Two-dimensional Boussinesq Equations Research Articles

The reproducing kernel particle Petrov–Galerkin method for solving two-dimensional nonstationary incompressible Boussinesq equations

The meshless techniques are improved to simulate a wide range of the physical models. Two common forms of the meshless methods are strong and (local) weak forms. In the current paper, we employ the local weak form technique to simulate the two-dimensional non-stationary Boussinesq equations. In the present method the trial functions have been selected from the shape functions of the RKPM. Also, the test function is based upon a Ck function. At first, the time variable has been discretized via a finite difference scheme and then the space direction has been approximated by using the MLPG procedure. In this procedure, by employing the continuity equation, the two-dimensional non-stationary Boussinesq equations have been transformed to a pressure Poisson equation. After solving the obtained pressure Poisson equation, the velocity of fluid in the x- and y-directions and also the temperature can be updated, directly. Numerical examples confirm the ability of the developed technique.

A global regularity result for the 2D Boussinesq equations with critical dissipation

This paper examines the global regularity problem on the two-dimensional incompressible Boussinesq equations with fractional dissipation, given by $\Lambda^\alpha u$ in the velocity equation and by $\Lambda^\beta \theta$ in the temperature equation, where $\Lambda=\sqrt{-\Delta}$ denotes the Zygmund operator. We establish the global existence and smoothness of classical solutions when $(\alpha,\beta)$ is in the critical range: $\alpha>\frac{\sqrt{1777}-23}{24} =0.798103..$, $\beta>0$ and $\alpha+ \beta =1$. This result improves the previous work of Jiu, Miao, Wu and Zhang \cite{JMWZ} which obtained the global regularity for $\alpha> \frac{23-\sqrt{145}}{12} \approx 0.9132$, $\beta>0$ and $\alpha+ \beta =1$.

Global well-posedness to the cauchy problem of two-dimensional density-dependent boussinesq equations with large initial data and vacuum

This paper concerns the Cauchy problem of the two-dimensional density-dependent Boussinesq equations on the whole space \begin{document}$ \mathbb{R}^{2} $\end{document} with zero density at infinity. We prove that there exists a unique global strong solution provided the initial density and the initial temperature decay not too slow at infinity. In particular, the initial data can be arbitrarily large and the initial density may contain vacuum states and even have compact support. Moreover, there is no need to require any Cho-Choe-Kim type compatibility conditions. Our proof relies on the delicate weighted estimates and a lemma due to Coifman-Lions-Meyer-Semmes [J. Math. Pures Appl., 72 (1993), pp. 247-286].

Global well-posedness for the 2D Boussinesq equations with a velocity damping term

In this paper, we prove global well-posedness of smooth solutions to the two-dimensional incompressible Boussinesq equations with only a velocity damping term when the initial data is close to an nontrivial equilibrium state $(0,x_2)$. As a by-product, under this equilibrium state, our result gives a positive answer to the question proposed by [ACWX] (see P.3597).

Two-dimensional Boussinesq equations applied to channel flows: deducing and applying the equations

ABSTRACT A basic hypothesis adopted for theoretical formulation of fluid flows is the hydrostatic pressure distribution. However, many researchers have pointed out that this simplification can lead to errors, in cases such as dam break flow. Discrepancy between computational solution and the experiment is attributed to the pressure distribution. These findings are not new, but it is not presented any formulation in the literature that considers the non-hydrostatic pressure distribution in 2D flow. This article deduces the Boussinesq Equations as an evolution of the Shallow Water Equations with the hypothesis of non-hydrostatic pressure distribution in the vertical direction. XYZ Orthogonal Cartesian System is used, considering the influence of channel bed slope and head losses of flow. It is presented the non-hydrostatical correction in the Boussinesq equation in two dimension using Fourier series. The solution uses Runge-Kutta Discontinuous Galerkin Method and the formulation is applied to a cylindrical dam-break.

A reduced-order extrapolated Crank–Nicolson finite spectral element method based on POD for the 2D non-stationary Boussinesq equations

In this paper, we mainly utilize proper orthogonal decomposition (POD) to reduce the order for the coefficient vector of the classical Crank–Nicolson finite spectral element (CCNFSE) method of the two-dimensional (2D) non-stationary Boussinesq equations about vorticity-stream functions so that the reduced-order method maintains all the advantages of the CCNFSE method. Toward this end, we first establish a CCNFSE format with the second-order time accuracy for the two-dimensional (2D) non-stationary Boussinesq equations about vorticity-stream functions and analyze the existence, stability, and convergence of the CCNFSE solutions. And then, by POD, we establish a reduced-order extrapolated Crank–Nicolson finite spectral element (ROECNFSE) method and analyze the existence, stability, and convergence of the ROECNFSE solutions as well as offer the flowchart for seeking ROECNFSE solutions. Finally, we use two sets of numerical experiments to validate that the numerical computational consequences are accorded with the theoretical ones such that the effectiveness and feasibility of the ROECNFSE method are further verified. Both theory and method in this paper is new and completely different from the existing reduced-order methods.

