15 results
Short-wavelength instabilities in a finite-amplitude plane inertial wave
- D.S. Abhiram, Manikandan Mathur
-
- Journal:
- Journal of Fluid Mechanics / Volume 982 / 10 March 2024
- Published online by Cambridge University Press:
- 13 March 2024, A22
-
- Article
-
- You have access Access
- Open access
- HTML
- Export citation
-
We perform a linear stability analysis of a finite-amplitude plane inertial wave (of frequency $\omega$ in the range $0\le \omega \le f$, where $f$ is the Coriolis frequency) by considering the inviscid evolution of three-dimensional (3-D), small-amplitude, short-wavelength perturbations. Characterizing the base flow plane inertial wave by its non-dimensional amplitude $A$ and the angle $\varPhi$ that its wavevector makes with the horizontal axis, the local stability equations are solved over the entire range of perturbation wavevector orientations. At sufficiently small $A$, 3-D parametric subharmonic instability (PSI) is the only instability mechanism, with the most unstable perturbation wavevector making an angle close to $60^{\circ }$ with the inertial wave plane. In addition, the most unstable perturbation is shear-aligned with the inertial wave in the inertial wave plane. Further, at large $\varPhi$, i.e. $\omega \approx f,$ there exists a wide range of perturbation wavevectors whose growth rate is comparable to the maximum growth rate. As $A$ is increased, theoretical PSI estimates become less relevant in describing the instability characteristics, and the dominant instability transitions to a two-dimensional (2-D) shear-aligned instability, which is shown to be driven by third-order resonance. The transition from 3-D PSI to a 2-D shear-aligned instability is shown to be reasonably captured by two different criteria, one based on the nonlinear time scale in the inertial wave and the other being a Rossby-number-based one.
Local stability analysis of homogeneous and stratified Kelvin–Helmholtz vortices
- H.M. Aravind, Thomas Dubos, Manikandan Mathur
-
- Journal:
- Journal of Fluid Mechanics / Volume 943 / 25 July 2022
- Published online by Cambridge University Press:
- 09 June 2022, A18
-
- Article
-
- You have access Access
- Open access
- HTML
- Export citation
-
We perform a three-dimensional short-wavelength linear stability analysis of numerically simulated two-dimensional Kelvin–Helmholtz vortices in homogeneous and stratified environments at a fixed Reynolds number of $Re = 300$. For the homogeneous case, the elliptic instability at the vortex core dominates at early times, before being taken over by the hyperbolic instability at the vortex edge. For the stratified case of Richardson number $ Ri = 0.08$, the early-time instabilities comprise a dominant elliptic instability at the core and a hyperbolic instability influenced strongly by stratification at the vortex edge. At intermediate times, the local approach shows a new branch of (convective) instability that emerges at the vortex core and subsequently moves towards the vortex edge. A few more convective instability bands appear at the vortex core and move away, before coalescing to form the most unstable region inside the vortex periphery at large times. In addition, the stagnation point instability is also recovered outside the periphery of the vortex at intermediate times. The dominant instability characteristics from the local approach are shown to be in good qualitative agreement with the results based on global instability studies for both homogeneous and stratified cases. A systematic study of the dependence of the dominant instability characteristics on $ Ri $ is then presented. While $ Ri = 0.1$ is identified as most unstable (with convective instability being dominant), another growth rate maximum at $ Ri = 0.025$ is not far behind (with the hyperbolic instability influenced by stratification being dominant). Finally, the local stability approach is shown to predict the potential orientation of the flow structures that would result from hyperbolic and convective instabilities, which is found to be consistent with three-dimensional numerical simulations reported previously.
