Two-dimensional Boussinesq Equations Research Articles

Spinning phenomena and energetics of spherically pulsating patterns in stratified fluids

The nonlinear solutions of the two-dimensional Boussinesq equations describing internal waves in rotating stratified fluids were obtained as group invariant solutions. The latter nonlinear solutions correspond to the rotation transformation preserving the form of the original nonlinear equations of motion. It is shown that the obtained class of exact solutions can be associated with the spherically pulsating patterns observed in uniformly stratified fluids. It is also shown that the obtained rotationally symmetric solutions are bounded functions that can be visualized as spinning patterns in stratified fluids. It is also shown that the rotational transformation provides the energy conservation law together with other conservation laws for which the spinning phenomena is observed. The effects of nonlinearity and the Earth's rotation on such a phenomenon are also discussed.

Thermosolutal convection at infinite prandtl number with or without rotation: Bifurcation and stability in physical space

We examine the nature of the thermosolutal convection with or without rotation in the infinite Prandtl number regime, which is applicable to magma chambers. The onset of bifurcation and the structure of the bifurcated solutions in this double diffusion problem are analyzed. The stress-free boundary condition is imposed at the top and bottom plates confining the fluid. For the rotation free case, two-dimensional Boussinesq equations are considered and we prove that there are bifurcating solutions from the basic solution and that the bifurcated solutions consist of only one cycle of steady state solutions that are homeomorphic to S1. By thoroughly investigating the structure and transitions of the solutions of the thermosolutal convection problem in physical space, we confirm that the bifurcated solutions are indeed structurally stable. In the presence of rotation, we consider three-dimensional Boussinesq equations and we can get similar results as of the rotation free case. We also see how intensively the rotation inhibits the onset of convective motion. In turn, this will corroborate and justify the suggested results with the physical findings about the presence of roll structure.

Initial Boundary Value Problem for Two-Dimensional Viscous Boussinesq Equations

We study the initial boundary value problem of two-dimensional viscous Boussinesq equations over a bounded domain with smooth boundary. We show that the equations have a unique classical solution for H 3 initial data and the no-slip boundary condition. In addition, we show that the kinetic energy is uniformly bounded in time.

A micro flowmeter based on the velocity measurement of a locally accelerated thermal flow in an upwardly directed Hagen–Poiseuille flow

A simple measurement method of micro volumetric flow rate with high precision and low cost is proposed for industrial use. Poiseuille flow is obtained within a very short entrance length, especially for low Reynolds number flows. A locally accelerated thermal flow is produced by a small electric heater which is placed in the center part of an upwardly directed Poiseuille flow. It was found that the mean velocity of locally accelerated flow V a between the heater and the thermal sensor, which accelerates the local maximum velocity of the base Poiseuille flow, is directly proportional to the volumetric flow rate Q v in a pipe within low Reynolds number flows. Q v can be obtained by using the linear relationship between V a and Q v . The validity of measuring the flow rate below 1 mL/min was confirmed for both water and air flows experimentally and numerically. Numerical simulation was performed using unsteady two-dimensional Boussinesq equations in order to validate the measurement principle of this new micro flowmeter, and also to determine the optimized flowmeter design, such as the most suitable sensor position and the amount of Joule heating.

Rational and Periodic Wave Solutions of Two-Dimensional Boussinesq Equation

Two new exact, rational and periodic wave solutions are derived for the two-dimensional Boussinesq equation. For the first solution it is obtained by performing an appropriate limiting procedure on the soliton solutions obtained by Hirota bilinear method. The second one in terms of Riemann theta function is explicitly presented by virtue of Hirota bilinear method and its asymptotic property is also analyzed in detail. Moreover, it is of interest to note that classical soliton solutions can be reduced from the periodic wave solutions.

Two-dimensional Boussinesq equation in a disc and anisotropic Sobolev spaces

The two-dimensional damped Boussinesq equation with a forcing term is considered in a unit disc. It governs forced, small, nonlinear oscillations of a thin elastic membrane in the presence of viscosity. The eigenfunction expansion method is used for constructing global-in-time solutions of the initial-boundary-value problem in question. Specially designed anisotropic Sobolev spaces are introduced in order to reflect the effect of nonlinear smoothing in the angular coordinate. Existence and uniqueness in these spaces are proved on the basis of the construction. To cite this article: V. Varlamov, C. R. Mecanique 335 (2007).

