JOURNAL OF GEOPHYSICAL RESEARCH, VOL. ???, XXXX, DOI:10.1029/,

Internal Waves in Laboratory Experiments

B. R. Sutherland1, T. Dauxois2 T. Peacock3

Abstract. Recent methods developed to generate and to analyze internal waves in lab-oratory experiments have revealed important insights that sometimes guide and some-times challenge theoretical predictions and the results of numerical simulations. Here wefocus upon observations of internal waves generated by oscillatory or uniform flow overa solid boundary in a continuously stratified fluid. In the laboratory such flows are of-ten examined by moving the boundary in an otherwise stationary ambient. This can bedone by oscillating or towing an object, or by sequentially oscillating a series of platesto simulate a translating or standing-wave-like boundary. The generated waves are some-times visualized and measured using a non-intrusive optical method called synthetic schlieren.This takes advantage of the changes of the refractive index with salinity which differ-entially bends light passing through salt-stratified fluid, a property that hampers the useof of more standard non-intrusive observation methods such as particle image velocime-try. From schlieren images, filtering methods can be applied to distinguish between left-ward and rightward as well as upward and downward propagating two-dimensional waves.Inverse tomographic techniques can be employed to measure the structure and ampli-tude of axisymmetric and relatively simple three-dimensional disturbances.

1. Introduction

Since the realization by physical oceanographers thattransport and mixing by internal waves are an importantcomponent of the thermohaline circulation, there has beena resurgence in interest in their dynamics [Polzin et al.,1997; Munk and Wunsch, 1998; Ledwell et al., 2000]. Con-sequent studies have been designed to examine mechanismsfor wave generation and interaction with topography. Rela-tively early theoretical studies examined the means by whichenergy from the moon forcing the barotropic tide might beconverted into internal wave energy as a consequence of os-cillatory stratified flow over topography [Balmforth et al.,2002; Smith and Young , 2002; Legg , 2004]. In essence, thiswork extended the results of previous theoretical and labo-ratory research into internal waves generated by oscillatingcylinders [Mowbray and Rarity , 1967; Thomas and Steven-son, 1972; Voisin, 1991, 1994; Hurley , 1997; Hurley andKeady , 1997]. More recently, attention has turned to fastertime-scale processes in which large amplitude internal wavepackets are generated during one cycle of the tide. Thiswork extends earlier studies of steady uniformly stratifiedflow over topography (e.g. see [Baines, 1995]) to includeconsideration of non-uniform stratification and large ampli-tude topography. As such the primary interest has beenupon the generation, propagation and dissipation of inter-nal solitary waves [Pinkel , 2000; Klymak and Gregg , 2004;Klymak et al., 2006; Li and Farmer , 2011; Alford et al.,2011].

In general, laboratory experiments have entered a re-naissance thanks to digitization technology, advancement inlasers and computer-controlled equipment, and due to in-creases in computational memory and speed, which have

1Departments of Physics and of Earth & AtmosphericSciences, University of Alberta, Edmonton, Alberta, Canada.

2Laboratoire de Physique, Ecole Normale Superieure,Lyon, France.

3Department of Mechanical Engineering, MassachusettsInstitute of Technology, Cambridge, MA, USA

Copyright 2012 by the American Geophysical Union.0148-0227/12/$9.00

created valuable new analysis tools such as particle imagevelocimetry (PIV) and laser-induced fluorescence (LIF). Asa result, it is now possible to make non-intrusive measure-ments of velocity and concentration in two and even threedimensions. These tools have provided new insights intoproblems involving turbulence and mixing that remain achallenge in computational fluid dynamics.

However, the study of stratified fluids remains an exper-imental challenge because light typically refracts differentlythrough fluids of varying density. This can distort and smearthe apparent positions of particles used in PIV and so leadto spurious predictions of flow speeds. On the other hand,the very fact that density and refractive index are relatedhas provided other means to examine non-intrusively thestructure of stratified fluid flow.

One visualization method used in laboratory experimentsof salt-stratified fluids is called shadowgraph. In this, a lightsource placed far behind the test section shines through thestratified fluid landing upon a translucent surface, like My-lar. At interfaces where the density rapidly changes the lightfocuses and defocuses as it bends relatively more or less whilepassing through fluid of varying salinity and, hence, varyingrefractive index. If density variations due to internal wavesare gradual, focusing may not be evident. Shadowgraphproves particularly useful in the examination of approxi-mately two-layer fluids in which case light focusing at theinterface can be used to track the motion of interfacial waves.For internal waves in uniformly stratified fluid, shadowgraphis particularly effective in the examination of internal wavesthat are close to breaking, as shown in Figure 1. In this ex-periment [Koop and McGee, 1986], sinusoidal topography istowed leftward beneath a shear flow whose speed increasesleftward with height. At mid-depth in the experiments thewaves encounter a critical level, where the background flowspeed is close to the towing speed of the hills.

Another method taking advantage of the relationshipbetween refractive index and density is called “schlieren”[Schardin, 1942; Settles, 2001]. In the traditional approach,light reflected from a parabolic mirror passes through a testsection before striking a second parabolic mirror that refo-cuses the light. A knife-edge at the focus acts as a filter onspurious signals thus revealing index-of-refraction dependentstructures within the test section.

