SCIENTIFIC PROGRAMS AND ACTIVITIES

Monday, May 20
Fields Institute, Room 230

"Big-Amplitude" Charney-Hasegawa-Mima simulation: video
"Small-Amplitude" Charney-Hasegawa-Mima simulation: video
"Medium-Amplitude" Charney-Hasegawa-Mima simulation: video

Peter Janssen, European Center for Medium-Range Weather Forecasts (presentation slides)
Effect of sea state on upper-ocean mixing

We will discuss issues related to time-periodic and traveling waves for the vortex sheet with surface tension and for the water wave with surface tension. Results include computations of nontrivially time-periodic solutions for the full equations of motion for the vortex sheet with surface tension, and computations and proof of existence of traveling waves (which are trivially time-periodic). The traveling waves to be discussed include large amplitude waves, such as waves with multi-valued height. If time permits, computation and analysis for a simple model system will be discussed.

This includes joint work with Jon Wilkening, Benjamin Akers, J. Douglas Wright, Mark Kondrla, Michael Valle, and possibly C. Eugene Wayne.

On the genaration and propagation of tsunamis

This lecture is spilt into two parts: in the first one I will discuss a rough model for the formation of a tsunami and in the second one I will present a deduction of a couple of KdV equations as the normal form of the equations for the water wave problem.

In the first part I will model the earthquake creating a tsumani by a boundary condition for the water wave problem and deduce the characteristic of the generated wave. In the second part I will start from the Hamiltonian formulation of the water wave problem, based on the use of the Dirichlet Neumann operator, and use normal form techniques to deduce an effective equation fo the propagation of the waves. The effective equation will turn out to coincide with a couple of independent KdV equations.

KAM for quasi-linear KdV equations

We prove the existence and the stability of Cantor families of quasi-periodic, small amplitude, solutions of quasi-linear autonomous Hamiltonian and reversible KdV equations. The proof is based on a Nash-Moser scheme, the search of an approximate inverse for the linearized operators and a Birkhoff normal form argument.

Which wave system is more turbulent: strongly or weakly nonlinear?

In a turbulent nonlinear wave system, there is usually a complex structure of energy transfers between modes of oscillations. In systems of finite size whose governing equation has quadratic or higher nonlinearity (e.g. planetary Rossby waves, water gravity/capillary waves), the wavevectors that interact most efficiently appear in groups of three (so-called triads). These groups tend to form “clusters”, which are networks of triads connected via one- and two-common-mode connections.

The cluster representation of turbulent interactions is particularly well suited for modelling turbulence theory and for comparison with pseudo-spectral numerical simulations, due to the discreteness of the wavevector spectrum in these approaches.

I will present two results that shed light on the actual physical mechanisms that are responsible for energy transfer and cascades in turbulence. First, the efficiency of transfers in turbulent cascades is maximised at a nonlinearity level that is intermediate between weakly nonlinear and fully nonlinear. This goes against the common belief that high nonlinearity implies stronger turbulence. Second, clusters formed by non-resonant triads are the rule more than the exception.

(Work in collaboration with Brenda Quinn)

We demonstrate theoretically and numerically the zonal-flow/wave turbulence feedback mechanism in Charney-Hasegawa-Mima turbulence forced by a small scale instability. Zonal flows are generated by a secondary modulational instability of the waves which are directly driven by the primary instability. The shear generated by the zonal flows then suppresses the small scale turbulence thereby arresting the energy injection into the system. This process can be described using nonlocal wave turbulence theory. Finally, the arrest of the energy input results in saturation of the zonal flows at a level which can be estimated from the theory and the system reaches stationarity even without large scale damping.

(Joint work with S. Nazarenko and B. Quinn)

Two examples will be discussed in order to illustrate how new estimates for singular integrals help us to obtain blow up, in finite time, for transport equations related to the Quasi-Geostrophic system, but also how some classical instruments of fluid dynamics can be used to understand conical Fourier multipliers.

We consider equations of Vlasov type, with periodic boundary conditions in space and small initial data. We introduce simple Hamiltonian nonlinear transformations allowing to control the long time behavior of these equations. We prove that the dynamics can be reduced to the free linear equation with a modified initial data over very long times. As a consequence, we obtain Landau damping results over polynomially long times with respect to the size of the perturbation, for initial data with finite regularity.

