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Journal articles on the topic 'Approximation de Boussinesq quadratique'

Author: Grafiati

Published: 7 December 2024

Last updated: 31 July 2025

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1

Young, William R. "Dynamic Enthalpy, Conservative Temperature, and the Seawater Boussinesq Approximation." Journal of Physical Oceanography 40, no. 2 (2010): 394–400. http://dx.doi.org/10.1175/2009jpo4294.1.

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Abstract A new seawater Boussinesq system is introduced, and it is shown that this approximation to the equations of motion of a compressible binary solution has an energy conservation law that is a consistent approximation to the Bernoulli equation of the full system. The seawater Boussinesq approximation simplifies the mass conservation equation to ∇ · u = 0, employs the nonlinear equation of state of seawater to obtain the buoyancy force, and uses the conservative temperature introduced by McDougall as a thermal variable. The conserved energy consists of the kinetic energy plus the Boussine

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2

Wood, T. S., and P. J. Bushby. "Oscillatory convection and limitations of the Boussinesq approximation." Journal of Fluid Mechanics 803 (August 30, 2016): 502–15. http://dx.doi.org/10.1017/jfm.2016.511.

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We determine the asymptotic conditions under which the Boussinesq approximation is valid for oscillatory convection in a rapidly rotating fluid. In the astrophysically relevant parameter regime of small Prandtl number, we show that the Boussinesq prediction for the onset of convection is valid only under much more restrictive conditions than those that are usually assumed. In the case of an ideal gas, we recover the Boussinesq results only if the ratio of the domain height to a typical scale height is much smaller than the Prandtl number. This requires an extremely shallow domain in the astrop

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3

RAJAGOPAL, K. R., M. RUZICKA, and A. R. SRINIVASA. "ON THE OBERBECK-BOUSSINESQ APPROXIMATION." Mathematical Models and Methods in Applied Sciences 06, no. 08 (1996): 1157–67. http://dx.doi.org/10.1142/s0218202596000481.

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This paper deals with a derivation (using a perturbation technique) of an approximation, due to Oberbeck8,9 and Boussinesq,1 to describe the thermal response of linearly viscous fluids that are mechanically incompressible but thermally compressible. The present approach uses a nondimensionalization suggested by Chandrasekhar2 and utilizing the ratio of two characteristic velocities as a measure of smallness, systematically derives the Oberbeck-Boussinesq approximation as a third-order perturbation. In the present approach, the material is subjected to the constraint that the volume change is d

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4

Barletta, Antonio, Michele Celli, and D. Andrew S. Rees. "On the Use and Misuse of the Oberbeck–Boussinesq Approximation." Physics 5, no. 1 (2023): 298–309. http://dx.doi.org/10.3390/physics5010022.

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The Oberbeck–Boussinesq approximation is the most commonly employed theoretical scheme for the study of natural or mixed convection flows. However, the misunderstanding of this approximated framework is a possibility that may cause the emergence of paradoxes or, at least, incorrect conclusions. In this paper, the basic features of the Oberbeck–Boussinesq approximation are briefly recalled and three simple examples where this theoretical scheme may be misused are provided. Such misuses of the approximation lead to erroneous conclusions that, in the examples presented in this note, entail violat

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5

Téchené, Jean-Jacques. "Les aspects fondamentaux de l'admissibilité en approximation quadratique d'applications linéaires." Linear Algebra and its Applications 264 (October 1997): 389–419. http://dx.doi.org/10.1016/s0024-3795(96)00405-3.

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6

Gomes, Diogo A., and Claudia Valls. "Approximation of ill-posed boussinesq equations." Dynamical Systems 19, no. 4 (2004): 345–57. http://dx.doi.org/10.1080/1468936042000269587.

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7

Baron-Pertuz, Cristian-Fabian, Ana-Magnolia Marin-Ramirez, and Ruben-Dario Ortiz-Ortiz. "An approximation to the Boussinesq equations." International Journal of Mathematical Analysis 8 (2014): 2433–37. http://dx.doi.org/10.12988/ijma.2014.48274.

