# Publications

## Large–scale, realistic laboratory modeling of M2 internal tide generation at the Luzon Strait

### M. J. Mercier, L. Gostiaux, K. Helfrich, J. Sommeria, S. Viboud, H. Didelle, S. J. Ghaemsaidi, T. Dauxois, & T. Peacock.

Geophysical Research Letters, 40, 1-6 (2013)

We have realized a large scale experimental study of internal tide generation by complex 3D topography, giving insight into the origin of internal solitary waves in the South China Sea. Our laboratory study modeling the Luzon Strait, and realized at the Coriolis turntable (Grenoble), the world's largest rotating table for GFD experiments, have demonstrated that despite having a complex three-dimensional geometry, the Luzon Strait radiates in the South China Sea a coherent, mode-1 dominated, weakly nonlinear M2 internal tide oriented in a west-northwest direction, which is prone to steepening.

Article [link]

## Lagrangian Coherent Structures: the hidden skeleton of fluid flows

### Peacock, T. & Haller, G.

Physics Today, 66, 41 (2013)

This article presents a review of the state-of-the-art in defining and detecting Lagrangian Coherent Structures (LCSs) in unsteady, two-dimensional fluid flows, with applications to ocean surface transport prediction.

Article [link]

## 3D Stereoscopic PIV visualization of the axisymmetric conical internal wave field generated by an oscillating sphere

### S. J. Ghaemsaidi, & T. Peacock.

Experiments in Fluids, 54, 1454 (2013)

To date, experimental studies of internal wave velocity fields have been limited to two-dimensional investigations of planar or axisymmetric systems. Here we present results of the first three-dimensional stereoscopic PIV visualizations of an internal wave field. The experiments utilize the canonical arrangement of a vertically oscillating sphere, which enables rigorous comparison with recently published theoretical results. The excellent level of agreement between experiment and theory demonstrates the utility of using stereoscopic PIV to study three-dimensional internal waves. Furthermore, the ability to measure all three components of the velocity field gives an alternative perspective on the significance of harmonics generated via nonlinear processes in the vicinity of the oscillating sphere.

Article [link]

## Tidally generated internal-wave attractors between double ridges

### P. Echeverri, T. Yokossi, N. J. Balmforth & T. Peacock

Journal of Fluid Mechanics, 669, 354 (2011)

A study is presented of the generation of internal tides by barotropic tidal flow over topography in the shape of a double ridge. An iterative map is constructed to expedite the search for the closed ray paths that form wave attractors in this geometry. The map connects the positions along a ray path of consecutive reflections from the surface, which is double-valued owing to the presence of both left- and right-going waves, but which can be made into a genuine one-dimensional map using a checkerboarding algorithm. Calculations are then presented for the steady-state scattering of internal tides from the barotropic tide above the double ridges. The calculations exploit a Green function technique that distributes sources along the topography to generate the scattering, and discretizes in space to calculate the source density via a standard matrix inversion. When attractors are present, the numerical procedure appears to fail, displaying no convergence with the number of grid points used in the spatial discretizations, indicating a failure of the Green function solution. With the addition of dissipation into the problem, these difficulties are avoided, leading to convergent numerical solutions. The paper concludes with a comparison between theory and a laboratory experiment.

Article [link]

## Internal tide generation by arbitrary two-dimensional topography

### P. Echeverri & T. Peacock

Journal of Fluid Mechanics, 659, 247 (2010)

To date, analytical models of internal tide generation by two-dimensional ridges have considered only idealized shapes. Here, we advance the Green function approach to address the generation of internal tides by two-dimensional topography of arbitrary shape, employing the Wentzel-Kramers-Brillouin (WKB) approximation to consider the impact of non-uniform stratifications. This allows for a more accurate analytical estimation of tidal conversion rates. Studies of single and double ridges reveal that the conversion rate and the nature of the radiated internal tide can be sensitive to the topographic shape, particularly around criticality and when there is interference between wave fields generated by neighbouring ridges. The method is then applied to the study of two important internal tide generation sites, the Hawaiian and Luzon Ridges, where it captures key features of the generation process.