Mowbray and Rarity [1967] were the first to use tradi-tional schlieren methods to visualize internal waves gener-ated by a cylinder oscillating at a fixed frequency, ω. Pro-vided ω was sufficiently small, they observed that the waves

1

X - 2 SUTHERLAND ET AL.: INTERNAL WAVES

Figure 1. Internal wave breaking near a critical layeras visualized by shadowgraph. Extracted from Figure 7of Koop and McGee [1986].

Figure 2. Pattern of internal waves generated by an os-cillating cylinder as visualized by conventional schlierenmethods. Reproduced from Plate 1(6) of Mowbray andRarity [1967].

emanated vertically and horizontally from the cylinder in across pattern, as shown in Figure 2.

In this experiment, the fluid was a uniformly stratifiedsalt solution whose density decreased linearly with height.The stratification can be represented by the buoyancy fre-quency, N , defined in the Boussinesq approximation by

N2 = −g

ρ0

dρ

dz. (1)

Here ρ(z) is the ambient density, ρ0 is the characteristic den-sity (e.g. that for fresh water at room temperature) and gis the acceleration of gravity. In agreement with predicteddispersion relation of internal waves, Mowbray and Rarity[1967] found that the arms of the cross-pattern of wavesformed a fixed angle, Θ, from the vertical which was relatedto the ratio of ω to N by |Θ| = cos(ω/N).

Colour filters have allowed schlieren to be more quantita-tive [Howes, 1984; Teoh et al., 1997]. But the expense and

Figure 3. a) Side-view looking through tank filled withsalt-stratified fluid with model inverted sinusoidal hills atthe surface and an image of horizontal black and whitelines placed behind the tank. b) Side-view after hills havebeen towed slowly a distance of one hill width. c) Qual-itative synthetic schlieren image produced by taking theabsolute value of the difference of the digitized imagesshown in a) and b).

physical constraints imposed by the need for well-alignedpairs of parabolic mirrors has limited the use of schlierenuntil recently.

Schlieren technology has advanced enormously since themid-1990s. As a result of digitization technology and com-puters, “synthetic schlieren” was developed as an inexpen-sive, versatile and - most importantly - quantitative toolfor sensitively measuring density perturbations in stratifiedfluids.

SUTHERLAND ET AL.: INTERNAL WAVES X - 3

In what follows we examine how synthetic schlieren hasbeen used to test theory and to develop new insights intothe dynamics of internal waves. In the process we reviewan analysis method for separating out waves propagating indifferent directions and describe a recently developed mech-anism for generating waves that does not suffer some of thedrawbacks of oscillating or towed rigid objects.

Section 2 briefly discusses how synthetic schlieren visu-alizes disturbances in a fluid through contrasting snapshotstaken by a digital camera looking through the fluid at ablack-and-white image of lines or dots. If the disturbancesare small, the displacements of objects in the image can becomputed and from these the magnitude of the disturbancecomputed. This is described in Section 3 with the assump-tion that the disturbance in the tank is uniform across theline-of-sight. The treatment of axisymmetric and fully threedimensional disturbances is described in Section 4. The useof PIV for measuring internal waves is described in Section 5,in which the use of schlieren to correct for particle displace-ments is also described. Future directions are described inSection 6.

2. Qualitative Use of Synthetic Schlieren

Synthetic schlieren [Dalziel et al., 2000] makes away withthe need for parabolic mirrors to straighten and refocus alocalized light source. Instead, a camera is focused upon animage behind a tank filled with salt-stratified fluid.1 Distur-bances in the fluid displace isopycnal surfaces and so locallychange the refractive index of the salt water through whichlight passes from a point on the image through the tank tothe camera. The image apparently distorts as a result.

For example, Figure 3 shows how qualitative syntheticschlieren observes distortions of an image of horizontal blackand white lines resulting when a set of model sinusoidal“hills” are towed from left-to-right over the surface of a tankfilled with uniformly stratified fluid.

In the initial image, shown in Figure 3a, the hills im-mersed in the ambient are apparent near the top of theframe. The black and white lines are not in the tank, how-ever. The image is situated approximately 10 cm behind thetank. After the hills are set in motion, various disturbancesin the ambient can be seen as a result of the distortion of theimage (Figure 3b). In the lee of each hill, boundary layerseparation results in large perturbations that warp and blurthe lines. Further below the hills, the eye can barely makeout smaller undulations of lines in the image.

These alterations can be enhanced through digital imageprocessing. Each snapshot can be represented as an array ofpixels with each pixel given a number corresponding to itsintensity (e.g. 0 for black, 1 for white, and in between forgray). The image in Figure 3c is produced by taking the dif-ference of the digitized snapshot in b) from that in a), thentaking the absolute value and multiplying the result by anenhancement factor, typically 10. Thus even small changesto the image become obvious.

One advantage of synthetic schlieren is that its sensitiv-ity can be increased by widening the distance between thetest-section and the image behind it. For example, it is easyto observe heat rising off one’s hand if the image is severalmeters away.

3. Spanwise-Uniform Disturbances

3.1. Quantitative Synthetic Schlieren

When light passes through a medium whose refractiveindex changes in space, it is deflected in a manner well-predicted by Snell’s Law. In the particular case of stably

stratified fluid in which the density, ρ, and hence refractiveindex, n, change with height z, the path of light passingin the y direction through the fluid at a small angle to thevertical from the y-axis is given by [Sutherland et al., 1999]

d2z

dy2≃

1

n0

∂n

∂z. (2)

Here n0 is the characteristic refractive index of the fluid (e.g.n0 = 1.3330).