On separating plumes from boundary layers in turbulent convection
- Prafulla P. Shevkar, R. Vishnu, Sanal K. Mohanan, Vipin Koothur, Manikandan Mathur, Baburaj A. Puthenveettil
-
- Journal:
- Journal of Fluid Mechanics / Volume 941 / 25 June 2022
- Published online by Cambridge University Press:
- 25 April 2022, A5
-
- Article
-
- You have access Access
- Open access
- HTML
- Export citation
-
We present a simple, novel kinematic criterion – that uses only the horizontal velocity fields and is free of arbitrary thresholds – to separate line plumes from local boundary layers in a plane close to the hot plate in turbulent convection. We first show that the horizontal divergence of the horizontal velocity field ($\boldsymbol {\nabla _H} \boldsymbol {\cdot } \boldsymbol {u}$) has negative and positive values in two-dimensional (2D), laminar similarity solutions of plumes and boundary layers, respectively. Following this observation, based on the understanding that fluid elements predominantly undergo horizontal shear in the boundary layers and vertical shear in the plumes, we propose that the dominant eigenvalue ($\lambda _D$) of the 2D strain rate tensor is negative inside the plumes and positive inside the boundary layers. Using velocity fields from our experiments, we then show that plumes can indeed be extracted as regions of negative $\lambda _D$, which are identical to the regions with negative $\boldsymbol {\nabla _H} \boldsymbol {\cdot } \boldsymbol {u}$. Exploring the connection of these plume structures to Lagrangian coherent structures (LCS) in the instantaneous limit, we show that the centrelines of such plume regions are captured by attracting LCS that do not have dominant repelling LCS in their vicinity. Classifying the flow near the hot plate based on the distribution of eigenvalues of the 2D strain rate tensor, we then show that the effect of shear due to the large-scale flow is felt more in regions close to where the local boundary layers turn into plumes. The lengths and areas of the plume regions, detected by the $\boldsymbol {\nabla _H}\boldsymbol {\cdot }\boldsymbol {u}$ criterion applied to our experimental and computational velocity fields, are then shown to agree with our theoretical estimates from scaling arguments. Using velocity fields from numerical simulations, we then show that the $\boldsymbol {\nabla _H}\boldsymbol {\cdot }\boldsymbol {u}$ criterion detects all the upwellings, while the available criteria based on temperature and flux thresholds miss some of these upwellings. The plumes detected by the $\boldsymbol {\nabla _H}\boldsymbol {\cdot }\boldsymbol {u}$ criterion are also shown to be thicker at Prandtl numbers ($Pr$) greater than one, expectedly so, due to the thicker velocity boundary layers of the plumes at $Pr>1$.
Triadic resonances in internal wave modes with background shear
- Ramana Patibandla, Manikandan Mathur, Anubhab Roy
-
- Journal:
- Journal of Fluid Mechanics / Volume 929 / 25 December 2021
- Published online by Cambridge University Press:
- 19 October 2021, A10
-
- Article
-
- You have access Access
- Open access
- HTML
- Export citation
-
In this paper, we use asymptotic theory and numerical methods to study resonant triad interactions among discrete internal wave modes at a fixed frequency ($\omega$) in a two-dimensional, uniformly stratified shear flow. Motivated by linear internal wave generation mechanisms in the ocean, we assume the primary wave field as a linear superposition of various horizontally propagating vertical modes at a fixed frequency $\omega$. The weakly nonlinear solution associated with the primary wave field is shown to comprise superharmonic (frequency $2\omega$) and zero frequency wave fields, with the focus of this study being on the former. When two interacting primary modes $m$ and $n$ are in triadic resonance with a superharmonic mode $q$, it results in the divergence of the corresponding superharmonic secondary wave amplitude. For a given modal interaction $(m, n)$, the superharmonic wave amplitude is plotted on the plane of primary wave frequency $\omega$ and Richardson number $Ri$, and the locus of divergence locations shows how the resonance locations are influenced by background shear. In the limit of weak background shear ($Ri\to \infty$), using an asymptotic theory, we show that the horizontal wavenumber condition $k_m + k_n = k_q$ is sufficient for triadic resonance, in contrast to the requirement of an additional vertical mode number condition ($q = |m-n|$) in the case of no shear. As a result, the number of resonances for an arbitrarily weak shear is significantly larger than that for no shear. The new resonances that occur in the presence of shear include the possibilities of resonance due to self-interaction and resonances that occur at the seemingly trivial limit of $\omega \approx 0$, both of which are not possible in the no shear limit. Our weak shear limit conclusions are relevant for other inhomogeneities such as non-uniformity in stratification as well, thus providing an understanding of several recent studies that have highlighted superharmonic generation in non-uniform stratifications. Extending our study to finite shear (finite $Ri$) in an ocean-like exponential shear flow profile, we show that for cograde–cograde interactions, a significant number of divergence curves that start at $Ri\to \infty$ will not extend below a cutoff $Ri$$\sim O(1)$. In contrast, for retrograde–retrograde interactions, the divergence curves extend all the way from $Ri\to \infty$ to $Ri = 0.5$. For mixed interactions, new divergence curves appear at $\omega = 0$ for $Ri\sim O(10)$ and extend to other primary wave frequencies for smaller $Ri$. Consequently, the total ($\text {cograde} + \text {retrograde} + \text {mixed}$) number of resonant triads is of the same order for small $Ri\approx 0.5$ as in the limit of weak shear ($Ri\to \infty$), although it attains a maximum at $Ri\sim O(10)$.
