Generation, propagation, and breaking of an internal wave beamHeather A. Clark and Bruce R. Sutherlanda�

Department of Physics, University of Alberta, Edmonton, Alberta T6G 2G7, Canada

�Received 26 October 2009; accepted 28 May 2010; published online 14 July 2010�

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 ofNaCl or NaI. The oscillatory forcing leads to the development of turbulence in the region boundingthe cylinder. This turbulence is found to be the primary source of the observed quasimonochromaticwave beams, whose characteristics at early times differ from theoretical predictions andexperimental investigations of waves generated by small-amplitude cylinder oscillations. Inparticular, their wavelength is set by the Ozmidov scale rather than the size of the cylinder. Thewave frequency is set by the buoyancy frequency N if the cylinder frequency is larger or much lessthan N. 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 process isexamined 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 instability.This conclusion is supported by fully nonlinear numerical simulations of the evolution of atemporally, although not spatially, monochromatic internal wave beam. © 2010 American Instituteof Physics. �doi:10.1063/1.3455432�

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 fluidsby the internal waves radiating from localized sources. Ob-servations, modeling, and experiments have been used tostudy in detail several generation mechanisms and the prop-erties of the resulting waves. In particular, topographic forc-ing by tidal flow over features of the ocean floor is observedto be a major source of oceanic internal waves,1–3 which aresubsequently responsible for significant diapycnal mixing.4

Similarly, flow over mountains may generate moderate- tolarge-amplitude atmospheric waves,5 the turbulent break-down of which has been observed directly through in-flightmeasurements.6 Fritts and Alexander7 reviewed the genera-tion of atmospheric internal waves by several primarysources, including topography, convection, shear, geo-strophic adjustment, and wave-wave interactions. In general,the existence of such disturbances can have a significant non-local effect on the mean flow through the propagation andbreaking of internal waves.

While turbulent flows are involved to varying extent ingeophysical sources of internal waves, the turbulent genera-tion process is currently not well understood. It has beenobserved in experiments8–10 and numerical simulations11 thatturbulent sources generate waves in a narrow frequencyrange relative to the background buoyancy frequency. In arecent numerical study, Taylor and Sarkar12 found that theinternal waves generated by oceanic bottom boundary layer

turbulence propagated at angles between 35° and 60° fromthe vertical. It was shown that linear differential viscous de-cay could produce the observed spectral peak and decay inwave amplitudes. However, the proposed model may not bean adequate explanation in the case of larger amplitudewaves.12 A key difference between their numerics and labo-ratory experiments is the use of large-eddy simulation for thenumerical boundary layer and the resulting loss of resolutionof the finescale turbulence. Although a viscous model was inagreement with the numerical results, experiments show theimmediate generation of narrow-band internal waves on atimescale that is less than the viscous timescale required fordifferential decay.

In this paper we present the results of a study of internalwaves generated by large-amplitude oscillations of a circularcylinder in strongly stratified fluids. We classify the forcingas large amplitude because in all cases the half peak-to-peakamplitude of the cylinder displacement was on the order of,but less than, the radius of the cylinder. Moderate-amplitudeforcing in Boussinesq fluids has been investigated in labora-tory experiments by Sutherland et al.13 and Sutherland andLinden,14 for circular and elliptical cylinders, respectively. Ingeneral, good qualitative agreement was found between theexperiments and the linear, viscous, Boussinesq theory ofHurley and Keady.15 However, the beam width was consis-tently underpredicted because the theory neglects the forma-tion of a 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 thissense, the extension of previous experimental work to in-a�Author to whom correspondence should be addressed.

PHYSICS OF FLUIDS 22, 076601 �2010�

1070-6631/2010/22�7�/076601/16/$30.00 © 2010 American Institute of Physics22, 076601-1

clude large-amplitude effects also alters the generationprocess.

As well as standard sodium chloride �NaCl� solutions, anew experimental technique16 that is applied in this work isthe use of sodium iodide �NaI� to produce stronger stratifi-cations than those in conventional tank experiments.Whereas typical stratifications have a density change on theorder of 5% over the depth of the fluid, the density differ-ences in this study are approximately 20% for solutions ofNaCl and 50% for NaI. The use in this work of significantvariations in the background density profile is motivated bythe phenomenon of anelastic growth in amplitude forupward-propagating atmospheric waves.17 When the verticaldistance traversed by a propagating wavepacket is a signifi-cant fraction of the local density scale height, non-Boussinesq effects result in an increase in wave amplitudewith height. As a result, nonlinear effects may have a morepronounced influence on the evolution of such internalwaves. Previous work18 has shown that weakly nonlineareffects modify the amplitude growth of internal wavepackets,thus affecting the location and intensity of wave breaking,should it occur. In the current experimental work the wavesare generated at finite amplitude and the vertical scale ofpropagation is such that non-Boussinesq growth may bemeasurable. The use of both NaCl and NaI solutions allowsfor observation of the wave behavior over different fractionsof a density scale height, since the vertical scale of propaga-tion remains the same while the strength of the stratificationis 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 asymptoticanalysis that nonlinear effects are relatively insignificant foran isolated beam with slow along-beam modulations in auniform, Boussinesq stratification. Dispersive and viscouseffects were found to be the dominant factors in determiningthe propagation of isolated beams. A subsequent paper20 in-vestigated the role of nonlinearity in situations where thereexists a region of interaction, namely, the reflection of awave beam from a slope or the collision of two beams. Insuch cases it was found that nonlinear effects result in thegeneration of higher-harmonic beams that propagate out ofthe interaction region and into the far field. In this study wefind that nonlinear effects have a significant influence on theevolution of the finite-amplitude waves, resulting in wavebreaking for a single beam in the absence of critical layers.Rather than the slow modulations considered by Tabaei andAkylas,19 we observe that the wave breaks down due to para-metric subharmonic instability �PSI�. This mechanism forwave breakdown has been studied analytically and in labo-ratory experiments for unbounded plane internal waves andwave modes in a rectangular domain.21–24 Ours is the firstexamination of PSI occurring for an internal wavebeam incontinuously stratified fluid.

A brief summary of the analytic results of Hurley andKeady15 for small-amplitude cylinder oscillations is given inSec. II. In Sec. III we describe the experimental apparatusand the implementation of the synthetic schlieren technique

in strongly stratified solutions. We also explain in detail theanalysis of wave frequency, wavenumber, and amplitudefrom the schlieren processed data. The final content of Sec.III is a description of the qualitative criterion for the onset ofwave breakdown. In Sec. IV we present the results of thequantitative analysis of the wave field for coherent beamstructures. A discussion of the characteristics of the observedinstabilities and subsequent wave breakdown is the subjectof Sec. V, where we also present our findings from numericalsimulations of a localized wave beam. We summarize theresults of the study in Sec. VI.

II. THEORY

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

N2�z� = −g

�̄

d�̄

dz, �1�

where g is the acceleration due to gravity and �̄�z� is thebackground density profile as a function of height. Given thenon-Boussinesq form of the equation as shown above, theconstruction of a near-exponential density profile in the ex-periments yields a uniform stratification with anapproximately constant background buoyancy frequency de-noted by N0.

Although our experiments use large density gradients,we may treat the fluid as Boussinesq over our region of focusfor the generation of the beams because density variationsare relatively small over the vertical extent of the cylinder.Previous experimental work on internal wave generation byoscillating cylinders13,14 has been compared to the analyticsolution of Hurley and Keady15 for the waves generated bysmall-amplitude oscillations of a cylinder in a viscous, uni-form, Boussinesq stratification. For the purposes of compar-ing their predictions to our large-amplitude experiments, herewe present their final equations15 recast in a modified set ofvariables as they relate to our analyses.

