Formulation of the equations of the humid atmosphere in the context of variational inequalities

Témam, Roger; Wu, Kakuen John

doi:10.1016/j.jfa.2015.02.010

Cited by 8 publications

(31 citation statements)

References 17 publications

(37 reference statements)

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“…Remark 4.5. Theorem 4.4 implies the existence of a solution for the system (2.23)-(2.25) which was proved in [31] by a different method.…”

Section: A Priori Estimatesmentioning

confidence: 95%

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Numerical Approximation of a Variational Inequality Related to the Humid Atmosphere

Témam¹,

Wang²

2017

SIAM J. Numer. Anal.

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We investigate the numerical approximation of solutions to some variational inequalities modeling the humid atmosphere when the saturation of water vapor in the air is taken into account. In order to overcome the difficulties caused by the constraints on the humidity q (0 ≤ q ≤ 1) and the discontinuity in the variational inequalities, we construct a penalized and regularized implicit Euler method. We manage to show that the approximation functions associated with the numerical scheme converge to the solutions of the variational inequalities through deriving various delicate a priori estimates and by using compactness arguments.

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“…Remark 4.5. Theorem 4.4 implies the existence of a solution for the system (2.23)-(2.25) which was proved in [31] by a different method.…”

Section: A Priori Estimatesmentioning

confidence: 95%

“…in M, and let t 1 > 0 be fixed. The weak formulation of problem (2.1)-(2.8) proposed in [31] is as follows:…”

Section: Functional Formulation We Denote As Usualmentioning

confidence: 99%

Numerical Approximation of a Variational Inequality Related to the Humid Atmosphere

Témam¹,

Wang²

2017

SIAM J. Numer. Anal.

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“…As in Temam and Tribbia (2013), here we will restrict ourselves to the spatially homogeneous case, leaving the general case, which involves more heavy mathematical tools, to the article Temam and Wu (2015). However, as in Temam and Tribbia (2013), the guiding ideas are the same in both situations.…”

Section: Mathematical Reformulationmentioning

confidence: 99%

The equations of moist advection: a unilateral problem

Témam

Tribbia

2015

Quart J Royal Meteoro Soc

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In an earlier article, the authors discussed the uniqueness of solutions for moist advection problems. This article dealt with a simplified case in which the saturation specific humidity qs—which is a slightly varying function of temperature—was assumed to be constant. In trying to extend the results of Temam and Tribbia to the case where qs is not constant, it appeared that the equations of moist advection in this case were not coherent in the extreme cases where the atmosphere is totally dry or totally humid. The aim of this article is to describe this difficulty and propose a mathematically satisfying solution in the context of so‐called variational inequalities.

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“…In the article [TWu15] the authors assumed again that the velocity u is prescribed but they also assumed, for simplicity, that the saturation concentration q vs is constant. They then observed that the basic equations for q and T as reported e.g.…”

Section: Introductionmentioning

confidence: 99%

“…in the classical references [Hal71,HW80,RY89] are inconsistent for the limit values q " 0 and q " 1. This difficulty was reported in the geophysics literature in [TT16], and, in [TWu15] and [TT16] the authors propose to resolve the contradiction in the equations by introducing a unilateral equation (an inequality) valid for q " 0 and q " 1, and then the whole problem is set as a variational inequality. For general results on variational inequalities and their utilization in mechanics and physics, see e.g.…”

Section: Introductionmentioning

confidence: 99%

The equations of the multi-phase humid atmosphere expressed as a quasi variational inequality

et al. 2018

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In this article we propose a new formulation of the equations of the humid atmosphere with a multi-phase saturation generalizing thus the model introduced in [TWu15] and [TWa15]. More precisely, we consider the more realistic situation where the humid atmosphere comprises these components, namely water vapor, liquid water and cloud condensates and furthermore the saturation concentration is not constant. With the additional constraint that the vapor mass ratio q v is less than the saturation concentration q vs , which depends itself on the state, we are led, from the mathematical point of view, to introduce and handle a system of equations and inequations involving some quasi-variational inequalities for which we prove the existence of solutions.

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Formulation of the equations of the humid atmosphere in the context of variational inequalities

Cited by 8 publications

References 17 publications

Numerical Approximation of a Variational Inequality Related to the Humid Atmosphere

Numerical Approximation of a Variational Inequality Related to the Humid Atmosphere

The equations of moist advection: a unilateral problem

The equations of the multi-phase humid atmosphere expressed as a quasi variational inequality

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