ICTS

In this lecture, I will present in a first part the different types of classical waves and instabilities that can occur in astro and geophysical flows. Inertial waves, caused by the rotation of the fluid, will first be introduced as well as their 2D version called Rossby waves. Then, it will be shown how a density stratification of the fluid can make internal gravity waves appear. In each case and in the case where both rotation and stratification are present, the dispersion relations of the waves will be derived. A direct consequence of the presence of inertial waves in flows is their possible resonance with for instance, the elliptic deformation of the rotating container in which they propagate: this resonance will give rise in this case to the so-called elliptic or tidal instability that may appear in celestial bodies [1]. A differential rotation will then be added on the flow. The classical Rayleigh criterium for the centrifugal instability is naturally recovered in the case of an homogeneous fluid but it will be shown that a new instability, called the strato-rotational instability (SRI), can occur in a Taylor-Couette device when the fluid is stratified [5, 4] (see figure 1-b)). Again, this instability arises because of the resonance of internal gravity waves which are in this case trapped close to the boundaries and Doppler shifted, allowing two counter propagating waves to become stationary and mutually resonant. More generally this wave interaction process identifies a class of instability which is characteristic of shear flows (e.g. [6]) as discovered for instance in the unstratified plane Couette flow in the shallow water approximation [7], or more recently in the stratified plane Couette [8] or in the stratified plane Poiseuille [9]. Finally, it will be shown how the application of a magnetic field can create Alfven waves in a rotating electrically conducting fluid and in which conditions, the magneto-rotational instability (MRI) can grow [10]. This instability is believed to destabilize proto-planetary accretion disks.

I will focus in a second part of the lecture, on laboratory experiments devoted to the study of vortices in rotating stratified flows. The first study describes the shape and aspect ratio of vortices that depends not only on the Coriolis parameter and buoyancy (or Brunt–Väisälaä) frequency of the background flow, but also on the buoyancy frequency within the vortex and on the Rossby number of the vortex. This law is valid for both cyclones and anticyclones. In the experiment, anticyclones are generated by injecting fluid into a rotating tank filled with linearly stratified salt water. The law for  is not only validated by our experiments, but is also shown to be consistent with observations of the aspect ratios of Atlantic meddies and Jupiter’s Great Red Spot and Oval BA. The relationship for  is also derived and examined numerically in a numerical study [12]. A second experimental study concerns the coalescence of two lenticular anticyclones in a linearly stratified rotating fluid [13]. This type of events,classically met in the oceans has also been observed in the Jovian atmosphere. Our results show that the merging critical distance between the vortices depends drastically on their Rossby radius of deformation. This is in complete agreement with previous numerical modelling of vortex coalescence. We have also observed that mergers involve threedimensional processes as the vortices intertwine together possibly because of the presence of an elliptic instability that tilts the vortex cores. They are also accompanied by the emission of vorticity filaments and internal gravity waves radiation although we cannot prove that in our experiments these waves are solely due to the merging process.

Références :
[1] M. Le Bars, D. Cébron, P. Le Gal, Flows Driven by Libration, Precession, and Tides, Annual Review of Fluid Mechanics, 47:1, 163-193, 2015.
[2] L. Lacaze, P. Le Gal, S. Le Diz`es, Elliptical instability in a rotating spheroid Journal of Fluid Mechanics 505, 1-22, 2004.

[3] C. Eloy, P. Le Gal, S. Le Dizès, Experimental Study of the Multipolar Vortex Instability Phys. Rev. Lett. 85, 3400, 2000.

[4] M. Le Bars, P. Le Gal, Experimental analysis of the stratorotational instability in a cylindrical Couette flow Phys. Rev. Lett. 99 (6), 064502, 2007.

