Papers
Global bifurcations to subcritical magnetorotational dynamo action in Keplerian shear flow
 A. Riols, F. Rincon, C. Cossu, G. Lesur, P.Y. Longaretti, G. I. Ogilvie, J. Herault

 Published online by Cambridge University Press:
 14 August 2013, pp. 145

 Article
 Export citation

Magnetorotational dynamo action in Keplerian shear flow is a threedimensional nonlinear magnetohydrodynamic process, the study of which is relevant to the understanding of accretion processes and magnetic field generation in astrophysics. Transition to this form of dynamo action is subcritical and shares many characteristics with transition to turbulence in nonrotating hydrodynamic shear flows. This suggests that these different fluid systems become active through similar generic bifurcation mechanisms, which in both cases have eluded detailed understanding so far. In this paper, we build on recent work on the two problems to investigate numerically the bifurcation mechanisms at work in the incompressible Keplerian magnetorotational dynamo problem in the shearing box framework. Using numerical techniques imported from dynamical systems research, we show that the onset of chaotic dynamo action at magnetic Prandtl numbers larger than unity is primarily associated with global homoclinic and heteroclinic bifurcations of nonlinear magnetorotational dynamo cycles born out of saddlenode bifurcations. These global bifurcations are found to be supplemented by local bifurcations of cycles marking the beginning of perioddoubling cascades. The results suggest that nonlinear magnetorotational dynamo cycles provide the pathway to injection of both kinetic and magnetic energy for the problem of transition to turbulence and dynamo action in incompressible magnetohydrodynamic Keplerian shear flow in the absence of an externally imposed magnetic field. Studying the nonlinear physics and bifurcations of these cycles in different regimes and configurations may subsequently help to understand better the physical conditions of excitation of magnetohydrodynamic turbulence and instabilitydriven dynamos in a variety of astrophysical systems and laboratory experiments. The detailed characterization of global bifurcations provided for this threedimensional subcritical fluid dynamics problem may also prove useful for the problem of transition to turbulence in hydrodynamic shear flows.
Turbulence spectra in smooth and roughwall pipe flow at extreme Reynolds numbers
 B. J. Rosenberg, M. Hultmark, M. Vallikivi, S. C. C. Bailey, A. J. Smits

 Published online by Cambridge University Press:
 14 August 2013, pp. 4663

 Article
 Export citation

Wellresolved streamwise velocity spectra are reported for smooth and roughwall turbulent pipe flow over a large range of Reynolds numbers. The turbulence structure far from the wall is seen to be unaffected by the roughness, in accordance with Townsend’s Reynolds number similarity hypothesis. Moreover, the energy spectra within the turbulent wall region follow the classical inner and outer scaling behaviour. While an overlap region between the two scalings and the associated ${ k}_{x}^{ 1} $ law are observed near ${R}^{+ } \approx 3000$, the ${ k}_{x}^{ 1} $ behaviour is obfuscated at higher Reynolds numbers due to the evolving energy content of the large scales (the verylargescale motions, or VLSMs). We apply a semiempirical correction (del Álamo & Jiménez, J. Fluid Mech., vol. 640, 2009, pp. 5–26) to the experimental data to estimate how Taylor’s frozen field hypothesis distorts the pseudospatial spectra inferred from timeresolved measurements. While the correction tends to suppress the long wavelength peak in the logarithmic layer spectrum, the peak nonetheless appears to be a robust feature of pipe flow at high Reynolds number. The inertial subrange develops around ${R}^{+ } \gt 2000$ where the characteristic ${ k}_{x}^{ 5/ 3} $ region is evident, which, for high Reynolds numbers, persists in the wake and logarithmic regions. In the logarithmic region, the streamwise wavelength of the VLSM peak scales with distance from the wall, which is in contrast to boundary layers, where the superstructures have been shown to scale with boundary layer thickness throughout the entire shear layer. Moreover, the similarity in the streamwise wavelength scaling of the large and verylargescale motions supports the notion that the two are physically interdependent.
Diffusiophoresis of colloidal particles in neutral solute gradients at finite Péclet number
 Aditya S. Khair