Exact Solutions with Variable Coefficient Function Forms for Conformable Fractional Partial Differential Equations by an Auxiliary Equation Method

In this paper, an auxiliary equation method is introduced for seeking exact solutions expressed in variable coefficient function forms for fractional partial differential equations, where the concerned fractional derivative is defined by the conformable fractional derivative. By the use of certain fractional transformation, the fractional derivative in the equations can be converted into integer order case with respect to a new variable. As for applications, we apply this method to the time fractional two-dimensional Boussinesq equation and the space-time fractional (2+1)-dimensional breaking soliton equation. As a result, some exact solutions including variable coefficient function solutions as well as solitary wave solutions for the two equations are found.

Mesoscale Ascent in Nocturnal Low-Level Jets

A theory for gentle but persistent mesoscale ascent in the lower troposphere is developed in which the vertical motion arises as an inertia–gravity wave response to the sudden decrease of turbulent mixing in a horizontally heterogeneous convective boundary layer (CBL). The zone of ascent is centered on the local maximum of a laterally varying buoyancy field (warm tongue in the CBL). The shutdown also triggers a Blackadar-type inertial oscillation and associated low-level jet (LLJ). These nocturnal motions are studied analytically using the linearized two-dimensional Boussinesq equations of motion, thermal energy, and mass conservation for an inviscid stably stratified fluid, with the initial state described by a zero-order jump model of a CBL. The vertical velocity revealed by the analytical solution increases with the amplitude of the buoyancy variation, CBL depth, and wavenumber of the buoyancy variation (larger vertical velocity for smaller-scale variations). Stable stratification in the free atmosphere has a lid effect, with a larger buoyancy frequency associated with a smaller vertical velocity. For the parameter values typical of the southern Great Plains warm season, the peak vertical velocity is ~3–10 cm s−1, with parcels rising ~0.3–1 km over the ~6–8-h duration of the ascent phase. Data from the 2015 Plains Elevated Convection at Night (PECAN) field project were used as a qualitative check on the hypothesis that the same mechanism that triggers nocturnal LLJs from CBLs can induce gentle but persistent ascent in the presence of a warm tongue.

Global regularity of the two-dimensional Boussinesq equations without diffusivity in bounded domains

We address the well-posedness for the two-dimensional Boussinesq equations with zero diffusivity in bounded domains. We prove global in time regularity for rough initial data: the initial velocity has ϵ fractional derivatives in Lq and the initial temperature is in Lq, for some q>2 and ϵ>0 arbitrarily small.

The 2D Boussinesq equations with fractional horizontal dissipation and thermal diffusion

This paper examines the global regularity problem on the two-dimensional (2D) incompressible Boussinesq equations with fractional horizontal dissipation and thermal diffusion. The goal is to establish the global existence and regularity for the Boussinesq equations with minimal dissipation and thermal diffusion. By working with this general 1D fractional Laplacian dissipation, we are no longer constrained to the standard partial dissipation and this study will help understand the issue on how much dissipation is necessary for the global regularity. Due to the nonlocality of these 1D fractional operators, some of the standard energy estimate techniques such as integration by parts no longer apply and new tools including several anisotropic embedding and interpolation inequalities involving fractional derivatives are derived. These tools allow us to obtain very sharp upper bounds for the nonlinearities.

Long time behavior of the two-dimensional Boussinesq equations without buoyancy diffusion

We study the global well-posedness and stability/instability of perturbations near a special type of hydrostatic equilibrium associated with the 2D Boussinesq equations without buoyancy (a.k.a. thermal) diffusion on a bounded domain subject to stress-free boundary conditions. The boundary of the domain is not necessarily smooth and may have corners such as in the case of rectangles. We achieve three goals. First, we establish the global-in-time existence and uniqueness of large-amplitude classical solutions. Efforts are made to reduce the regularity assumptions on the initial data. Second, we obtain the large-time asymptotics of the full nonlinear perturbation. In particular, we show that the kinetic energy and the first order derivatives of the velocity field converge to zero as time goes to infinity, regardless of the magnitude of the initial data, and the flow stratifies in the vertical direction in a weak topology. Third, we prove the linear stability of the hydrostatic equilibrium T(y) satisfying T′(y)=α>0, and the linear instability of periodic perturbations when T′(y)=α<0. Numerical simulations are supplemented to corroborate the analytical results and predict some phenomena that are not proved.The authors are pleased to dedicate this paper to Professor Edriss Saleh Titi on the occasion of his 60th birthday. Professor Titi’s myriad research contributions – including contributions to the problem constituting the focus of this work – and leadership in the mathematical fluid dynamics community serve as an inspiration to his students, collaborators and colleagues.

