In this paper, we consider the three-dimensional rotating Boussinesq equations (the “primitive” of geophysical fluid flows). Inspired by Christodoulou and Lindblad (Pure Appl Math 53:1536–1602, 2000), establish a priori estimates Sobolev norms for free boundary problem inviscid under Taylor-type sign condition on initial boundary. Using same method, can also obtain incompressible MHD system wit...

Standard perturbation methods are applied to Euler’s equations of motion governing the capillary-gravity shallow water waves to derive a general higher-order Boussinesq equation involving the small-amplitude parameter, α = a/h0, and long-wavelength parameter, β = (h0/l), where a and l are the actual amplitude and wavelength of the surface wave, and h0 is the height of the undisturbed water surf...

We show the possibility of temporally quasiperiodic waves at the onset of thermal convection in a thin horizontal layer of slowly rotating zero-Prandtl-number Boussinesq fluid confined between stress-free conducting boundaries. Two independent frequencies emerge due to an interaction between straight rolls and waves along these rolls in the presence of Coriolis force, if the Taylor number is ra...

We study numerically a recently introduced formulation of incompressible Newtonian fluid equations in vorticity – helical density and velocity – Bernoulli pressure variables. Unlike most numerical methods based on vorticity equations, the current approach provides discrete solutions with mass conservation, divergencefree vorticity, and accurate kinetic energy balance in a simple and natural way...

Fluid Flow via Reduced Order Modeling1 John A. Burns, Belinda B. King Center for Optimal Design and Control Interdisciplinary Center for Applied Mathematics Virginia Polytechnic Institute and State University Blacksburg, VA 24061{0531 Diana Rubio Center for Research in Scienti c Computation North Carolina State University Raleigh, NC 27695{8205 Abstract In this paper, we study the e ect of mode...

A new numerical finite difference code has been developed to solve a thermal convection of a Boussinesq fluid with infinite Prandtl number in a three-dimensional spherical shell. A kind of the overset (Chimera) grid named “YinYang grid” is used for the spatial discretization. The grid naturally avoids the pole problems which are inevitable in the latitude-longitude grids. The code is applied to...

It is shown that the anelastic Oberbeck-Boussinesq system is a small Mach, small Péclet and small Froude number limit of the complete Navier-Stokes-Fourier system for gases with large specific heat at constant volume. This result is obtained on an arbitrary large time interval. The proof allows an intrinsic view into the process of separation of fast oscillating acoustic waves, governed by a Li...

A numerical study of Rayleigh-Benard convection in an infinite fluid layer (Pr=0.71) is performed using large eddy simulation (LES) of the Navier-Stokes equations with the Boussinesq approximation. We present results in the ’hard turbulence’ regime (2.10 < Ra < 2.10). The LES modelling uses the mixed scale diffusivity model, that we have originally developed in the case of the differentially he...

The onset of square convective cells in fluid layers heated from below is investigated. Amplitude equations are deduced from the Boussinesq equations and a standard stability analysis is performed. Square cells are shown to be preferred when the instability is mainly capillarity driven. The influence of the Prandtl and Biot numbers are examined. At small Pr, the Biot number has not very much in...

This work presents a derivation of the energy density and energy flux of surface waves modeled by the so-called Kaup system, a variant of the Boussinesq system. The derivation is based on reconstruction of the velocity field and the pressure in the fluid column below the free surface, and is an extension of a method recently proposed in [2]. In the present note, it is shown that the total energ...