 Published online by Cambridge University Press:
 14 August 2013, pp. 6494

 Article
 Export citation

The role of neutral solute advection on the diffusiophoretic motion of colloidal particles is quantified. Theoretical analyses of this phenomenon usually assume that the solute concentration evolves solely via diffusion; that is, the Péclet number ($\mathit{Pe}$) for solute transport is identically zero. This leads to the conclusion that the translational diffusiophoretic velocity of a colloid is independent of its size, shape, and orientation with respect to the imposed solute gradient, provided that the colloid has uniform surface properties and that the length scale of interaction between the solute and the particle surface is much smaller than the particle size (Morrison, J. Colloid Interface Sci. vol. 34, 1970, p. 210). For a single spherical colloid, we show that the particle velocity decreases monotonically with increasing $\mathit{Pe}$. Moreover, the solute concentration and fluid flow around the colloid become markedly fore–aft asymmetric as $\mathit{Pe}$ is increased. Next, an asymptotic expansion at small $\mathit{Pe}$ predicts that solute advection leads to relative phoretic motion between two identical spherical colloids, which ultimately align in a plane normal to the imposed gradient (there is no relative motion at $\mathit{Pe}= 0$). Finally, asymptotic analysis of the diffusiophoretic motion of a slightly nonspherical colloid at small $\mathit{Pe}$ demonstrates that advection leads to a shape and orientationdependent particle velocity, in contrast to the insensitivity of the velocity to shape and orientation at $\mathit{Pe}= 0$.
Effects of mean shear on the local turbulent entrainment process
 Marc Wolf, M. Holzner, B. Lüthi, D. Krug, W. Kinzelbach, A. Tsinober

 Published online by Cambridge University Press:
 14 August 2013, pp. 95116

 Article
 Export citation

We report on effects of mean shear on the turbulent entrainment process, focusing in particular on their relation to smallscale processes in the proximity of the turbulent/nonturbulent interface (TNTI). Threedimensional particle tracking velocimetry (3DPTV) measurements of an axisymmetric jet are compared to data from a direct numerical simulation (DNS) of a zeromeanshear (ZMS) flow. First, conditional statistics relative to the interface position are investigated in a pseudoEulerian view (i.e. in a fixed frame relative to the interface position) and in a Lagrangian view. We find that in a pseudoEulerian frame of reference, both vorticity fluctuations and mean shear contribute to the vorticity jump at the boundary between irrotational and turbulent regions. In contrast, the Lagrangian evolution of enstrophy along trajectories crossing the entrainment interface is almost exclusively dominated by vorticity fluctuations, at least during the first Kolmogorov time scales after passing the interface. A mapping between distance to the instantaneous interface versus conditional time along the trajectory shows that entraining particles remain initially close to the TNTI and therefore attain lower average enstrophy values. The ratio between the rate of change of enstrophy in the two frames of references defines the local entrainment velocity ${v}_{n} =  (\mathrm{D} {\omega }^{2} / \mathrm{D} t)/ (\partial {\omega }^{2} / \partial {\hat {x} }_{n} )$, where ${\omega }^{2} $ is enstrophy and ${\hat {x} }_{n} $ is the coordinate normal to the TNTI. The quantity ${v}_{n} $ is decomposed into mean and fluctuating components and it is found that mean shear enhances the local entrainment velocity via inviscid and viscous effects. Further, the analysis substantiates that for all investigated flow configurations the local entrainment velocity depends considerably on the geometrical shape of the interface. Depending on the surface shape, different smallscale mechanisms are dominant for the local entrainment process, i.e. viscous effects for convex shapes and vortex stretching for concave shapes, looking from the turbulent region towards the convoluted boundary. Moreover, turbulent fluctuations display a stronger dependence on the shape of the interface than mean shear effects.
Experiments on gravity currents propagating on different bottom slopes
 Albert Dai