Convolutions of Rayleigh functions and their application to semi-linear equations in circular domains

Rayleigh functions σ l ( ν ) are defined as series in inverse powers of the Bessel function zeros λ ν , n ≠ 0 , σ l ( ν ) = ∑ n = 1 ∞ 1 λ ν , n 2 l , where l = 1 , 2 , … ; ν is the index of the Bessel function J ν ( x ) and n = 1 , 2 , … is the number of the zeros. Convolutions of Rayleigh functions with respect to the Bessel index, R l ( m ) = ∑ k = − ∞ ∞ σ l ( | m − k | ) σ l ( | k | ) for l = 1 , 2 , … ; m = 0 , ± 1 , ± 2 , … , are needed for constructing global-in-time solutions of semi-linear evolution equations in circular domains [V. Varlamov, On the spatially two-dimensional Boussinesq equation in a circular domain, Nonlinear Anal. 46 (2001) 699–725; V. Varlamov, Convolution of Rayleigh functions with respect to the Bessel index, J. Math. Anal. Appl. 306 (2005) 413–424]. The study of this new family of special functions was initiated in [V. Varlamov, Convolution of Rayleigh functions with respect to the Bessel index, J. Math. Anal. Appl. 306 (2005) 413–424], where the properties of R 1 ( m ) were investigated. In the present work a general representation of R l ( m ) in terms of σ l ( ν ) is deduced. On the basis of this a representation for the function R 2 ( m ) is obtained in terms of the ψ-function. An asymptotic expansion is computed for R 2 ( m ) as | m | → ∞ . Such asymptotics are needed for establishing function spaces for solutions of semi-linear equations in bounded domains with periodicity conditions in one coordinate. As an example of application of R l ( m ) a forced Boussinesq equation u t t − 2 b Δ u t = − α Δ 2 u + Δ u + β Δ ( u 2 ) + f with α , b = const > 0 and β = const ∈ R is considered in a unit disc with homogeneous boundary and initial data. Construction of its global-in-time solutions involves the use of the functions R 1 ( m ) and R 2 ( m ) which are responsible for the nonlinear smoothing effect.

On Global Well‐Posedness of the Lagrangian Averaged Euler Equations

We study the global well-posedness of the Lagrangian averaged Euler equations in three dimensions. We show that a necessary and sufficient condition for the global existence is that the bounded mean oscillation of the stream function is integrable in time. We also derive a sufficient condition in terms of the total variation of certain level set functions, which guarantees the global existence. Furthermore, we obtain the global existence of the averaged two-dimensional (2D) Boussinesq equations and the Lagrangian averaged 2D quasi-geostrophic equations in finite Sobolev space in the absence of viscosity or dissipation.

Forced nonlinear oscillations of elastic membranes

An initial-boundary-value problem is considered for the spatially two-dimensional damped Boussinesq equation with a forcing term. The problem in question is argued to serve as a model for describing small nonlinear oscillations of a circular membrane under the influence of acoustic pressure. Eigenfunction expansion method is used for constructing the global-in-time solution of the problem in question. Existence and uniqueness follow from the construction. Long-time asymptotics is computed on the basis of the obtained series representation. Numerical simulations are conducted. The algorithm shows excellent convergence properties.

Initial and boundary value problems for two-dimensional non-hydrostatic boussinesq equations

Based on the theory of stratification, the well-posedness of the initial and boundary value problems for the system of two-dimensional non-hydrostatic Boussinesq equations was discussed. The sufficient and necessary conditions of the existence and uniqueness for the solution of the equations were given for some representative initial and boundary value problems. Several special cases were discussed.

Variants of the two-dimensional boussinesq equation with compactons, solitons, and periodic solutions

Variants of the two-dimensional Boussinesq equation with positive and negative exponents are studied. The sine-cosine ansatz is fruitfully used to carry out the analysis. Exact solutions of different physical structures: compactons, solitary patterns, solitons, and periodic solutions, are obtained. The quantitative change in the physical structure of the solutions is shown to depend mainly on the exponent of the wave function u( x, t) and on the ratio a/b of the derivatives of u( x, t).

Standard forms of elliptic integrals and their applications to nonlinear evolution equations

Abstract Many nonlinear evolution equations can be converted into first- or high-order ordinary differential equations. In this paper, we illustrate a connection between the elliptic integrals and some nonlinear evolution equations such as the two-dimensional Boussinesq equation, and demonstrate that traveling wave solutions and periodic wave solutions to these nonlinear equations can be obtained by means of the standard forms of elliptic integrals directly.