In uniformly stratified fluid, the vertical gradient of therefractive index can be related to the vertical density gradi-ent:

∂n

∂z=

dn

dρ

∂ρT∂z

. (3)

Here ρT denotes the sum of the ambient density ρ(z) andthe perturbation density ρ(~x, t).

In computing image displacements, it is sometimes moreintuitive to compute them in terms of the squared buoy-ancy frequency rather than density gradients. The localstratification resulting from both the background and per-turbation density is expressed by the total squared buoyancyfrequency:

NT2 = −

g

ρ0

∂ρT∂z

. (4)

Thus vertical variations of the refractive index can be writ-ten in terms of NT by

∂n

∂z= −n0γNT

2, (5)

in which the coefficient γ is defined so that

γ ≡1

g

ρ0n0

dn

dρ≃ 1.878× 10−4 s2/cm. (6)

The rightmost empirical approximation assumes relativelyweak concentrations of sodium chloride solutions [Weast ,1981].

Combining equations (2) and (5), we find that light fol-lows a parabolic path when passing through a uniform strat-ification at a scant angle from the horizontal such that

z(y) = zi + y tanφi −1

2γNT

2y2, (7)

in which zi is the height and φi is the angle at which thelight ray enters the tank, In deriving (7), we have assumedthat NT is independent of y, which is the case for spanwise-uniform disturbances in a tank. This assumption will be re-laxed below in the consideration of axisymmetric and fullythree-dimensional disturbances.

Synthetic schlieren is usually employed to measure per-turbations to the ambient. Directly, it measures how thesquared buoyancy frequency changes as a result of the com-pression and stretching of isopycnal surfaces. Over a dis-tance y = LT , light is deflected vertically by

∆z = −1

2γ∆N2 LT

2, (8)

in which

∆N2 ≡ NT2 −N2 = −

g

ρ0

∂ρ

∂z(9)

is the change in the squared buoyancy due to the densityperturbation, ρ.

For example, if internal waves compress isopycnals so thatthe ambient N2 locally increases by 10% from 1.0 s−2 to

X - 4 SUTHERLAND ET AL.: INTERNAL WAVES

1.1 s−2, then the light deflects by 0.04mm crossing a 20 cmwide tank. This is a small but discernible amount for adigital camera with sufficiently high resolution.

The apparent deflection is larger if the image is placedsome distance behind the tank. Not only is the light de-flected downward if the stratification increases, but the an-gle of the light ray at the tank-wall changes. So the imagedisplacement magnifies linearly as the image is moved fur-ther away.

Assuming the tank walls are negligibly thin, one can pre-dict the total displacement of light from an object a distanceLo from one side of the tank to a camera on the other sideof the tank to be

∆z(∆N2) ≃ −1

2γ∆N2 LT

2 −n0

na

γ∆N2 LoLT , (10)

in which na is the refractive index of air and φ0 is the anglefrom the horizontal at which light enters the camera fromthe object.

In the example above, if we now suppose the image is20 cm behind the tank, then the displacement of the lightpath is 0.14mm - much more easily discernible. If one pixelof the camera has a vertical resolution of 0.5mm, then thedisturbance in the tank will shift the image by about a thirdof a pixel, which can easily be observed by the change of in-tensity of light emanating from the edge of a line.

Through (10), we have found the forward equation inwhich known changes to the stratification enables us to pre-dict the vertical displacement of a point in an object asseen by a camera looking through the stratified fluid. Inthe derivation of (7), because we have assumed the distur-bance is spanwise-uniform it is a trivial matter to invert (10)so that an observed vertical displacement in an image canpredict the change in the stratification:

∆N2 ≃ −∆z1

γ

[1

2LT

2 + LoLT

n0

na

]−1

. (11)

If part of an image appears to deflect downwards, it meansthat the stratification between it and the camera has locallybecome stronger.

Figure 4. Pattern of internal waves generated by anoscillating cylinder, in the upper left-hand corner, as vi-sualized by qualitative synthetic schlieren applied to apattern of equally spaced dots behind the tank. Adaptedfrom Figure 7c of Dalziel et al. [2000].

This result is straightforwardly applied to the circum-stance in which a camera looks through a tank at an imageof horizontal black and white lines, as is the case in Figure 3.Even if density perturbations result in image distortions thatshift the lines by a fraction of their width (which, in fact,is ideal if you wish easily to compute the line displacement)then (11) immediately predicts the change to the squaredbuoyancy frequency.

More processing is required if the line is displaced signifi-cantly. The calculations of displacements is even more diffi-cult if the lines become magnified or contracted because sec-ond gradients of the refractive index become significant (inwhich case shadowgraph becomes the more useful, if qual-itative, tool). The method breaks down entirely, as withinthe valleys of the model hills in Figure 3b, when the linesblur due to three-dimensional mixing.

The examination of an image of lines provides a relativelyeasy method to calculate small vertical displacements in animage and hence ∆N2. But a far more informative, thoughcomputationally more intensive, application uses an imageof dots [Dalziel et al., 2000]. For example, Figure 4 shows a

Figure 5. Quantitative synthetic schlieren applied tothe circumstance shown in Figure 4, in which a) the hor-izontal displacement of dots behind the tank is used tocompute the (assumed spanwise-uniform) horizontal per-turbation density gradient of fluid in the tank, as indi-cated by intensity of the gray scale. In b) vertical dotdisplacements are used to compute the vertical densitygradient. In c) the two components of the density gradi-ent are integrated to find the perturbation density field.Reproduced from Figure 13 of Dalziel et al. [2000].