 Published online by Cambridge University Press:
 14 August 2013, pp. 117141

 Article
 Export citation

Experiments for gravity currents generated from an instantaneous buoyancy source propagating on an inclined boundary in the slope angle range ${0}^{\circ } \leq \theta \leq {9}^{\circ } $ are reported. While the flow patterns for gravity currents on $\theta = {6}^{\circ } , {9}^{\circ } $ are qualitatively different from those on $\theta = {0}^{\circ } $, similarities are observed in the acceleration phase for the flow patterns between $\theta = {2}^{\circ } $ and $\theta = {6}^{\circ } , {9}^{\circ } $ and in the deceleration phase, the patterns for gravity currents on $\theta = {2}^{\circ } $ are found similar to those on $\theta = {0}^{\circ } $. Previously, it was known that the front location history in the deceleration phase obeys a powerrelationship, which is essentially an asymptotic form of the solution to thermal theory. We showed that this powerrelationship applies only in the early stage of the deceleration phase, and when gravity currents propagate into the later stage of the deceleration phase, viscous effects become more important and the front location data deviate from this relationship. When the powerrelationship applies, it is found that at $\theta = {9}^{\circ } $, ${u}_{f} {({x}_{f} + {x}_{0} )}^{1/ 2} / {{ B}_{0}^{\prime } }^{1/ 2} \approx 2. 8{ 8}_{ 0. 17}^{+ 0. 19} $, which changes to $2. 8{ 6}_{ 0. 13}^{+ 0. 13} $ at $\theta = {6}^{\circ } $, $2. 5{ 4}_{ 0. 07}^{+ 0. 08} $ at $\theta = {2}^{\circ } $, and $1. 5{ 1}_{ 0. 07}^{+ 0. 07} $ on a horizontal boundary, where ${u}_{f} $ is the front velocity, $({x}_{f} + {x}_{0} )$ is the front location measured from the virtual origin, and ${ B}_{0}^{\prime } $ is the released buoyancy. Our results indicate that in the slope angle range ${6}^{\circ } \leq \theta \leq {9}^{\circ } $, the asymptotic relationship between the front velocity and front location in the deceleration phase is not sensitive to the variation of slope angle. In the late deceleration phase when the front location data deviate from the powerrelationship, we found that the flow patterns for $\theta = {6}^{\circ } , {9}^{\circ } $ are dramatically different from those for $\theta = {0}^{\circ } , {2}^{\circ } $. For high slope angles, i.e. $\theta = {6}^{\circ } , {9}^{\circ } $, the edge of the gravity current head experiences a large upheaval and enrolment by ambient fluid towards the end of the deceleration phase, while for low slope angles, i.e. $\theta = {0}^{\circ } , {2}^{\circ } $, the gravity current head maintains a more streamlined shape without violent mixing with ambient fluid throughout the course of gravity current propagation. Our findings indicate two plausible routes to the finale of a gravity current event.
Rise in optimized capillary channels
 B. Figliuzzi, C. R. Buie

 Published online by Cambridge University Press:
 14 August 2013, pp. 142161

 Article
 Export citation

Many technological applications rely on the phenomenon of wicking flow induced by capillarity. However, despite a continuing interest in the subject, the influence of the capillary geometry on the wicking dynamics remains underexploited. In numerous applications, the ability to promote wicking in a capillary is a key issue. In this article, a model describing the capillary rise of a liquid in a capillary of varying circular crosssection is presented. The wicking dynamics is described by an ordinary differential equation with a term dependent upon the shape of the capillary channel. Using optimal control theory, we were able to design optimized capillaries which promote faster wicking than uniform cylinders. Numerical simulations show that the height of the rising liquid was up to 50 % greater with the optimized shapes than with a uniform cylinder of optimal radius. Experiments on specially designed capillaries with silicone oil show a good agreement with the theory. The methods presented can be useful in the design and optimization of systems employing capillarydriven transport including microheat pipes or oil extracting devices.
Modelling the suppression of viscous fingering in elasticwalled HeleShaw cells
 Draga PihlerPuzović, Raphaël Périllat, Matthew Russell, Anne Juel, Matthias Heil

 Published online by Cambridge University Press:
 14 August 2013, pp. 162183

 Article
 Export citation

Recent experiments by PihlerPuzovic et al. (Phys. Rev. Lett., vol. 108, 2012, article 074502) have shown that the onset of viscous fingering in circular HeleShaw cells in which an air bubble displaces a viscous fluid is delayed considerably when the top boundary of the cell is replaced by an elastic membrane. Nonaxisymmetric instabilities are only observed at much larger flow rates, and the largeamplitude fingers that develop are fundamentally different from the highly branched fingers in rigidwalled cells. We explain the mechanism for the suppression of the instability using a combination of linear stability analysis and direct numerical simulations, based on a theoretical model that couples a depthaveraged lubrication equation for the fluid flow to the Föppl–von Kármán equations, which describe the deformation of the elastic membrane. We show that fluid–structure interaction affects the instability primarily via two changes to the axisymmetric base flow: the axisymmetric inflation of the membrane prior to the onset of any instabilities slows down the expansion of the air bubble and forces the air–liquid interface to propagate into a converging fluidfilled gap. Both of these changes reduce the destabilizing viscous effects that drive the fingering instability in a rigidwalled cell. In contrast, capillary effects only play a very minor role in the suppression of the instability.
On the quest for a hyperbolic effectivefield model of disperse flows
 Daniel Lhuillier, ChihHao Chang, Theo G. Theofanous