For a cylinder oscillating vertically with angular fre-quency �c in a uniformly stratified fluid with backgroundbuoyancy frequency N0, such that �c�N0, four wave beamsemanate from the source region at a fixed angle to thevertical, given by

� = cos−1��c

N0� . �2�

The coordinate transformation from the �x ,z� plane to cross-beam and along-beam coordinates, �� ,r�, of the beam in thefirst quadrant is given by

� = − x cos � + z sin �, r = x sin � + z cos � . �3�

The coordinate transformations for other quadrants may beobtained using symmetry properties of the solution.13

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

076601-2 H. A. Clark and B. R. Sutherland Phys. Fluids 22, 076601 �2010�

� = �1

2Ac�cRce

−i�−�

Rc− i1 − ��/Rc�2� , ��/Rc� � 1,

0, ��/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=��x ,z�e−i�ct. From this, the velocity fields and otherfields of interest can be determined. In particular,ur=� /�� is the along-beam component of velocity, and thepressure is given by

p = − �i�c�0 tan �� , �5�

where �0 is a characteristic density. Thus, the period-averaged power transmitted along the beam per along-cylinder distance is

P = �−Rc

Rc

�urp�d� =

8�0Ac

2�c3Rc

2 tan � . �6�

III. EXPERIMENTAL SETUP AND ANALYSISMETHODS

We have performed a series of experiments using anoscillating cylinder to generate waves in strong stratifica-tions. As illustrated schematically in Fig. 1, a cylinder ofradius Rc was forced by a variable-speed motor to producevertical oscillations with half peak-to-peak amplitude Ac andangular frequency �c in a rectangular acrylic tank. In order

to obtain disturbances that were uniform in the spanwise �y�direction, the length of the cylinder was 2 mm less than theinner separation of the tank walls. The tank of dimensionsWt=122.3 cm and Lt=15.5 cm was filled to a depth ofHt�55 cm. For both types of stratification, experimentswere performed with the cylinder centered approximately12 cm above the bottom of the tank or approximately 8 cmbelow the fluid surface. This allows for a comparison be-tween the characteristics of upward- and downward-propagating waves. For all experiments the cylinder was lo-cated approximately 80 cm ��2Wt /3� from the left side ofthe tank. The spatial region for quantitative analysis was re-stricted to the left side of the cylinder, as we assume sym-metry of the wave properties about the vertical axis. As inthe case of small-amplitude forcing, four wave beams wereobserved emanating from the source region. Hereafter wewill refer to the “primary beam” and the “reflected beam” asthey are shown in Fig. 1. The terms are used similarly for thecase with the cylinder near the top of the tank, but the beamreflection occurs off of the fluid surface rather than the bot-tom of the tank.

Approximately exponential density stratifications weremade from solutions of either NaCl or NaI, as described byClark and Sutherland.16 For each subset of experiments thedensity profile was measured with a conductivity probe andthe data were fitted with an exponential function of the form

�̄�z� = ��z0�exp�− �z − z0�/H� , �7�

where H is the density scale height and z0 is the smallestmeasured vertical coordinate. H�270 cm for NaCl stratifi-cations, and H�130 cm for experiments using NaI. Thesevalues yield background buoyancy �angular� frequencies ofN0�1.9 rad /s and N0�2.7 rad /s for NaCl and NaI, respec-tively, according to Eq. �1�.

A. Synthetic schlieren technique

For the quantitative analysis of the internal wave field,we implemented the synthetic schlieren technique as de-picted in Fig. 1�b�. Due to the large variations in backgrounddensity over the tank depth for both NaCl and NaI stratifica-tions, we have used a modified form of the syntheticschlieren equations for non-Boussinesq fluids.16 Thus wetake into account the full vertical dependence of the densityprofile and the corresponding index of refraction for eachtype of stratification.

From measurements of the apparent vertical displace-ment �z of an image of horizontal black and white linesplaced behind the tank �see Fig. 2�a��, we may calculate theperturbation to the squared buoyancy frequency �N2 due towaves within the tank.13 In this work we focus on the timederivative of this field, denoted by Nt

2, given by

Nt2 � − zt

1

�1

2Lt

2 + Ltn̄�Lp

np+

Ls

na��−1

. �8�

The lengths Ls=15.5 cm and Lp=1.7 cm are shown in Fig.1�b�. Here n̄�z� is the index of refraction profile for thebackground stratification, while np=1.49 and na=1.0 arethe indices of refraction of the acrylic tank walls and air,

x

z

Ht

Wt

Rc

2Ac, ωc

primarybeam

reflectedbeam

(a) Front View

y

ztank imagescreen

camera

lights

Lp LpLt Ls

(b) Side View

FIG. 1. �a� Schematic diagram of the suspended cylinder oscillating withconstant frequency and amplitude. An approximate resulting wave beampattern is shown. �b� A side view of the synthetic schlieren apparatus. Thedashed line represents the path of a light ray from the screen to the camera.

076601-3 Generation, propagation, and breaking Phys. Fluids 22, 076601 �2010�

respectively. The parameter ��z� is evaluated in terms of thebackground density and index of refraction profiles as16

� =1

g

�̄

n̄�a1 + a2��̄ − �0�� , �9�

where �0=0.99823 g /cm3 is the density of fresh water and

a1 = 0.2458 cm3/g,

a2 = − 0.1208 cm6/g2 for NaCl,

a1 = 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 1/30 s.The images were digitized using the DIGIMAGE �Ref. 25�software package with a spatial resolution of approximately0.12 cm. For each experiment, vertical time series were madefrom the digitized video footage at 24 equally spaced hori-zontal locations over a total horizontal distance of approxi-mately 40–50 cm, each corresponding to a width of 1 pixel.DIGIMAGE was used to extract the time series images andfurther processing to obtain Nt

2�z , t� was performed with aseparate computer program. As shown in Fig. 2�b�, we re-construct a spatial snapshot at any desired time from the setof vertical time series images. In this figure, vertical wave-numbers greater than 1.0 cm−1 have been filtered in order toremove small-scale spatial noise. Interpolation is also per-formed between grid points in the horizontal direction tosmooth the image that results initially from the computa-tional reconstruction of the snapshot. We calculate the enve-lope �Nt

2� as shown in Fig. 2�c�, from 2 times the root meansquare of 16 snapshots evenly spaced in time over one waveperiod14 to obtain information about the average wave am-plitude and beam structure in space and time. The directionof the increasing positive cross-beam coordinate � is super-imposed on the primary beam in Figs. 2�b� and 2�c�. In Fig.2 and the spatial plots that follow in this paper, the coordi-nate origin has been shifted to correspond to the equilibriumposition of the cylinder center.

B. Measurements of wave frequency

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

2 for t� �5Tc ,15Tc�, where Tc isthe oscillatory period of the cylinder. For several experi-ments with low forcing frequencies, the experimental dataspan fewer than 15 periods, so the time window for fre-quency measurements was reduced. The standard time inter-val was chosen so that the waves were well developed andreasonably steady. Figure 3 shows an example of the analysisprocess for one experiment in a NaI stratification with N0

�2.7 rad /s, Rc=2.98 cm, Ac=2.0 cm, and �c=1.53 rad /s.We constructed the horizontal time series of the data, as plot-ted in Fig. 3�a�, at a vertical distance of 5Rc from the equi-librium vertical position of the cylinder. Eleven equallyspaced profiles over the region x� �−6Rc ,−5Rc�, which isbetween the thick vertical lines in Fig. 3�a�, were fast Fouriertransformed �FFT� in time to obtain power spectra in fre-quency space. The spatial region was chosen to be closeenough to the cylinder for the waves to be indicative of theproperties at generation, but outside of the turbulent patch sothat meaningful synthetic schlieren measurements were pos-sible. This interval was found to be the most appropriatechoice to apply over all experiments. The field for one rep-resentative profile is plotted in Fig. 3�b�, with the corre-sponding spectrum shown in Fig. 3�c�. The average of the 11spectra was computed before finding the location of the peakin frequency, which we denote by �igw. For an improvedmeasurement, �igw was obtained from a parabolic fit to thethree discrete points that defined the peak of the spectrum.The uncertainty �igw in the measurement was taken to bethe half-width of the parabolic curve. In general, the waveand cylinder frequencies need not be equal because of theturbulent generation process. A comparison between mea-sured wave and cylinder frequencies is the topic of Sec.IV A.