[5] I. Yavneh, J.C. McWilliams, J. C. Molemaker, M. Jeroen, Non-axisymmetric instability of centrifugally stable stratified Taylor-Couette flow, Journal of Fluid Mechanics 448, 1-21, 2001.
[6] P. G. Baines, H. Mitsudera, On the mechanism of shear flow instabilities, Journal of Fluid Mechanics 276, 327342, 1994.
[7] T. Satomura, An investigation of shear instability in a shallow water, Journal of the Meteorological Society of Japan. Ser. II 59 (1), 148-167, 1981.
[8] G. Facchini, B. Favier, P. Le Gal, M. Wang, M. Le Bars, The linear instability of the stratified plane Couette flow, Journal of Fluid Mechanics 853, 205-234, 2018.
[9] P. Le Gal, U. Harlander, I.D. Borcia, S. Le Dizès, J. Chen, B. Favier, Instability of vertically stratified horizontal plane Poiseuille flow, Journal of Fluid Mechanics 907, 2021.
[10] S.A. Balbus, J.F. Hawley, Instability, turbulence, and enhanced transport in accretion disks, Rev. Mod. Phys. 70, 153, 1998.

[11] O. Aubert, M. Le Bars, P. Le Gal, P. S. Marcus, The universal aspect ratio of vortices in rotating stratified flows: experiments and observations, Journal of Fluid Mechanics 706, 34-45,2012.
[12] P. Hassanzadeh, P. Marcus, P. Le Gal . The universal aspect ratio of vortices in rotating stratified flows: theory and simulation, J. Fluid Mech., vol. 706, 2012, pp. 46–57, 2012
[13] A. Orozco Estrada, Raúl C Cruz Gómez, A Cros, P Le Gal, ] Coalescence of lenticular anticyclones in a linearly stratified rotating fluid, Geophysical & Astrophysical Fluid Dynamics 114, 4-5, 504-523, 2020.

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We study dimple and jet formation from an axisymmetric wave trough, in the inviscid, irrotational regime for pure capillary and capillary-gravity waves in a confined cylindrical geometry. The initial perturbation is taken in the form of an eigenmode to the linear problem [1, 2] . It is numerically observed (from numerical solutions to the incompressible Euler equations with surface tension) that at moderate to large perturbation amplitude, the wave trough evolution deviates substantially from linear theory, in particular showing transfer of energy to higher wavenumbers [1]. This energy transfer coincides with the formation of an initially localised flat region near the axis of symmetry which subsequently forms a dimple like structure [2]. The dimple serves as a precursor to a  jet at larger perturbation amplitude [2, 3]. This jet can shoot upwards sharply, rising much higher than the initial perturbation and is reminiscent of similar jets occurring in bubble cavity collapse at a free surface [ 4, 5]. We compare our numerical observations with second order (capillary-gravity case) and third order (pure capillary case) theory, showing that the latter can capture the shape of the dimple very well, at moderate perturbation amplitude. In the strongly nonlinear regime and for pure capillary waves, we find that the interface time evolution, locally develops self-similarly following the well-known Keller-Miksis similarity scales [6]. 

1. Axisymmetric viscous interfacial oscillations – theory and simulations, Farsoiya, Mayya & Dasgupta, J. Fluid Mech., vol. 826, 2017.

