Three-dimensional Boussinesq Research Articles

Global smooth axisymmetric solutions of the damped Boussinesq equations with zero thermal diffusion

In this paper we study the Cauchy problem for the three-dimensional damped Boussinesq system with zero thermal diffusion. We show that the solution of this system with 1\leq\beta\leq\frac{7}{3} is globally well posed if the initial data is axisymmetric without swirl.

Global in time well-posedness of a three-dimensional periodic regularized Boussinesq system

Abstract Global in time weak solution to a regularized periodic three-dimensional Boussinesq system is proved to exist in energy spaces. This solution depends continuously on the initial data. In particular, it is unique. The main novelty is the global in time aspect of this solution. The proofs use the coupling between the temperature and the velocity of the fluid, energy methods, and compactness argument.

Suppression of lift-up effect in the 3D Boussinesq equations around a stably stratified Couette flow

In this paper, we establish linear enhanced dissipation results for the three-dimensional Boussinesq equations around a stably stratified Couette flow, in the viscous and thermally diffusive setting. The dissipation rates are faster compared to those observed in the homogeneous Navier-Stokes equations, in light of the interplay between velocity and temperature, driven by buoyant forces. Our approach involves introducing a change of variables grounded in a Fourier space symmetrization framework. This change elucidates the energy structure inherent in the system. Specifically, we handle non-streaks modes through an augmented energy functional, while streaks modes are amenable to explicit solutions. This explicit treatment reveals the oscillatory nature of shear modes, providing the elimination of the well-known three-dimensional instability mechanism known as the “lift-up effect”.

Global well-posedness of three-dimensional incompressible Boussinesq system with temperature-dependent viscosity

In this paper, we focus on the global well-posedness of solutions to three-dimensional incompressible Boussinesq equations with temperature-dependent viscosity under the smallness assumption of initial velocity fields u0 in the critical space Ḃ3,10. The key ingredients here lie in the decomposition of the velocity fields and the regularity theory of the Stokes system, which are crucial to get rid of the smallness restricition of the initial temperature θ0. In addition, we mention that the improved decay estimates in time is also necessary.

Exploring the excess of cloud condensation nuclei and rain suppression using a minimal three-dimensional Boussinesq model with bulk cloud microphysics

Over the years, there have been discussions about the possibility of air pollution affecting the process of rain formation. In this study, we have developed a simplified model that represents the atmospheric dynamics and cloud microphysics to explore the impact of pollution on rain formation. We used an existing three-dimensional minimal model consisting of five equations, for which we added a simple bulk parametrization that represents the role of cloud condensation nuclei (CCN) in cloud formation processes. We conducted numerical tests using two CCN profiles, with either one or two accumulation layers and modified their abundance to explore the effects of different CCN concentrations and distributions. We conducted four numerical tests corresponding to the two aforementioned profiles with polluted and low-polluted scenarios. The numerical simulations suggested that a layer with high CCN concentration close to the surface tends to suppress precipitation, while the same concentration distributed over two layers tends to enhance the efficiency of rain formation. The simulations also showed that CCN particles far from the surface produced higher cloud tops and longer events, consistent with previous research. Although the model includes a stable representation of precipitating turbulent convection with bulk cloud microphysics, we expect its simplicity and conservation properties to allow for deeper theoretical analyses that can help us better understand the physical processes involved in the studied phenomenon. We hope this model will serve as a tool to explore different aerosol-related scenarios within the context of minimal models.

Periodic solution and asymptotic behavior of the three-dimensional sixth-order Boussinesq equation in shallow water waves

Periodic solution and asymptotic behavior of the three-dimensional sixth-order Boussinesq equation in shallow water waves

A hybrid discrete exterior calculus and finite difference method for Boussinesq convection in spherical shells

We present a new hybrid discrete exterior calculus (DEC) and finite difference (FD) method to simulate fully three-dimensional Boussinesq convection in spherical shells subject to internal heating and basal heating, relevant in the planetary and stellar phenomenon. We employ DEC to compute the surface spherical flows, taking advantage of its unique features including coordinate system independence to preserve the spherical geometry, while we discretize the radial direction using FD method. The grid employed for this novel method is free of problems like the coordinate singularity, grid non-convergence near the poles, and the overlap regions. We have developed a parallel in-house code using the PETSc framework to verify the hybrid DEC-FD formulation and demonstrate convergence. We have performed a series of numerical tests which include quantification of the critical Rayleigh numbers for spherical shells characterized by aspect ratios ranging from 0.2 to 0.8, simulation of robust convective patterns in addition to stationary giant spiral roll covering all the spherical surface in moderately thin shells near the weakly nonlinear regime, and the quantification of Nusselt and Reynolds numbers for basally heated spherical shells.