 Published online by Cambridge University Press:
 14 August 2013, pp. 184194

 Article
 Export citation

The cornerstone of multiphase flow applications in engineering practice is a scientific construct that translates the basic laws of fluid mechanics into a set of governing equations for effective interpenetrating continua, the effectivefield (or twofluid) model. Over more than half a century of development this model has taken many forms but all of them fail in a way that was known from the very beginning: mathematical illposedness. The aim of this paper is to refocus awareness of this problem from a unified fundamental perspective that clarifies the manner in which such failures took place and to suggest the means for a final closure.
Mixing in a vortex breakdown flow
 P. Meunier, K. Hourigan

 Published online by Cambridge University Press:
 14 August 2013, pp. 195222

 Article
 Export citation

In this paper we present experimental and theoretical results on the mixing inside a cylinder with a rotating lid. The helical flow that is created by the rotation of the disc is well known to exhibit a vortex breakdown bubble over a finite range of Reynolds numbers. The mixing properties of the flow are analysed quantitatively by measuring the exponential decay of the variance as a function of time. This homogenization time is extremely sensitive to the asymmetries of the flow, which are introduced by tilting the rotating or the stationary disc and accurately measured using particle image velocimetry (PIV). In the absence of vortex breakdown, the homogenization time is strongly decreased (by a factor of 10) with only a moderate tilt angle of the rotating lid (of the order of $1{5}^{\circ } $). This phenomenon can be explained by the presence of small radial jets at the periphery which create a strong convective mixing. A simple model of exchange flow between the periphery and the bulk correctly predicts the scaling laws for the homogenization time. In the presence of vortex breakdown, the scalar is trapped inside the vortex breakdown bubble, and thus increases substantially the time needed for homogenization. Curiously, the tilt of the rotating lid has a weak effect on the mixing, but a small tilt of the stationary disc (of the order of ${2}^{\circ } $) strongly decreases (by a factor of 10) the homogenization time. Even more surprising is that the homogenization time diverges when the size of the bubble vanishes. All of these features are recovered by applying the Melnikov theory to calculate the volume of the lobes that exit the bubble. It is the first time that this technique has been applied to a threedimensional stationary flow with a nonaxisymmetric perturbation and compared with experimental results, although it has been applied often to twodimensional flows with a periodic perturbation.
Towards modular analysis of tropicalcyclone structure: the boundary layer
 Francis Fendell, Paritosh Mokhasi

 Published online by Cambridge University Press:
 14 August 2013, pp. 223258

 Article
 Export citation

In the early 1970s, George Carrier and coworkers undertook a modular approach to modelling the internal thermofluiddynamics of tropical cyclones of tropicaldepressionorgreater intensity. A novel, relatively simplistic, approximate analysis of the vortex, idealized as axisymmetric, was carried out in the asymptotic limit of large Reynolds number, so that inviscid and diffusive subdomains of the structure were distinguished. Little subsequent work has followed this line of investigation. The indifference has proven problematic because accurate prediction of tropicalcycling intensity remains a challenge for operational forecasting, despite decades of effort at direct integration of comprehensive boundary/initialvalue formulations. A contributing factor is that, to achieve solution in real time, such computational treatment of the entire vortex invariably resorts to coarse gridding, and key features remain inadequately resolved. Accordingly, here the modular approach is revisited, with the assistance of: recent observational insights; greatly enhanced computerprocessing power; and convenient computational software, which facilitates implementation of a semianalytic, seminumerical methodology. Focus is largely, but not exclusively, on the dynamics and energetics occurring in the nominally kilometrethick, oceansurfacecontiguous boundary layer, especially on influx to the boundary layer and efflux therefrom. The modular approach not only permits the boundary layer, which develops its own highly significant substructure under the highspeed portion of the inviscid vortex, to be wellresolved, but also allows the layer to be investigated in the context of the other tropicalcyclonestructure subdivisions.
Turbulent transport of a highSchmidtnumber scalar near an air–water interface
 Evan A. Variano, Edwin A. Cowen

