuscript. Thiswork has been supported by the Natural Sciences and Engi-neering Research Council (NSERC), the Canadian Founda-tion for Climate and Atmospheric Sciences (CFCAS) andan Alberta Ingenuity Fund Graduate Scholarship.

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