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8

PRUŠA, VÍT, and K. R. RAJAGOPAL. "ON MODELS FOR VISCOELASTIC MATERIALS THAT ARE MECHANICALLY INCOMPRESSIBLE AND THERMALLY COMPRESSIBLE OR EXPANSIBLE AND THEIR OBERBECK–BOUSSINESQ TYPE APPROXIMATIONS." Mathematical Models and Methods in Applied Sciences 23, no. 10 (2013): 1761–94. http://dx.doi.org/10.1142/s0218202513500516.

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Viscoelastic fluid like materials that are mechanically incompressible but are compressible or expansible with respect to thermal stimuli are of interest in various applications ranging from geophysics and polymer processing to glass manufacturing. Here we develop a thermodynamical framework for the modeling of such materials. First we illustrate the basic ideas in the simpler case of a viscous fluid, and after that we use the notion of natural configuration and the concept of the maximization of the entropy production, and we develop a model for a Maxwell type viscoelastic fluid that is mecha

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9

Siddiqui, Perwez. "Density Modelling in Mixed Convection Flow in a Vertical Parallel Plate Channel." International Journal of Heat and Technology 39, no. 4 (2021): 1294–304. http://dx.doi.org/10.18280/ijht.390428.

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In this paper, a novel way of modelling the density in buoyancy term of mixed convection flow problem is presented using equation of state and Boussinesq approximation without first-order approximation of density with respect to temperature. The presented density model is used to investigate the laminar mixed convection flow in a vertical parallel plate channel under symmetric constant wall heat flux. The results obtained are compared with the results obtained using first-order approximation of density with Boussinesq approximation, and also compared with the results obtained using variable th

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10

Lopez, Jose M., Francisco Marques, and Marc Avila. "The Boussinesq approximation in rapidly rotating flows." Journal of Fluid Mechanics 737 (November 15, 2013): 56–77. http://dx.doi.org/10.1017/jfm.2013.558.

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AbstractIn commonly used formulations of the Boussinesq approximation centrifugal buoyancy effects related to differential rotation, as well as strong vortices in the flow, are neglected. However, these may play an important role in rapidly rotating flows, such as in astrophysical and geophysical applications, and also in turbulent convection. Here we provide a straightforward approach resulting in a Boussinesq-type approximation that consistently accounts for centrifugal effects. Its application to the accretion-disc problem is discussed. We numerically compare the new approach to the typical

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11

Grandi, Diego, and Arianna Passerini. "On the Oberbeck–Boussinesq approximation for gases." International Journal of Non-Linear Mechanics 134 (September 2021): 103738. http://dx.doi.org/10.1016/j.ijnonlinmec.2021.103738.

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12

Danchin, Raphaël, and Lingbing He. "The Oberbeck–Boussinesq approximation in critical spaces." Asymptotic Analysis 84, no. 1-2 (2013): 61–102. http://dx.doi.org/10.3233/asy-131170.

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13

Churilov, S. M. "Holmboe instability beyond the Boussinesq approximation revisited." Physics of Fluids 31, no. 9 (2019): 094104. http://dx.doi.org/10.1063/1.5115826.

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14

Demuren, Ayodeji, and Holger Grotjans. "Buoyancy-Driven Flows—Beyond the Boussinesq Approximation." Numerical Heat Transfer, Part B: Fundamentals 56, no. 1 (2009): 1–22. http://dx.doi.org/10.1080/10407790902970080.

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15

Passerini, Arianna, and Gudrun Thäter. "Boussinesq-type approximation for second-grade fluids." International Journal of Non-Linear Mechanics 40, no. 6 (2005): 821–31. http://dx.doi.org/10.1016/j.ijnonlinmec.2004.07.019.