SUTHERLAND ET AL.: INTERNAL WAVES X - 5

qualitative synthetic schlieren image produced with a beamof internal waves from an oscillating cylinder that passesin front of an image of regularly spaced black circles on awhite background. The difference image is shown, analogousto that in Figure 3.

The cat-eye-like patterns have a black portion near thecenter of the dot surrounded on two sides by white, the lat-ter indicating how the dot shifted both horizontally as wellas vertically as a result of internal waves passing betweenthe camera and image. The angle of the resulting ellipticaldisturbance gives both components of the perturbation den-sity gradient. (This may be expressed as the gradient of theperturbation squared buoyancy frequency, with components∆N2

x and ∆N2z ≡ ∆N2.)

Alternately, if the image of dots is randomly distributed,then particle image velocimetry techniques can be used tomeasure horizontal and vertical displacements of portions ofthe image [Dalziel et al., 2000]. Assuming the disturbancein the ambient is uniform along the line-of-sight, each dis-placement field can be used to find the perturbation densitygradient. This can be integrated to compute the perturba-tion density field, as shown in Figure 5.

3.2. Separating Up from Down and Left from Right

A powerful analysis tool, related to Hilbert Transforms,can be used to distinguish upward from downward propagat-

Figure 6. Change in the background squared buoy-ancy frequency due to internal waves generated by an os-cillating cylinder, as measured by quantitative syntheticschlieren. Reproduced from Figure 2a of Mercier et al.[2008].

Figure 7. The four arms of the cross shown in Figure 6determined by Fourier implementation of the HilbertTransform. Reproduced from Figure 3 of Mercier et al.[2008].

ing internal waves and simultaneously distinguish leftwardfrom rightward propagating waves [Mercier et al., 2008].

For example, consider Figure 6, which shows the ∆N2

field computed using synthetic schlieren for an oscillatingcylinder experiment. Imagining a Cartesian grid superim-posed on the wave field with the origin at the centre of thedisturbance, the four arms of the cross consist of upwardpropagating waves emanating rightward and leftward in thefirst and second quadrants, respectively, and of downwardpropagating waves emanating leftward and rightward in thethird and fourth quadrants, respectively. Hilbert Trans-forms can extract each arm of the cross, as shown in Fig-ure 7.

In application, the method does not formally compute theCauchy Principle Value integral associated with a HilbertTransform. Instead it employs filtering of Fourier Trans-formed images, specifically those of time-series of the distur-bances. For example, consider synthetic schlieren employingan image of horizontal black and white lines behind a tank.During the evolution of a spanwise-uniform disturbance onecan compute ∆N2(x, z, t), in which z is vertical, x is thealong-tank co-ordinate and t is time. Fixing an arbitraryhorizontal location X, one can construct the vertical timeseries ∆N2(z, t;x = X). Taking the double fast-Fouriertransform first in t and then in z gives the complex series

coefficients ∆N2(kz, ω), in which kz is the vertical wavenum-ber and ω is the frequency.

Figure 8. Predicted transmission coefficient formonochromatic internal waves of frequency ω and hor-izontal wavenumber kx, incident upon a region of depthL where the ambient buoyancy frequency is N1 instead ofN0. Adapted from Figure 1 of Sutherland and Yewchuk[2004].

X - 6 SUTHERLAND ET AL.: INTERNAL WAVES

To extract upward propagating disturbances, we use thefact that the group velocity of internal waves is positive if

the vertical wavenumber is negative. So we set ∆N2 to zeroif kz is positive and leave the field untouched otherwise.Inverse Fourier transforming then produces a filtered field∆N2

↑(z, t;x = X) with only upward propagating distur-bances. This process can be repeated at different horizontallocations until the entire evolution field of upward propa-gating disturbances is reproduced: ∆N2

↑(x, z, t).We can similarly extract rightward-propagating waves

by Fourier transforming horizontal time series at succes-sive z = Z, setting the coefficients of negative horizontalwavenumbers to zero and then inverse transforming. Theresult of applying this to ∆N2

↑, for example, gives the upand rightward wavebeam shown in the top-left panel of inFigure 7.

3.3. Partial Transmission and Reflection

The use of the Hilbert Transform method described abovehas proven particularly useful in the study of internal wavepropagation in non-uniformly stratified fluid. The intuitiveunderstanding of their propagation is based upon ray theory,which assumes the small-amplitude wavepackets are quasi-monochromatic and that the background varies slowly com-pared to the wavelength (e.g. see Lighthill [1978], Sutherland[2010]). In particular, in stationary fluid this predicts thatwaves reflect from a level where the background buoyancyfrequency is less the wave frequency.

Following a theoretical approach analogous to that usedin thin-film optics for light or in quantum mechanics forelectrons, Sutherland and Yewchuk [2004] showed that in-ternal waves can partially transmit (i.e. “tunnel”) througha weakly stratified layer provided it is thin compared to thehorizontal wavelength of the incident waves. For piecewiseconstant profiles of the background buoyancy frequency,they predicted the transmission coefficient, T , as a functionof the relative frequency of the waves, ω/N0, and their rela-tive horizontal wavenumber, kxL, in which N0 is the far-fieldbuoyancy frequency and L is the depth of the thin strati-fied layer with buoyancy frequency N1. The predictions areshown in Figure 8 for the cases of an unstratified layer, aweakly stratified layer and a strongly stratified layer.