C. Measurements of wavenumber

Beginning with a snapshot of the Nt2 field, the image was

reflected about x=0 and rotated such that the lines of con-stant phase of the waves appeared horizontal. Through thisprocess the horizontal and vertical coordinates of the image

−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

FIG. 2. Synthetic schlieren is applied to yield the instantaneous value of Nt2, plotted in �b� at a time corresponding to the raw image �a�. The envelope, �Nt

2�,over one wave period is shown in �c�. In each case the coordinate origin is at the equilibrium position of the cylinder. For the experiment shown, theparameters are N0=2.7 rad /s, �c=1.96 rad /s, Rc=2.98 cm, and Ac=1.5 cm.

076601-4 H. A. Clark and B. R. Sutherland Phys. Fluids 22, 076601 �2010�

become r and �, respectively, as plotted in Fig. 4�a� for atypical experiment with N0=2.7 rad /s, �c=1.96 rad /s,Rc=2.98 cm, and Ac=2.0 cm. The snapshot shown corre-sponds to a time t� �4Tigw,5Tigw�. Note that the r axis asshown does not start from zero. For small values of the radialcoordinate the schlieren processed data are unreliable due tothe presence of the cylinder and the surrounding three-dimensional turbulence. We have restricted our analysis to��0 because of potential interference between the primarybeam and the reflected beam in the lower flank, for which��0. Although interference is not predicted by theory, theturbulent generation process results in more diffuse bound-aries for the experimental beams. The heavy dashed lines inFig. 4�a� correspond to the edges of the original schlierenimage that has undergone reflection and rotation. The ampli-tude is not necessarily zero in the upper corners of Fig. 4�a�,but the region for data acquisition did not include the areasoutside of the dashed curves. In some cases it was evidentthat a clockwise rotation by the angle /2−�= /2−cos−1��igw /N0� resulted in beams that were not horizontal

despite time series images confirming that the waves hadreached quasisteady state. Although we have been unable todetermine the cause of this observation, in these cases a cor-rection of up to �0.1 rad was applied to the rotation angleso that a vertical profile through the image would be close toperpendicular to the phase lines. A spatially averaged profile,plotted as the solid curve in Fig. 4�b�, was computed from 11evenly spaced profiles over r� �5Rc ,6Rc�, which are repre-sented by the dashed vertical lines in Fig. 4�a�. The dashedcurves in Fig. 4�b� are the envelope of the r-averaged Nt

2

profiles over one wave period, with the lower curve being thereflection of the positive amplitudes obtained from the rms.The profile shown at a particular phase moderately over-shoots the envelope because the experimental signal includesnoise and so is not perfectly sinusoidal in time. We thencalculated the power spectrum, shown in Fig. 4�c�, of theFourier transformed data.

In all cases we observed a distribution of power over arange of wavenumbers, which is expected for a beam ofinternal waves. However, here we focus on the magnitude 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

FIG. 3. �a� A horizontal time series of the Nt2 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 corresponding power spectrum of the Fourier transform in time plotted in �c�. The forcingand buoyancy frequencies are marked on the horizontal axis.

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

FIG. 4. Contours of Nt2 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. The r-averaged profile is given by the solid curve in �b�, with the envelope over onewave 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 ofcross-beam wavenumber.

076601-5 Generation, propagation, and breaking Phys. Fluids 22, 076601 �2010�

the wavenumber with the maximum associated power in theFourier spectrum and denote it as k�

� . For each spectrum asshown in Fig. 4�c�, the peak value was found from a qua-dratic fit to the three points with the maximum amplitudes.The resulting peak values from 16 evenly spaced snapshotswere then averaged in time for t� �4Tigw,5Tigw�. The waveperiod was calculated from the measured wave frequency asTigw=2 /�igw. This interval in time was chosen because thebeams were well developed and significant distortions due towave instabilities were not yet present. The uncertainty in themeasurement of the cross-beam wavenumber k�

� was takento be the standard deviation determined from the averagingprocess. This procedure provides a characteristic value for ananalysis of the wavelengthscale and is used in further calcu-lations that involve the polarization relations. We acknowl-edge that this treatment is a simplification, and in using thisvalue we do not intend to imply that the wave field is entirelymonochromatic in space. However, the characteristics of theFourier wavenumber spectra are consistent with the selectionof the dominant wavenumber to describe the dynamics.

D. Measurements of wave amplitude

For the analysis of wave amplitudes, we have focusedon the Nt

2 field for times t� �4Tigw,5Tigw� and we computethe envelope as described in Sec. III A. In order to capturethe properties of the beam, we have computed a profile inthe � direction, averaged over a radial coordinate ofr� �10 cm,20 cm�. Although we work with the envelope,�Nt

2�, for this analysis, some “patchiness” is still evident inthe final image that we use for further analysis. Averaging inthe r-direction reduces some of the variability in the signalthat is an artifact of processing rather than an indication ofthe wave structure. The lower bound of the spatial intervalwas chosen such that the measurements would be outside ofthe turbulent boundary layer around the cylinder. The char-acteristic amplitude, denoted by ANt

2, was taken as the maxi-mum value extracted from the r-averaged profile, with itsuncertainty given by the standard deviation.

E. Wave breakdown: Qualitative observations

While synthetic schlieren was used to obtain quantitativemeasurements of the wave field in the quasisteady regime,the technique was not applicable to the strongly disturbedflow that was evident at the onset of instability. Meaningfuldata are obtained from synthetic schlieren measurementsonly in the case of spanwise-uniform waves of moderate am-plitude. This property precludes the use of quantitative syn-thetic schlieren when the background image becomes highlydistorted due to the development of three-dimensional flow.In order to gain insight into the mechanism for the observedinstabilities, a qualitative examination was performed of theraw video footage for a subset of experiments. The collectionof experiments included both NaCl and NaI stratifications, aswell as the full range of cylinder radius, amplitude, and os-cillation frequency. A consistent, although qualitative, crite-rion was chosen to mark the onset of significant disturbancesto the beam structure; we will refer to this phenomenon aswave breakdown. The horizontal black and white lines of theschlieren image were monitored visually for the first occur-rence of the lines appearing to be oriented vertically at alocation outside of the turbulent region bounding the cylin-der. This is an indication that the displacement of isopycnalsis so extreme that the image behind the tank is magnified andsignificantly distorted. Although our criterion is not an indi-cation of wave overturning, we found that the image blurredaround this location shortly after the lines becomes verticalwhich is an indication of the wave breakdown through thedevelopment of small-scale three-dimensional structures.Our criterion thus provides an objective measure of the lo-cation and onset time of the start of wave breakdown.

An example of the visual characteristics of the schlierenimage is shown in Fig. 5 for an experiment in a NaCl strati-fication with N0=1.9 rad /s, �c=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 Fig. 5�a� to Fig. 5�c�. Atypical image resulting from a coherent wave beam is shownin Fig. 5�a�, which includes visible deflections of the lines

−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)

FIG. 5. Closeup view of synthetic schlieren background showing evolution of image characteristics. �a� Typical distortions of the image due to waves. �b� Theimage when satisfying the criterion for breakdown. �c� Loss of resolution of the lines.