Diffusive effects in local instabilities of a baroclinic axisymmetric vortex
- Suraj Singh, Manikandan Mathur
-
- Journal:
- Journal of Fluid Mechanics / Volume 928 / 10 December 2021
- Published online by Cambridge University Press:
- 15 October 2021, A14
-
- Article
-
- You have access Access
- Open access
- HTML
- Export citation
-
We present a local stability analysis of an idealized model of the stratified vortices that appear in geophysical settings. The base flow comprises an axisymmetric vortex with background rotation and an out-of-plane stable stratification, and a radial stratification in the thermal wind balance with the out-of-plane momentum gradient. Solving the local stability equations along fluid particle trajectories in the base flow, the dependence of short-wavelength instabilities on the Schmidt number $Sc$ (ratio between momentum and mass diffusivities) is studied, in the presence of curvature effects. In the diffusion-free limit, the well-known symmetric instability is recovered. In the viscous, double-diffusive regime, instability characteristics are shown to depend on three non-dimensional parameters (including $Sc$), and two different instabilities are identified: (i) a monotonic instability (same as symmetric instability at $Sc = 1$), and (ii) an oscillatory instability (absent at $Sc = 1$). Separating the base flow and perturbation characteristics, two each of base flow and perturbation parameters (apart from $Sc$) are identified, and the entire parameter space is explored for the aforementioned instabilities. In comparison with $Sc = 1$, monotonic and oscillatory instabilities are shown to significantly expand the instability region in the space of base flow parameters as $Sc$ moves away from unity. Neutral stability boundaries on the plane of $Sc$ and a modified gradient Richardson number are then identified for both these instabilities. In the absence of curvature effects, our results are shown to be consistent with previous studies based on normal mode analysis, thus establishing that the local stability approach is well suited to capturing symmetric and double-diffusive instabilities. The paper concludes with a discussion of curvature effects, and the likelihood of monotonic and oscillatory instabilities in typical oceanic settings.
Experimental study of the effects of droplet number density on turbulence-driven polydisperse droplet size growth
- M. Shyam Kumar, Manikandan Mathur, S.R. Chakravarthy
-
- Journal:
- Journal of Fluid Mechanics / Volume 917 / 25 June 2021
- Published online by Cambridge University Press:
- 23 April 2021, A12
-
- Article
- Export citation
-
Interaction of polydisperse droplets in a turbulent air flow features prominently in a wide range of phenomena, such as warm rain initiation as an example. In the current study, we present an experimental investigation on the effects of initial droplet field characteristics on the maximum droplet size growth. By performing experiments in a vertically oriented air flow facility, the air flow turbulence was able to be controlled through the mean flow velocity and an active turbulence generator. The initial droplet field characteristics (droplet diameter range of 0–120 $\mathrm {\mu }$m) were varied using spray nozzles of different flow numbers. Based on quantitative measurements of the droplet size distribution at various spatial locations using phase Doppler interferometry (PDI), we estimated the droplet size growth rate $R$ as a function of turbulence intensity $I$, initial droplet number density $\rho _N$ and initial mean droplet size $\bar {D}$. For each ($\rho _N$, $\bar {D}$), we observed the occurrence of an optimum turbulence intensity $I^*$, with the corresponding maximum droplet size growth rate being $R^*$. Two different trends were observed. When $\rho _N$ and $\bar {D}$ were simultaneously increased and decreased, respectively, their competing influences resulted in small variations in $R^*$. In contrast, when $\bar {D}$ was held constant with a corresponding Stokes number $St$ smaller than unity, there existed a threshold $\rho _N$ above which $R^*$ increased rapidly with $\rho _N$. These trends were then understood through long-distance microscopy (LDM) measurements. Beyond the aforementioned threshold $\rho _N$, the fraction of uncorrelated small-sized $(St<1)$ droplet pairs was found to rapidly increase with $\rho _N$. Further detailed analysis of droplet tracking in the LDM images identified that the velocity fluctuations in the small-sized droplet pairs being induced by close encounters with inertial droplets was the underlying mechanism for the rapid increase of $R^*$ with $\rho _N$. This mechanism potentially explains how droplet collisions can be enhanced in small droplets if the droplet field is sufficiently polydisperse.