This is a joint work with Frédéric Rousset (Univ. Rennes 1).

Surface gravity waves--wind waves and swell--can affect the upper ocean in a number of ways. The Craik-Leibovich Boussinesq (CLB) equations are an asymptotic approximation to the fluid equations that filter out the processes leading to surface gravity and sound waves, but preserve the Stokes drift coupling between surface gravity waves and flow. The CLB equations are amenable to Large Eddy Simulations of Langmuir (wave-driven) Turbulence and analysis. I will present recent work with my colleagues studying the effects of Stokes drift in the CLB equations. Surprising and unsurprising results for laminar flow balances, turbulent fluxes, and coupling between turbulence and submesoscale flow will be discussed. Important remaining questions will be highlighted.

KAM theorem for multidimensional PDEs

I will present a quick overview of the KAM results proved in the context of nonlinear PDEs. In particular I will detail the recent result that I have obtained in collaboration with H. Eliasson and S. Kuksin for multidimensional PDEs and an application to the Beam equation and the nonlinear wave equation.

Nonlinear interactions of water waves with river ice

In the first part of our presentation we will discuss some new analytical results regarding the dynamics of river ice wave motion near a breaking front. We will present closed form analytical solutions of ice velocity as a function of time for several values of the breaking front speed and river bank resistance. The solutions are derived by solving an Abel equation of the second kind analytically. Several photos and videos will be presented to illustrate salient features of the dynamics of river ice breakup waves.

In the second part we will present a nonlinear study of the interaction of floating ice cover with shallow water waves. The ice cover is assumed to be a relatively thin, uniform elastic plate. The nonlinear propagation of waves is analyzed using a coupled system of three time-dependent and nonlinear partial differential equations(PDEs). These governing equations describing fluid continuity, momentum, and ice-cover response are reduced to a fifth order Korteweg-de Vries equation (FKDV), which is a well know evolutionary PDE. It is shown analytically that when the time evolution and nonlinear wave steepening are balanced by wave dispersion due to ice cover bending and inertia of the ice cover and the axial force, the FKDV model equation predicts solitary waves which propagate with a permanent shape and constant speed. Closed-form cnoidal solutions and solitary wave solutions are obtained for any order of the nonlinear term and for any given values of the coefficients of the cubic and quintic dispersive terms. Analytical expressions for three conservation laws and for three invariants of motion that represent the conservation of mass, momentum and energy for solitary wave solutions are also derived.

Coherent frequency profiles for the periodic nonlinear Schrodinger equation

Inspired by the general paradigm of weak turbulence theory, we consider the 2D cubic nonlinear Schrodinger equation with periodic boundary conditions. In an appropriate "large box limit", we derive a continuum equation on $\R^2$, whose solutions serve as approximate profiles (or envelopes) for the frequency modes of the cubic NLS equation. The derived equation turns out to satisfy many surprising symmetries and conservation laws, as well as several families of explicit solutions.
(This is joint work with Erwan Faou (INRIA, France) and Pierre Germain (Courant Institute, NYU)).

Finite-time blow-up for the inviscid Primitive equation

The Primitive equations are of great use in weather prediction. In large oceanic and atmospheric dynamic models, the viscous Primitive equations can be derived from Boussinesq equations using the so called hydrostatic balance approximation. In this talk we show that, contrarily to the viscous case, for certain class of initial data the corresponding smooth solutions of the inviscid primitive equations blow up in ?finite time. The proof is based on a reduction of the equations to a 1D model. More related results about the 1D model will also be discussed.

These are joint works with C. Cao, K. Nakanishi and E. S. Titi.

Peter A.E.M Janssen
European Center for Medium-Range Weather Forecasts

Effect of sea state on upper-ocean mixing

I will briefly discuss sea state effects, such as Stokes-Coriolis force and enhanced mixing by wave breaking on the evolution of the sea surface temperature (SST). In particular I will give a 'simple' derivation of the Stokes drift and I will point out the role of the wave-induced surface drift. Furthermore, I will report work of a number of my collegues who performed on simulations over a 30 year period. It was found that such sea state effects may have a considerable impact on the mean SST field.