Article [link]

## New wave generation

### M. J. Mercier, D. Martinand, M. Mathur, L. Gostiaux, T. Peacock & T. Dauxois

Journal of Fluid Mechanics, 657, 308 (2010)

We present the results of a combined experimental and numerical study of the generation of internal waves using the novel internal wave generator design of Gostiaux et al. (Exp. Fluids, vol. 42, 2007, pp. 123–130). This mechanism, which involves a tunable source composed of oscillating plates, has so far been used for a few fundamental studies of internal waves, but its full potential is yet to be realized. Our study reveals that this approach is capable of producing a wide variety of two-dimensional wave fields, including plane waves, wave beams and discrete vertical modes in finite-depth stratifications. The effects of discretization by a finite number of plates, forcing amplitude and angle of propagation are investigated, and it is found that the method is remarkably efficient at generating a complete wave field despite forcing only one velocity component in a controllable manner. We furthermore find that the nature of the radiated wave field is well predicted using Fourier transforms of the spatial structure of the wave generator.

Article [link]

## Propulsion generated by diffusion driven flow

### M. R. Allshouse, M. F. Barad & T. Peacock

Nature Physics, 6, 516 (2010)

Buoyancy-driven flow, which is flow driven by spatial variations in fluid density, lies at the heart of a variety of physical processes, including mineral transport in rocks, the melting of icebergs and the migration of tectonic plates. Here we show that buoyancy-driven flows can also generate propulsion. Specifically, we find that when a neutrally buoyant wedge-shaped object floats in a density-stratified fluid, the diffusion-driven flow at its sloping boundaries generated by molecular diffusion produces a macroscopic sideways thrust. Computer simulations reveal that thrust results from diffusion-driven flow creating a region of low pressure at the front, relative to the rear of an object. This discovery has implications for transport processes in regions of varying fluid density, such as marine snow aggregation at ocean pycnoclines, and wherever there is a temperature difference between immersed objects and the surrounding fluid, such as particles in volcanic clouds.

Article [link]

## An internal wave interferometer

### M. Mathur & T. Peacock

Physical Review Letters, 104, 118501 (2010)

Internal waves are a ubiquitous and significant means of momentum and energy transport in the oceans, atmosphere and astrophysical bodies. Here, we show that internal wave propagation in nonuniform density stratifications, which are prevalent throughout nature, has a direct mathematical analogy with the classical optical problem of a multiple beam light interferometer. We rigorously establish this correspondence, and furthermore provide the first experimental demonstration of an internal wave interferometer, based on the theory of resonant transmission of internal waves.

Article [link]

## A laboratory study of low-mode internal tide scattering by finite-amplitude topography

### T. Peacock, M. J. Mercier, S. Vibaud, H. Didelle & T. Dauxois

Physics of Fluids, 21, 121702 (2010)

We present the first laboratory experimental results concerning the scattering of a low-mode internal tide by topography. Experiments performed at the Coriolis Platform in Grenoble used a recently-conceived internal wave generator as a means of producing a high-quality mode-1 wave field. The evolution of the wave field in the absence and presence of a supercritical Gaussian was studied by performing spatiotemporal modal decompositions of velocity field data obtained using Particle Image Velocimetry (PIV). The results support predictions that large-amplitude supercritical topography produces significant reflection of the internal tide and transfer of energy from low to high modes.

Article [link]

Preprint [link]

## Lagrangian coherent structures and internal tide attractors

### W. E. Tang & T. Peacock

CHAOS, 20, 017508 (2010)

Internal gravity wave energy propagates along wave characteristics and is widely believed to be responsible for elevated mixing in the ocean when waves break. Recently, studies have revealed the possibility of internal wave attractors in confined basins and between multiple submarine ridges that have steep slopes, providing a new mechanism for strong mixing to occur away from ocean boundaries. These recent studies focus on linear stratification in ideal laboratory conditions. In this paper we present an efficient mathematical tool based on ideas about Lagrangian Coherent Structures to extract internal wave attractors in non-uniformly stratified fluids. This remedies the inherent difficulty of constructing a return map where there is complex topography and skewing of internal wave characteristics due to a changing buoyancy frequency.

Article [link]

In addition to this article, there is a short review article of research on Lagrangian Coherent Structures from the same edition of the journal CHAOS.