076601-6 H. A. Clark and B. R. Sutherland Phys. Fluids 22, 076601 �2010�

from their undisturbed orientation. The region is shown inFig. 5�b�; 4 s ��1Tigw� later with clear qualitative changesoccurring 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 s��15Tigw� from the start of the oscillations of the cylinder.The image in Fig. 5�c�, taken 7 s ��1.7Tigw� after the time ofFig. 5�b�, shows blurring and the inability of the camera toresolve each separate line due to the development of fullythree-dimensional structures in the flow. We interpret thisevolution of the raw image as evidence of the evolution ofthe beam instability, but these features alone do not providesignificant insight into the instability mechanism. We providea discussion of potential causes for wave breakdown inSec. V B.

For each experiment, several frames separated by 1 swere captured from the video recording, starting at approxi-mately the time of breakdown. In each case, the still imageswere used to estimate the time and location at which break-down occurred, which could then be compared for differentforcing parameters of the cylinder. We emphasize that thisanalysis focuses on the first occurrence of the image over-turning shown in Fig. 5. Such overturning often was subse-quently observed at different locations in the tank for latertimes in the experiment.

IV. WAVE STRUCTURE AND TRANSPORT

A. Wave frequencies

In Fig. 6 we compare the normalized wave frequency�igw /N0 to the normalized forcing frequency �c /N0. In Fig.6�a� the average power spectrum for each experiment isshown with an offset on the horizontal axis corresponding tothe value of �c /N0. The spectra are plotted on a linear scaleand have been rescaled such that the maximum is the samearbitrary value for all experiments. Multiple spectra shown atthe same value of �c /N0 represent experiments with thesame forcing frequency but different cylinder radii and forc-ing amplitudes. Here we focus on the dependence of thewave frequency upon the forcing frequency only, since lineartheory predicts a direct correspondence between these quan-tities. Thus, in the plot we do not make distinctions based onRc or Ac. Note that each spectrum displays similar content tothat shown in Fig. 3, although the spectra for Fig. 6 are theaverage results and the �igw axis has been normalized by thebuoyancy frequency. We include the presentation of the fre-quency spectra in this form to provide additional informationabout the shape of the spectra for varying forcing frequency.For example, an increase in the width of the spectral peaks isevident for cylinder frequencies that are approaching orabove the buoyancy frequency. In addition, at near-criticalfrequencies increased variation can be seen among multiplespectra at the same forcing frequency, as in the case of�c /N0�0.87, and the wave power is found at frequenciesother than the forcing value. In Fig. 6�b� the value of �igw

with peak power is plotted with a solid circle. The opencircles correspond to secondary, lower amplitude peaks inthe frequency spectrum. The dashed line with a slope of 1

corresponds to the prediction of linear theory that the wavefrequency is equal to the forcing frequency. Two additionaldashed lines corresponding to frequency superharmonics arealso shown in Fig. 6�b�. In the low forcing frequency experi-ments, the majority of the power in the measured spectrumwas at twice the forcing value with a peak also occurring atthe forcing frequency. For the lowest forcing frequency, an-other small-amplitude peak can be seen in the spectrum atthree times the forcing value, but frequency tripling is notpermitted in other experiments due to the upper limit of thebuoyancy frequency. Also according to linear theory, nowaves can be generated directly by the cylinder for �c aboveN0, which is indicated on the plot by the dotted vertical line.Although the spectra are broader for forcing frequenciesabove the buoyancy frequency, we nonetheless observedpropagating internal waves with �igw�0.5N0 in these experi-ments due to generation by the oscillatory turbulent patch.

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)

FIG. 6. �a� Profiles of the normalized power spectrum as a function ofnormalized wave frequency �igw /N0. Each profile corresponds to a differentexperiment in which the cylinder frequency is �c. Each power spectrum ishorizontally offset by �c /N0 as indicated by the horizontal axis. The spectraare normalized by the maximum power and scaled for visual clarity. Thepeaks from each of these curves are plotted in �b� with solid circles. Theopen circles show the locations of secondary peaks in the spectra for thelowest frequency cases. The dashed line corresponds to the linear theoryprediction that the wave frequency should match the forcing frequency if thelatter is below N0. A typical horizontal error bar is shown in the upperleft-hand corner.

076601-7 Generation, propagation, and breaking Phys. Fluids 22, 076601 �2010�

The observed relative frequency corresponds to propagationat approximately 60° from the vertical. This result is near thelargest angle of the ranges for turbulently generated wavesfound numerically by Taylor and Sarkar12 �35°–60°� and ex-perimentally by Dohan and Sutherland9 �42°–55°�.

In summary, we find that the dominant frequency ofwaves generated by the oscillatory turbulent patch surround-ing the cylinder tends to lie between 0.5N0 and 0.8N0

whether the cylinder itself oscillates with very low or veryhigh frequency. Agreement with the theoretical prediction isclosest for the range �c /N0� �0.5,0.8�, where we see thatthe experimental data lie on the theoretical curve within theerror bars 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 facilitatingour analysis and interpretation of the data.

B. Cross-beam wavelengths

In the linear regime the length scale of the waves iscompletely determined by the size of the cylinder.15 In thecurrent experiments, for which Ac is of the same order as Rc,we anticipate that the large-amplitude forcing may influencethe wavenumber. In Fig. 7�a� the inverse of the characteristiccross-beam wavenumber, 1 /k�

� , is plotted as a function of thecylinder radius. From the plot we observe that the pointscorresponding to the smallest values of oscillation frequencyare separated from the remaining experiments. We concludethat Rc alone is not an adequate predictor of the wavenum-ber, particularly for small oscillation frequencies.

Due to the turbulent nature of the wave generation pro-cess, we propose that the Ozmidov scale LO should set thevalue of 1 /k�

� . The Ozmidov scale is a measure of the verti-cal extent of the largest turbulent eddies that have sufficientkinetic energy to overturn in a given stratification. In termsof the turbulent dissipation rate � and the buoyancyfrequency,26

LO = �1/2N−3/2. �10�

To estimate �, we assume that the energy of the transmittedwaves is small in comparison with the total energy input by

the cylinder, so that the turbulent dissipation rate may beapproximated by the rate of energy input per unit mass.

From observations of the unprocessed experimental data,we find that when the cylinder begins to oscillate, fluid in thebounding region above or below the cylinder is displacedand flows around the cylinder to the opposite side. This ap-pears to be the primary process occurring in the generation ofthe turbulent patch, and it is also consistent with our obser-vation that in quasisteady state the oscillatory turbulence isout of phase with the cylinder. We consider the area in thex−z plane, A, of fluid that is displaced by the motion of thecylinder with amplitude Ac during a quarter cycle. A directcalculation of this area through integration yields

A = AcRc1 −Ac

2

4Rc2 + Rc

2 − 2 sin−1�1 −Ac

2

4Rc2�� .

�11�

Assuming that Ac�Rc, which is the case for our experi-ments, this expression may be approximated by A�AcRc. Toestimate the characteristic speed of the displaced fluid, weassume that it travels a vertical distance of 2Rc in a timescaleof �c

−1, so that v�2�cRc. From these calculations, we findthat the rate of kinetic energy input per unit length along thecylinder for the displaced fluid is approximately �0�AcRc���2�cRc�2�c. To obtain the input rate of energy per unitmass, we divide this quantity by the fluid density and thearea of the region of energy input, which to leading order isRc

2. Thus, we estimate

� ��AcRc��2�cRc�2�c

Rc2 � AcRc�c

3, �12�

and so from Eq. �10�,

LO � AcRc��c

N�3/2

. �13�

In Fig. 7�b�, �k���−1 is plotted as a function of LO. We do

not expect perfect collapse of the data because of the com-plicated structure of the fluid motion surrounding the cylin-der. However, Fig. 7�b� demonstrates that a clearer trend

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)

FIG. 7. The inverse of the cross-beam wavenumber as a function of �a� Rc and �b� the Ozmidov scale LO. A typical horizontal error bar is shown in the upperleft-hand corner of �b�. In the legend, the direction of the arrows corresponds to the direction of propagation of the primary beam. Symbols are designatedaccording to the relative forcing frequency for each experiment.