Time scales and structures of wave interaction

Presently two models for computing energy spectra in weakly nonlinear dispersive media are known: kinetic wave turbulence theory, using a statistical description of an energy cascade over a continuous spectrum (K-cascade), and the D-model, describing resonant clusters and energy cascades (D-cascade) in a deterministic way as interaction of distinct modes.
In this talk we give an overview of these structures and their properties and a list of criteria, which model of an energy cascade should be used in the analysis of a given experiment, using water waves as an example. Applying time scale analysis to weakly nonlinear wave systems modeled by the focusing nonlinear Schrodinger equation, we demonstrate that K-cascade and D-cascade are not competing processes but rather two processes taking place at different time scales, at different characteristic levels of nonlinearity and based on different physical mechanisms.

Applying those criteria to data known from various experiments with water waves we find, that the energy cascades observed occurs at short characteristic times compatible only with a D-cascade.

A Conservative Adaptive Wavelet Method for the Rotating Shallow Water Equations on the Sphere

The fundamental computational challenge for climate and weather models is to efficiently and accurately resolve the vast range of space and time scales that characterize atmosphere and ocean flows. Not only do these scales span many orders of magnitude, the minimum dynamically active scale is also highly intermittent in both time and space. In this talk we introduce an innovative wavelet-based approach to dynamically adjust the local grid resolution to maintain a uniform specified error tolerance. The wavelet multiscale method is used to make dynamically adaptive the TRiSK model (Ringler et al. 2010) for the rotating shallow water equations on the sphere. We have carefully designed the inter-scale restriction and prolongation operators to retain the mimetic properties that are the main strength of this model. The wavelet method is computationally efficient and allows for straightforward parallelization using MPI. We will show verification results from the suite of smooth test cases proposed by Williamson (1991), and a more recent nonlinear test case suggested by Galewsky (2004): an unstable mid-latitude zonal jet. To investigate the ability of the method to handle boundary layers in ocean flows, we will also show an example of flow past an island using penalized boundary conditions. This adaptive "dynamical core" serves as the foundation on which to build a complete climate or weather model.

Alex Korotkevich
University of New Mexico

Inverse cascade of gravity waves in the presence of condensate: numerical simulation.

We performed simulation of the isotropic turbulence of gravity waves with the pumping narrow in frequency domain. Observed formation of the inverse cascade and condensate in low frequencies. Currently observed slopes of the inverse cascade are close to n_k ~ k^{-3.15}, which differ significantly from theoretically predicted n_k ~ k^{-23/6} ~ k^{-3.83}. In order to investigate the origin of this discrepancy, the dispersion relation for gravity waves was measured directly. Simple qualitative explanation of the results has been given.

Nonlinear-Optics-like Behaviour in Water Waves

A sufficiently high intensity beam of light in a medium whose refractive index is intensity dependent (such as air or water) will exhibit self focussing until higher order effects, noise, or plasma generation come into play. The cross-sectional profile of the focussed beam depends on the initial profile. It turns out that a very similar phenomenon occurs in a patch of capillary-gravity water waves until nonlinearity arrests the focussing and the patch breaks up into a complex set of localised structures. The connection between the two problems is the focussing 2+1 NLS equation. Whilst water under normal conditions may be too viscous for the phenomena to be observed, computations suggest that the behaviour should be observable in mercury.

Theoretical challenges in Wave Turbulence

Wave Turbulence has a long an successful history and by now it is well accepted as an effective approach for describing physical phenomena across a wide range of applications from quantum to cosmological scales. However, there remain few theoretical challenges concerning rigorous justifications of the assumptions and techniques used in Wave Turbulence, overcomming which would allow to establish Wave Turbulence as a mathematical subject. In my talk I will describe an approach dealing with milti-mode statistics in Wave Turbulence one of the major goals of which is to justify that the assumed statistical properties survive over the nonlinear evolution time.

An instability of wave turbulence as the source of radiating coherent pulses

I discuss the recent finding that wave turbulence can be unstable in certain (usually one dimensional) systems by an instability that breaks spatial homogeneity. This triggers a turbulent transport of energy by radiating pulses. The direct energy cascade is provided by adiabatically evolving pulses, the inverse cascade is due to the excitation of radiation. The spectrum is steeper than the Kolmogorov-Zakharov spectrum of wave turbulence.