Counter to intuition based upon ray theory, one sees inparticular that waves can partially reflect from a stronglystratified layer even though their frequency is always smallerthan the background buoyancy frequency.

Of course, the phenomena is well-known in optics. In-deed, Mathur and Peacock [2010] made the analogy between

Figure 9. Schlieren image showing incident internalwaves in non-uniformly stratified media, partially trans-mitting and reflecting with partial trapping in a regionof locally enhanced stratification. Adapted from Figure3 of Mathur and Peacock [2010].

internal waves and light showing that they behave like aFabry-Perot multiple-beam interferometer. The resultingresonance of internal waves in a localized region of enhancedstratification was demonstrated in laboratory experiments,shown in Figure 9.

The transmission of a small-amplitude wavepacketthrough arbitrary stratification and background flow canbe computed through the solution of the Taylor-Goldsteinequation [Nault and Sutherland , 2007, 2008]. These resultswere compared with laboratory experiments of internal wavebeams incident upon a weakly stratified layer [Gregory andSutherland , 2010]. Using the Hilbert Transform method, theincident, reflected and transmitted waves could be distin-guished and so transmission and reflection coefficients couldbe computed. Theory was found to be consistent with theexperiments, but there was great sensitivity of the predictedtransmission coefficient to the details of the stratification.

4. Non-Spanwise-Uniform Disturbances

The quantitative uses of synthetic schlieren describedabove assumed that any disturbances in the stratified fluidwere uniform across the width of the tank. With this as-sumption, it was straightforward to relate displacements ofimages to changes in stratification through equation (10).It was likewise trivial to invert this equation and so inferchanges in the stratification knowing the measured displace-ments, as in equation (11).

If disturbances are not spanwise-uniform, one can stilluse Snell’s law to write down expressions for the apparentdisplacement of an image due to light passing through fluidwith known varying density (hence with known varying re-fractive index). The challenge is to invert this formula tofind the change in stratification for given observed displace-ments.

For axisymmetric disturbances about a vertical axis, theprocedure amounts to inverting a square matrix to deter-mine ∇ρ from observed displacements of an image. Forfully three dimensional disturbances, tomographic inversion

dy02

dy01

dy00

dy12

dy11dy22

dr= dx

(∆N2)0

(∆N2)1

(∆N2)2

−2 dx −dx 0 dx 2 dx

∆N2 = 0

∆z(0) ∆z(dx) ∆z(2 dx)

x

y

Figure 10. Discretization used to represent apparentdisplacements in an image behind a tank to axisymmet-ric disturbances within the tank. The disturbances arerepresented in terms of changes to the squared buoyancyfrequency ∆N2, which is assumed constant on annuli ofwidth dr = dx, in which dx is the horizontal (e.g. pixel)resolution of the observed vertical displacements ∆z ofthe object image.

SUTHERLAND ET AL.: INTERNAL WAVES X - 7

techniques are needed to reconstruct ∇ρ from displacementsobserved from multiple perspectives.

4.1. Axisymmetric Synthetic Schlieren

We consider the simplest case of reconstructing ∆N2 fromobserved vertical displacements of an image, ∆z [Onu et al.,2003]. First we consider the vertical displacements at a fixedtime and at a fixed height, so that ∆z(x) is taken to be afunction only of the along-tank distance, x. We seek thecorresponding value of ∆N2(r), which is assumed to be ax-isymmetric, varying with radius r.

The inversion problem begins with representing the alongtank direction by n + 1 discrete points xi = i dx for i =0 . . . n, and by discretizing the radial disturbances by con-centric rings of inner radius rj = j dr. So that the inver-sion problem is well-posed, we take dr ≡ dx and we setj = 0 . . . n, in which j = 0 signifies the innermost circle.The correspondence of the x and r co-ordinate systems isshown in Figure 10.

We assume that ∆N2 is constant within each annulus inthe central circle. And so we denote (∆N2)0 = ∆N2 for0 ≤ r < dr/2, (∆N2)1 = ∆N2 for dr/2 ≤ r < 3 dr/2,(∆N2)2 = ∆N2 for 3 dr/2 ≤ r < 5 dr, etc. Outside the out-ermost ring we assume the ambient is undisturbed so that∆N2 = 0.

We now consider the path of light passing in the y-direction from the far side of the tank through the distur-bance field to the side of the tank nearest the camera (i.e.from top to bottom of the schematic in Figure 10.

Given values of ∆N2 in each ring, we integrate equa-tions (2) and (5) by iteratively computing the path of lightz(y) crossing each annulus. Doing so requires computingin advance the distance, dyij , that light from location xi

crosses the j’th annulus (with the zeroth “annulus” beingthe central circle). Although they could be computed an-alytically, these geometrical distances are straightforwardlydetermined by a numerical algorithm.

The result of the forward problem is a matrix set of equa-tions:

~∆z = G ~∆N2, (12)

in which ~∆z is the transpose of (∆z(0),∆z(x1), . . . ,∆z(xn)),~∆N2 is the transpose of ((∆N2)0, (∆N2)1, . . . , (∆N2)n),

and G is a square matrix composed of the distances dyijand coefficient, γ, defined by (6).