## Low-mode internal tide generation: an experimental and numerical investigation

### P. Echeverri, M. R. Flynn, K. B. Winters & T. Peacock

Journal of Fluid Mechanics, 636, 91-108 (2009)

We analyse the low-mode structure of internal tides generated in laboratory experiments and numerical simulations by a two-dimensional ridge in a channel of finite depth. The height of the ridge is approximately half of the channel depth and the regimes considered span sub- to supercritical topography. For small tidal excursions, of the order of 1% of the topographic width, our results agree well with linear theory. For larger tidal excursions, we find that the scaled mode-1 conversion rate decreases by less than 15%, in spite of nonlinear phenomena that break down the familiar wave-beam structure. For this topographic configuration, most of the linear baroclinic energy flux is associated with the mode 1 tide, so our experiments reveal that nonlinear behaviour does not significantly affect the barotropic to baroclinic energy conversion in this regime, which is relevant to large-scale ocean ridges. This may not be the case, however, for smaller scale ridges that generate a response dominated by higher modes.

Article [link]

Preprint [link]

## Internal wave beam propagation in nonuniform stratifications

### M. Mathur & T. Peacock

Journal of Fluid Mechanics, 638, 133-152 (2009)

In addition to being observable in laboratory experiments, internal wave beams are reported in geophysical settings, which are characterized by non-uniform density stratifications. Here, we perform a combined theoretical and experimental study of the propagation of internal wave beams in non-uniform density stratifications. Transmission and reflection coefficients, which can differ greatly for different physical quantities, are determined for sharp density-gradient interfaces and finite-width transition regions, accounting for viscous dissipation. Thereafter, we consider even more complex stratifications to model geophysical scenarios. We show that wave beam ducting can occur under conditions that do not necessitate evanescent layers, obtaining close agreement between theory and quantitative laboratory experiments. The results are also used to explain recent field observations of a vanishing wave beam at the Keana Ridge, Hawaii.

Article [link]

Preprint [link]

## Enhanced drag of a sphere settling in a stratified fluid at small Reynolds number

### K. Y. Yick, C. R. Torres, T. Peacock & R. Stocker

Journal of Fluid Mechanics, 632, 49-68 (2009)

We present a combined experimental and numerical investigation of a sphere settling in a linearly stratified fluid at small Reynolds numbers. Using time-lapse photography and numerical modeling, we observed and quantified an increase in drag due to stratification. Microscale synthetic schlieren revealed that a settling sphere draws lighter fluid downwards, resulting in a density wake extending tens of particle radii. Analysis of the flow and density fields shows that the added drag results from the buoyancy of the fluid in a region surrounding the sphere, while the bulk of the wake does not influence drag. A scaling argument is provided to rationalize the observations. The enhanced drag can increase settling times in natural aquatic environments, affecting retention of particles at density interfaces and vertical fluxes of organic matter.

Article [link]

Preprint [link]

## Tidal conversion by supercritical topography

### N. J. Balmforth & T. Peacock

Journal of Physical Oceanography, 39, 1965-1974 (2009)

Calculations are presented of the rate of energy conversion of the barotropic tide into internal gravity waves above topography on the ocean floor. The ocean is treated as infinitely deep, and the topography consists of periodic obstructions; a Green function method is used to construct the scattered wavefield. The calculations extend previous results for subcritical topography (wherein waves propagate along rays whose slopes exceed that of the topography everywhere), by allowing the obstacles to be arbitrarily steep, or supercritical (so waves propagate at shallower angles than the topographic slopes and are scattered both up and down). A complicated pattern is found for the dependence of energy conversion on the ratio of maximum topographic slope to wave slope, and the ratio of obstacle amplitude and separation. This results from a sequence of constructive and destructive interferences between scattered waves, that has implications for computing tidal conversion rates on a global scale.

Article [link]

## Experimental and numerical investigation of the kinematic theory of unsteady separation

### M. Weldon, T. Peacock, G. B. Jacobs, M. Helu & G. Haller

Journal of Fluid Mechanics, 611, 1-11 (2008)

We present the results of a combined experimental and numerical study of flow separation in the unsteady two-dimensional rotor-oscillator flow. Experimentally detected material spikes are directly compared to separation profiles predicted from numerical shear-stress and pressure data, using a recent kinematic theory of unsteady separation. For steady, periodic, quasi-periodic and random forcing, fixed separation is observed, and experimental observations and theoretical predictions are in close agreement. The transition from fixed to moving separation is also reported

Article [link]

Preprint [link]