076601-8 H. A. Clark and B. R. Sutherland Phys. Fluids 22, 076601 �2010�

emerges through a comparison between the wavelengthscaleand a length scale of turbulence. In particular, the separationof the data according to frequency, as mentioned above forFig. 7�a�, is reduced in Fig. 7�b�. Here we note that the val-ues of LO that we calculate based on Eq. �13� are betweenapproximately 1 and 2 cm. Although LO is crudely estimated,the relationship between �k�

��−1 and LO is of order of 1, whichmakes a direct correspondence between these quantitiesmore physically plausible.

The estimated values of LO are consistent with the ver-tical extent of turbulent patches surrounding the cylinder, asshown in Fig. 8. This shows an unprocessed syntheticschlieren image at a time approximately four wave periodsfrom the start of the cylinder oscillation, which correspondsto the lower bound of the time interval for the wavenumberanalysis, as described in Sec. III C. Fully three-dimensionaldisturbances in a stratified fluid scatter light passing throughthe tank from the background image of black and white lines.We identify the resulting region where the image is blurredas being turbulent. Figure 8 shows that the turbulent bound-ary layer above the cylinder is of comparable extent to theOzmidov scale estimated using Eq. �13�. This result is typi-cal of all the large-amplitude cylinder oscillation experi-ments we have performed. Outside the turbulent boundarylayer, the image of black and white lines is distorted but notblurred, indicating the presence of spanwise-coherent inter-nal waves emanating from the oscillatory turbulent patch.

Thus it is reasonable to suppose that LO gives a measureof the largest length scales of the turbulent patch that acts asa source for the internal waves. Some of the scatter in thedata shown in Fig. 7 may be attributed to the value of k�

� .Since we have retained only the peak wavenumber in our

analysis, the results for each experiment may be affecteddifferently by this simplification depending on the true wave-number distribution.

C. Wave amplitudes

Although we measured the amplitude ANt2 directly, we

may use the Boussinesq polarization relations to find theamplitude of the vertical displacement of the waves, givenby

A� =ANt

2

N03kx sin �

=ANt

2

N03k�

� cos � sin �. �14�

We normalize A� by the horizontal wavelength, �x=2 /kx, toprovide a more physically meaningful interpretation of thedata. The final values shown in Fig. 9 were calculated as

A�

�x=

ANt2

2N03 sin �

, �15�

which have been plotted as a function of Ac /LO. As shown inFig. 9, there is a clear separation in the ratio A� /�x for NaCland NaI experiments, with an approximately constant ratio of�2.5% for all NaI experiments. In previous work9,10 withNaCl stratifications it was observed across experiments thatA� /�x collapsed to a value in the range of 2%–4%, regardlessof the forcing amplitude. We do not have a clear explanationfor the observed increase in the ratio for NaCl experiments.The effect of the smaller value of N0 for NaCl stratificationsis magnified by the cubic power of N0 in Eq. �14�. However,there is no obvious physical reason to anticipate that thisdifference between the amplitude ratios should be based onstratification alone. We have estimated the Reynolds numberbased on the cylinder diameter, oscillation amplitude, fre-quency, and a characteristic kinematic viscosity of the fluid.Viscosity measurements were performed using an AntonPaar DMA 500 density meter for solutions of NaI with den-sities between fresh water and approximately 1.6 g /cm3. Forthe experiments of focus here, we estimate Reynolds num-

−12 −9 −6 −3x [cm]

−5

0

5

z[c

m]

LO

t � 4Tigw

0−10

10

FIG. 8. Snapshot of the unprocessed synthetic schlieren image in the near-cylinder region for an experiment with N0=1.9 rad /s, �c=1.53 rad /s,Rc=2.98 cm, and Ac=2.0 cm. The coordinate origin corresponds to thecenter of the cylinder. Fully three-dimensional turbulence surrounding thecylinder blurs the image of black and white lines behind the tank. Theestimated Ozmidov scale, LO�1.76 cm, is shown in relation to the turbu-lent layer at 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

FIG. 9. Vertical displacement amplitude A� normalized by horizontal wave-length �x shown as a function of normalized cylinder amplitude. Upward�downward�-pointing arrows denote measurements of upward �downward�-propagating waves.

076601-9 Generation, propagation, and breaking Phys. Fluids 22, 076601 �2010�

bers of approximately 1500–2000 for both types of stratifi-cation. Thus, viscosity differences between solutions of NaIand NaCl do not account for the results shown in Fig. 9. Theseparation of the results for NaCl and NaI may indicate thatthe wave behavior is outside the applicability of the polar-ization relations, which were derived under the assumptionsof linear theory. There also exists the possibility that a quan-tity other than A� /�x, with a different dependence on N0,remains relatively constant despite changes in the forcingamplitude. We have presented the results as shown to allowfor comparison within the context of previous studies.

For all calculations in the analysis of the wave genera-tion, we have assumed that the fluid can be treated as Bouss-inesq because the extent of vertical propagation is small incomparison with the density scale height. As shown in Fig. 9,the vertical error bars are sufficiently large as to produce aregion of overlap between the data for upward- anddownward-propagating waves. If trends in the data are sig-nificant, the slightly reduced amplitudes for downward-propagating cases may indicate that the region of observationis at the threshold of non-Boussinesq asymmetry between thedirections of vertical propagation. However, we cannot con-clude that the data shown contain evidence of non-Boussinesq wave behavior.

D. Wave power

We use the measurements of the characteristic wave-number and amplitude, as described in the previous subsec-tions, to calculate the average energy flux of the primarybeam, and hence the wave power. The present analysis isrestricted to experiments for which k�

� /k�� �0.1 and

ANt2 /ANt

2 �0.2 simultaneously, so that the values used forthe calculation of 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 k�:

�FE� =1

2�0

ANt2

2

N03 cos � sin �k�

3 . �16�

The use of this expression in our analysis requires thatmodes other than k� do not contribute significantly to the Nt

2

field. Based on the breadth of the wavenumber spectra, wehave modified the above expression to account for the con-tributions 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

2�0

1

N03 cos � sin �

�n

An2

kn3 . �17�

The images used in the present analysis were processedusing the same procedure as for the wavenumber analysis, asdescribed in Sec. III C. Signal attenuation with increasing �and a geometric effect caused by the image rotation resultedin a region of zero amplitude for the largest values of �.Based on the properties of the FFT algorithm that we haveemployed, we account for this in our analysis by scaling theamplitude of a mode according to the ratio L� /L, where L� is

a measure of the beam width and L is the length of the spatialdomain for the Fourier transform. In this case the amplitudeAn that one would obtain from the transform is related to theamplitude of the real signal, A, through

An = �L�

L�A . �18�

We find that the squared amplitude is given in terms of thepower by

An2 = � 2

L�/L�2

Pn, �19�

in which Pn represents the squared magnitude of the Fouriercoefficient of the nth mode of the Nt

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

tiply the vertical energy flux by the area of a horizontal crosssection through the beam, LcLx=LcL� /cos �, in which Lc isthe length of the cylinder. Through this step and the substi-tution of Eq. �19� into Eq. �17�, we arrive at the expressionfor the total measured power of the experimental beam,

Pexpt =2�0LcL

2

N03 cos2 � sin �L�

�n

Pn

kn3 . �20�

In order to use the above expression we also require a quan-titative method of determining the beam width L�. We ex-press L� in terms of a multiple of a characteristic wavelength�� as

L� = ��� =2�

k�� . �21�

Substituting this expression into Eq. �18� with An→A� cor-responding to the amplitude of the waves with kn→k�

� , weobtain

� =A�

A� k�

�L

� =

1

�

P�

A� k�

�L

�2

. �22�

Thus, we find

� =P�

1/4

A� k�

�L

� . �23�

For our estimate of �, we take the power of the mode forwhich kn is closest to k�

� , and we use the characteristic am-plitude ANt

2 as described in Sec. III D. This yields values of�� �1.6,2.2� across all experiments. Comparison with thestructure of the original Nt

2 profiles in the � direction showsthat this range of � is reasonable when we consider that �characterizes the number of wavelengths contained in theprimary beam.