2. Jetting in finite amplitude, capillary-gravity waves, Basak, Farsoiya & Dasgupta, J. Fluid Mech., vol. 909, 2020.

3. Dimples, jets and self-similarity in nonlinear, capillary waves, Kayal, Basak & Dasgupta, 2022 (To be submitted)

4. Jet formation in bursting bubbles at a free surface, Duchemin, Josserand & Zaleski, Phys. Fluids, 14(9), 2002.

5. Bubble bursting: universal cavity and jet profiles, Lai, Eggers & Deike, Phys. Rev. Lett 121 (14), 144501, 2018.

6. Surface tension driven flows, Keller & Miksis, SIAM J. Applied Maths, vol. 43, 1983.

At the ocean’s O(1-10)km scale motions, a gradient-wind balanced flow with an elliptic streamline parametrically excites internal inertia-gravity waves through ageostrophic anticyclonic instability (AAI). This study numerically investigates the breaking of internal waves and the following turbulence dissipation resulting from the AAI. In our simulation, we periodically distort the calculation domain following the streamlines of an elliptic vortex and integrate the equations of motion using a Fourier spectral method. This technique enables us to exclude the overall structure of the large-scale flow field from the computation and concentrate on resolving the small-scale waves and turbulence. From a series of numerical experiments, we identify two different scenarios of wave breaking. First, when the instability growth rate is high, the primary wave amplitude excited by AAI quickly goes far beyond the overturning threshold and directly breaks. The final state is thus strongly nonlinear quasi-isotropic turbulence. Second, if the growth rate is relatively low, weak wave-wave interactions begin to redistribute energy across frequency space before the primary wave reaches a breaking limit. Then, after a sufficiently long time, the system approaches a Garrett-Munk-like stationary internal wave spectrum, in which waves break at finer vertical scales. In both experimental conditions, the growth and decay time scales of the primary wave energy are well correlated. However, since the primary wave amplitude saturates in one scenario but not in the other, the energy dissipation rate exhibits two types of scaling properties. For the unsaturated case, the maximum amplitude of the primary wave is roughly proportional to the square root of its growth rate, as expected from the weak turbulence theory.

In close exoplanetary systems, tidal interactions are known to shape the orbital architecture of the system, modify star and planet spins, and have an impact on the internal structure of the bodies through tidal heating. Most stars around which planets have been discovered are low-mass stars and thus feature a convective envelope, as is also expected in giant gaseous planets like Hot-Jupiter. The dissipation of tidal flows, and more specifically the dissipation of tidal inertial waves (restored by the Coriolis acceleration) can be particularly important in the convective envelopes, especially in the early stage of the life of a star. In parallel, the nonlinear self-interaction of inertial waves is known to affect the structure of the background flow by triggering differential rotation in convective shells, as shown in numerical and experimental hydrodynamical studies.  In this context, we investigate how the addition of nonlinearities affects the tidal flow properties, the energy and angular momentum balances, thanks to 3D hydrodynamic nonlinear simulations of tides, in an adiabatic and incompressible convective shell. Unlike previous studies, we chose a realistic way to tidally excite inertial waves, through an effective body force taking into account the remaining action of the equilibrium tide (the quasi-hydrostatic tidal flow component) on inertial waves. By doing so, we are able to identify and assess the importance of the different type of nonlinearities involving or not tidal (non-)wavelike components, and thus discuss unphysical contributions leading to the unexpected angular momentum evolution which has been seen in some previous simulations. By removing the unphysical terms, we demonstrate that differential rotation still develops in the shell due to the non uniform deposition of angular momentum, when shear layers (straight structures where the waves are focused) are activated inside the shell. Moreover, we show new results for nonlinear tidal dissipation estimates, and demonstrate that the development of sheared zonal flows is responsible for causing a departure in the dissipation from linear predictions. We also highlight unstable cases where triadic resonances are triggered, which further complicates the picture we have of the impact of nonlinear terms on tidal flows and dissipation.

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The physics of turbulent buoyant flows--stably stratified and turbulent convection--is still not fully understood. In a recent work on stably-stratified turbulence, we provided numerical evidence of Bologiano-Obukhov scaling for Richardson number of the order of unity. In addition, using energy flux arguments, we showed that turbulent convection exhibits Kolmogorov-like spectrum, not Bolgiano-Obukhov spectrum, due to the unstable nature of the flow. Thus, these two phenomenologies very different. I will briefly introduce the role of rotation on energy spectrum. Ref: M. K. Verma, Physics of Buoyant Flows: From Instabilities to Turbulence, World Scientific, Singapore (2018). M. K. Verma, A. Kumar, and A. Pandey, Phenomenology of buoyancy-driven turbulence: recent results, New J. Phys., 19, 025012 (2017).