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16

Nadolin, K. A. "Boussinesq approximation in the Rayleigh-Benard problem." Fluid Dynamics 30, no. 5 (1995): 645–51. http://dx.doi.org/10.1007/bf02079380.

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17

Castro, Ángel, Diego Córdoba, and Daniel Lear. "On the asymptotic stability of stratified solutions for the 2D Boussinesq equations with a velocity damping term." Mathematical Models and Methods in Applied Sciences 29, no. 07 (2019): 1227–77. http://dx.doi.org/10.1142/s0218202519500210.

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We consider the 2D Boussinesq equations with a velocity damping term in a strip domain, with impermeable walls. In this physical scenario, where the Boussinesq approximation is accurate when density or temperature variations are small, our main result is the asymptotic stability for a specific type of perturbations of a stratified solution.

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18

Lamb, K. G. "Energetics of internal solitary waves in a background sheared current." Nonlinear Processes in Geophysics 17, no. 5 (2010): 553–68. http://dx.doi.org/10.5194/npg-17-553-2010.

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Abstract. The energetics of internal waves in the presence of a background sheared current is explored via numerical simulations for four different situations based on oceanographic conditions: the nonlinear interaction of two internal solitary waves; an internal solitary wave shoaling through a turning point; internal solitary wave reflection from a sloping boundary and a deep-water internal seiche trapped in a deep basin. In the simulations with variable water depth using the Boussinesq approximation the combination of a background sheared current, bathymetry and a rigid lid results in a cha

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19

Selmi, Ridha. "Global Well-Posedness and Convergence Results for the 3D-Regularized Boussinesq System." Canadian Journal of Mathematics 64, no. 6 (2012): 1415–35. http://dx.doi.org/10.4153/cjm-2012-013-5.

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Abstract Analytical study of the regularization of the Boussinesq systemis performed in frequency space using Fourier theory. Existence and uniqueness of weak solutions with minimum regularity requirement are proved. Convergence results of the unique weak solution of the regularized Boussinesq system to a weak Leray-Hopf solution of the Boussinesq system are established as the regularizing parameter α vanishes. The proofs are done in the frequency space and use energy methods, the Arselà-Ascoli compactness theorem and a Friedrichs-like approximation scheme.

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20

El-Gendi, Mahmoud M., and Abdelraheem M. Aly. "Numerical simulation of natural convection using unsteady compressible Navier-stokes equations." International Journal of Numerical Methods for Heat & Fluid Flow 27, no. 11 (2017): 2508–27. http://dx.doi.org/10.1108/hff-10-2016-0376.

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Purpose Boussinesq approximation is widely used in solving natural convection problems, but it has severe practical limitations. Using Boussinesq approximation, the temperature difference should be less than 28.6 K. The purpose of this study is to get rid of Boussinesq approximation and simulates the natural convection problems using an unsteady compressible Navier-Stokes solver. The gravity force is included in the source term. Three temperature differences are used namely 20 K, 700 K and 2000 K. Design/methodology/approach The calculations are carried out on the square and sinusoidal cavitie

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21

Greatbatch, Richard J., Youyu Lu, and Yi Cai. "Relaxing the Boussinesq Approximation in Ocean Circulation Models." Journal of Atmospheric and Oceanic Technology 18, no. 11 (2001): 1911–23. http://dx.doi.org/10.1175/1520-0426(2001)018<1911:rtbaio>2.0.co;2.

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22

Zeytounian, Radyadour Kh. "Joseph Boussinesq and his approximation: a contemporary view." Comptes Rendus Mécanique 331, no. 8 (2003): 575–86. http://dx.doi.org/10.1016/s1631-0721(03)00120-7.

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23

Anufriev, Alexander P., and Pavel Hejda. "Earth's core convection: Boussinesq approximation or incompressible approach?" Geophysical & Astrophysical Fluid Dynamics 104, no. 1 (2010): 65–83. http://dx.doi.org/10.1080/03091920903517863.