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

076601-10 H. A. Clark and B. R. Sutherland Phys. Fluids 22, 076601 �2010�

Pthy =

8�0� N0

2

�igw2 − 1�1/2

�igwRc2�Ac�c�2Lc. �24�

The use of both �igw and �c in the calculation of Pthy arisesfrom the conversion of different parameters in Eq. �6� to thevariables in our notation. The product of Ac�c is the magni-tude of the maximum velocity of the oscillating cylinder,whereas the single power of �igw is a result of our distinctionbetween the properties of the waves and the cylinder. Wehave expressed Pthy in terms of the wave frequency ratherthan the angle from the vertical because frequency wasthe directly measured quantity. For calculations of bothPexpt and Pthy we have used characteristic densities of�0=1.25 g /cm3 for NaI stratifications and �0=1.1 g /cm3

for NaCl stratifications. These values were estimated fromthe experimental measurements of the density at the verticallevel of our analysis of wavenumber and amplitude.

Figure 10 shows a plot of the experimentally measuredpower versus the theoretical prediction. The large verticalerror bars on the data points are a result of adding significantcontributions from several of the variables in Eq. �20�.Namely, the largest contributions to the final error were dueto the uncertainties in �, L�, and Pn, the last of which wasestimated from the standard deviation of the time-averagedspectrum.

The experimental measurements and theoretical predic-tions are similar for small values of Pthy, but the two valuesdeviate more with increasing forcing intensity. In general, weexpect the experimental values to be less than the theoreti-cally predicted power because the coupling of the cylinder tothe internal waves is affected by the development of the tur-bulent boundary layer. This behavior is observed for all ex-periments, but the low values of Pexpt for large Pthy alsosuggest that we may be observing a saturation of the wavefield. Where the theory predicts an increasing rate of energytransport by waves, we hypothesize that much of the forcingenergy is lost to turbulent kinetic energy in the boundingregion of the cylinder. Here we may use the expression for �,Eq. �12�, to estimate the total turbulent dissipation rate. Theproduct of �0� with the characteristic volume of displaced

fluid, �AcRcLc, yields an approximate dissipation rate of�0�AcRc�2Lc�c

3. Using characteristic experimental values, weobtain an estimate of �2000 erg /s. Therefore, it is reason-able to observe smaller wave powers than predicted bytheory. In particular, the estimated dissipation rate is compa-rable to the discrepancy between the values of Pthy and Pexpt

as the forcing intensity is increased.

V. WAVE INSTABILITIES AND BREAKING

A. In situ probe measurements

Synthetic schlieren provides a means of measuring quan-titatively the structure and amplitude of internal waves inspace and time. In the previous sections we have demon-strated the application of synthetic schlieren to experimentsin strongly stratified fluids. We have used a separate quanti-tative technique for one characteristic experiment, in whichmeasurements were made of the waves at a fixed location asthey evolved in time. This provides an independent meansthrough which we may observe the establishment of thewave field and its subsequent breakdown. Additional infor-mation is gained through this technique since quantitativesynthetic schlieren measurements are not possible as thetransition to instability occurs, as discussed in Sec. III E.

A conductivity probe was used to perform a verticaltraverse of the background stratification in the region of in-terest, with approximately 42 measurements of voltage V pervertical centimeter. The probe was then placed at fixed loca-tions in space for a series of three experiments in which acylinder with Rc=4.43 cm and Ac=2.0 cm was oscillatingwith frequency �c=1.96 rad /s approximately 10 cm abovethe bottom of the tank. The probe provided measurements ofthe voltage with a resolution in time of �t=0.05 s. The hori-zontal location of the probe was approximately 30 cm fromthe center of the cylinder while the vertical coordinate wasset at 45, 40, and 35 cm successively above the bottom of thetank.

The time series measurements of voltage were translatedinto densities using the function �̄�V� obtained from a linearfit to four discrete measurements of voltage for densities inthe range of �0.998,1.30� g /cm3. With waves in the fluid,the perturbation density at each vertical level was computedby subtracting the initial background density. The wave ver-tical displacement � was then calculated according to

� = −d�̄

dz� , �25�

where � is the perturbation density and the background den-sity gradient at each of the three vertical levels was foundthrough

d�̄

dz=

dV

dt

d�̄

dV

dt

dz. �26�

Time series of the vertical displacement are shown inFig. 11 for each vertical level. The motor driving the oscil-lations of the cylinder was turned on at t=30 s for Fig. 11�a�and t=20 s for Figs. 11�b� and 11�c� and was turned off att�150 s for each experiment. In all cases, we observe at

200

400

600

800

1000

Pex

pt

[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)

FIG. 10. Experimentally measured power vs theoretically predicted powerof the primary beam. 1 erg=1 g cm2 s−2. Note that the horizontal and ver-tical scales differ significantly.

076601-11 Generation, propagation, and breaking Phys. Fluids 22, 076601 �2010�

first a regular, periodic signal in time, with growth in ampli-tude as the initial transient reaches and passes the location ofthe probe. Based on the background stratification and theforcing frequency, we expect close agreement between thecylinder and wave frequencies because �c /N0� �0.5,0.8�, asdiscussed in Sec. IV A. A closer examination of the earlytimes in Fig. 11 yields a wave period that is similar to thecylinder period of Tc=2 /�c=3.2 s. For each plot the ver-tical axis is the same to allow for direct comparison of thewave amplitudes at varying location in the vertical. For anexperiment with the same forcing parameters as the experi-ment under discussion, the estimated vertical displacementamplitude �as described in Sec. IV C� was A�=0.5 cm. Al-though the spatial locations of the measurements differ, theconditions should be the most similar to those at z=35 cm inthe present analysis. The appropriate time for comparison ofthe wave amplitudes is approximately 4 wave periods afterthe oscillations of the cylinder began, which is t�33 s.While we do not expect exact agreement between the mea-surements because of the different techniques of instanta-neous measurement and averaging over space and time, themagnitude of the vertical displacement is similar to the valueof 0.5 cm obtained using synthetic schlieren. As shown inFig. 11, after a finite time the signal becomes irregular andthe wave amplitude varies erratically. We consider thischange in the regularity of the signal the signature of theonset of instability. The occurrence of large wave amplitudesor irregularity 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 Sec. III E.