 Published online by Cambridge University Press:
 14 August 2013, pp. 259287

 Article
 Export citation

We measure solute transport near a turbulent air–water interface at which there is zero mean shear. The interface is stirred by highReynoldsnumber homogeneous isotropic turbulence generated far below the surface, and solute transport into the water is driven by an imposed concentration gradient. The air–water interface is held at a constant concentration much higher than that in the bulk of the water by maintaining pure ${\mathrm{CO} }_{2} $ gas above a water tank that has been initially purged of dissolved ${\mathrm{CO} }_{2} $. We measure velocity and concentration fluctuations below the air–water interface, from the viscous sublayer to the middle of the ‘source region’ where the effects of the surface are first felt. Our laboratory measurement technique uses quantitative imaging to collect simultaneous concentration and velocity fields, which are measured at a resolution that reveals the dynamics in the turbulent inertial subrange. Twopoint statistics reveal the spatial structure of velocity and concentration fluctuations, and are examined as a function of depth beneath the air–water interface. There is a clear dominance of large scales at all depths for all quantities, but the relative importance of scales changes markedly with proximity to the interface. Quadrant analysis of the turbulent scalar flux shows a fourway balance of flux components far from the interface, which near the interface evolves into a twoway balance between motions that are raising and lowering parcels of lowconcentration fluid.
Transition delay in a boundary layer flow using active control
 Onofrio Semeraro, Shervin Bagheri, Luca Brandt, Dan S. Henningson

 Published online by Cambridge University Press:
 15 August 2013, pp. 288311

 Article

 You have access Access
 Open access
 Export citation

Active linear control is applied to delay the onset of laminar–turbulent transition in the boundary layer over a flat plate. The analysis is carried out by numerical simulations of the nonlinear, transitional regime. A threedimensional, localized initial condition triggering Tollmien–Schlichting waves of finite amplitude is used to numerically simulate the transition to turbulence. Linear quadratic Gaussian controllers based on reducedorder models of the linearized Navier–Stokes equations are designed, where the wall sensors and the actuators are localized in space. A parametric analysis is carried out in the nonlinear regime, for different disturbance amplitudes, by investigating the effects of the actuation on the flow due to different distributions of the localized actuators along the spanwise direction, different sizes of the actuators and the effort of the controllers. We identify the range of parameters where the controllers are effective and highlight the limits of the device for high amplitudes and strong control action. Despite the fully linear control approach, it is shown that the device is effective in delaying the onset of laminar–turbulent transition in the presence of packets characterized by amplitudes $a\approx 1\hspace{0.167em} \% $ of the free stream velocity at the actuator location. Up to these amplitudes, it is found that a proper choice of the actuators positively affects the performance of the controller. For a transitional case, $a\approx 0. 20\hspace{0.167em} \% $, we show a transition delay of $\Delta {\mathit{Re}}_{x} = 3. 0\times 1{0}^{5} $.
Role of coherent structures in multiple selfsimilar states of turbulent planar wakes
 JeanPierre Hickey, Fazle Hussain, Xiaohua Wu

 Published online by Cambridge University Press:
 22 August 2013, pp. 312363

 Article
 Export citation

We study the nature of archetypal, incompressible, planar splitterplate wakes, specifically the effects of the exit boundary layer state on multiple approximate selfsimilarity. Temporally developing direct numerical simulations, at a Reynolds number of 1500 based on the volumeflux defect, are performed to investigate three distinct wake evolution scenarios: Kelvin–Helmholtz transition, bypass transition in an asymmetric wake, and an initially fully turbulent wake. The differences in the evolution and farwake statistics are analysed in detail. The individual approximately selfsimilar states exhibit a relative variation of up to 48 % in the spread rate, in secondorder statistics, and in peak values of the energy budget terms. The multiplicity of selfsimilar states is tied to the nonuniversality of the largescale coherent structures. These structures maintain the memory of the initial conditions. In the far wake, two distinct spanwisecoherent motions are identified: (i) staggered, segregated spanwise rollers on either side of the centreplane, dominant in wakes transitioning via antisymmetric instability modes; and, (ii) larger spanwise rollers spanning across the centreplane, emerging in the absence of a nearwake characteristic length scale. The latter structure is characterized by strong spanwise coherence, crosswake velocity correlations and a larger entrainment rate caused by deep pockets of irrotational fluid within the folds of the turbulent/nonturbulent interface. The midsized structures, primarily vortical rods, are generic for all initial conditions and are inclined at ∼$\pm 3{3}^{\circ } $ to the downstream, shallower than the preferential $\pm 4{5}^{\circ } $ inclination of the vorticity vector. The spread rate is driven by the innerwake dynamics, more specifically the advective flux of spanwise vorticity across the centreplane, which depends on the largescale coherent structures.
Lowdimensional models for compressible temporally developing shear layers
 Bashar R. Qawasmeh, Mingjun Wei