Ergodicity for the Dissipative Boussinesq Equations with Random Forcing

We study the stationary measure for the two-dimensional Boussinesq equation with random forcing. We prove the ergodicity for the two-dimensional stochastically forced Boussinesq equation. We also study the Galerkin truncations of the three-dimensional Boussinesq equations under degenerate stochastic forcing. We follow closely the previous results on the stochastically forced Navier–Stokes equations.

Constructing exact solutions for two-dimensional nonlinear dispersion Boussinesq equation. II: Solitary pattern solutions

In the first paper (I) we have shown that two-dimensional nonlinear dispersion Boussinesq equations, B( m, n, k, p), admitted abundant compact structures for different parameters m, n, k, p. In this paper we shall further investigate the solitary pattern solutions of B( m, n, k, p) equations. As a consequence, it is shown that these equations also possess many solitary pattern solutions expressed in terms of the hyperbolic sine and cosine functions.

On the spatially two-dimensional Boussinesq equation in a circular domain

On the spatially two-dimensional Boussinesq equation in a circular domain

KP description of unidirectional long waves. The model case

The Kadomtsev–Petviashvili (KP) equation can be formally derived as an envelope equation for three-dimensional unidirectional water waves in the limit of long waves. As a first step towards a mathematical justification, we consider here a two-dimensional Boussinesq equation, which is a realistic model for three-dimensional water waves. Using rigorous estimates, we show that part of the dynamics of the KP equation can be found approximately in the two-dimensional Boussinesq equation. On the other hand, there exist initial data for the KP equation such that the corresponding solutions of the two-dimensional Boussinesq equation behave in no way according to the KP prediction. We expect that similar results hold for the three-dimensional water wave problem too.

Soliton propagation in a wide channel with an uneven bottom

Nonlinear waves of small amplitude in wide horizontal channels are considered. The channel depth is assumed to be a function weakly dependent on the transverse coordinate. To describe the waves, the two-dimensional Boussinesq equations in the form obtained in this paper are used. Stationary solutions in the form of a soliton followed by a set of sinusoidal waves are found. The phase velocity of these waves in the channel direction is equal to the soliton velocity.

A finite element analysis of mach reflection by using the Boussinesq equation

SUMMARY The numerical analysis of ‘Mach reflection’, which is the reflection of an obliquely incident solitary wave by a vertical wall, is presented. For the mathematical model of the analysis, the two-dimensional Boussinesq equation is used. In order to solve the equation in space, the finite element method based on the linear triangular element and the conventional Galerkin method is applied. The combination of explicit and semi-implicit schemes is employed for the time integration. Moreover, one of the treatments for the open boundary condition, in which the analytical solution of the linearized Boussinesq equation in the outside domain is linked to the discrete values of velocity and water elevation in the inside domain, is applied for the modeling of the Mach reflection problem. © 1998 John Wiley & Sons, Ltd.

A Modified Leap-Frog Scheme for Linear Shallow-Water Equations

A new leap-frog finite difference scheme is proposed to solve the linear shallow-water equations. The accuracy of propagation speed in diagonal direction is improved through the inclusion of frequency dispersion correction terms in linear shallow-water equations. The two-dimensional linear Boussinesq equations are recovered when the proper spatial grid size and time step size are chosen. As a result, more accurate simulation can be made for the transoceanic propagation of tsunamis. The scheme is stable when the Courant number is less than 0.866.

A note on an extra constraint found in some multidimensional extensions of integrable equations

In an earlier paper, Johnson generalized the (integrable) Boussinesq equation to 2 (space) +1 (time) dimensions, and found that the generalized equation admits a 3-soliton solution only if an auxiliary condition is satisfied [R.S. Johnson, “A two-dimensional Boussinesq equation for water waves and some of its solutions”, J. Fluid Mech., to appear.]. We show that the condition has a simple meaning: it guarantees that the solution is time-independent in a moving coordinate system. Further, it implies a change of variables that reduces the generalized equation by at least one dimension, usually to an integrable equation in 1 + 1 dimensions. Finally, we give some other related examples where equivalent conditions have been found by other authors in other multidimensional generalizations of integrable equations.