Inverting G, we can then determine the disturbance fieldknowing vertical displacements along a horizontal line:

~∆N2 = G−1 ~∆z. (13)

If the image is placed well behind the tank, the com-ponents of G are somewhat more complicated because onemust consider the angle at which light enters the tank fromthe image as well as the vertical displacement of light. Theextra terms may be added to components of G, akin tothe inclusion of the second term in equation (10) for thespanwise-uniform problem.

Note that computing ∆N2(r) need only be done usingimage displacements rightward of the centre of the distur-bance. Independently, one can compute ∆N2(r) using im-age displacements leftward of centre. Thus comparing rightand left gives a check on the accuracy of the assumptionthat the disturbance field was indeed axisymmetric.

Of course, the process of computing ∆N2 at a particularheight can be repeated at different heights so as to recon-struct a ‘snapshot’ of ∆N2(r, z). If the image is of dots in-stead of lines, one can compute horizontal as well as verticalcomponents of the density gradient through this methodol-ogy.

An early application of axisymmetric schlieren examinedthe internal wave field surrounding a vertically oscillating

sphere in uniformly stratified fluid [Onu et al., 2003; Flynnet al., 2003]. For example, Figure 11a shows the observedapparent vertical displacement of an image once the spherehad oscillated three times. No data were computed in thelower left-hand corner where the image was obscured bythe sphere. The corresponding ∆N2 field is shown in Fig-ure 11b. As anticipated by theory, the along-beam ampli-tude decayed rapidly with distance from the center of thesphere as the conical wave beam expanded radially aboutthe z-axis. The theory well-predicted the amplitude of thewave-cones provided the sphere was sufficiently small. Oth-erwise it overpredicted the amplitude, presumably becauseof it neglected dynamics occurring with the boundary layersurrounding the sphere [Flynn et al., 2003].

This observation reveals a particularly useful aspect ofthe use of schlieren. Although the amplitude decays, thehorizontal extent of the disturbance widens with distancefrom the origin. As a result, the vertical displacement sig-nal does not weaken with distance away from the source.Indeed, the value of ∆z in Figure 11a is largest near the topright-hand corner of the image. Hence schlieren can extractsignals over noise where in situ probe measurements or at-tempts to observe the motion of embedded particles mightfail.

Since the development of the technique, it has since beenused to measure the laminar wake behind a falling sphere[Yick et al., 2007] and internal waves above a plume in astratified fluid - a model for internal wave generation by

Figure 11. a) Apparent vertical displacement, ∆z(x, z)of horizontal lines in an image behind a tank in whicha sphere (situated to the bottom left) oscillates in uni-formly stratified fluid. b) Corresponding change insquared buoyancy frequency, ∆N2(r, z), due to inter-nal waves computed through axisymmetric syntheticschlieren. Adapted from Figures 2b and 4a of Onu et al.[2003].

X - 8 SUTHERLAND ET AL.: INTERNAL WAVES

convective storms through the mechanical oscillator effect[Ansong and Sutherland , 2010].

4.2. Inverse Tomography

If the disturbance is fully three dimensional, then theproblem of using of synthetic schlieren to reconstruct thedensity gradient field from observed displacements of a sin-gle image is ill-posed: without invoking symmetry, it is im-possible to reconstruct a three-dimensional object from itsshadow. With multiple perspectives, however, it is pos-sible to reconstruct and approximate the structure of thedisturbance. In the medical use of magnetic resonanceimaging, the method of tomographic reconstruction is well-established. Making use of refractive index variations withair temperature, tomographic inversion has been used tomeasure the density of a supersonically expanding jet [Farisand Byer , 1988] and of two interacting jets [Goldhahn andSeume, 1988]. The latter was the first to employ themethodology of synthetic schlieren - recording the appar-

Figure 12. Image displacements recorded by differentperspectives looking horizontally through an oscillatingparaboloid filled with uniformly stratified fluid. Repro-duced from Figure 2 of Hazewinkel et al. [2011].

Figure 13. Tomographic reconstruction of internal wavefield inside an oscillating parabolic domain computedfrom many views of image displacements like those inFigure 12. Reproduced from Figure 4a of Hazewinkelet al. [2011].

ent displacement of an image of random dots - to determinethe displacement of light rays.

Two approaches have since been taken to apply tomo-graphic methods for the measurement of internal waves us-ing synthetic schlieren. The Fourier-convolution approach ofFaris and Byer [1988] and Goldhahn and Seume [1988] wasused by Hazewinkel et al. [2011] in their study of internalwave attractors in a parabolic basin. The experiment itselfwas an extension of earlier studies into the formation of in-ternal wave beams in spanwise uniform, non-rectangular do-mains that oscillate at a fixed frequency [Maas et al., 1997;Hazewinkel et al., 2008]. Because internal waves at a givenfrequency propagate at a fixed angle to the vertical, slop-ing side-walls in the domain tend to focus the disturbancesinto a beam whose path effectively acts as an “attractor”for internal waves [Maas and Lam, 1995].

When stratified fluid within a paraboloid was oscillated,looking through the tank at different angles around the hor-izontal revealed attractor-like patterns in the observed dis-placement of images behind the tank. Four such images areshown in Figure 12. The information in these and severalmore images taken at different perspectives were combinedthrough a convolution of their Fourier decompositions. Theinverse transform of the result revealed the three dimen-sional structure of the attractor, as shown in Figure 13.