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We investigate the scaling form of appropriate time-scales extracted from time-dependent correlation functions in rotating, turbulent flows. In particular, we obtain precise estimates of the dynamic exponents zp, associated with the time-scales, and their relation with the more commonly measured equal-time exponents ζp. These theoretical predictions, obtained by using the multifractal formalism, are validated through extensive numerical simulations of a shell model for such rotating flows. [Ref.: ArXiv: 2112.06475]

Internal Waves (IWs) are generated at the thermocline/pycnocline depth of the oceans. In some ocean bodies, like the Bay of Bengal (BoB) and Andaman Sea where a large amount of freshwater is received from nearby major rivers, highly stratified waters produce a shallow pycnocline which prompts the generation and propagation of IWs over longer distances (a few hundreds of kilometers). Most of the IWs energy dissipates into turbulent mixing. The BoB and Andaman Sea are dominated by semi-diurnal tides throughout the year, and IWs of internal tidal period (of both semi-diurnal and diurnal) are reported based on observations, sea level data, model simulations, satellite observations (such as Synthetic Aperture Radar data and Ocean Color Imageries) in the coastal BoB and Andaman Sea. Stronger surface winds of the seasonal monsoons (i.e. the Northeast and Southwest monsoons) can induce higher surface waves, higher precipitation, and affects the thermocline/pycnocline variability. The strength of IW motions can cause hazardous stress to offshore oil rigs, navigating ships, commercial fishing, and other underwater operations. In this study, we used NASA’s Soil Moisture Active Passive (SMAP) salinity to understand the relationship between freshwater dynamics and IWs. We have also used the NASA’s Estimating the Circulation and Climate of the Ocean (ECCO) project’s 1/48° LLC4320 hourly simulations of salinity estimates during September 2011 – November 2012 in the BoB, and time-series of upper ocean temperature and salinity data from buoys in the BoB and Andaman Sea. We selected 6 boxes of each 3x3 degree to analyze the data for the characteristics of IWs. The hourly time series salinity data were subjected to 51- and 39-hour period high pass filters (depending up on the local inertial period at each box) and the resultant high pass filtered time series of SSS show large variation (over 0.1 psu) in April-May, September, December in the central Andaman Sea and BoB. Our results reveal dominant semi-diurnal (12.42 hour), diurnal (24 hour) and low frequency (>36-hour period) IWs of higher wave power and spectral density in the northern Andaman Sea, northern Bay, and off Sri Lanka. The analysis of buoy data over the 200 m water column in the boxes reveal that the low frequency IWs are generated at the base of the mixed layer and the internal tidal period IWs are generated in the thermocline. The spectral energy distribution in the wavenumber vs. frequency domain reveals the southwestward propagation of low frequency IWs from the central Andaman Sea to Sri Lanka coast (from box A to box C) with higher speeds (0.87 m/s) in the Andaman Sea to 0.19 m/s west of the Andaman Islands and increased to 0.34 m/s off Sri Lanka. Similarly, IWs of tidal period propagated northwestward from the northern Andaman Sea (box D to box F) to northwestern BoB with speeds of 0.41 m/s in the northern Andaman Sea to 0.27 m/s west of the northern Andaman Islands and slightly increased to 0.39 m/s. This increase might be due to the propagation of IWs into the regions of salinity stratified waters and hence increased stratification in the upper layers (off Sri Lanka and northwestern Bay) due to the advection of low salinity waters respectively by the westward flowing Northeast Monsoon Current (NEC) in the southern BoB and Ganges-Brahmaputra river discharges in the northern Bay at the end of SW monsoon season. This study shows that enhanced stratification would strengthen the amplitude of the propagating IWs and hence their energy. Low frequency IWs (of varying period 36 hour to 49 hour in the central Andaman Sea to southern BoB) are dominantly seen generated in the salinity stratified mixed layer and above the seasonal thermocline (BD12 and BD14 data), and their imprints reach sea surface and are captured well by the high resolution ECCO surface salinity. The internal tides (IWs of tidal period of both semi-diurnal and diurnal) are seen generated in the seasonal thermocline/pycnocline and dominantly seen in the northern Andaman Sea and northern BoB (boxes D to F), and propagated towards sea surface and are well captured in the ECCO surface salinity.