## Stability of a stratified fluid with a vertically moving sidewall

### F. Blanchette, T. Peacock & R. Cousin

Journal of Fluid Mechanics, 609, 305-317 (2008)

We present the results of a combined theoretical and experimental study of the stability of a uniformly stratified fluid bounded by a side-wall moving vertically with constant velocity. This arrangement is perhaps the simplest in which boundary effects can drive instability and, potentially, layering in a stratified fluid. Our investigations reveal that for a given stratification and diffusivity of the stratifying agent, the side-wall boundary-layer flow becomes linearly unstable when the wall velocity exceeds a critical value. The onset of instability is clearly observed in the experiments, and there is good quantitative agreement with some predictions of the linear stability analysis.

Article [link]

## Going with (or against) the flow

### T. Peacock & E. Bradley

Science, 320 (5881), 1302-1303 (2008)

Our daily lives are governed by flow control. We not only move through fluids, such as air and water, but we also depend on fluid flows for transport and mixing on all scales, from the blood flow that brings oxygen to our cells to geophysical convection and turbulence in the oceans and atmosphere. Flow control therefore lies at the heart of a panoply of scientific problems, such as improving the fuel efficiency of cars or controlling pollution dispersion in the environment. The ever-increasing importance of these issues is driving new theoretical and technological advances in this diverse and challenging research field.

Article [link]

Preprint available on request

## Experimental investigation of internal tide generation by two-dimensional topography

### T. Peacock, P. Echeverri, P. & N.J. Balmforth

Journal of Physical Oceanography, 38 (1), 235-242 (2008)

Experimental results of internal tide generation by two-dimensional topography are presented. The synthetic Schlieren technique is used to study the wave fields generated by a Gaussian bump and a knife edge. The data compare well to theoretical predictions, supporting the use of these models to predict tidal conversion rates. In the experiments, viscosity plays an important role in smoothing the wave fields, which heals the singularities that can appear in inviscid theory and suppresses secondary instabilities of the experimental wave field.

Article [link]

Preprint available on request

## Internal waves across the Pacific

### M. H. Alford, J. A. MacKinnon, Z. Zhao, R. Pinkel, J. Klymak & T. Peacock

Geophysical Research Letters 34 (24), L24601 (2007)

The long-range propagation of the semidiurnal internal tide northward from the Hawaiian ridge and its susceptibility to parametric subharmonic instability (PSI) at the "critical latitude were examined in spring 2006 with intensive shipboard and moored observations spanning 25-37°N along a tidal beam. Velocity and shear at the critical latitude were dominated by intense vertically-standing, inertially-rotating bands of several hundred meters vertical wavelength. These occurred in bursts following spring tide, contrasting sharply with the downward-propagating, wind-generated features seen at other latitudes. Our observations indicate that PSI occurs in the ocean with sufficient intensity to substantially alter the inertial shear field at and equatorward of the critical latitude, but that it does not appreciably disrupt the propagation of the tide.

Article [link]

Preprint available on request

## Uncovering the lagrangian skeleton of turbulence

### M. Mathur, G. Haller, T. Peacock, J. E. Ruppert-Felsot & H.L. Swinney

Physical Review Letters, 98 (14), Art. No. 144502 (2007)

We present a technique that uncovers the Lagrangian building blocks of turbulence, and apply this technique to a quasi-two-dimensional turbulent flow experiment. Our analysis identifies an intricate network of attracting and repelling material lines. This chaotic tangle, the Lagrangian skeleton of turbulence, shows a level of complexity found previously only in theoretical and numerical examples of strange attractors. We quantify the strength (hyperbolicity) of each material line in the skeleton and demonstrate dramatically different mixing properties in different parts of the tangle.

Article [link]

Preprint available on request

## Microscale Synthetic Schlieren

### K.Y. Yick, R. Stocker & T. Peacock

Experiments in Fluids, 42 (1), 41-48 (2007)

We develop the axisymmetric Synthetic Schlieren technique to study the wake of a microscale sphere settling through a density stratification. A video-microscope was used to magnify and image apparent displacements of a micron-sized random-dot pattern. Due to the nature of the wake, density gradient perturbations in the horizontal greatly exceed those in the vertical, requiring modification of previously developed axisymmetric techniques. We present results for 780 and 383 μm spheres, and describe the limiting role of noise in the system for a 157 μm sphere. This technique can be instrumental in understanding a range of ecological and environmental oceanic processes on the microscale.

