Dispersive hydrodynamics and oceanography: from experiments to theory   27 août - 1 septembre 2017, Les Houches (France)

# Résumés des exposés

## Conférenciers pléniers (séances de 2x1h)

Anne de Bouard: Long time behavior of stochastic dissipative-dispersive equations

We will introduce the theory allowing to define solutions of dissipative or dispersive partial differential equations with stochastic forcing, and review some results about the long time behavior (invariant measure and convergence to equilibrium) of dissipative equations with noise. We will then show how the theory can be applied in some cases to dissipative stochastic equations involving also dispersive (Hamiltonian) dynamics, with a particular example modeling the dynamics of Bose Einstein condensates close to critical temperature.

Gennady El: Dispersive Hydrodynamics and Modulation Theory

Dispersive hydrodynamics is concerned with fluid motion in which internal friction, e.g., viscosity, is very small or negligible relative to wave dispersion. In general, large-scale inhomogeneities in dispersive hydrodynamic media evolve into a combination of compression and expansion waves. Compression waves lead to gradient catastrophe regularised by the emergence of highly oscillatory, unsteady, nonlinear wavetrains termed dispersive shock waves (DSWs). Physical manifestations of DSWs include surface undular bores in rivers, internal bores in the ocean and in the atmosphere as well as nonlinear diffraction patterns in laser and atom (matter wave) optics.

The mathematical description of DSWs involves a synthesis of methods from hyperbolic quasi-linear systems, asymptotics and soliton theory.

One of the principal tools is nonlinear wave modulation theory, often referred to as Whitham averaging. In these lectures I shall outline the modulation theory of DSWs and consider some applications of this theory to water waves and fibre optics.

The lectures are based on a recent joint review article with Mark Hoefer, Physica D, 333, 11 - 65 (2016).

John Hunter: The Hamiltonian formalism and nonlinear surface wave

In the absence of dissipation and damping, almost all wave motions have a Hamiltonian structure, although it may be disguised by the noncanonical nature of the physical variables. A Hamiltonian structure imposes significant constraints on the form of the wave equations and provides a useful criterion for the selection of good model equations. We will review the Hamiltonian formalism for nonlinear waves and describe its application to surface waves modeled by nonlocal evolution equations. Examples include water waves and waves on vorticity and SQG fronts.

Evelyne Miot: An introduction to vortex dynamics in two-dimensional or three-dimensional incompressible flows

The aim of these lectures will be to study some singular dynamics arising in fluid mechanics, called vortex dynamics. For two-dimensional incompressible, inviscid fluids, such solutions correspond to a vorticity behaving as a sum of Dirac masses centered
at points called point vortices, while in the three-dimensional case the vorticity is concentrated on curves called vortex filaments. In
the 2D case we will present some mathematical results asserting that the point vortices evolve according to an Hamiltonian system of ordinary differential equations (point vortex system), for which we will study some basic mathematical properties. We will then introduce the binormal curvature flow equation, which formally governs the motion of (single) vortex filaments. Finally, we will present a system of simplified equations proposed by Klein, Majda and Damodaran to describe the interaction of several almost parallel vortex filaments.

Sunao Murashige: Numerical and theoretical studies of nonlinear water waves using the hodograph transformation

We consider large-amplitude motion of water waves as nonlinear dispersive phenomena in fluid mechanics. For steady and irrotational plane motion, it has been known that the hodograph transformation method, namely conformal mapping of the flow domain with suitable change of independent variables, can be applied to fully nonlinear motion of water waves.

In this talk, it is shown that this method can be extended to (i) fully nonlinear computation of unsteady motion such as wave breaking, and (ii) derivation of some mathematical models for large-amplitude motion of which wave profile can overhang. Also some merits and demerits of this method are made clear.

Kraig Winters: Fun and games with the dispersion relation for rotating internal gravity waves

Dispersive internal gravity waves, i.e. waves that propagate in the interior of a rotating stratified fluid such as the ocean or atmosphere, play important roles in the transfer of energy from generation regions to the far field and in the transformation of energy from large to small scales where nonlinearities can act to initiate turbulent mixing and dissipation.