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24

Bowker, Jordan A., David W. Hughes, and Evy Kersalé. "Incorporating velocity shear into the magneto-Boussinesq approximation." Geophysical & Astrophysical Fluid Dynamics 108, no. 5 (2014): 553–67. http://dx.doi.org/10.1080/03091929.2014.917296.

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25

Durran, Dale R., and Akio Arakawa. "Generalizing the Boussinesq approximation to stratified compressible flow." Comptes Rendus Mécanique 335, no. 9-10 (2007): 655–64. http://dx.doi.org/10.1016/j.crme.2007.08.010.

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26

Kagei, Yoshiyuki, and Michael Růžička. "The Oberbeck–Boussinesq approximation as a constitutive limit." Continuum Mechanics and Thermodynamics 28, no. 5 (2015): 1411–19. http://dx.doi.org/10.1007/s00161-015-0483-9.

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27

Duchêne, Vincent. "Boussinesq/Boussinesq systems for internal waves with a free surface, and the KdV approximation." ESAIM: Mathematical Modelling and Numerical Analysis 46, no. 1 (2011): 145–85. http://dx.doi.org/10.1051/m2an/2011037.

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28

Dhar, A. K., and K. P. Das. "Stability analysis from fourth order evolution equation for small but finite amplitude interfacial waves in the presence of a basic current shear." Journal of the Australian Mathematical Society. Series B. Applied Mathematics 35, no. 3 (1994): 348–65. http://dx.doi.org/10.1017/s0334270000009346.

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AbstractA fourth-order nonlinear evolution equation is derived for a wave propagating at the interface of two superposed fluids of infinite depths in the presence of a basic current shear. On the basis of this equation a stability analysis is made for a uniform wave train. Discussions are given for both an air-water interface and a Boussinesq approximation. Significant deviations are noticed from the results obtained from the third-order evolution equation, which is the nonlinear Schrödinger equation. In the Boussinesq approximation, it has been possible to compare the present results with the

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29

Zhang, Yao, Andrew Brian Kennedy, Joannes Westerink, Nishant Panda, and Client Dawson. "NEW BOUSSINESQ SYSTEM FOR NONLINEAR WATER WAVES." Coastal Engineering Proceedings 1, no. 33 (2012): 4. http://dx.doi.org/10.9753/icce.v33.waves.4.

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In this paper, a new Boussinesq water wave theory is derived which can simulate highly dispersive nonlinear waves, their depth-varying velocities, and wave-induced currents, from very deep, but still finite, depths through the surf zone to the shoreline.. Boussinesq scaling is employed. We removed the irrotationality assumption by using polynomial basis functions for velocity profile which are inserted into basic equations of motion. Keep terms up to the desired approximation level and solve the coupled weighted residual system together with vertically integrated mass equation. The computation

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30

Melinand, Benjamin. "Long wave approximation for water waves under a Coriolis forcing and the Ostrovsky equation." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 148, no. 6 (2018): 1201–37. http://dx.doi.org/10.1017/s0308210518000136.

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This paper is devoted to the study of the long wave approximation for water waves under the influence of the gravity and a Coriolis forcing. We start by deriving a generalization of the Boussinesq equations in one (spatial) dimension and we rigorously justify them as an asymptotic model of water wave equations. These new Boussinesq equations are not the classical Boussinesq equations: a new term due to the vorticity and the Coriolis forcing appears that cannot be neglected. We study the Boussinesq regime and derive and fully justify different asymptotic models when the bottom is flat: a linear

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31

Ahmad, Samaila K., Basant k. Jha, Ayyub M. Hussaini, and Muhammad M. Altine. "MHD Free-Convective Couette flow in a Vertical Porous Microchannel Using Non-Linear Boussinesq Approximation." Journal of Applied Science, Information and Computing 3, no. 2 (2022): 25–34. http://dx.doi.org/10.59568/jasic-2022-3-2-05.