Article [link]

## Optimizing diffusion-driven flow in a fissure

### R. Heitz, T. Peacock & R. Stocker

Physics of Fluids, 17, Art. No. 128104 (2005)

Diffusion-driven flow arises when a stably stratified fluid is bounded by an inclined wall. For a stratified fluid in an inclined fissure, in which fluid is confined to a gap between two inclined parallel walls, the flow field is determined by the gap width and the angle of inclination. When the gap width is much wider than the buoyancy layer thickness, the problem reduces to that of a semi-infinite fluid. As the gap width decreases, interaction between the boundary layer flows on the upper and lower walls increasingly influences the velocity profile, affecting transport within the fissure. We have obtained supporting experimental results that show these trends, demonstrating the existence of optimum conditions for dye transport.

Article [link]

Preprint [link]

## Visualization of nonlinear effects in internal wave beam reflection

### T. Peacock & A. Tabaei

Physics of Fluids, 17, Art. No. 061702 (2005)

Recent theoretical and numerical investigations predict that localized nonlinear effects in the overlapping region of an incoming and reflected internal wave beam can radiate higher-harmonic beams. We present the first set of experimental visualizations, obtained using the digital Schlieren method, that confirm the existence of radiated higher-harmonic beams. For arrangements in which the angle of propagation of the second harmonic exceeds the slope angle, radiated beams are visualized. When the propagation angle of the second harmonic deceeds the slope angle no radiated beams are detected, as the associated density gradient perturbations are too weak for the experimental method. The case of a critical slope is also reported.

Article [link]

Preprint [link]

## The stratified Boycott effect

### T. Peacock, F. Blanchette & J.W.M Bush

Journal of Fluid Mechanics, 529, 33-49 (2005)

We present the results of an experimental investigation of the flows generated by monodisperse particles settling at low Reynolds number in a stably stratified ambient with an inclined sidewall. In this configuration, upwelling beneath the inclined wall associated with the Boycott effect is opposed by the ambient density stratification. The evolution of the system is determined by the relative magnitudes of the container depth, and the neutral buoyancy height. For sufficiently weak stratification, the Boycott layer transports dense fluid from the bottom to the top of the system; subsequently, the upper clear layer of dense saline fluid is mixed by convection. For sufficiently strong stratification, layering occurs.

Article [link]

Preprint available on request

## The effect of rotation on conical wave beams in a stratified fluid

### T. Peacock & P. Weidman

Experiments in Fluids, 39, 32-37 (2005)

Experiments are conducted to test extant theory on the effect of uniform rotation on the angle of conical beam wave propagation excited by a sphere vertically oscillating in a density stratified fluid. The near-constant Brunt-Väisälä frequency stratification N produced in situ in a rotating cylindrical tank exhibits no effect of residual motion for the range of Froude numbers investigated. Good agreement between experiment and theory is found using the synthetic schlieren visualization technique. In particular, the cut-off for wave propagation below which waves do not propagate, is clearly observed.

Article [link]

Preprint available on request

## An experimental investigation of the angular dependence of diffusion-driven flow

### T. Peacock, R. Stocker & J. Aristoff

Physics of Fluids, 16, 3503-3505 (2004)

We present experimental results on diffusion-driven flow along an inclined wall in a stably stratified fluid. The experiments focus on the dependence of the velocity in the buoyancy layer on the angle of inclination. The increase of velocity with decreasing angle is in agreement with theory for larger angles. For small angles, where current theory breaks down, the velocity tends to zero and an angle of maximum velocity exists.

Article [link]

Preprint [link]

## Forcing a planar jet flow using MEMS

### T. Peacock, E. Bradley, J. Hertzberg & Y.C. Lee

Experiments in Fluids, 37, 22-28 (2004)

We present the results of an experimental study in which a planar laminar jet of air was forced by an array of micro-electromechanical systems (MEMS) micro-actuators. In the absence of forcing, the velocity profile of the experimental jet matched the classic analytic solution. Driving actuators on either side of the jet in-phase or anti-phase, respectively, excited the symmetric or anti-symmetric mode of instability of the jet. Asymmetric forcing, using MEMS actuators on only one side of the jet, was also investigated.