A different approach follows that of the matrix inversionmethod used to measure axisymmetric disturbances [De-camp et al., 2008]. At a fixed vertical level the observedimage displacements could be represented by a vector with2n entries, in which n is the number of pixels and the valueis doubled to account for horizontal as well as vertical dis-placements. The perturbation density field (from which thedensity gradient is computed) could be discretized either inCartesian or polar co-ordinates involving N ≡ nx × ny orN ≡ nr×nθ points, respectively. For localized disturbances,the latter approach is found to be more effective.

The forward problem can thus be written as a coupledset of 2n equations in N unknowns. This is cast in matrixform analogous to equation (12):

~∆ ≡(

~∆xT, ~∆z

T)T

= G ~ρ, (14)

in which the differentiation operators acting on elements ofρ, to give∇ρ are buried inside the components ofG, which isa 2n by N rectangular matrix. The resolution of the distur-bance is chosen so that there are more unknowns than equa-tions. The typical method to solve this system of equations

Figure 14. Camshaft and schematic cross-section show-ing how the rotation of the shaft results in the back andforth oscillation of flat plates. Reproduced from Figure2 of Gostiaux et al. [2007].

SUTHERLAND ET AL.: INTERNAL WAVES X - 9

is to multiply through by the transpose, GT , thus recast-ing the problem as N equations in N unknowns. Becausethe N by N sparse matrix GTG is singular, it is typical toshift its eigenvalues by a so-called regularization parameterµ [Zhdanov , 2002]. Hence, the forward problem is written

GT ~∆ =

(G

TG+ µI

)~ρ, (15)

in which I is the identity matrix.Rather than compute the inverse of the matrix multiply-

ing ~ρ on the right-hand side of (15), it is efficient to solveiteratively using the Bi-Conjugate Gradient method [Goluband Loan, 1996].

This approach was tested against idealized disturbancesby Decamp et al. [2008] who showed that a polar grid is bestused provided the number of sectors is not a multiple of thenumber of perspectives. Even with just 6 perspectives, acosine-times-Gaussian disturbance was well reproduced ona grid with 33 sectors and 40 rings.

Applying this method to internal wave fields generated inthe laboratory has proved challenging in part because of therequirement to have multiple perspectives. In Hazewinkelet al. [2011], the tank had curved side-walls, whose influ-ence upon the path of light rays could be accounted for. Inattempts to study non-axisymmetric waves generated (e.g.by a horizontally towed object) in a square tank, at most2 perspectives at 90 degrees might be recorded simultane-ously, each with a camera on one side and the image on theother. To gain more perspectives, the experiment must berepeated but the generation mechanism re-oriented withinthe tank to give the cameras a different perspective. Thismethod requires perfect repeatability. Small changes canlead to large errors in the computation of ∇ρ.

5. Particle Image Velocimetry

Here we review progress made in the examination of in-ternal waves using particle image velocimetry (PIV). In thismethod, small particles are illuminated by a laser light sheet.Their displacements (or, more precisely, displacements ofpatches of particles in a window) are tracked between pulsesof the laser. The technique has revolutionized laboratoryexperiments by providing a non-intrusive method that mea-sures velocity at all points in the plane of the light sheet.If multiple light sheets are used, the flow field can be re-constructed in three dimensions to within the resolution setby the separation between successive light-sheets and thedigital camera.

However, using PIV in the study of internal waves pro-vides additional challenges. Because light bends as it passesthrough stratified fluid the position of particles in the flowcan be misrepresented. If the stratification is not too strongand disturbances in the fluid are not too large, then thedistortion due to refractive index changes can be ignored.Otherwise, schlieren can be used to predict the distortionand so provide a correction to the digitized image of parti-cles before they are processed to compute displacements.

Some of these advancements are discussed below. Butfirst we describe another innovation in the study of internalwaves in the laboratory. A new mechanism for generatinginternal waves has proven a versatile tool for the study oftheir propagation and dispersion.

5.1. A Novel Wave Generator

Typical methods for generating internal waves in thelaboratory include oscillating a rigid body at constant fre-quency or towing a body horizontally at a constant speed.The former has the disadvantage that it creates four wavebeams, as in Figure 2, or at least two if oscillated against aside boundary. Towed objects along a top or bottom bound-ary produce unidirectional waves, but towing piles up the

stratified fluid ahead of the object forming what is calleda “columnar mode”. As a result, the end-wall of the tankcan influence the dynamics of flow over the obstacle [Baines,1995].

A new mechanism for the generation of internal wavesavoids these deficiencies [Gostiaux et al., 2007]. In it ver-tically stacked flat plates periodically move back and forthproviding forcing on the stratified fluid from the side. If theforcing is driven by a rotating spiral camshaft, as in Fig-ure 14, the plates effectively move collectively as a verticallypropagating wave whose vertical wavelength and amplitudeis set by the geometry of the camshaft and whose frequencyis set by the rotation rate.

The mechanism thus acts like towed topography exceptthat the translation of the periodic boundary is vertical. Itdoes not generate columnar modes upstream. Nor is their

Figure 15. a) Left-to-right propagating vertical mode-1 internal waves generated using horizontally oscillatingflat plates, as in Gostiaux et al. [2007]. b) Internal wavebeam downstream of hill placed in front of the oncom-ing mode-1 waves. Adapted from Figures 2a and 3a ofPeacock et al. [2009].

