Related papers: Conservative Integrators for a Toy Model of Weak T…
Turbulence -- ubiquitous in nature and engineering alike [1-5] -- is traditionally viewed as an intrinsically inertial phenomenon, emerging only when the Reynolds number (Re), which quantifies the ratio of inertial to dissipative forces…
In elastic-wave turbulence, strong turbulence appears in small wave numbers while weak turbulence does in large wave numbers. Energy transfers in the coexistence of these turbulent states are numerically investigated in both of the Fourier…
A model Hamiltonian dynamical system has been derived to study frequency cascades in the cubic defocusing nonlinear Schr\"odinger equation on the torus. Here, we explore the framework for exploring a canonical ensemble formulation of the…
We experimentally explore solutions to a model Hamiltonian dynamical system derived in Colliander et al., 2012, to study frequency cascades in the cubic defocusing nonlinear Schr\"odinger equation on the torus. Our results include a…
We consider the cubic defocusing nonlinear Schr\"odinger equation on the two dimensional torus. We exhibit smooth solutions for which the support of the conserved energy moves to higher Fourier modes. This weakly turbulent behavior is…
We explore further the rarefaction wave-like solutions recently discussed in work of the second author with J. Colliander, T. Oh and G. Simpson for a model Hamiltonian dynamical system derived by Colliander-Keel-Staffilani-Takaoka-Tao to…
We report numerical investigations of wave turbulence in a vibrating plate. The possibility to implement advanced measurement techniques and long time numerical simulations makes this system extremely valuable for wave turbulence studies.…
We introduce a novel toy model for shear flows, exploiting the spatial intermittency and the scale separation between large-scale flows and small-scale structures. The model is highly sparse, focusing exclusively on the most intense…
Recent remarkable progress in computing power and numerical analysis is enabling us to fill a gap in the dynamical systems approach to turbulence. One of the significant advances in this respect has been the numerical discovery of simple…
We consider long simulations of 2D Kolmogorov turbulence body-forced by $\sin4y \ex$ on the torus $(x,y) \in [0,2\pi]^2$ with the purpose of extracting simple invariant sets or `exact recurrent flows' embedded in this turbulence. Each…
For nonlinear dispersive systems, the nonlinear Schr\"odinger (NLS) equation can usually be derived as a formal approximation equation describing slow spatial and temporal modulations of the envelope of a spatially and temporally…
This paper constructs a rigorous mathematical framework for investigating laminar-turbulent transition induced by weak singularities of incompressible Navier-Stokes (NS) equations. By integrating the energy identity of Leray weak solutions…
In the framework of the focusing Nonlinear Schrodinger (NLS) equation we study numerically the nonlinear stage of the modulation instability (MI) of the condensate. As expected, the development of the MI leads to formation of "integrable…
Using weak wave turbulence theory analysis, we distinguish three main regimes for 2D stratified fluids in the dimensionless parameter space defined by the Froude number and the Reynolds number: discrete wave turbulence, weak wave…
We introduce a simplified model for wave turbulence theory -- the Wick NLS, of which the main feature is the absence of all self-interactions in the correlation expansions of its solutions. For this model, we derive several wave kinetic…
The search for solutions to the theory of weakly non-linear internal gravity wave turbulence is an active research topic. It is notably stimulated by the fact that this regime could drive fine-scale ocean dynamics for which the…
General scenario of turbulence theory is proposed and applied to streaky wall-bounded turbulence. This scenario introduces a new field of transverse waves. Significance of the theory rests on a mathematical theorem associated with the…
Turbulence follows a few well-known organizational principles, rooted in conservation laws. One such principle states that a system conserving two sign-definite invariants self-organizes into large-scale structures. Ordinary…
A particular interest on two-dimensional turbulence is the inverse energy cascade from small to large sales, which leads to an energy condensation accompanied by the formation of large-scale vortical structures. Indeed, such a phenomenon is…
We propose and rigorously analyze a novel family of explicit low-regularity exponential integrators for the nonlinear Schr\"odinger (NLS) equation, based on a time-relaxation framework. The methods combine a resonance-based scheme for the…