The ocean and atmosphere can be thought of as variable media within which internal waves propagate and the limits of the dispersion relation, under various approximations, define theoretical wave guides within which these waves can propagate. The topology of these wave guides depends on the ambient density stratification and latitude owing to the strong constraint imposed by the locally vertical component of the earth’s rotation.

In these lectures, we’ll consider two distinct problems. The first involves atmospheric flow over a mountain ridge and we’ll consider how the upstream propagation of certain waves act to determine the nature of intense downslope flow in the lee of the ridge. The problem will be motivated by the results of a laboratory experiment and a theory will be constructed and refined to explain the results and make predictions of dynamically similar flows. The predictions will then be compared with observations of an Alpine foehn flow and direct numerical simulations.

The second example focuses on the mechanisms by which the spatial scales and temporal frequencies of radiating near-inertial internal waves are selected as mid-latitude storms blow across the ocean surface. These storms generate not only radiating waves but also near-surface jet-like flows, and a rich field of interacting eddies that play important dynamical roles in the wave excitation. We will focus on two mechanisms that lead to the surprising result that storm-driven near inertial waves can propagate poleward as well as equator-ward.

## Conférenciers invités (séances de 1h)

Alexander Chesnokov: Dispersive and hyperbolic models of breaking waves and mixing on shallow water

The wave breaking is a complex phenomenon accompanied by a strong vorticity generation and air entrainment. Breaking of water surface waves may occur anywhere that the amplitude is sufficient, however, it is particularly common on beaches because wave heights are amplified in the region of shallower water. If a wave approaching the coast is long and the variation of the coastal slope is gradual, spilling breakers usually appear. They are characterized by the presence of a finite turbulent zone spilling down the face of the wave (Longuet-Higgins & Turner 1974; Duncan 2001). At the toe of this turbulent zone, the wave slope changes sharply, resulting in flow separation and vorticity creation. The breaking waves entrain air into the water by forming `whitewater' where an intensive dissipation occurs. Such a turbulent zone has a strong influence on the wave evolution.

The main goal of the talk is to present a method for constructing mathematical models of breaking waves. The proposed approach is based on a two-layer modelling, where the upper turbulent layer is considered within the framework of shear shallow water flows (Teshukov, 2007), while the lower layer is potential and can be described by the Green--Naghdi or Sen-Venan type models. The interaction between the layers is taken into account through a natural mixing process. Experimental data on the structure of the turbulent flow area under breaking waves show that the boundary between the turbulent and potential regions are clearly visible (Lin & Rockwell 1994; Misra et al. 2008).

The hydrostatic model of multi-layer flow proposed by Liapidevskii & Chesnokov (2014) correctly describes the vorticity generation and turbulent bores if the Froude number is sufficiently large. This model was generalized by Gavrilyuk, Liapidevskii & Chesnokov (2016) taking into account the non-hydrostatic (dispersive) effect in the lower layer. Both models neglect the air entrainment. In particular, the obtained dispersive model describes the transition from an undular bore to a breaking (monotone) bore when the Froud number is between 1.3 and 1.4. The shoaling and breaking of a solitary wave propagating in a long channel of mild slope are also well predicted by the model. It is in a good agreement with experimental data by Hsiao et al. Further (Gavrilyuk, Liapidevskii & Chesnokov, 2017) we extend our model taking into account the effect of air entrainment. As a result we derive a new two-layer model for the interaction between a bubbly shear layer and long internal waves over topography. We perform non-stationary calculations to describe the formation of bores, and both periodic and damped oscillations. A hydrostatic approximation of this model is compared to the full non-hydrostatic system. Even if a quantitative description by means of the both models is quite similar, the qualitative behaviour of the wave fronts is different. In particular, a non-monotonic behaviour of bores has been found by using the dispersive model, while it is monotonic in the hydrostatic model. The analogy between mathematical modelling of two-layer bubbly flows and internal waves is drawn. This allows us, although implicitly, to compare the theoretical and experimental results relating to the generation of internal waves of large amplitude (Nash & Moum, 2005; Harris & Decker, 2017).

This talk is based on the joint works with Sergey Gavrilyuk and Valery Liapidevskii.

