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...

The Boussinesq equation is a mathematics model of thermohydraulics, which consists of equations of fluid and temperature in the Boussinesq approximation.The deterministic case has been studied systematically by many authors (e.g., see [1– 3]). However, in many practical circumstances, small irregularity has to be taken into account.Thus, it is necessary to add to the equation a random force, wh...

This paper is concerned with the mixed formulation of the Boussinesq equations in two-dimensional domains and its numerical approximation. The paper deals first with existence and uniqueness results, as well as the description of the regularity of any solution. The problem is then approximated by a mixed finite element method, where the gradient of the velocity and the gradient of the temperatu...

We consider Euler equations with stratified background state that is valid for internal water waves. The solution of the initial-boundary problem for Boussinesq approximation in the waveguide mode is presented in terms of the stream function. The orthogonal eigenfunctions describe a vertical shape of the internal wave modes and satisfy a Sturm-Liouville problem. The horizontal profile is define...

The unsteady natural convection flow of an incompressible viscous fluid near a vertical plate that applies an arbitrary shear stress to the fluid is studied using the Laplace transform technique. The fluid flow is due to both the shear and the heating of the plate. Closed-form expressions for velocity and temperature are established under the usual Boussinesq approximation. For illustration pur...

This paper aims to renew interest on mixed thermal convection research and to emphasize three issues that arise from the present analysis: (i) a clear definition of the reference temperature in the Boussinesq approximation; (ii) a practical delimitation of the three convective modes, which are the forced convection (FC), mixed convection (MC) and natural (or free) convection (NC); (iii) and, fi...

The effects of randomly irregular bathymetry on the propagation of interfacial gravity waves are studied. Modelling sea water by a two-layer fluid of different densities, weakly nonlinear waves much longer than the sea depth but comparable to the horizontal scale of bathymetry are treated by Boussinesq approximation and multiplescale analysis. For transient wave pulses, the governing equation f...

We present an asymptotic form of the Boussinesq equations in the limit of small Péclet numbers i.e. when the time scale of motions is much larger than the time scale of thermal diffusion. We find that, in this limit, the effects of thermal diffusion and stable stratification combine in a single physical process. This process is an anisotropic dissipation (not effective for horizontal motions) w...

We derive a semi-geostrophic variational balance model for the three-dimensional Euler–Boussinesq equations on nontraditional f-plane under rigid lid approximation. The is obtained by small Rossby number expansion in Hamilton principle, with no other approximations made. allow fully non-hydrostatic flow and do not neglect horizontal components of Coriolis parameter; that is, we make so-called “...

In turbulent Rayleigh-Bénard convection (a fluid between two parallel horizontal plates and heated from below) a transition was predicted to occur [1,2] with increasing Rayleigh number Ra from a ‘‘classical’’ state with laminar boundary layers (BLs) to an ‘‘ultimate’’ state with turbulent BLs. Recently, this transition was found in measurements of the Nusselt number Nu and Reynolds number Re as...