 Published online by Cambridge University Press:
 15 August 2013, pp. 364393

 Article
 Export citation

A methodology to achieve extremelylowdimensional models for temporally developing shear layers is extended from incompressible flows to weakly compressible flows. The key idea is to first remove the slow variation (i.e. viscous growth of shear layers) through symmetry reduction, so that the model reduction using proper orthogonal decomposition (POD)Galerkin projection in the symmetryreduced space becomes more efficient. However, for the approach to work for compressible flows, thermodynamic variables need to be retained. We choose the isentropic Navier–Stokes equations for the simplicity and the availability of a welldefined inner product for total energy. To capture basic dynamics, the compressible lowdimensional model requires only two POD modes for each frequency. Thus, a twomode model is capable of representing singlefrequency dynamics such as vortex rollup, and a fourmode model is capable of representing the nonlinear dynamics involving a fundamental frequency and its subharmonic, such as vortex pairing and merging. The compressible model shows similar behaviour and accuracy as the incompressible model. However, because of the consistency of the inner product defined for POD and for projection in the current compressible model, the orthogonality is kept and it results in simple formulation. More importantly, the inclusion of compressibility opens an entirely new avenue for the discussion of compressibility effect and possible description of aeroacoustics and thermodynamics. Finally, the model is extended to different flow parameters without additional numerical simulation. The extension of the compressible fourmode model includes different Mach numbers and Reynolds numbers. We can clearly observe the change in the nonlinear interaction of modes at two frequencies and the associated promotion or delay of vortex pairing by varying compressibility and viscosity. The dynamic response of the lowdimensional model to different flow parameters is consistent with the vortex dynamics observed in experiments and numerical simulation.
Riccatiless approach for optimal control and estimation: an application to twodimensional boundary layers
 Onofrio Semeraro, Jan O. Pralits, Clarence W. Rowley, Dan S. Henningson

 Published online by Cambridge University Press:
 15 August 2013, pp. 394417

 Article

 You have access Access
 Open access
 Export citation

The control of Tollmien–Schlichting waves in a twodimensional boundary layer is analysed using numerical simulations. Fulldimensional optimal controllers are used in combination with a setup of spatially localized inputs (actuator and disturbance) and outputs (sensors). The adjoint of the directadjoint (ADA) algorithm, recently proposed by Pralits & Luchini (In Seventh IUTAM Symposium on Laminar–Turbulent Transition (ed. P. Schlatter & D. S. Henningson), vol. 18, 2010, Springer), is used to efficiently compute an optimal controller known as a linear quadratic regulator; the method is iterative and allows one to bypass the solution of the corresponding Riccati equation, which is infeasible for highdimensional systems. We show that an analogous iteration can be made for the estimation problem; the dual algorithm is referred to as adjoint of the adjointdirect (AAD). By combining the solutions of the estimation and control problem, fulldimensional linear quadratic Gaussian controllers are obtained and used for the attenuation of the disturbances arising in the boundary layer flow. The fulldimensional controllers turn out to be an excellent benchmark for evaluating the performance of the optimal control/estimation design based on reducedorder models. We show under which conditions the two strategies are in perfect agreement by focusing on the issues arising when feedback configurations are considered. An analysis of the finiteamplitude disturbances is also carried out by addressing the limitations of the optimal controllers, the role of the estimation, and the robustness to the nonlinearities arising in the flow of the control design.
Layering and turbulence surrounding an anticyclonic oceanic vortex: in situ observations and quasigeostrophic numerical simulations
 Bach Lien Hua, Claire Ménesguen, Sylvie Le Gentil, Richard Schopp, Bruno Marsset, Hidenori Aiki