Mark Hoefer: Solitonic Dispersive Hydrodynamics

Long wavelength, hydrodynamic theories abound in physics, from fluids to optics, condensed matter to quantum mechanics, and beyond.  Such theories describe expansion and compression waves until breaking.  For many media, such as the interfacial dynamics between two fluids, the physics at shorter wavelengths are dispersive rather than dissipative, hence dispersive hydrodynamic theories are used to describe spatially extended simple waves and dispersive shock waves (DSWs).  Another celebrated feature of these media is the localized solitary wave or soliton but a hydrodynamic theory incorporating solitons is lacking.  Here we introduce a new, general soliton-mean field theory and use it to describe the interaction of microscopic solitons with macroscopic hydrodynamics in experiment.  This theory invokes the scale separation between extended hydrodynamic states and solitons in order to reveal a system of decoupled modulation equations for the mean field, the soliton’s amplitude, and a phase.  Due to the existence of two adiabatic invariants for the modulation system, the theory predicts that solitons are trapped by or transmitted through hydrodynamic states.  Continuity of the modulation solution implies that the result of solitons incident upon smooth expansion waves or compressive, highly oscillatory DSWs is the same, an effect termed hydrodynamic reciprocity.  We demonstrate quantitative agreement with experiments on viscous fluid conduits, with broader implications for, e.g., geophysical flows, nonlinear optics, and superfluids.

Leo Maas : Wave attractors

In dynamical systems, an attractor is a subset of phase space towards which a system evolves, regardless of its initial conditions. In geophysical and astrophysical fluids, rotation and density stratification create anisotropic equilibria. Perturbations to these equilibrium states are present as internal waves. Internal waves, reflecting at boundaries that are sloping with respect to the direction of gravity or rotation axis, are focused, and propagate towards a subset of real space at socalled wave attractors. Theory and experiments elucidate the nature and ubiquity of wave attractors.

Charlotte Perrin: Lagrangian approach to one-dimensional constrained systems

In this talk I will introduce and study two constrained systems which may appear in fluid mechanics in the modelling of mixtures (constraint on the maximal volume fraction) or of partially free surface flows (constraint on the maximal height of the flow). I will develop a Lagrangian approach, based on one-dimensional optimal transport tools, which enables to obtain original existence results. I will finally show that this approach can be also used from a numerical point of view.

Bruno Voisin : A linear theory of the generation of monochromatic internal waves

Internal gravity waves propagate in density-stratified fluids owing to the restoring effect of buoyancy. They are anisotropic and dispersive, with a propagation equation that is elliptic in the frequency range of evanescent waves, and hyperbolic in the range of propagating waves. A Green's function approach of their generation is proposed, that separates the problem into two distinct ones: the representation of the generation process by a distribution of singularities, obtained by the boundary integral method; and the calculation of the waves radiated by this distribution, using Fourier analysis. The first problem is the most difficult of the two, involving anisotropic coordinate stretching in the evanescent range followed by causal analytic continuation into the propagating range. The approach is implemented for two-dimensional elliptical forcing and three-dimensional spheroidal forcing. The interpretation of the results in terms of added mass is discussed, together with the implications of added mass for the calculation of the conversion rate of barotropic forcing into baroclinic internal tides at the ocean bottom. The application of the analysis to the recent discovery of internal wave focusing by ring forcing is presented, and compared with novel experiments involving oscillating tori at LEGI and an annular wave generator at the physics laboratory of ENS de Lyon.

Picture caption: Amplification factor, in the vertical plane, for the waves generated by Gaussian forcing at a horizontal circular annulus of vertical axis and aspect ratio $\epsilon = 10$. Amplification reaches a maximum $\Gamma(3/4)/2^{1/4}\sqrt{\epsilon} \approx 3.26$ at the focus.