Global Sobolev regular solution for Boussinesq system

Abstract This article is concerned with the study of the initial value problem for the three-dimensional viscous Boussinesq system in the thin domain Ω ≔ R 2 × ( 0 , R ) \Omega := {{\mathbb{R}}}^{2}\times \left(0,R) . We construct a global finite energy Sobolev regularity solution ( v , ρ ) ∈ H s ( Ω ) × H s ( Ω ) \left({\bf{v}},\rho )\in {H}^{s}\left(\Omega )\times {{\mathbb{H}}}^{s}\left(\Omega ) with the small initial data in the Sobolev space H s + 2 ( Ω ) × H s + 2 ( Ω ) {H}^{s+2}\left(\Omega )\times {{\mathbb{H}}}^{s+2}\left(\Omega ) . Some features of this article are the following: (i) we do not require the initial data to be axisymmetric; (ii) the Sobolev exponent s s can be an arbitrary big positive integer; and (iii) the explicit asymptotic expansion formulas of Sobolev regular solution is given. The key point of the proof depends on the structure of the perturbation system by means of a suitable initial approximation function of the Nash-Moser iteration scheme.

Transmission and reflection of three-dimensional Boussinesq internal gravity wave packets in nonuniform retrograde shear flow

Simulated fully localized internal gravity wave packets with moderately large initial amplitude exhibit finite transmission across a reflection level, due to the combined effects of the wave-induced shear locally cancelling the retrograde background shear, modulational instability, and the nonlinear generation and evolution of secondary wave packets.

Determining Map, Data Assimilation and an Observable Regularity Criterion for the Three-Dimensional Boussinesq System

In this paper, we consider the analysis of a nudging based algorithm for data assimilation for the three-dimensional Boussinesq system, which we call the AOT system. A rigorous analysis of this algorithm for well-posed dissipative partial differential equations was first provided by Azouani, Olson and Titi (J Nonlinear Sci 24:277–304, 2014); thus justifying our terminology for the associated nudging system. We provide a sufficient condition, based solely on the observed velocity data obtained from a Leray-Hopf weak solution, for the global well-posedness, regularity and most crucially, the asymptotic tracking property of solutions of the associated (three-dimensional) nudging system. It is to be noted that neither regularity nor any knowledge of a uniform \({\mathbb {H}}^1\)-norm bound is a priori assumed on the solution of the original three-dimensional Boussinesq system from which the observations are obtained. As a corollary of our result, we obtain a novel observable regularity criterion based on finitely many observational data. Our condition also guarantees the construction of the Lipschitz continuous determining map, which is known to play a crucial role in the construction of the so-called determining form and in statistical data assimilation.

On the 3D Incompressible Boussinesq Equations in a Class of Variant Spherical Coordinates

This paper investigates the global stabilizing effects of the geometry of the domain at which the flow locates and of the geometric structure of the solution to the incompressible flows by studying the three-dimensional (3D) incompressible, viscosity, and diffusivity Boussinesq system in spherical coordinates. We establish the global existence and uniqueness of the smooth solution to the Cauchy problem for a full 3D incompressible Boussinesq system in a class of variant spherical coordinates for a class of smooth large initial data. We also construct one class of nonempty bounded domains in the three-dimensional space ℝ 3 , in which the initial boundary value problem for the full 3D Boussinesq system in a class of variant spherical coordinates with a class of large smooth initial data with swirl has a unique global strong or smooth solution with exponential decay rate in time.

Stability and large-time behavior of the inviscid Boussinesq system for the micropolar fluid with damping

In this paper, we consider the three-dimensional inviscid Boussinesq system for the micropolar fluid in porous media. We proved the global well-posedness and large time behavior of solutions in the whole space R3. Precisely, when the H3-norm of initial data is small, but the higher-order derivatives can be arbitrarily large, the system is globally well-posed by the pure energy method. Moreover, by a set of mature negative Sobolev and Besov space interpolation methods, the Lp − L2 (1 ≤ p ≤ 2) type of the optimal time decay rates are obtained without any smallness assumption on the Lp norm of the initial data. Our results mathematically explain the stability of the system in an unbounded domain.

On the analyticity and gevrey regularity of solutions to the three-dimensional inviscid Boussinesq equations in a half space

On the analyticity and gevrey regularity of solutions to the three-dimensional inviscid Boussinesq equations in a half space

Stability and optimal decay for a system of 3D anisotropic Boussinesq equations

This paper focuses on a system of three-dimensional (3D) Boussinesq equations modeling anisotropic buoyancy-driven fluids. The goal here is to solve the stability and large-time behavior problem on perturbations near the hydrostatic balance, a prominent equilibrium in fluid dynamics, atmospherics and astrophysics. Due to the lack of the vertical kinematic dissipation and the horizontal thermal diffusion, this stability problem is difficult. When the spatial domain is with being a 1D periodic box, this paper establishes the desired stability for fluids with certain symmetries. The approach here is to distinguish the vertical averages of the velocity and temperature from their corresponding oscillation parts. In addition, the oscillation parts are shown to decay exponentially to zero in time.