B. Qualitative analysis of synthetic schlieren

As discussed in previous sections, we observed thebreakdown of propagating beams through both quantitativeand qualitative techniques. Here we provide the results offurther investigations of the wave breakdown and the asso-ciated instability mechanism. First we note that the measurednormalized vertical displacement amplitudes A� /�x, as pre-sented in Sec. IV 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 Sec. III E, yielded severalsignificant results. Instability was not observed in every ex-periment; in some cases, the breakdown criterion was notsatisfied at any time that the cylinder was oscillating. It wascommon to these experiments that the forcing frequency wasvery near to or above the background buoyancy frequency, orthe amplitude of oscillation was the smallest of our param-eter range. This result for high-frequency forcing is consis-tent with the quantitative synthetic schlieren measurements,in which the wave signal was weaker and less coherent thanfor midrange forcing frequencies. Thus, we should not nec-essarily expect significant growth and transition to instabilityfor the waves generated by high-frequency forcing. The oc-currence of wave breakdown for both upward- anddownward-propagating beams confirms that non-Boussinesqgrowth of upward-propagating waves is not responsible forthe behavior. Another trend in the observations is that thetime of wave breakdown varied significantly across experi-ments. For the same cylinder radius and amplitude, a changein the forcing frequency yielded an opposite change in theobserved time of breakdown, i.e., a decrease �increase� infrequency resulted in a later �earlier� breakdown. This is aphysically reasonable consequence of the change in the time-scale for beam development. The effects of cylinder radiusand amplitude 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 Fig. 12, that could lead to the break-down of waves as observed. We expect that nonlinear effectsare the most significant in regions of beam self-interaction,such as when the primary beam reflects off of the surface, orin a beam-beam interaction that could arise through multiplereflections off of the tank walls and the fluid surface. Thesesituations are depicted in Figs. 12�a� and 12�b�, respectively.We also consider the possibility of the breakdown of a freelypropagating, noninterfering primary beam, as shown in Fig.12�c�. Breakdown in a region of beam reflection is the moststraightforward to identify from observations because of theproximity to the surface or the bottom of the tank. In the caseof beam-beam interactions at mid-depth, we expect that thelocation of the breakdown would vary significantly accord-ing to the wave frequency and the corresponding angle of

−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]

(c)

0 180−1.5

1.5

FIG. 11. Time series of the vertical displacement measured atz=45,40,35 cm in �a�, �b�, and �c�, respectively, with the same cylinder andforcing parameters in each case.

076601-12 H. A. Clark and B. R. Sutherland Phys. Fluids 22, 076601 �2010�

propagation. Through geometrical considerations, the regionof interference would move farther from the position of thecylinder with decreasing 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 Fig. 13. The experiments have been separatedinto upward- and downward-propagating beams in Figs.13�a� and 13�b�, respectively, with further distinctions madeaccording 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 of�c /N0 according to Eq. �2�, by which we can place approxi-mate bounds 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 fre-quency range that was used for quantitative analyses. In Fig.13, a marker indicates the location of wave breakdown foreach experiment. We observe that as a group, the markers aredisplaced somewhat upward in Fig. 13�a� and downward inFig. 13�b� relative to a line through the center of the cylinder.This is consistent with our observations from quantitativesynthetic schlieren that the primary beam was not centeredabout the equilibrium position of the cylinder. Such an effectis evident in Fig. 4�a�, in which the rotated beam is displacedfrom a line through the center of the cylinder. Therefore, thelocations of the markers agree well with our expectation thatthe path of the primary beam emanates from a turbulentpatch above and below the cylinder. The horizontal coordi-nates of breakdown were clustered within approximately20 cm from the center of the cylinder, and there were noapparent trends in this location based on forcing parameters.The initial wave breakdown occurred at a vertical coordinate

that does not correspond to a region of surface or bottomwave reflection, nor does the horizontal distance change sig-nificantly with frequency as it would in the case of interact-ing beams illustrated in Fig. 12. As an additional qualitativedemonstration, a larger view of the synthetic schlieren back-ground image during a breakdown event is shown in Fig. 14.The experiment is the same as that shown in Fig. 5, with theexpanded view corresponding to the image in Fig. 5�b�. Theimage in Fig. 14 is typical of the observations across experi-ments with different forcing parameters and stratifications.Significant distortions and blurring of the image occur closeto the cylinder, as discussed in previous sections, and thewave breakdown is evident at �x ,z���−11 cm,28 cm�. Thevertical lines at x�−15,−36 cm are due to junctions in thebackground image and they do not influence the analysis orthe interpretation of results. From Fig. 14 we note that vis-ible distortions of the background image outside of the tur-bulent patch are confined to the locations of the primary andreflected wave beams as determined from quantitative syn-thetic schlieren analysis at earlier times in the experiment. Ifpresent, reflected beams from the right side of the tank wouldresult in disturbances in the upper right-hand corner of theimage. Although the absence of a visually detectable signaldoes not imply the absence of small-amplitude waves, thisexample demonstrates that reflected waves from the side-walls or free surface were not apparent at the time and loca-tion of breakdown. Considering all of the observations dis-cussed above, we conclude that beam-beam interactions arenot responsible for the breakdown. Rather, the waves con-tained in a single beam undergo a transition to instabilityindependently of interactions with boundaries or otherbeams. A candidate mechanism for the breakdown of an iso-

(a) (b) (c)

FIG. 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. The domain and the cylinder size and position are shown to scale.

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)

FIG. 13. Schematic of experimental apparatus �to scale� with a marker denoting the location of wave breakdown for each experiment as indicated in thelegend. Results for upward- and downward-propagating primary beams are shown in �a� and �b�, respectively.

076601-13 Generation, propagation, and breaking Phys. Fluids 22, 076601 �2010�

lated beam is PSI, whereby energy is transferred to waves oflower frequency and higher wavenumber than the primarydisturbance.27,28,24 This hypothesis provides the motivationfor the numerical simulations described in Sec. V C.

C. Numerical simulations

A two-dimensional, fully nonlinear, Boussinesq code18

was used to simulate the evolution of an internal wave beam.Previous work18,29 has focused on plane waves and spatiallylocalized wavepackets, often with uniform structure in thealong-stream direction. The simulations presented here arenot an attempt to model accurately all of the characteristicsof the oscillating cylinder experiments. Our objective in per-forming simulations was to investigate potential instabilitiesin an established beam of finite width. For this study thespatial domain was horizontally and vertically periodic witha resolution of 128 by 512 points in the x and z directions,respectively. The code uses finite differencing in the verticalwith periodic upper and lower boundary conditions and wasrun with 64 spectral modes in the horizontal direction. Whilea horizontally plane wave structure can be resolved with farfewer modes, the finite beam width in the current studymeans that increased horizontal resolution was required foradequate sampling across the signal.

We have initialized the simulations with a perturbationin the form of plane wave structure in the cross-beam ���direction with a Gaussian envelope to determine the beamwidth. In order to satisfy the doubly periodic boundary con-ditions, the full disturbance consisted of a superposition ofthree identical beams separated by a fixed distance. Thebeams decayed sufficiently rapidly with � to prevent an in-crease in amplitude of the neighboring beams due to super-position. Given a domain x� �0,L�, z� �0,H�, the beamseparation was given by

�s =LH

L2 + H2. �27�

This distance guaranteed the periodicity of the structure bypositioning the center line of the two secondary beams at theappropriate corners of the domain. The center line of theprimary beam was from corner to corner of the domain, re-gardless of the dimensions L and H. These parameters deter-mined the angle of the beam to the vertical direction, andhence the frequency �igw, through the relations

� = tan−1� L

H� = cos−1��igw

N0� . �28�

The maximum amplitude of the perturbation, cross-beamwavenumber, and standard deviation of the Gaussian enve-lope, denoted by a0, k�, and �0, respectively, are freeparameters. The initial structure of the streamfunction is thengiven by

��,t = 0� = a0�exp− �2

2�02 �cos�k���

+ exp− �� − �s�2

2�02 �cos�k��� − �s��

+ exp− �� + �s�2

2�02 �cos�k��� + �s��� . �29�

For each spatial coordinate pair �x ,z� in the domain, atransformation to the � coordinate was performed using Eq.�3�. The value of was calculated for the resulting value of� according to Eq. �29� and was then assigned at the originalgrid point. Small-amplitude randomly generated noise wasalso superimposed on the field over the entire domain to seedany physical instabilities evenly.