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An analytical solution for free convection flow of an electrically conducting fluid in a vertical micro-porous-channel, in the existence of transversely applied magnetic-field and nonlinear Boussinesq approximation is carried out in this article. The governing equations representing stated objective are obtained and solved analytically using method of undetermiend coefficients and direct integration. Pictorial and tabularlar representions of solutions obtained are carried out, so as to ascertain the role of various governing parameters entering flow formation. During the course of numerical si

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32

Bannon, Peter R., Jeffrey M. Chagnon, and Richard P. James. "Mass Conservation and the Anelastic Approximation." Monthly Weather Review 134, no. 10 (2006): 2989–3005. http://dx.doi.org/10.1175/mwr3228.1.

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Abstract Numerical anelastic models solve a diagnostic elliptic equation for the pressure field using derivative boundary conditions. The pressure is therefore determined to within a function proportional to the base-state density field with arbitrary amplitude. This ambiguity is removed by requiring that the total mass be conserved in the model. This approach enables one to determine the correct temperature field that is required for the microphysical calculations. This correct, mass-conserving anelastic model predicts a temperature field that is an accurate approximation to that of a compres

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33

Zhang, Lifang, Jianmin Zhang, Yong Peng, Jiangyang Pan, and Zhongxian Peng. "Numerical Simulation of Flow and Temperature Fields in a Deep Stratified Reservoir Using Water-Separating Curtain." International Journal of Environmental Research and Public Health 16, no. 24 (2019): 5143. http://dx.doi.org/10.3390/ijerph16245143.

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In this work, the flow and temperature fields of a thermally stratified reservoir under different settings of a water-separating curtain are simulated by using the standard k-ε turbulence model. In the simulation, two different equations of state including Boussinesq approximation and the density-temperature function have been used and compared. This study shows that Boussinesq approximation is more time-saving, and the density-temperature function has higher computational accuracy. Thus, the standard k-ε turbulence model with two equations of state is applied to study the effect of adding a w

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34

Mahanty, Debarati, Reeba Babu, and B. Mahanthesh. "Theoretical and analytical analysis of convective heat transport of radiated micropolar fluid over a vertical plate under nonlinear Boussinesq approximation." Multidiscipline Modeling in Materials and Structures 16, no. 5 (2020): 915–36. http://dx.doi.org/10.1108/mmms-05-2019-0099.

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PurposeIn heat transfer problems, if the temperature difference is not sufficiently so small then the linear Boussinesq approximation is not adequate to describe thermal analysis. Also, nonlinear density variation with respect to temperature/concentration has a significant impact on heat and fluid flow characteristics. Because of this reason, the impact of nonlinear density variation in the buoyancy force term cannot be neglected. Therefore in this paper, the unsteady flow and heat transfer of radiating magneto-micropolar fluid by considering nonlinear Boussinesq approximation is investigated

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35

ROTUNNO, R., J. B. KLEMP, G. H. BRYAN, and D. J. MURAKI. "Models of non-Boussinesq lock-exchange flow." Journal of Fluid Mechanics 675 (April 8, 2011): 1–26. http://dx.doi.org/10.1017/jfm.2010.648.

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Nearly all analytical models of lock-exchange flow are based on the shallow-water approximation. Since the latter approximation fails at the leading edges of the mutually intruding fluids of lock-exchange flow, solutions to the shallow-water equations can be obtained only through the specification of front conditions. In the present paper, analytic solutions to the shallow-water equations for non-Boussinesq lock-exchange flow are given for front conditions deriving from free-boundary arguments. Analytic solutions are also derived for other proposed front conditions – conditions which appear to

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36

Gravanis, Elias, Evangelos Akylas, and Ernestos Nikolas Sarris. "Approximate Solutions for Horizontal Unconfined Aquifers in the Buildup Phase." Water 16, no. 7 (2024): 1031. http://dx.doi.org/10.3390/w16071031.