## Exposés courts (20 min)

Cosmin Burtea: Long time existence results for the abcd systems with bore-type initial data

Abstract en PDF

Fernando Cortez: Blow-up for the $b$- family Equation

The $b$-family equation u_t -u_{xxt} + (b+1) u u_x=b u_{x} u_{xx}+ u u_{xxx} is introduced by D.D. Holm and M.F Struley, which describes the balance between the convection and the stretching for small viscosity in the dynamic of one-dimensional nonlinear waves in fluids. For appropriate values for $b$ includes well known models, such as Camassa-Holm equation or the Degasperis-Procesis equation. We establish a local-in-space blowup criterion.

Clémentine Courtès: Ondes progressives pour l'équation de KdV-KS

L'équation de Korteweg-de Vries-Kuramoto-Sivashinsky (KdV-KS) est une équation dispersive-diffusive pouvant modéliser un écoulement mince de fluide visqueux sur un plan incliné. Elle dérive des équations de Saint-Venant visqueuses à nombre de Froude super-critique. Nous nous intéressons à l'existence d'ondes progressives pour des cas dégénérés de l'équation de KdV-KS.
Lorsque la diffusion d'ordre quatre s'annule, il s'agit de l'équation de Korteweg-de Vries-Burgers. Il y a existence d'une onde progressive oscillante dans le cas d'une forte dispersion, ou monotone, dans le régime de grande diffusion.
Lorsque la diffusion d'ordre deux s'annule, nous montrons l'existence d'ondes progressives de petite amplitude.

Joint work with Frédéric Rousset.

Rafael Granero-Belinchon: New models for the Rayleigh-Taylor instability

The Rayleigh-Taylor instability is an instability of an interface between two fluids of different densities which occurs when the heavier fluid is placed on top of the lighter fluid under the effect of gravity. In this talk I will present a couple of new models for the Rayleigh-Taylor instability for two dimensional irrotational Euler flows. One of the models (known as the h-model) considers the case where the interface between the fluids is given by a small graph. The other model (known as the z-model) allows the interface to be an arbitrary curve (not necessarily a graph). In particular, I will compare the results of these models with the well-established Rocket-rig experiment by Read and Youngs. Finally, I will address some rigorous mathematical results for these models.

Joint work with Steve Shkoller.

Kseniya Ivanova: Multi–dimensional shear shallow water flows

The mathematical model of shear shallow water flows of uniform density is studied. This is a
2D hyperbolic non-conservative system of equations which is reminiscent of a generic Reynolds-
averaged model of barotropic turbulent flows. The model has three families of characteristics
corresponding to the propagation of surface waves, shear waves and average flow (contact
characteristics). The system is non-conservative : for six unknowns (the fluid depth, two
components of the depth averaged horizontal velocity, and three independent components of
the symmetric Reynolds stress tensor) one has only five conservation laws (conservation of
mass, momentum, energy and mathematical ‘entropy'). A splitting procedure for solving such
a system is proposed allowing us to define a weak solution. Each split subsystem contains only
one family of waves (either surface or shear waves) and contact characteristics. The accuracy of
such an approach is tested on exact 2D solutions describing the flows where the velocity is linear
with respect to the space variables, and 1D solutions. The capacity of the model to describe the
full transition observed in the formation of roll waves : from uniform flow to one-dimensional
roll waves, and, finally, to 2D transverse ‘fingering' of roll wave profiles is shown.

Joint work with Sergey Gavrilyuk and Nicolas Favrie.

Maria Kazakova: Discrete transparent boundary conditions for the linearised Green-Naghdi equations

We consider the linearised Green-Naghdi one-dimensional system of equations which models the small amplitude and long waves propagation on the free water surface. While the original system is set on the whole space, usually for practical applications the area of study is restricted and we should fix the articial boundary conditions. We propose continuous and discrete explicit boundary conditions. The dissipation property is shown and the validity of the technique is proved numerically as well.

Joint work with Pascal Noble.

Leonardo Kosloff: Local theory for the surface quasi-geostrophic equation in an exterior domain

We study the exterior problem with Dirichlet boundary conditions for the SQG equation via a spectral representation of the fractional Laplacian (−∆)^s , 0 < s < 1, based on a generalization of the Fourier transform for exterior domains.