Effects of Schmidt number on the short-wavelength instabilities in stratified vortices
- Suraj Singh, Manikandan Mathur
-
- Journal:
- Journal of Fluid Mechanics / Volume 867 / 25 May 2019
- Published online by Cambridge University Press:
- 28 March 2019, pp. 765-803
-
- Article
- Export citation
-
We present a local stability analysis to investigate the effects of differential diffusion between momentum and density (quantified by the Schmidt number $Sc$) on the three-dimensional, short-wavelength instabilities in planar vortices with a uniform stable stratification along the vorticity axis. Assuming small diffusion in both momentum and density, but arbitrary values for $Sc$, we present a detailed analytical/numerical analysis for three different classes of base flows: (i) an axisymmetric vortex, (ii) an elliptical vortex and (iii) the flow in the neighbourhood of a hyperbolic stagnation point. While a centrifugally stable axisymmetric vortex remains stable for any $Sc$, it is shown that $Sc$ can have significant effects in a centrifugally unstable axisymmetric vortex: the range of unstable perturbations increases when $Sc$ is taken away from unity, with the extent of increase being larger for $Sc\ll 1$ than for $Sc\gg 1$. Additionally, for $Sc>1$, we report the possibility of oscillatory instability. In an elliptical vortex with a stable stratification, $Sc\neq 1$ is shown to non-trivially influence the three different inviscid instabilities (subharmonic, fundamental and superharmonic) that have been previously reported, and also introduce a new branch of oscillatory instability that is not present at $Sc=1$. The unstable parameter space for the subharmonic (instability IA) and fundamental (instability IB) inviscid instabilities are shown to be significantly increased for $Sc<1$ and $Sc>1$, respectively. Importantly, for sufficiently small and large $Sc$, respectively, the maximum growth rate for instabilities IA and IB occurs away from the inviscid limit. The new oscillatory instability (instability III) is shown to occur only for sufficiently small $Sc<1$, the signature of which is nevertheless present with zero growth rate in the inviscid limit. The Schmidt number is then shown to play no role in the evolution of transverse perturbations on the flow around a hyperbolic stagnation point with a stable stratification. We conclude by discussing the physical length scales associated with the $Sc\neq 1$ instabilities, and their potential relevance in various realistic settings.
Three-dimensional small-scale instabilities of plane internal gravity waves
- Sasan John Ghaemsaidi, Manikandan Mathur
-
- Journal:
- Journal of Fluid Mechanics / Volume 863 / 25 March 2019
- Published online by Cambridge University Press:
- 29 January 2019, pp. 702-729
-
- Article
- Export citation
-
We study the evolution of three-dimensional (3-D), small-scale, small-amplitude perturbations on a plane internal gravity wave using the local stability approach. The plane internal wave is characterised by its non-dimensional amplitude, $A$, and the angle the group velocity vector makes with gravity, $\unicode[STIX]{x1D6F7}$. For a given $(A,\unicode[STIX]{x1D6F7})$, the local stability equations are solved on the periodic fluid particle trajectories to obtain growth rates for all two-dimensional (2-D) and 3-D perturbation wave vectors. For small $A$, the local stability approach recovers previous results of 2-D parametric subharmonic instability (PSI) while offering new insights into 3-D PSI. Higher-order triadic resonances, and associated deviations from them, are also observed at small $A$. Moreover, for small $A$, purely transverse instabilities resulting from parametric resonance are shown to occur at select values of $\unicode[STIX]{x1D6F7}$. The possibility of a non-resonant instability mechanism for transverse perturbations at finite $A$ allows us to derive a heuristic, modified gravitational instability criterion. We then study the extension of small $A$ to finite $A$ internal wave instabilities, where we recover and build upon existing knowledge of small-scale, small-amplitude internal wave instabilities. Four distinct regions of the $(A,\unicode[STIX]{x1D6F7})$-plane based on the dominant instability modes are identified: 2-D PSI, 3-D oblique, quasi-2-D shear-aligned, and 3-D transverse. Our study demonstrates the local stability approach as a physically insightful and computationally efficient tool, with potentially broad utility for studies that are based on other theoretical approaches and numerical simulations of small-scale instabilities of internal waves in various settings.