B. Rumpf, A.C. Newell, V.E. Zakharov, PRL 103, 074502 (2009) A.C. Newell, B. Rumpf, V.E. Zakharov, PRL 109, 194502 (2012) B. Rumpf, A.C. Newell, PLA 377, 1260 (2013)

Large deviations from a stationary measure for a class of dissipative PDE's with random kicks

We study a class of dissipative PDE's perturbed by a random kick force. It is well known that if the random perturbation is sufficiently non-degenerate, then the Markov process associated with the problem in question has a unique stationary distribution, which is exponentially mixing. In addition, the strong law of large numbers and the central limit theorem are true. We are now interested in probabilities of deviations for the time average of continuous functionals from their spatial average with respect to the stationary distribution. Our main result shows that the occupation measures of solutions satisfy the LDP with a good rate function. The proof is based on Kifer's criterium for LDP, a Lyapunov-Schmidt type reduction, and a general result on long-time behaviour of generalised Markov semigroups.

This is a joint work with V. Jaksic, V. Nersesyan, and C.-A. Pillet.

Towards probability distribution of wave heights in the ocean from first principles

The ultimate aim of studies of random wind waves is to predict probability density function of wave characteristics, primarily wave height, at any given place and time. Within the framework of wave turbulence paradigm the evolution of wave spectra is described by the kinetic (Hasselmann) equation derived from first principles in the sixties and now routinely employed in operational forecasting. In contrast, in present the probability density function is found using some empirical formulae.

We study long-term nonlinear evolution of typical random wind waves which are characterized by broad-banded spectra and quasi-Gaussian statistics. We find the departure of wave statistics from Gaussianity from first principles using higher-order statistical momenta (skewness and kurtosis) as a measure of this departure. Non-zero values of kurtosis mean an increase or decrease of extreme wave probability (compared to that in a Gaussian sea), which is important for assessing the risk of freak waves and other applications. The approach is as follows. Non-Gaussianity of a weakly nonlinear random wave field has two components. The first one is due to nonlinear wave-wave interactions. We refer to this component as `dynamic', since it is linked to wave field evolution. The other component is due to bound harmonics. It is non-zero for every wave field with finite amplitude, contributes both to skewness and kurtosis of gravity water waves, and can be determined entirely from the instantaneous spectrum of surface elevation. We calculate the dynamic kurtosis by two different methods. First, by performing a DNS simulation of wind-generated random wave fields, using a specially designed algorithm, based on the Zakharov equation for water waves. Second, using the integral formulae found by Janssen (2003). In all generic situations, the contribution to kurtosis due to wave interactions is shown to be small compared to the bound harmonics contribution. This crucial observation enables us to determine higher momenta by calculating the bound harmonics part directly from spectra using asymptotic expressions. Thus, the departure of evolving wave fields from Gaussianity is explicitly contained in the instantaneous wave spectra. This enables us to broaden significantly the capability of the existing systems for wave forecasting: in addition to simulation of spectra it becomes possible to find also higher momenta and, hence, the probability density function. We found that the contributions due to bound harmonics to both skewness and kurtosis are significant for oceanic waves, and non-zero kurtosis (typically in the range 0.1-0.3) implies a tangible increase of freak wave probability.

For random wave fields generated by steady or slowly varying wind and for swell the derived large-time asymptotics of skewness and kurtosis predict power law decay of the moments. The exponents of these laws are determined by the degree of homogeneity of the interaction coefficients. For all self-similar regimes the kurtosis decays twice as fast as the skewness. These formulae complement the known large-time asymptotics for spectral evolution prescribed by the Hasselmann equation. The results are verified by the DNS of random wave fields based on the Zakharov equation. The predicted asymptotic behaviour is shown to be very robust: it holds both for steady and gusty winds.

From observations very little is known about the higher moments of sea waves statistics. For observational model of wave spectra (JONSWAP) we derived simple formulae for skewness and kurtosis valid for a very broad range of parameters.

Metastability and the Navier-Stokes equations

The study of stable, or stationary, states of a physical system is a well established field of applied mathematics. Less well known or understood are ``metastable'' states. Such states are a signal that multiple time scales are important in the problem - for instance, one associated with the emergence of the metastable state, one associated with the evolution along the family of such states, and one associated with the emergence of the asymptotic states. I will describe a dynamical systems based approach to metastable behavior in the two-dimensional Navier-Stokes equation.