Figure 16. Image of random dots distorted by strongstratification at a density interface in an approximatelytwo-layer fluid. Reproduced from Figure 6c of Dalzielet al. [2007].

X - 10 SUTHERLAND ET AL.: INTERNAL WAVES

boundary layer separation behind their crests. Because theboundary displacements are horizontal large amplitude forc-ing is less inclined to result in mixing of the stratified fluid.

This technique has been used in a variety of circumstancesthat have revealed important processes in the evolution ofinternal waves.

5.2. Internal wave vertical modes

Satellite altimetry [Egbert and Ray , 2000] has recentlybeen employed to observe the generation of oceanic inter-nal waves by tidal flow over the continental margin andsubmarine sills. These have revealed the generation of lowvertical-mode internal waves, associated with undulations ofthe thermocline. As well as theory and numerical simula-tions [Balmforth et al., 2002; Smith and Young , 2002; Legg ,2004], laboratory experiments have been performed to ex-amine the generation of internal waves by Gaussian hill thatoscillated horizontally back and forth with fixed frequencyand amplitude [Peacock et al., 2008; Echeverri et al., 2009].

An outstanding question is how such low modes trans-fer their energy to smaller scales (high modes) so that theyultimately dissipate and mix the ocean.

The laboratory experiments by Peacock et al. [2009]examined one mechanism through which this may occur.When a low mode internal wave is incident upon topog-raphy, the sloping sides of the hill refocus the energy intobeams. This is shown in Figure 15.

The waves are created using the mechanism of oscillatingstacked horizontal plates described above. Here, however,the rotating shaft is not spiral but is constructed with a ver-tically sinusoidal variation in a fixed plane so that rotatingthe shaft produces a mode-2 wave in a uniformly stratifiedmedium. In the absence of topography, PIV is used to revealthe mode-2 wave structure, which at one point in the phaseexhibits forward motion at the surface and bottom and ret-rograde motion at mid-depth. The flow directions reverse ahalf-period later.

When this incident wavefield encounters a Gaussian hill,the structure of the wavefield changes significantly down-stream. Just as an oscillating body creates internal wavebeams in a stratified fluid, oscillations resulting from the in-cident low mode internal waves creates beam-like structuresdownstream. Thus energy from low modes is efficiently con-verted into higher modes. These ideas have recently beenextended to the examination of internal waves incident uponthe continental shelf [Klymak et al., 2011].

5.3. Adjustments using Synthetic Schlieren

In the above experiments, the perturbations due to waveswere sufficiently small that changes to the density gradientsinsignificantly distorted the apparent position of particles.Thus PIV analysis could be employed with no need for cor-rections.

In the study of solitary waves by Dalziel et al. [2007], thedirect application of PIV was hindered by distortions result-ing from the sharp density gradient at the interface betweenthe fresh and underlying salty water. For example, Fig-ure 16 shows the smearing and significant apparent particledisplacement at a sharp density gradient. This is not dueto the vertical motion of the wave. It results from photonsbetween the laser light sheet and observer being deflected asthey pass through the interface.

Dalziel et al. [2007] addressed the issue by using schlierento measure the density gradient and then using this infor-mation to correct for the apparent in situ particle displace-ment. The experimental configuration strobed between thecamera recording particle positions in a laser light sheet andit recording images of random dots behind the tank. Thiseffectively rendered the schlieren and PIV measurements si-multaneous. The result is shown in Figure 17. The cor-rected PIV image gives values of velocity and the schlierenmeasurements predicted the density. The combined resultspowerfully measured the gradient Richardson number andso assessed the stability of internal solitary waves.

6. Conclusions

Several technological innovations have provided new toolsfor the study of internal waves in the laboratory. Here wehave focused mostly upon the use of synthetic schlieren as anon-intrusive way to measure perturbation density gradientsdue to internal waves in continously stratified media.

With increasing observations of internal solitary wavesin the ocean, there is more interest in examing this phe-nomenon in the laboratory [Grue et al., 2000; Carr et al.,2008]. Although synthetic schlieren can work together withPIV to help correct apparent particle displacements withinthe flow [Dalziel et al., 2007], it is not ideally suited to thestudy of interfacial waves. This is because the large curva-ture of the density field at the interface bends light to sucha degree that an image behind the tank is distorted to muchto compute apparent displacements.

Of course, one can track the motion of the interface byinjecting dye there while the ambient is being established.Shadowgraph is also a useful tool. Other more innovatedmethods include the use of ultrasonic probes which mea-sure the interface displacement by recording the travel-timeof sound vertically through the ambient between a trans-mitter and receiver at fixed depth straddling the interface[Michallet and Barthelemy , 1997, 1998]. One can try to elim-inate particle distortions in PIV by adding another fluid tothe ambient (e.g. alcohol) that cancels the refractive indexchange due to salinity, but this can also lead to problemswith double diffusive behaviour.

Notes

1. Synthetic schlieren has also been called “background orientedSchlieren” by Meier [2002], who has used it to visualize andmeasure shock waves in air.

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T. Dauxois, Laboratoire de Physique, Ecole NormaleSuperieure, 46 allee d’Italie, Lyon, France, 69007 (Thierry.Dauxois@ens-lyon.fr)

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