The kinematic genesis of vortex formation due to finite rotation of a plate in still fluid
- M. Jimreeves David, Manikandan Mathur, R. N. Govardhan, J. H. Arakeri
-
- Journal:
- Journal of Fluid Mechanics / Volume 839 / 25 March 2018
- Published online by Cambridge University Press:
- 02 February 2018, pp. 489-524
-
- Article
- Export citation
-
We present a combined experimental and numerical study of an idealized model of the propulsive stroke of the turning manoeuvre in fish. Specifically, we use the framework of Lagrangian coherent structures (LCSs) to describe the kinematics of the flow that results from a thin plate performing a large angle rotation about its tip in still fluid. Temporally and spatially well-resolved velocity fields are obtained using a two-dimensional, incompressible finite-volume solver, and are validated by comparisons with experimentally measured velocity fields and alternate numerical simulations. We then implement the recently proposed variational theory of LCSs to extract the hyperbolic and elliptic LCSs in the numerically generated velocity fields. Detailed LCS analysis is performed for a plate motion profile described by $\dot{\unicode[STIX]{x1D703}}(t)=\unicode[STIX]{x1D6FA}_{max}\sin ^{2}(\unicode[STIX]{x1D714}t)$ during $0\leqslant t\leqslant t_{o}$ and zero otherwise. The stopping time $t_{o}$ is given by $t_{o}=\unicode[STIX]{x03C0}/\unicode[STIX]{x1D714}=10~\text{s}$, the value of $\unicode[STIX]{x1D6FA}_{max}$ chosen to give a stopping angle of $\unicode[STIX]{x1D703}_{max}=90^{\circ }$, resulting in a Reynolds number $Re=c^{2}\unicode[STIX]{x1D6FA}_{max}/\unicode[STIX]{x1D708}=785.4$, where $c$ is the plate chord length and $\unicode[STIX]{x1D708}=10^{-6}~\text{m}^{2}~\text{s}^{-1}$ the kinematic viscosity of water. The flow comprises a starting and a stopping vortex, resulting in a pair of oppositely signed vortices of unequal strengths that move away from the plate in a direction closely aligned with the final plate orientation at $t/t_{o}\approx 2$. The hyperbolic LCSs are shown to encompass the fluid material that is advected away from the plate for $t>t_{o}$, henceforth referred to as the advected bulk. The starting and stopping vortices, identified using elliptic LCSs and hence more objective than Eulerian vortex detection methods, constitute only around two thirds of the advected bulk area. The advected bulk is traced back to $t=0$ to identify five distinct lobes of fluid that eventually form the advected bulk, and hence map the long-term fate of various regions in the fluid at $t=0$. The five different lobes of fluid are then shown to be delineated by repelling LCS boundaries at $t=0$. The linear momentum of the advected bulk region is shown to account for approximately half of the total impulse experienced by the plate in the direction of its final orientation, thus establishing its dynamical significance. We provide direct experimental evidence for the kinematic relevance of hyperbolic and elliptic LCSs using novel dye visualization experiments, and also show that attracting hyperbolic LCSs provide objective characterization of the spiral structures often observed in vortical flows. We conclude by showing that qualitatively similar LCSs persist for several other plate motion profiles and stopping angles as well.
