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 ...

Considered here are Boussinesq systems of equations of surface water wave theory over a variable bottom. A simplified such Boussinesq system is derived and solved numerically by the standard Galerkin-finite element method. We study by numerical means the generation of tsunami waves due to bottom deformation and we compare the results with analytical solutions of the linearized Euler equations. ...

We apply a multiple–time version of the reductive perturbation method to study long waves as governed by the Boussinesq model equation. By requiring the absence of secular producing terms in each order of the perturbative scheme, we show that the solitary–wave of the Boussinesq equation can be written as a solitary–wave satisfying simultaneously all equations of the KdV hierarchy, each one in a...

We investigate the linear stability of unstably strati ed Poiseuille ow between two horizontal parallel plates under non-Boussinesq conditions. It is shown, that Squire's transformation can be used to reduce the three-dimensional stability problem to an equivalent two-dimensional one. The eigenvalue problem, consisting of the generalized Orr-Sommerfeld equations, is solved numerically using an ...

We propose a modification of the classical Boussinesq approximation for buoyancy-driven flows of viscous, incompressible fluids in situations where viscous heating cannot be neglected. This modification is motivated by unresolved issues regarding the global solvability of the original system. A very simple model problem leads to a coupled system of two parabolic equations with a source term inv...

We analyze an optimal control problem for the stationary two-dimensional Boussinesq system with controls taken in space of regular Borel measures. Such measure-valued are known to produce sparse solutions. First-order and second-order necessary sufficient optimality conditions established. Following optimize-then-discretize strategy, corresponding finite element approximation will be solved by ...

The convection of matter in the Earth's upper mantle is considered, which Oberbeck–Boussinesq approximation due to thermogravitational differentiation. Within framework this approximation, a 2-D numerical simulation convective flows medium was performed. equation for temperature follows from entropy balance relation, where, taking into account variable viscosity system, there an effect energy d...

A source function method for the generation of waves internal to Boussinesq model grid boundaries (Wei G, Kirby JT, Sinha A. Generation of waves in Boussinesq models using a source function method. Coastal Engng 1999;36:271–299) is generalized to account for the presence of a strong current, for application to wave-current interaction problems. The method is also modified to eliminate waves pro...

We prove the global well-posedness of the viscous incompressible Boussinesq equations in two spatial dimensions for general initial data in Hm with m ≥ 3. It is known that when both the velocity and the density equations have finite positive viscosity, the Boussinesq system does not develop finite time singularities. We consider here the challenging case when viscosity enters only in the veloci...