 Published online by Cambridge University Press:
 21 August 2013, pp. 418442

 Article
 Export citation

Evidence of persistent layering, with a vertical stacking of sharp variations in temperature, has been presented recently at the vertical and lateral periphery of energetic oceanic vortices through seismic imaging of the water column. The stacking has vertical scales ranging from a few metres up to 100 m and a lateral spatial coherence of several tens of kilometres comparable with the vortex horizontal size. Inside this layering, in situ data display a $[{ k}_{h}^{ 5/ 3} { k}_{h}^{ 2} ] $ scaling law of horizontal scales for two different quantities, temperature and a proxy for its vertical derivative, but for two different ranges of wavelengths, between 5 and 50 km for temperature and between 500 m and 5 km for its vertical gradient. In this study, we explore the dynamics underlying the layering formation mechanism, through the slow dynamics captured by quasigeostrophic equations. Threedimensional highresolution numerical simulations of the destabilization of a lensshaped vortex confirm that the vertical stacking of sharp jumps in density at its periphery is the threedimensional analogue of the preferential windup of potential vorticity near a critical radius, a phenomenon which has been documented for barotropic vortices. For a smallBurger (flat) lens vortex, baroclinic instability ensures a sustained growth rate of sharp jumps in temperature near the critical levels of the leading unstable modes. Such results can be obtained for a background stratification which is due to temperature only and does not require the existence of salt anomalies. Aloft and beneath the vortex core, numerical simulations well reproduce the $[{ k}_{h}^{ 5/ 3} { k}_{h}^{ 2} ] $ scaling law of horizontal scales for the vertical derivative of temperature that is observed in situ inside the layering, whatever the background stratification. Such a result stems from the tracerlike behaviour of the vortex stretching component and previous studies have shown that spectra of tracer fields can be steeper than $ 1$, namely in $ 5/ 3$ or $ 2$, if the advection field is very compact spatially, with a $ 5/ 3$ slope corresponding to a spiral advection of the tracer. Such a scaling law could thus be of geometric origin. As for the kinetic and potential energy, the ${ k}_{h}^{ 5/ 3} $ scaling law can be reproduced numerically and is enhanced when the background stratification profile is strongly variable, involving sharp jumps in potential vorticity such as those observed in situ. This raises the possibility of another plausible mechanism leading to a $ 5/ 3$ scaling law, namely surfacequasigeostrophic (SQG)like dynamics, although our setup is more complex than the idealized SQG framework. Energy and enstrophy fluxes have been diagnosed in the numerical quasigeostrophic simulations. The results emphasize a strong production of energy in the oceanic submesoscales range and a kinetic and potential energy flux from mesoscale to submesoscales range near the critical levels. Such horizontal submesoscale production, which is correlated to the accumulation of thin vertical scales inside the layering, thus has a significant slow dynamical component, wellcaptured by quasigeostrophy.
On the nonlinear destabilization of stably stratified shear flow
 Nadia Mkhinini, Thomas Dubos, Philippe Drobinski

 Published online by Cambridge University Press:
 15 August 2013, pp. 443460

 Article
 Export citation

A weakly nonlinear analysis of the bifurcation of the stratified Ekman boundarylayer flow near a critical bulk Richardson number is conducted and compared to a similar analysis of a continuously stratified parallel shear flow subject to Kelvin–Helmholtz instability. Previous work based on asymptotic expansions and predicting supercritical bifurcation at Prandtl number $Pr\lt 1$ and subcritical bifurcation at $Pr\gt 1$ for the parallel base flow is confirmed numerically and through fully nonlinear temporal simulations. When applied to the nonparallel Ekman flow, weakly nonlinear analysis and fully nonlinear calculations confirm that the nature of the bifurcation is dominantly controlled by $Pr$, although a sharp threshold at $Pr= 1$ is not found. In both flows the underlying physical mechanism is that the mean flow adjusts so as to induce a viscous (respectively diffusive) flux of momentum (respectively buoyancy) that balances the vertical flux induced by the developing instability, leading to a weakening of the mean shear and mean stratification. The competition between the former nonlinear feedback, which tends to be stabilizing, and the latter, which is destabilizing and strongly amplified as $Pr$ increases, determines the supercritical or subcritical character of the bifurcation. That essentially the same competition is at play in both the parallel shear flow and the Ekman flow suggests that the underlying mechanism is valid for complex, nonparallel stratified shear flows.
The effect of small viscosity and diffusivity on the marginal stability of stably stratified shear flows
 S. A. Thorpe, W. D. Smyth, Lin Li