I revisit two papers from 2000 which claim to separate waves and turbulence based on Lagrangian frequency. In this framework stratified flows are modeled as the sum of nearly isotropic turbulence with Lagrangian frequencies greater than N and anisotropic internal waves with Lagrangian frequencies less than N. The horizontal velocities are larger than the vertical velocities for the internal wave component but approximately equal for the turbulent component. A ‘‘wave–turbulence’’ (W–T) transition based on Richardson number is predicted: At high Richardson numbers, mixing is controlled by interactions between internal wave modes. At Richardson numbers of order 1, mixing is controlled by instabilities of the large-scale wave modes. The transition occurs when the energy of the turbulence reaches the level of the internal waves, or, equivalently, when the bandwidth of internal waves becomes small. At energies below that of the W–T transition, the dissipation rate varies as the energy squared; above the transition the dependence is linear. Traditional turbulence closure models, which ignore internal waves, can be accurate only at energies above the W–T transition; at lower energies wave-wave interactions, typically not included in such models, are the dominant dynamics. This formulation does not have a place for "stratified turbulence" either as a modified form of 3D turbulence or as quasi-2D turbulence strongly constrained by stratification. Recent work suggests that these motions may play an important part in horizontal and vertical mixing in the ocean.

Motivated by recent experiments, we obtain the torque on an arbitrary aspect ratio rigid spheroid sedimenting in a linearly stratified viscous fluid ambient. We show theoretically, for the first time, that an ambient stratification leads to the emergence of a
longside-on settling regime even for small Reynolds (Re) and Richardson (Riv) numbers. Our analysis helps demarcate regimes of broadside-on and longside-on settling in a parameter space consisting of Re, Riv and the spheroid aspect ratio . For large Peclet numbers (based on the diffusivity of the stratifying agent), one may characterize the spheroid orientation dynamics in terms of a pair of threshold curves in the Riv/Re 3/2- plane that separate a longside-on settling regime above from a broadside-on settling regime below, with intermediate equilibrium orientations in between. The predictions are broadly consistent with experimental observations.
Time permitting, I will briefly discuss other recent work pertaining to (1) the far-field flow induced by a translating particle, and itsimplicationsfor the recent proposal of a biogenic ocean-mixing mechanism; (2) the possibility of drag reduction for sedimenting drops.

We report recent results on the generation, propagation and dissipation of internal tides using a high-resolution numerical model in a region over the northern Mid-Atlantic Ridge near the Azores. The lifecycle of internal tides is described. Major sites for the generation of internal tides are identified and exhibit strong disparities between different locations, both in wave amplitude and length scale. In particular, large amplitude isolated seamounts generate low-mode internal tide, while rough topographic irregularities over the ridge are associated with smaller wavelength modes. As a result, wide portions of the ridge dissipate more energy in the internal tide field than they gain from the barotropic tide conversion, a surprising result which evidences the scattering of low-mode internal tide into higher modes, up to dissipation. Based on these results, we characterize the tidally-induced deep-ocean mixing in this region.

The Asian summer monsoon has a planetary-scale, westward propagating “quasi-biweekly” mode of variability with 10-25 day period. Six years of observations from open ocean moorings at 18oN, 89.5oE in the north Bay of Bengal reveal distinct quasi-biweekly oscillations in sea surface salinity (SSS) during summer and autumn, with peak-to-peak amplitude of 3-8 psu. These large-amplitude oscillations are not due to quasi-biweekly variations of surface freshwater flux or river runoff. We show from moored wind, salinity and current data, satellite SSS, and reanalyses, that surface wind changes associated with the quasi-biweekly mode of the monsoon and embedded weather-scale systems, drive the SSS and coastal sea level oscillations in the summer of 2015. When winds are calm, mesoscale ocean eddies transport Ganga-Brahmaputra-Meghna river water southward to the mooring, salinity falls and the ocean mixed layer shallows to 1-10 m. During active (cloudy, windy) spells of the quasi-biweekly monsoon mode, directly wind-forced surface Ekman-like currents carry river water away to the east and north, leading to increased salinity at the mooring, as well as 0.1-0.5 m rise in sea level along the northeastern boundary of the bay.

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