For a beam in the first quadrant, the vertical componentof the group velocity is positive. Therefore, in order to obtainthe correct signs of the horizontal and vertical wavenumberskx and kz, they were calculated as

kx = �k��cos �, kz = − �k��sin � , �30�

where k��0.The parameters that determine the flow were chosen to

model the experimental conditions. In all cases, the back-ground velocity was zero and N0 was the value determinedfrom the density profile, as described in Sec. III. For a givenexperiment, the characteristic cross-beam wavenumber k�

� ,frequency �igw, and vertical displacement amplitude A� wereknown. The initial streamfunction amplitude was determinedthrough polarization relations as

a0 =�igw

kxA� =

�igw

�k�� �cos �

A�, �31�

such that a0�0. The value of A� was determined similarlyusing a polarization relation, so the final value of a0 shouldbe considered an estimate due to the propagation of uncer-tainties in our experimental measurements. We also have an

−30 −20 −10x [cm]

−10

0

10

20

30

z[c

m]

−40 0

FIG. 14. Snapshot of an experiment at the time of wave breakdown.�x ,z�= �0,0� corresponds to the equilibrium position of the cylinder. The topand bottom of the image are located approximately at the free surface andtank bottom, respectively. Experimental parameters are the same as thosegiven for Fig. 5.

076601-14 H. A. Clark and B. R. Sutherland Phys. Fluids 22, 076601 �2010�

approximate measure of the beam width from experiments interms of the parameter �, given by Eq. �23�. For the numeri-cal beam initialization, we have attempted to obtain approxi-mately two wavelengths across the width of the beam forconsistency with observations and the measurements of �.Contours of the vorticity field, �, are shown at initializationin Fig. 15�a�. Note that the extrema of the contour range arethe same for each panel.

Although experimental parameters are used to initializethe simulations, the plots in Fig. 15 are cast in nondimen-sional form, emphasizing, for example, that the resultsshould scale in time as N−1.

For all simulations that were initialized using parameterscomparable to experimental conditions, an instability devel-oped along the central beam after an initial period of regularpropagation of phase lines through the beam at a constantangle. The onset of the instability occurred along the centerline of the beam, where the initial amplitude was largest, andthe transition appeared visually to occur along the entirelength of the beam simultaneously, as shown in Fig. 15�b�.Therefore, we have confidence that the instability is physicaland is not caused by boundary effects in the numerical for-mulation. Within the beam structure, waves began to developat a larger angle to the vertical direction, and hence a lowerfrequency, than the initial disturbance. A cascade of energyto smaller scales was also observed. As in other studies ofPSI for internal waves,24 we find that the disturbance whichgrows fastest from the background noise is that with fre-quency half that of the wavebeam, as illustrated in Fig. 15�c�.The phase lines of the waves including the developed insta-bility align well with the expected direction for the subhar-monic, thereby supporting the conclusion that PSI was theprimary mechanism for the breakdown of the wave beams inthe numerical context. In general, the instability grew in am-plitude until overturning began to occur, after which thesimulations broke down rapidly. For waves with smaller ini-tial amplitudes, the simulations ran for the full time of thecorresponding 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 McEwan,30 who noted that densitymicrostructure became evident surprisingly rapidly in re-gions outside of breaking due to PSI.

The use of experimentally realistic parameters to initial-ize the simulations facilitates a comparison between the ob-served time of PSI onset in the simulations and the experi-mentally observed time of significant visual distortions of theschlieren image, as described in Sec. V B. Although we can-not specify what physical effect was occurring at the time ofobservation, the hypothesis that PSI arose in the experimentsmay be supported or refuted through an order-of-magnitudecomparison with the results of the simulations. Runs wereperformed with parameters modeling five characteristic ex-periments in two different stratifications with a range of forc-ing frequencies. We have found that the estimated time of theonset of PSI in the simulations differs from the experimentalbreakdown time by a maximum of a factor of 2. This isreasonable agreement if one considers that the simulationsserve to model approximately some of the characteristics ofthe experimentally measured waves. Using the numerics, wehave verified that instabilities became evident in a physicallyreasonable timescale given physically realistic input param-eters. There were cases in which the experimental time wasapproximately 30% less than the time from simulations andvice versa, so no systematic pattern emerged for these par-ticular runs. The similarity of the experimental and numeri-cal timescales for the development of instabilities serves assupport for the hypothesis that PSI was the cause for break-down of the beams 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 measurements ofwave frequency, wavenumber, and amplitude were made us-ing a generalized form of synthetic schlieren that takes intoaccount the full vertical profile of the density and index ofrefraction for fluids with significant density variations withheight. The large-amplitude forcing produced a turbulentboundary layer around the cylinder that modified the genera-

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

FIG. 15. Contours of the magnitude of the vorticity, ��s−1�, at �a� initialization, �b� approximately the onset time of the instability, and �c� a late time in theevolution of the wave field. The angles /2−�= /2−cos−1��igw /N0� and /2−��= /2−cos−1��igw /2N0� are shown in �c�.

076601-15 Generation, propagation, and breaking Phys. Fluids 22, 076601 �2010�

tion region and the resulting characteristics of the wavebeams.

It was found that the wave frequency was equal to theforcing frequency only for the interval �c /N0�0.5–0.8. Forsmall forcing frequencies, beams were generated at higherharmonics and with larger associated power than the primarybeam. With forcing frequencies above N0, the waves gener-ated by the localized turbulent patch had frequencies of ap-proximately 0.5N0. In all cases, the waves with the maximumassociated power were observed in the range �igw /N0

�0.5–0.8 regardless of the forcing frequency. These resultsindicate a preferred frequency range that is in agreementwith previous work on the turbulent generation of internalwaves.9,10,12 Wave frequency selection in the typical rangeoccurred despite a dominant frequency component due toforcing by the cylinder, of the turbulent source. Coherentquasimonochromatic beam structures were observed emanat-ing directly from the source without an intermediate spatialregion of waves with a broad frequency distribution. Thus,differential viscous decay does not account for the observa-tions of frequency selection in this study.

The Ozmidov scale, which characterizes the verticalscale of the eddies in stratified turbulence, was found to bemore predictive of the length scale of the waves than thecylinder radius alone. Also as a consequence of the turbulentgeneration mechanism, the wave amplitudes were found tobe an approximately constant fraction of the horizontalwavelength, which has been noted in previous experimentalstudies.10 However, the magnitude of this ratio differed forexperiments in NaCl and NaI stratifications. This observationis currently unexplained and requires further investigation.

Qualitative observations from unprocessed videos wereused to characterize the time and location of wave break-down in the experiments. With the motivation of examiningpotential instabilities for a temporally monochromatic wavebeam, fully nonlinear numerical simulations were performedwith experimentally realistic input parameters. The results ofthe simulations showed the development of PSI at timescomparable to the observed values for the experiments.Based on this outcome, we conclude that PSI of the isolatedprimary beam was responsible for the breakdown of thewaves in the experiments at relatively late times.

ACKNOWLEDGMENTS

The laboratory experiments were performed in the labo-ratory 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. This workhas been supported by the Natural Sciences and EngineeringResearch Council �NSERC�, the Canadian Foundation forClimate and Atmospheric Sciences �CFCAS�, and an AlbertaIngenuity Fund Graduate Scholarship.

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