Article [link]

Preprint available on request

## The Boycott effect in magma chambers

### F. Blanchette, T. Peacock, T. & J.W.M Bush

Geophysical Research Letters, 31, L05611 (2004)

We investigate the plausibility of the stratified Boycott effect as a source of layering in magma chambers. Crystal settling within the magma chamber will generate buoyant fluid near the sloping sidewalls whose vertical ascent may be limited by the ambient stratification associated with vertical gradients in SiO 2. The resulting flow may be marked by a layered structure, each layer taking the form of a convection cell spanning the lateral extent of the magma chamber. Using parameters relevant to magma chambers, we estimate that such convection cells would be established over a timescale of a month and have a depth on the order of 4m, which is roughly consistent with field observations of strata within solidified chambers.

Article [link]

Preprint available on request

## Solder assembled large MEMS flaps for fluid mixing

### Z. C. Ma, E. Bradley, T. Peacock, J. R. Hertzberg & Y. C. Lee

IEEE Transactions on Advanced Packaging, 26 (3), 268-276 (2003)

We describe surface-micromachined thermal actuator-based micro-electro-mechanical systems (MEMS) flaps with a length scale of 1,000 μm. These flaps were developed for the enhancement of fluid mixing in an experimental study of a planar air jet. The scales of the flow physics required the actuators to be much larger than the typical MEMS scales, and the experiment required an array of 10 flaps (1 cm in length) to be soldered onto a ceramic substrate with high precision. These unusual requirements made it difficult to achieve good assembly yields that could provide large deflections. To improve the yields and deflections, we modified the initial flap design by reducing the size of the solder pads, removing sharp corners, changing the number and the width of the actuator's hot arms, and strengthening the support beams. Flow velocity measurements showed that these MEMS flaps amplified the natural instabilities of the planar jet.

Article [link]

Preprint available on request

## The transition to turbulence in a microscopic fluid flow

### T. Peacock & T. Mullin

Physics of Fluids, 12, S8 (2000)

Article [link]

## Homoclinic bifurcations in a liquid crystal flow

### T. Peacock & T. Mullin

Journal of Fluid Mechanics, 432, 369-386 (2000)

The results of an experimental study of electrohydrodynamic convection in a liquid crystal are presented. Investigations concerned a small-aspect-ratio device so that finite geometry effects could be exploited to study the mechanisms by which complicated flows were organized. The results have been related to ideas on Shil'nikov dynamics and gluing bifurcations in low-dimensional dynamical systems.

Article [link]

Preprint available on request

## From low- to high-dimensional dynamics in a microscopic fluid flow

### T. Peacock, D.J. Binks & T. Mullin

Physical Review Letters, 82, 1446-1449 (1999)

The results of an experimental study of flow in a small aspect ratio liquid crystal cell are presented where the dimensions of the device are on the same scale as the width of a human hair. This system is found to display wide ranging dynamical behavior, from simple oscillations to seemingly homogeneous turbulence. Through a systematic study of the low-dimensional behavior we uncover an organizing center for the dynamics.

Article [link]

Preprint available on request

## Bifurcation phenomena in flows in a nematic liquid crystal

### T. Peacock, T. Mullin & D.J. Binks

International Journal of Bifurcations and Chaos, 9, 427-441 (1999)

Preprint available on request

## Hydrodynamic instabilities in nematic liquid crystals under oscillatory shear

### T. Mullin & T. Peacock

Proceedings of the Royal Society of London A, 455, 2635-2653 (1999)

We present the results of an experimental study of hydrodynamic instabilities in two classes of nematic liquid crystal material subjected to linear oscillatory shear. The materials are distinguished by their viscosity coefficient α3, which is negative in one case and positive in the other. The instabilities appear above a critical amplitude of the shear whose value also depends on the applied frequency. In the material with negative α3, the instability has the form of microscopic Williams domains, which align with the shear and are uniformly spread throughout the sample. The principal set of results relate to instabilities in a material with positive α3. They concern two qualitatively distinct macroscopic features that are on the length-scale of hundreds of layer thicknesses. One of them consists of a uniform distortion of the director, and the other contains microscopic structures in the form of Williams domains. It is proposed that an observed large scale mean flow is responsible for the modulation of the latter instability.

Article [link]

Preprint available on request