A partial differential equation related to a problem in atmospheric pollution

Ali, Ismail; Kalla, S. L.; Khajah, H. G.

doi:10.1016/s0895-7177(98)00169-1

Cited by 4 publications

(4 citation statements)

References 4 publications

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“…2 )Ψ(t), from which, applying the Grönwall inequality and using Corollary 1 we deduce 2 , which concludes the proof. Thus, we now have…”

Section: Therefore We Havesupporting

confidence: 53%

“…Each of the scaling equations defined below can be heuristically deduced from the typical physical dimension of the respective quantities. Specifically, we remind that viscosity constants have physical dimensions [ν xi ] = typical length 2 xi /typical time, and the same holds for diffusivity (see also [6], [7], [14]). In detail, we use the following scaling identities to arrive to the rescaled model.…”

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confidence: 99%

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The primitive equations of the polluted atmosphere as a weak and strong limit of the 3D Navier-Stokes equations in downwind-matching coordinates

Donatelli

Juhász

2022

DCDS

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<p style='text-indent:20px;'>A widely used approach to mathematically describe the atmosphere is to consider it as a geophysical fluid in a shallow domain and thus to model it using classical fluid dynamical equations combined with the explicit inclusion of an <inline-formula><tex-math id="M1">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula> parameter representing the small aspect ratio of the physical domain. In our previous paper [<xref ref-type="bibr" rid="b14">14</xref>] we proved a weak convergence theorem for the polluted atmosphere described by the Navier-Stokes equations extended by an advection-diffusion equation. We obtained a justification of the generalised hydrostatic limit model including the pollution effect described for the case of classical, east-north-upwards oriented local Cartesian coordinates. Here we give a two-fold improvement of this statement. Firstly, we consider a meteorologically more meaningful coordinate system, incorporate the analytical consequences of this coordinate change into the governing equations, and verify that the weak convergence still holds for this altered system. Secondly, still considering this new, so-called downwind-matching coordinate system, we prove an analogous strong convergence result, which we make complete by providing a closely related existence theorem as well.</p>

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“…2 )Ψ(t), from which, applying the Grönwall inequality and using Corollary 1 we deduce 2 , which concludes the proof. Thus, we now have…”

Section: Therefore We Havesupporting

confidence: 53%

mentioning

confidence: 99%

The primitive equations of the polluted atmosphere as a weak and strong limit of the 3D Navier-Stokes equations in downwind-matching coordinates

Donatelli

Juhász

2022

DCDS

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“…• The proof of b(U, Ũ , U # ) = −b(U, U # , Ũ ) for U, Ũ , U # ∈ V and Ũ or U # in V (2) can be derived by integration by parts as we have • To establish the improvement…”

Section: Discussionmentioning

confidence: 99%

Weak solution of the merged mathematical equations of the polluted atmosphere

Donatelli

Juhász

2019

Preprint

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Considered as a geophysical fluid, the polluted atmosphere shares the shallow domain characteristics with other natural large-scale fluids such as seas and oceans. This means that its domain is excessively greater horizontally than in the vertical dimension, leading to the classic hydrostatic approximation of the Navier-Stokes equations. The authors of the [6] article have proved a convergence theorem for this model with respect to the ocean, without considering pollution effects. The novelty of this present work is to provide a generalisation of their result translated to the atmosphere, extending the fluid velocity equations with an additional convection-diffusion equation representing pollutants in the atmosphere.

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“…Each of these following scaling equations can be heuristically deduced from the typical physical dimension of the quantities (for more details see [7], [6], [15]), unlike the previously presented equation (2), which is the effect of the coordinate choice itself. In detail, apart from the already introduced equation (2) we use the following scaling identities to arrive to the rescaled model.…”

Section: Notationsmentioning

confidence: 99%

The primitive equations of the polluted atmosphere as a weak and strong limit of the 3D Navier-Stokes equations in downwind-matching coordinates

Donatelli¹,

Juhász²

2020

Preprint

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A widely used approach to mathematically describe the atmosphere is to consider it as a geophysical fluid in a shallow domain -and thus to model it using classical fluid dynamical equations combined with the explicit inclusion of an ǫ parameter representing the small aspect ratio of the physical domain. In our previous paper [15] we proved a weak convergence theorem for the polluted atmosphere described by the Navier-Stokes equations extended by an advection-diffusion equation. We obtained a justification of the generalised hydrostatic limit model including the pollution effect described for the case of classical, east-north-upwards oriented local Cartesian coordinates. Here we give a two-fold improvement of this statement. Firstly, we consider a meteorologically more meaningful coordinate system, incorporate the analytical consequences of this coordinate change into the governing equations, and verify that the weak convergence still holds for this altered system. Secondly, still considering this new, so-called downwindmatching coordinate system, we prove an analogous strong convergence result, which we make complete by providing a closely related existence theorem as well.

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A partial differential equation related to a problem in atmospheric pollution

Cited by 4 publications

References 4 publications

The primitive equations of the polluted atmosphere as a weak and strong limit of the 3D Navier-Stokes equations in downwind-matching coordinates

The primitive equations of the polluted atmosphere as a weak and strong limit of the 3D Navier-Stokes equations in downwind-matching coordinates

Weak solution of the merged mathematical equations of the polluted atmosphere

The primitive equations of the polluted atmosphere as a weak and strong limit of the 3D Navier-Stokes equations in downwind-matching coordinates

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