Propagation and overturning of three-dimensional Boussinesq wave packets with rotation

Simulations of fully localized internal gravity wave packets with their induced mean flow superimposed reveal that waves overturn even for initial amplitudes significantly lower than the critical amplitudes predicted by linear theory, in contrast with previous results of one- and two-dimensional wave packets without rotation.

Inertia-gravity-wave scattering by three-dimensional geostrophic turbulence

In rotating stratified flows including in the atmosphere and ocean, inertia-gravity waves (IGWs) often coexist with geostrophically balanced turbulent flows. Advection and refraction by such flows lead to wave scattering, redistributing IGW energy in the position–wavenumber phase space. We give a detailed description of this process by deriving a kinetic equation governing the evolution of the IGW phase-space energy density. The derivation relies on the smallness of the Rossby number characterising the geostrophic flow, which is treated as a random field with known statistics, makes no assumption of spatial scale separation, and neglects wave–wave interactions. It extends previous work restricted to near-inertial waves, barotropic flows or waves much shorter than the flow scales. The kinetic equation describes energy transfers that are restricted to IGWs with the same frequency, as a result of the time scale separation between waves and flow. We formulate the kinetic equation on the constant-frequency surface – a double cone in wavenumber space – using polar spherical coordinates, and we examine the form of the two scattering cross-sections involved, which quantify energy transfers between IGWs with, respectively, the same and opposite directions of vertical propagation. The kinetic equation captures both the horizontal isotropisation and the cascade of energy across scales that result from scattering. We focus our attention on the latter to assess the predictions of the kinetic equation against direct simulations of the three-dimensional Boussinesq equations, finding good agreement.

Interaction of near-inertial waves with an anticyclonic vortex

AbstractAnticyclonic vortices focus and trap near-inertial waves so that near-inertial energy levels are elevated within the vortex core. Some aspects of this process, including the nonlinear modification of the vortex by the wave, are explained by the existence of trapped near-inertial eigenmodes. These vortex eigenmodes are easily excited by an initialwave with horizontal scale much larger than that of the vortex radius. We study this process using a wave-averaged model of near-inertial dynamics and compare its theoretical predictions with numerical solutions of the three-dimensional Boussinesq equations. In the linear approximation, the model predicts the eigenmode frequencies and spatial structures, and a near-inertial wave energy signature that is characterized by an approximately time-periodic, azimuthally invariant pattern. The wave-averaged model represents the nonlinear feedback of the waves on the vortex via a wave-induced contribution to the potential vorticity that is proportional to the Laplacian of the kinetic energy density of the waves. When this is taken into account, the modal frequency is predicted to increase linearly with the energy of the initial excitation. Both linear and nonlinear predictions agree convincingly with the Boussinesq results.

Stability near hydrostatic equilibrium to the three-dimensional Boussinesq equations with partial dissipation

This paper focuses on the stability problem for perturbations near a hydrostatic equilibrium associated with the three-dimensional Boussinesq equations with partial dissipation. We mainly study the global stability and large-time behavior of solutions to this system with only horizontal dissipation and obtain three results. The first establishes the global $$H^2$$ -stability, the second obtains the large-time asymptotic of the solutions, and the last assesses the asymptotic linear stability with explicit decay rates. Finally, we remark that these results can also be extended to the system with mixed partial dissipation.

Wave-averaged balance: a simple example

In the presence of inertia-gravity waves, the geostrophic and hydrostatic balance that characterises the slow dynamics of rapidly rotating, strongly stratified flows holds in a time-averaged sense and applies to the Lagrangian-mean velocity and buoyancy. We give an elementary derivation of this wave-averaged balance and illustrate its accuracy in numerical solutions of the three-dimensional Boussinesq equations, using a simple configuration in which vertically planar near-inertial waves interact with a barotropic anticylonic vortex. We further use the conservation of the wave-averaged potential vorticity to predict the change in the barotropic vortex induced by the waves.

Strong Solutions to 3D-Lagrangian Averaged Boussinesq System

Under suitable assumptions on the initial data, we prove the existence, uniqueness of the strong solutions to a regularized periodic three-dimensional Lagrangian averaged Boussinesq system, in a Sobolev spaces. Also, we establish the convergence results of this unique strong solution of this regularized Boussinesq system to a strong solution of the three-dimensional Boussinesq system, as the regularizing parameter vanishes.