We present an analysis of the primary bifurcations that occur in a mathematical model that uses the (three-dimensional) Navier-Stokes equations in the Boussinesq approximation to describe the flow of a near unity Prandtl number fluid (i.e. air) in the differentially heated rotating annulus. In particular, we investigate the double Hopf (Hopf-Hopf) bifurcations that occur along the axisymmetric ...

The Serre or Green and Naghdi equations are fully-nonlinear and weakly dispersive and have a built-in assumption of irrotationality. However, like the standard Boussinesq equations, also Serre’s equations are only valid for long waves in shallow waters. To allow applications in a greater range of h0/l, where h0 and l represent, respectively, depth and wavelength characteristics, a new set of ex...

New analytic estimates of the rate at which parametric subharmonic instability (PSI) transfers energy to high-vertical-wavenumber near-inertial oscillations are presented. These results are obtained by a heuristic argument which provides insight into the physical mechanism of PSI, and also by a systematic application of the method of multiple time scales to the Boussinesq equations linearized a...

In this paper, we introduce an interactive coastal wave simulation and visualization software, called Celeris. Celeris is an open source software which needs minimum preparation to run on a Windows machine. The software solves the extended Boussinesq equations using a hybrid finite volume–finite difference method and supports moving shoreline boundaries. The simulation and visualization are per...

We give an example of a well posed, finite energy, 2D incompressible active scalar equation with the same scaling as the surface quasi-geostrophic equation and prove that it can produce finite time singularities. In spite of its simplicity, this seems to be the first such example. Further, we construct explicit solutions of the 2D Boussinesq equations whose gradients grow exponentially in time ...

The research area of nonlinear partial differential equations (NLPDEs) has been very active for the past few decades. The study of the exact solutions of a nonlinear evolution equation (NLEE) plays an important role to understand the nonlinear physical phenomena which are described by these equations. In recent years several powerful and efficient methods have been developed for finding analyti...

We describe a pseudo-spectral numerical method to solve the systems of one-dimensional evolution equations for free surface waves in a homogeneous layer of an ideal fluid. We use the method to solve a system of one-dimensional integro-differential equations, first proposed by Ovsjannikov and later derived by Dyachenko, Zakharov, and Kuznetsov, to simulate the exact evolution of nonlinear free s...

We aim to improve the techniques to predict tsunami wave heights along the coast. The modeling of tsunamis with the shallow water equations has been very successful, but often shortcomings arise, for example because wave dispersion is neglected. To bypass the latter shortcoming, we use the (linearized) variational Boussinesq model derived by Klopman et al. [12]. Another shortcoming is that comp...

We present a review of the normal form theory for weakly dispersive nonlinear wave equations where the leading order phenomena can be described by the KdV equation. This is an infinite dimensional extension of the well-known Poincaré-Dulac normal form theory for ordinary differential equations. We also provide a detailed analysis of the interaction problem of solitary waves as an important appl...

This paper focuses on the 2D incompressible Boussinesq equations with fractional dissipation, given by Λαu in the velocity equation and by Λβθ in the temperature equation, where Λ= √−Δ denotes the Zygmund operator. Due to the vortex stretching and the lack of sufficient dissipation, the global regularity problem for the supercritical regime α+β<1 remains an outstanding problem. This paper prese...