Lump dynamics of a generalized two-dimensional Boussinesq equation in shallow water

The Boussinesq equation can describe wave motions in media with damping mechanism, e.g., the propagation of long waves in shallow water and the oscillations of nonlinear elastic strings. To study the propagation of gravity waves on the surface of water, a second spatial variable (say, y) is weakly dependent, and an alternative form of generalized two-dimensional Boussinesq equation is investigated in this paper. Four families of lump solutions are derived by searching for positive quadratic function solutions to the associated bilinear equation. To guarantee the analyticity and rational localization of the lumps, some conditions are posed on both the lump parameters and the coefficients of the generalized two-dimensional Boussinesq equation. Localized structures and energy distribution of the lumps are analyzed as well.

An improved regularity criterion for the 2D inviscid Boussinesq equations

In this short paper, we consider the two-dimensional inviscid Boussinesq equations and establish an improved regularity criterion.

On the improved blow-up criterion for the 2D zero diffusivity Boussinesq equations with temperature-dependent viscosity

This paper examines the two-dimensional zero diffusivity Boussinesq equations with temperature-dependent viscosity in the whole space. Whether or not its classical solutions global in time existence is a difficult problem and remains open. The main goal of this paper is to establish several blow-up criteria. More precisely, it is proved that is the maximal time interval if and only if the viscosity satisfies or the vorticity satisfies As a direct application of the above results, several other blow-up criteria in the Besov spaces and the multiplier spaces are also established.

Stability analysis of solutions for the sixth-order nonlinear Boussinesq water wave equations in two-dimensions and its applications

In the present study, we are concerned with the generalized Boussinesq equation including the singular sixth-order Boussinesq equation, which describes the bi-directional propagation of small amplitude and long capillary-gravity waves on the surface of shallow water for bond number less than but very close to 1/3. By using the extended auxiliary equation method, we obtained some new soliton like solutions for the two-dimensional sixth-order nonlinear Boussinesq equation with constant coefficients. These solutions include symmetrical, non-symmetrical kink solutions, solitary pattern solutions, Jacobi and Weierstrass elliptic function solutions and triangular function solutions. The stability analysis for these solutions is discussed.

Initial boundary value problem for two-dimensional viscous Boussinesq equations for MHD convection

This paper is concerned with the initial boundary value problem for two-dimensional viscous Boussinesq equations for MHD convection. We show that the system has a unique classical solution for $H^3$ initial data, and the non-slip boundary condition for velocity field and the perfectly conducting wall condition for magnetic field. In addition, we show that the kinetic energy is uniformly bounded in time.

Global well-posedness for the 2D Boussinesq equations with zero viscosity

We prove the global well-posedness of the two-dimensional Boussinesq equations with zero viscosity and positive diffusivity in bounded domains for rough initial data [u0∈L2, curlu0∈L∞ and θ0∈Bq,p2−2/p with p∈(1,∞), q∈(2,∞)]. Our method is based on the maximal regularity for heat equation.

On the global regularity of the 2D Boussinesq equations with fractional dissipation

In this paper, we consider the two-dimensional (2D) incompressible Boussinesq equations with fractional Laplacian dissipation and thermal diffusion. Attention is focused on the subcritical case when the velocity dissipation dominates. More precisely, we establish the global regularity result of the 2D Boussinesq equations in a new range of fractional powers of the Laplacian, namely 1−α<β<minα2,(3α−2)(α+2)10−7α,2−2α4α−3 with 0.783≈21−2178<α<1. Therefore, this result significantly improves the previous work [31] which obtained the global regularity result for 1−α<β<f(α) with 0.888≈6−64<α<1, where f(α)<1 is an explicit function.

On the regularity criteria of the 2D Boussinesq equations with partial dissipation

In this paper, we consider the two-dimensional (2D) Boussinesq equations with partial dissipation. The issue of whether the 2D Boussinesq equations always possess global (in time) classical solutions can be difficult when there is only partial dissipation. Therefore, the goal of this paper is to establish some regularity criteria of the corresponding systems.

New solutions for conformable fractional Boussinesq and combined KdV-mKdV equations using Jacobi elliptic function expansion method

In this paper, the Jacobi elliptic function expansion method is proposed for the first time to construct the exact solutions of the time conformable fractional two-dimensional Boussinesq equation and the combined KdV-mKdV equation. New exact solutions are found. This method is based on Jacobi elliptic functions. The results obtained confirm that the proposed method is an efficient technique for analytic treatment of a wide variety of nonlinear conformable time-fractional partial differential equations.