Internal wave resonant triads in finite-depth non-uniform stratifications
- Dheeraj Varma, Manikandan Mathur
-
- Journal:
- Journal of Fluid Mechanics / Volume 824 / 10 August 2017
- Published online by Cambridge University Press:
- 05 July 2017, pp. 286-311
-
- Article
- Export citation
-
We present a theoretical study of nonlinear effects that result from modal interactions in internal waves in a non-uniformly stratified finite-depth fluid with background rotation. A linear wave field containing modes $m$ and $n$ (of horizontal wavenumbers $k_{m}$ and $k_{n}$) at a fixed frequency $\unicode[STIX]{x1D714}$ results in two different terms in the steady-state weakly nonlinear solution: (i) a superharmonic wave of frequency $2\unicode[STIX]{x1D714}$, horizontal wavenumber $k_{m}+k_{n}$ and a vertical structure $\bar{h}_{mn}(z)$ and (ii) a time-independent term (Eulerian mean flow) with horizontal wavenumber $k_{m}-k_{n}$. For some $(m,n)$, $\bar{h}_{mn}(z)$ is infinitely large along specific curves on the $(\unicode[STIX]{x1D714}/N_{0},f/\unicode[STIX]{x1D714})$ plane, where $N_{0}$ and $f$ are the deep ocean stratification and the Coriolis frequency, respectively; these curves are referred to as divergence curves in the rest of this paper. In uniform stratifications, a unique divergence curve occurs on the $(\unicode[STIX]{x1D714}/N_{0},f/\unicode[STIX]{x1D714})$ plane for those $(m,n\neq m)$ that satisfy $(m/3)<n<(3m)$. In the presence of a pycnocline (whose strength is quantified by the maximum stratification $N_{max}$), divergence curves occur for several more modal interactions than those for a uniform stratification; furthermore, a given $(m,n)$ interaction can result in multiple divergence curves on the $(\unicode[STIX]{x1D714}/N_{0},f/\unicode[STIX]{x1D714})$ plane for a fixed $N_{max}/N_{0}$. Nearby high-mode interactions in a uniform stratification and any modal interaction in a non-uniform stratification with a sufficiently strong pycnocline are shown to result in near-horizontal divergence curves around $f/\unicode[STIX]{x1D714}\approx 1$, thus implying that strong nonlinear effects often occur as a result of interaction within triads containing two different modes at the near-inertial frequency. Notably, self-interaction of certain modes in a non-uniform stratification results in one or more divergence curves on the $(\unicode[STIX]{x1D714}/N_{0},f/\unicode[STIX]{x1D714})$ plane, thus suggesting that even arbitrarily small-amplitude individual modes cannot remain linear in a non-uniform stratification. We show that internal wave resonant triads containing modes $m$ and $n$ at frequency $\unicode[STIX]{x1D714}$ occur along the divergence curves, and their existence is guaranteed upon the satisfaction of two different criteria: (i) the horizontal component of the standard triadic resonance criterion $\boldsymbol{k}_{1}+\boldsymbol{k}_{2}+\boldsymbol{k}_{3}=0$ and (ii) a non-orthogonality criterion. For uniform stratifications, criterion (ii) reduces to the vertical component of the standard triadic resonance criterion. For non-uniform stratifications, criterion (ii) seems to be always satisfied whenever criterion (i) is satisfied, thus significantly increasing the number of modal interactions that result in strong nonlinear effects irrespective of the wave amplitudes. We then adapt our theoretical framework to identify resonant triads and hence provide insights into the generation of higher harmonics in two different oceanic scenarios: (i) low-mode internal tide propagating over small- or large-scale topography and (ii) an internal wave beam incident on a pycnocline in the upper ocean, for which our results are in qualitative agreement with the numerical study of Diamessis et al. (Dynam. Atmos. Oceans., vol. 66, 2014, pp. 110–137).
Centrifugal instability in non-axisymmetric vortices
- David Nagarathinam, A. Sameen, Manikandan Mathur
-
- Journal:
- Journal of Fluid Mechanics / Volume 769 / 25 April 2015
- Published online by Cambridge University Press:
- 13 March 2015, pp. 26-45
-
- Article
- Export citation
-
We study the centrifugal instability of non-axisymmetric vortices in the presence of an axial flow ($w$) and a background rotation (${\it\Omega}_{z}$) using the local stability approach. Analytically solving the local stability equations for an axisymmetric vortex with $w$ and ${\it\Omega}_{z}$, growth rates for wave vectors that are periodic upon evolution around a closed streamline are calculated. The resulting sufficient criterion for centrifugal instability in an axisymmetric vortex is then heuristically extended to non-axisymmetric vortices and written in terms of integral quantities and their derivatives with respect to the streamfunction on a streamline. The new criterion for non-axisymmetric vortices, which converges to the exact criterion of Bayly (Phys. Fluids, vol. 31, 1988, pp. 56–64) in the absence of background rotation and axial flow, is validated by comparisons with numerically calculated growth rates for two different anticyclonic vortices: the Stuart vortex (specified by the concentration parameter ${\it\rho},~0<{\it\rho}\leqslant 1$) and the Taylor–Green vortex (specified by the aspect ratio $E,~0<E\leqslant 1$). With no axial velocity and finite background rotation, the criterion predicts a lower and an upper threshold of $|{\it\Omega}_{z}|$ between which centrifugal instability is present. We further demonstrate that the criterion represents an improvement over the criterion of Sipp & Jacquin (Phys. Fluids, vol. 12, 2000, pp. 1740–1748). Finally, in the presence of both axial velocity and background rotation, the criterion is shown to be accurate for large enough ${\it\rho}$ and $E$.