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We construct approximate analytical solutions of the Boussinesq equation for horizontal unconfined aquifers in the buildup phase under constant recharge and zero-inflow conditions. We employ a variety of methods, which include wave solutions, self-similar solutions, and two classical linear approximations of the Boussinesq equation (linear and quadratic), to explore the behavior and performance of the solutions derived from each method against the Boussinesq equation and the exact (non-closed form) analytical solutions. We find that the wave approximation, which is of a conceptual nature, enca

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37

Artemov, Mikhail A., and Evgenii S. Baranovskii. "Solvability of the Boussinesq Approximation for Water Polymer Solutions." Mathematics 7, no. 7 (2019): 611. http://dx.doi.org/10.3390/math7070611.

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We consider nonlinear Boussinesq-type equations that model the heat transfer and steady viscous flows of weakly concentrated water solutions of polymers in a bounded three-dimensional domain with a heat source. On the boundary of the flow domain, the impermeability condition and a slip condition are provided. For the temperature field, we use a Robin boundary condition corresponding to the classical Newton law of cooling. By using the Galerkin method with special total sequences in suitable function spaces, we prove the existence of a weak solution to this boundary-value problem, assuming that

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38

Mikaelian, Karnig O. "Boussinesq approximation for Rayleigh-Taylor and Richtmyer-Meshkov instabilities." Physics of Fluids 26, no. 5 (2014): 054103. http://dx.doi.org/10.1063/1.4874881.

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39

Lannes, D., and Jean-Claude Saut. "Weakly transverse Boussinesq systems and the Kadomtsev–Petviashvili approximation." Nonlinearity 19, no. 12 (2006): 2853–75. http://dx.doi.org/10.1088/0951-7715/19/12/007.

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40

Paolucci, S., and D. R. Chenoweth. "Departures from the Boussinesq approximation in laminar Bénard convection." Physics of Fluids 30, no. 5 (1987): 1561. http://dx.doi.org/10.1063/1.866218.

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41

Katsaounis, Theodoros, Dimitrios Mitsotakis, and Georges Sadaka. "Boussinesq-Peregrine water wave models and their numerical approximation." Journal of Computational Physics 417 (September 2020): 109579. http://dx.doi.org/10.1016/j.jcp.2020.109579.

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42

Tarman, I. Hakan. "A Karhunen-Loève based Galerkin approximation to Boussinesq equation." Computer Methods in Applied Mechanics and Engineering 137, no. 3-4 (1996): 275–84. http://dx.doi.org/10.1016/s0045-7825(96)01048-1.

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43

Eden, Carsten. "Revisiting the Energetics of the Ocean in Boussinesq Approximation." Journal of Physical Oceanography 45, no. 3 (2015): 630–37. http://dx.doi.org/10.1175/jpo-d-14-0072.1.

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AbstractFollowing a suggestion by Tailleux, a consistent formulation of internal energy, the first law of thermodynamics, and the thermodynamic potentials for an ocean in Boussinesq approximation with a nonlinear equation of state is given. A modification of the pressure work in the first law is the only necessary modification from which all thermodynamic potentials and thermodynamic relations follow in a consistent way. This treatment of thermodynamics allows for a closed and explicit formulation of conservation equations for dynamic and potential reservoirs of both enthalpy and internal ener

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44

Manaa, Saad A., and Nergiz M. Mosa. "Adomian Decomposition and Successive Approximation Methods for Solving Kaup-Boussinesq System." Science Journal of University of Zakho 7, no. 3 (2019): 101–7. http://dx.doi.org/10.25271/sjuoz.2019.7.3.582.

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The Kaup-Boussinesq system has been solved numerically by using two methods, Successive approximation method (SAM) and Adomian decomposition method (ADM). Comparison between the two methods has been made and both can solve this kind of problems, also both methods are accurate and has faster convergence. The comparison showed that the Adomian decomposition method much more accurate than Successive approximation method.