We then implement a localized version of (−∆)^s due to Caffarelli and Silvestre, as improved by Stinga and Torrea. We give applications to the problem of existence of weak solutions of the two dimensional dissipative quasi-geostrophic equation and the large time decay of these solutions in the L^2 -norm through a modified version of the Fourier splitting technique due to M. Schonbek.

Lastly, we explore local well-posedness of this system in critical Sobolev spaces using the heat kernel representation of the fractional Dirichlet Laplacian. The approach is based on estimates for the Littlewood-Paley localizations which ensue from precise pointwise estimates for the heat kernel.

Joint work with Tomas Schonbek.

Zakhar Makridin: Multi-Dimensional Conservation Laws for Integrable Systems

We introduce and investigate a new phenomenon in the Theory of Integrable Systems – the concept of multi-dimensional conservation laws for two- and three-dimensional integrable systems. Existence of infinitely many local two-dimensional conservation laws is a well-known property of two-dimensional integrable systems. We show that pairs of commuting two-dimensional integrable systems possess infinitely many three-dimensional conservation laws. Examples: the Benney hydrodynamic chain, the Korteweg de Vries equation. Simultaneously three-dimensional integrable systems (like the Kadomtsev-Petviashvili equation) have infinitely many three-dimensional quasi-local conservation laws. We illustrate our approach considering the dispersionless limit of the Kadomtsev-Petviashvili equation and the Mikhalev equation. Applications in three-dimensional case: the theory of shock waves, the Whitham averaging approach.

Joint work with Maxim Pavlov.

Frédéric Marbach: High frequency analysis of Prandtl models

This talk concerns the mathematical analysis of two variants of the Prandtl equations: the interactive boundary layer model and the prescribed displacement thickness model.

Both models have been used extensively for numerical simulation of steady boundary layer flows and compare favorably to the usual Prandtl formulation, especially past a separation point. They rely on a change of point of view. The evolution equation is the same as in the usual Prandtl case, but the matching boundary condition which links the inviscid flow to the boundary layer is modified and involves a physically relevant quantity (namely, the displacement thickness).

We consider the unsteady version of these models and study their linear well-posedness. More precisely, we investigate the linear stability of shear flow solutions with respect to high frequency perturbations. We show that both models exhibit strong unrealistic instabilities, which are in particular distinct from the Tollmien-Schlichting waves.

Joint work with Anne-Laure Dalibard, David Gerard-Varet and Helge Dietert.

Sophie Marbach: What do we know about slippage ? From atomic solid-liquid interface to bulk flow properties

The nature of boundary conditions in hydrodynamics was widely debated in the 19th century. While centuries of experimental and theoretical studies have been in agreement with a zero slip length, since 20 years measurements of partial slip have been made.

The standard boundary condition used today to describe slippage was introduced by Navier in 1823, who states that the fluid velocity tangent to the surface is proportional to the rate of strain at the surface. This linear relationship defines a proportionality constant that has the units of a length and is referred to as the « slip length ».

The slip length increases with increasing contact angle of the fluid with the interface ([1]), and takes its origin in molecular structuration at the interface ([2]). Measurements and simulations have found that water moves through carbon nanotubes at exceptionally high rates owing to nearly frictionless interfaces. The exact mechanisms of water transport at the water-carbon interface continues to be debated, partly because of a lack of experimental results, and partly because none has met the considerable technical challenge of measuring the pressure driven flow rate through a single nanotube – only a few fL/s, e.g. 10^(-15) L/s. We established a reliable and sensitive method to determine with unprecedented sensitivity the flow rate through individual nanotubes (see [3,4]). Our measurements reveal unexpectedly large and radius-dependent surface slippage in carbon nanotubes, and no slippage in boron nitride nanotubes that are crystallographically similar to carbon nanotubes, but electronically different. This contrast between the two systems must originate from subtle differences in the atomic-scale details of their solid-liquid interfaces, and goes beyond the predictions made by molecular dynamics [5]. This illustrates that nanofluidics is the frontier at which the continuum picture of fluid mechanics meets the atomic nature of matter, and where simple descriptions at the molecular level fail to reproduce continuum effects. This research was published in Nature.