Effects of an axial flow on the centrifugal, elliptic and hyperbolic instabilities in Stuart vortices
- Manikandan Mathur, Sabine Ortiz, Thomas Dubos, Jean-Marc Chomaz
-
- Journal:
- Journal of Fluid Mechanics / Volume 758 / 10 November 2014
- Published online by Cambridge University Press:
- 10 October 2014, pp. 565-585
-
- Article
- Export citation
-
Linear stability of the Stuart vortices in the presence of an axial flow is studied. The local stability equations derived by Lifschitz & Hameiri (Phys. Fluids A, vol. 3 (11), 1991, pp. 2644–2651) are rewritten for a three-component (3C) two-dimensional (2D) base flow represented by a 2D streamfunction and an axial velocity that is a function of the streamfunction. We show that the local perturbations that describe an eigenmode of the flow should have wavevectors that are periodic upon their evolution around helical flow trajectories that are themselves periodic once projected on a plane perpendicular to the axial direction. Integrating the amplitude equations around periodic trajectories for wavevectors that are also periodic, it is found that the elliptic and hyperbolic instabilities, which are present without the axial velocity, disappear beyond a threshold value for the axial velocity strength. Furthermore, a threshold axial velocity strength, above which a new centrifugal instability branch is present, is identified. A heuristic criterion, which reduces to the Leibovich & Stewartson criterion in the limit of an axisymmetric vortex, for centrifugal instability in a non-axisymmetric vortex with an axial flow is then proposed. The new criterion, upon comparison with the numerical solutions of the local stability equations, is shown to describe the onset of centrifugal instability (and the corresponding growth rate) very accurately.
Soliton generation by internal tidal beams impinging on a pycnocline: laboratory experiments
- Matthieu J. Mercier, Manikandan Mathur, Louis Gostiaux, Theo Gerkema, Jorge M. Magalhães, José C. B. Da Silva, Thierry Dauxois
-
- Journal:
- Journal of Fluid Mechanics / Volume 704 / 10 August 2012
- Published online by Cambridge University Press:
- 29 June 2012, pp. 37-60
-
- Article
- Export citation
-
In this paper, we present the first laboratory experiments that show the generation of internal solitary waves by the impingement of a quasi-two-dimensional internal wave beam on a pycnocline. These experiments were inspired by observations of internal solitary waves in the deep ocean from synthetic aperture radar (SAR) imagery, where this so-called mechanism of ‘local generation’ was argued to be at work, here in the form of internal tidal beams hitting the thermocline. Nonlinear processes involved here are found to be of two kinds. First, we observe the generation of a mean flow and higher harmonics at the location where the principal beam reflects from the surface and pycnocline; their characteristics are examined using particle image velocimetry (PIV) measurements. Second, we observe internal solitary waves that appear in the pycnocline, detected with ultrasonic probes; they are further characterized by a bulge in the frequency spectrum, distinct from the higher harmonics. Finally, the relevance of our results for understanding ocean observations is discussed.
New wave generation
- MATTHIEU J. MERCIER, DENIS MARTINAND, MANIKANDAN MATHUR, LOUIS GOSTIAUX, THOMAS PEACOCK, THIERRY DAUXOIS
-
- Journal:
- Journal of Fluid Mechanics / Volume 657 / 25 August 2010
- Published online by Cambridge University Press:
- 19 July 2010, pp. 308-334
-
- Article
- Export citation
-
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.
Internal wave beam propagation in non-uniform stratifications
- MANIKANDAN MATHUR, THOMAS PEACOCK
-
- Journal:
- Journal of Fluid Mechanics / Volume 639 / 25 November 2009
- Published online by Cambridge University Press:
- 30 October 2009, pp. 133-152
-
- Article
- Export citation
-
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.