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45

Xie, Jing, Chen Miao, Zhi Li Gao, Jian Bing Shi, and Tai Wang. "Numerical Simulation of Air Curtain Used in the Cold Store through Different Models." Advanced Materials Research 749 (August 2013): 554–60. http://dx.doi.org/10.4028/www.scientific.net/amr.749.554.

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Air curtains are commonly used to cut off air flow between the cold store and hot environment, and reduce heat and mass transfer in order to maintain the low temperature in the cold store. CFD can be used to predict operation rule of air curtain used in the cold store intuitively and modelling is the most important part of numerical simulation of air curtain used in the cold store. In this study, different numerical models including standardmodel (with and without boussinesq approximation) and RSM (with and without boussinesq approximation) were used to simulate the temperature field and air f

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46

Korre, L., N. Brummell, and P. Garaud. "Boussinesq convection in a gaseous spherical shell." EAS Publications Series 82 (2019): 373–82. http://dx.doi.org/10.1051/eas/1982033.

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In this paper, we investigate the dynamics of convection in a spherical shell under the Boussinesq approximation but considering the compressibility which arises from a non zero adiabatic temperature gradient, a relevant quantity for gaseous objects such as stellar or planetary interiors. We find that depth-dependent superiadiabaticity, combined with the use of mixed boundary conditions (fixed flux/fixed temperature), gives rise to unexpected dynamics that were not previously reported.

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47

Wei, Ge, James T. Kirby, Stephan T. Grilli, and Ravishankar Subramanya. "A fully nonlinear Boussinesq model for surface waves. Part 1. Highly nonlinear unsteady waves." Journal of Fluid Mechanics 294 (July 10, 1995): 71–92. http://dx.doi.org/10.1017/s0022112095002813.

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Fully nonlinear extensions of Boussinesq equations are derived to simulate surface wave propagation in coastal regions. By using the velocity at a certain depth as a dependent variable (Nwogu 1993), the resulting equations have significantly improved linear dispersion properties in intermediate water depths when compared to standard Boussinesq approximations. Since no assumption of small nonlinearity is made, the equations can be applied to simulate strong wave interactions prior to wave breaking. A high-order numerical model based on the equations is developed and applied to the study of two

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48

Wang, Yinxia. "On the Cauchy problem for one dimension generalized Boussinesq equation." International Journal of Mathematics 26, no. 03 (2015): 1550023. http://dx.doi.org/10.1142/s0129167x15500238.

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In this paper, we study the Cauchy problem for one dimension generalized damped Boussinesq equation. First, global existence and decay estimate of solutions to this problem are established. Second, according to the detail analysis for solution operator the generalized damped Boussinesq equation, the nonlinear approximation to global solutions is established. Finally, we prove that the global solution u to our problem is asymptotic to the superposition of nonlinear diffusion waves expressed in terms of the self-similar solution of the viscous Burgers equation as time tends to infinity.

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Dewar, W. K., J. Schoonover, T. J. McDougall, and W. R. Young. "Semicompressible Ocean Dynamics." Journal of Physical Oceanography 45, no. 1 (2015): 149–56. http://dx.doi.org/10.1175/jpo-d-13-0268.1.

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AbstractThe equations of motion are reexamined with the objective of improving upon the Boussinesq approximation. The authors derive new equations that conserve energy, filter out sound waves, are more accurate than the Boussinesq set, and are computationally competitive with them. The new equations are partly enabled by exploiting a reversible exchange between internal and gravitational potential fluid energy. To improve upon these equations appears to require the inclusion of acoustics, at which point one should use full Navier–Stokes. This study recommends the new sets for testing in genera

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Geyer, Anna, and Ronald Quirchmayr. "Shallow water equations for equatorial tsunami waves." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 376, no. 2111 (2017): 20170100. http://dx.doi.org/10.1098/rsta.2017.0100.

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We present derivations of shallow water model equations of Korteweg–de Vries and Boussinesq type for equatorial tsunami waves in the f -plane approximation and discuss their applicability. This article is part of the theme issue ‘Nonlinear water waves’.

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