 Published online by Cambridge University Press:
 15 August 2013, pp. 461476

 Article
 Export citation

The effect of nonzero, but small, viscosity and diffusivity on the marginal stability of a stably stratified shear flow is examined by making perturbations around the neutral solution for an inviscid and nondiffusive flow. The results apply to turbulent flows in which horizontal and vertical turbulent transports of momentum and buoyancy are represented by eddy coefficients of viscosity and diffusivity that vary in the vertical ($z$) direction. General expressions are derived for the modified phase speed and the growth rate of small disturbances as a function of wavenumber. To first order in their coefficients, the effect on the phase speed of adding viscosity and diffusivity is zero. Growth rates are found for two mean flows when the horizontal or vertical coefficients of viscosity and diffusivity vary in $z$ in such a way that the rates can be found analytically. The first flow, denoted as a ‘Holmboe flow’, has a velocity and density interface: the mean horizontal velocity and the density are both proportional to $\tanh az$, where $a$ is proportional to the inverse of the interface thickness. The second, ‘Drazin flow’, has a similar velocity variation in $z$ but uniform density gradient. The analytical results compare favourably with numerical calculations. Small horizontal coefficients of viscosity and diffusivity may affect disturbances to the flow in opposite ways. Although the effect of uniform vertical coefficients of viscosity is to decrease the growth rates, and uniform vertical coefficients of diffusivity increase them, cases are found in which, with suitably chosen $z$ dependence, vertical coefficients of viscosity (or diffusivity) may cause a previously neutral disturbance to grow (or to diminish); viscosity may destabilize a stably stratified shear flow. The introduction of viscosity and diffusivity may consequently increase the critical Richardson number to a value exceeding $1/ 4$. While some patterns of behaviour are apparent, no simple rule appears to hold about whether flows that are neutral in the absence of these effects (viscosity or diffusivity) will be stabilized or destabilized when they are added. One such rule, namely our conjecture that viscosity is always stabilizing and that diffusivity is destabilizing, is explicitly refuted.
Entraining gravity currents
 Christopher G. Johnson, Andrew J. Hogg

 Published online by Cambridge University Press:
 19 August 2013, pp. 477508

 Article
 Export citation

Entrainment of ambient fluid into a gravity current, while often negligible in laboratoryscale flows, may become increasingly significant in largescale natural flows. We present a theoretical study of the effect of this entrainment by augmenting a shallow water model for gravity currents under a deep ambient with a simple empirical model for entrainment, based on experimental measurements of the fluid entrainment rate as a function of the bulk Richardson number. By analysing longtime similarity solutions of the model, we find that the decrease in entrainment coefficient at large Richardson number, due to the suppression of turbulent mixing by stable stratification, qualitatively affects the structure and growth rate of the solutions, compared to currents in which the entrainment is taken to be constant or negligible. In particular, mixing is most significant close to the front of the currents, leading to flows that are more dilute, deeper and slower than their nonentraining counterparts. The longtime solution of an inviscid entraining gravity current generated by a lockrelease of dense fluid is a similarity solution of the second kind, in which the current grows as a power of time that is dependent on the form of the entrainment law. With an entrainment law that fits the experimental measurements well, the length of currents in this entraining inviscid regime grows with time approximately as ${t}^{0. 447} $. For currents instigated by a constant buoyancy flux, a different solution structure exists in which the current length grows as ${t}^{4/ 5} $. In both cases, entrainment is most significant close to the current front.
A weighted residual method for twolayer nonNewtonian channel flows: steadystate results and their stability
 K. Alba, S. M. Taghavi, I. A. Frigaard

 Published online by Cambridge University Press:
 28 August 2013, pp. 509544

 Article
 Export citation

We study buoyant displacement flows in a plane channel with two fluids in the longwavelength limit in a stratified configuration. Weak inertial effects are accounted for by developing a weighted residual method. This gives a firstorder approximation to the interface height and flux functions in each layer. As the fluids are shearthinning and have a yield stress, to retain a formulation that can be resolved analytically requires the development of a system of special functions for the weight functions and various integrals related to the base flow. For displacement flows, the addition of inertia can either slightly increase or decrease the speed of the leading displacement front, which governs the displacement efficiency. A more subtle effect is that a wider range of interface heights are stretched between advancing fronts than without inertia. We study stability of these systems via both a linear temporal analysis and a numerical spatiotemporal method. To start with, the Orr–Sommerfeld equations are first derived for two generalized nonNewtonian fluids satisfying the Herschel–Bulkley model, and analytical expressions for growth rate and wave speed are obtained for the longwavelength limit. The predictions of linear analysis based on the weighted residual method shows excellent agreement with the Orr–Sommerfeld approach. For displacement flows in unstable parameter ranges we do observe growth of interfacial waves that saturate nonlinearly and disperse. The observed waves have similar characteristics to those observed experimentally in pipe flow displacements. Although the focus in this study is on displacement flows, the formulation laid out can be easily used for similar twolayer flows, e.g. coextrusion flows.