A unified asymptotic theory of the anelastic approximation in geophysical gases and liquids

Bois, Pierre-Antoine

doi:10.1016/j.mechrescom.2005.05.016

Cited by 10 publications

(15 citation statements)

References 7 publications

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“…To study (1) in the case of physical vacuum, i.e., (3), we will need to apply the following Hardy-type inequalities:…”

Section: Preliminariesmentioning

confidence: 99%

“…In particular, the anelastic approximation is used to describe the semi-compressible ocean dynamics (see, e.g., [7,8]), as well as the tornado-hurricane dynamics (see, e.g., [24,26]). We refer the readers to [1,2,10,[15][16][17]22,25] for related topics and comparisons of various models of the atmospheric and oceanic dynamics. We remark that the background density profile ρ in the anelastic relation div(ρu) = 0 is given by the resting state ∇P (ρ) = ρg e z , where P (ρ) denotes the pressure potential and g is the gravity acceleration.…”

Section: Introductionmentioning

confidence: 99%

“…While there are some recent In comparison to the Navier-Stokes system (see, e.g., [5]), the density profile interacts with the velocity field. To explain this statement, let us forget about the boundary and consider (1) in 2T n for a moment. Also we assume the density profile ρ is non-degenerate and smooth.…”

Section: Introductionmentioning

confidence: 99%

“…As one will see later, one will need to consider the interaction of the pressure term ρ∇p with the nonlinearity term ρu•∇u as well as the viscosity term Δu. We take advantage of the regular density profile in (2), and consider an associated problem in 2T n , whose solutions satisfies (1) in Ω. Then we employ an elementary approach in the Galerkin's approximation which takes into account the aforementioned interactions.…”

Section: Introductionmentioning

confidence: 99%

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Well-Posedness of Strong Solutions to the Anelastic Equations of Stratified Viscous Flows

Liu

Titi

2020

J. Math. Fluid Mech.

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We establish the local and global well-posedness of strong solutions to the two- and three-dimensional anelastic equations of stratified viscous flows. In this model, the interaction of the density profile with the velocity field is taken into account, and the density background profile is permitted to have physical vacuum singularity. The existing time of the solutions is infinite in two dimensions, with general initial data, and in three dimensions with small initial data.

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“…To study (1) in the case of physical vacuum, i.e., (3), we will need to apply the following Hardy-type inequalities:…”

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Well-Posedness of Strong Solutions to the Anelastic Equations of Stratified Viscous Flows

Liu

Titi

2020

J. Math. Fluid Mech.

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“…Let us therefore consider relations such as (5) and (6) that better approximate the true mass continuity equation in low-Mach number stratified compressible flow. The generalization of (5) to liquids is nevertheless of considerable interest, and is discussed by Bois [3].…”

Section: Comparing the Boussinesq Approximation For Liquids And Gasesmentioning

confidence: 99%

Generalizing the Boussinesq approximation to stratified compressible flow

Durran

Arakawa

2007

Comptes Rendus. Mécanique

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The simplifications required to apply the Boussinesq approximation to compressible flow are compared with those in an incompressible fluid. The larger degree of approximation required to describe mass conservation in a stratified compressible fluid with the Boussinesq continuity equation has led to the development of several different sets of "anelastic" equations that may be regarded as generalizations of the original Boussinesq approximation. These anelastic systems filter sound waves while allowing a more accurate representation of non-acoustic perturbations in compressible flows than can be obtained using the Boussinesq system. The energy conservation properties of several anelastic systems are compared under the assumption that the perturbations of the thermodynamic variables about a hydrostatically balanced reference state are small. The "pseudo-incompressible" system is shown to conserve total kinetic and anelastic dry static energy without requiring modification to any governing equation except the mass continuity equation. In contrast, other energy conservative anelastic systems also require additional approximations in other governing equations. The pseudo-incompressible system includes the effects of temperature changes on the density in the mass conservation equation, whereas this effect is neglected in other anelastic systems. A generalization of the pseudo-incompressible equation is presented and compared with the diagnostic continuity equation for quasi-hydrostatic flow in a transformed coordinate system in which the vertical coordinate is solely a function of pressure.

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Double diffusion, convection de Boussinesq et convection profonde en air atmosphérique pollué ou humide

Bois

2006

Comptes Rendus. Mécanique

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On donne la forme prise par les équations de diffusion moléculaire et on montre comment la détermination des coefficients de diffusion moléculaire de scalaires passifs inertes (polluants, humidité, etc.) dans l'air atmosphérique peut être réalisée en première approximation par des relevés de pression, température et densités dans le milieu au repos. Ces approximations sont valables lorsque l'on écrit des équations de convection peu profonde (équations de Boussinesq), quelle que soit la fréquence de Brunt-Väisälä (donc dans la stratosphère et la troposphère). Dans le cas de la convection profonde, possible uniquement dans la troposphère, la faible fréquence de Brunt-Väisälä affecte aussi l'équation de diffusion moléculaire, celle-ci modifiant de son côté la forme des équations de convection. Des évaluations plus fines des coefficients doivent également être faites, à partir de relevés statiques pour différentes distributions de température par exemple.

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A unified asymptotic theory of the anelastic approximation in geophysical gases and liquids

Cited by 10 publications

References 7 publications

Well-Posedness of Strong Solutions to the Anelastic Equations of Stratified Viscous Flows

Well-Posedness of Strong Solutions to the Anelastic Equations of Stratified Viscous Flows

Generalizing the Boussinesq approximation to stratified compressible flow

Double diffusion, convection de Boussinesq et convection profonde en air atmosphérique pollué ou humide

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