[1] From microfluidic application to nanofluidic phenomena issue, L. Bocquet, and E. Charlaix, Chem. Soc. Rev., 39, 1073-1095 (2010)

[2] On the Green-Kubo relationship for the liquid-solid friction coefficient, L. Bocquet, and J.-L. Barrat, J. Chem Phys. 139, 044704 (2013)

[3] Massive radius-dependent flow slippage in carbon nanotubes, E. Secchi, S. Marbach, A. Niguès, D. Stein, A. Siria and L. Bocquet, Nature537, 210-213 (2016)

[4] The Landau-Squire Plume, E. Secchi, S. Marbach, A. Niguès, A. Siria, and L. Bocquet, J. Fluid Mech. (to appear 2017)

[5] Molecular Origin of Fast Water Transport in Carbon Nanotube Membranes : Superlubricity versus curvature dependent friction, K. Falk, F. Sedlmeier, L. Joly, R. Netz, L. Bocquet, Nano Lett.10, 4067-4073, (2010)

Giovanni Ortenzi: Hydrodynamic models of stratified Euler fluids and boundary confinement effects

Confinement effects by rigid boundaries in the dynamics of ideal fluids are considered from the perspective of long-wave models, with the focus on the consequences of establishing contacts of material surfaces with the confining boundaries. When contact happens, we show that the model evolution can lead to the dependent variables developing singularities in finite time. The conditions and the nature of these singularities are illustrated in several cases, progressing from a single layer homogeneous fluid with a constant pressure free surface and flat bottom, to the case of a two-fluid system contained between two horizontal rigid plates and finally, through numerical simulations, to the full Euler stratified system.These illustrate the qualitative and quantitative predictions of the models within a set of examples chosen to illustrate the theoretical results.

Joint work with Roberto Camassa, Gregorio Falqui, Marco Pedroni and Colin Thomson.

Stefano Scrobogna: On the dynamic of low Froude number fluids

In this exposition we study the dynamic of non-homogeneous, viscid fluids subjected to a vertical stratification induced by gravity. We prove that the equations describing viscid fluids which are slightly perturbed around a vertically-decreasing density profile are globally well-posed if the Froude number is sufficiently small, and their dynamic is approximated by a two-dimensional stratified Navier-Stokes system with vertical diffusivity.

Irina Stepanova: On some types of instability of Hele-Shaw flows

In this talk the unsteady two-layer flows in a Hele-Shaw cell are under consideration from different points of view. We are interested in behavior of the interface between layers and development of different types of instability. Firstly, we study a class of shear flows characterized by arbitrary dependence of the horizontal velocity component and pressure in the vertical coordinate. It is shown that the interface of layers moving with different velocities is instable for short-wave perturbations and stable for long-wave ones. When the horizontal flow of liquid prevails the pressure changes weakly in the vertical direction. It allows to proceed to long-wave approximation and to study the layered flows governed by simplified one-dimensional models. Based on the reduced models, the position of the interface of the layers is calculated numerically and the results are compared by two models. It is shown that the simplified models correctly describe well-known fact about the instability of the interface between two liquids moving with different velocities and having different physical properties.

The second point of the talk is the quasi-periodic flows called the roll waves regime. The main attention is paid on the question how the roll waves arise and develop from small perturbations of the interface of two-layer Hele-Shaw flows under a lid. The Whitham conditions for the existence of the roll waves are formulated and tested. It is established that the roll waves propagate upward or downstream depending on the physical parameters of the problem. The theoretical results are confirmed by numerical calculation of the interface position. The diagrams of the roll waves are constructed. They show the relationship between the critical depth, at which the roll waves begin to develop, with the wave amplitude.

Thirdly, we suggest a kinematic-wave model of the viscous fingers growth based on an assumption about intermediate mixing layer formation between two layers of liquids with different viscosities and velocities. The model predicts well the propagation velocity of the viscous fingers to a sufficient degree of accuracy and do nor require large processing power. The model is verified by means of the comparison with well-know Koval model and confirmed by the numerical simulation in the framework of two-dimensional equations describing displacement process in the Hele-Shaw cell.

The work is joint with Alexander Chesnokov and Valery Liapidevskii. It is supported by Russian Science Foundation (project 15-11-20013).

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