Some Mathematical Problems in Geophysical Fluid Dynamics

Petcu, Mădălina; Témam, Roger; Ziane, Mohammed

doi:10.1016/s1570-8659(08)00212-3

Cited by 98 publications

(125 citation statements)

References 39 publications

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“…The basic mathematical structure of this system is relatively well understood (Petcu et al 2009). This is in stark contrast to the nonhydrostatic anelastic system, a standard formulation adopted for convective-scale modeling.…”

Section: Scientific Challenges Partial Differential Equation Problemmentioning

confidence: 99%

Scientific Challenges of Convective-Scale Numerical Weather Prediction

Yano

Ziemiański

Cullen

et al. 2018

Bulletin of the American Meteorological Society

122

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After extensive efforts over the course of a decade, convective-scale weather forecasts with horizontal grid spacings of 1–5 km are now operational at national weather services around the world, accompanied by ensemble prediction systems (EPSs). However, though already operational, the capacity of forecasts for this scale is still to be fully exploited by overcoming the fundamental difficulty in prediction: the fully three-dimensional and turbulent nature of the atmosphere. The prediction of this scale is totally different from that of the synoptic scale (103 km), with slowly evolving semigeostrophic dynamics and relatively long predictability on the order of a few days. Even theoretically, very little is understood about the convective scale compared to our extensive knowledge of the synoptic-scale weather regime as a partial differential equation system, as well as in terms of the fluid mechanics, predictability, uncertainties, and stochasticity. Furthermore, there is a requirement for a drastic modification of data assimilation methodologies, physics (e.g., microphysics), and parameterizations, as well as the numerics for use at the convective scale. We need to focus on more fundamental theoretical issues—the Liouville principle and Bayesian probability for probabilistic forecasts—and more fundamental turbulence research to provide robust numerics for the full variety of turbulent flows. The present essay reviews those basic theoretical challenges as comprehensibly as possible. The breadth of the problems that we face is a challenge in itself: an attempt to reduce these into a single critical agenda should be avoided.

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Section: Scientific Challenges Partial Differential Equation Problemmentioning

confidence: 99%

Scientific Challenges of Convective-Scale Numerical Weather Prediction

Yano

Ziemiański

Cullen

et al. 2018

Bulletin of the American Meteorological Society

122

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“…Due to their importance, the primitive equations has been studied analytically by many authors, see, e.g., [17,18,20,22,24] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Global Well‐Posedness of the Three‐Dimensional Primitive Equations with Only Horizontal Viscosity and Diffusion

Cao

Titi

2015

Comm Pure Appl Math

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In this paper, we consider the initial boundary value problem of the three‐dimensional primitive equations for planetary oceanic and atmospheric dynamics with only horizontal eddy viscosity in the horizontal momentum equations and only horizontal diffusion in the temperature equation. Global well‐posedness of the strong solution is established for any H2 initial data. An N‐dimensional logarithmic Sobolev embedding inequality, which bounds the L∞‐norm in terms of the Lq‐norms up to a logarithm of the Lp‐norm for p > N of the first‐order derivatives, and a system version of the classic Grönwall inequality are exploited to establish the required a~priori H2 estimates for global regularity.© 2016 Wiley Periodicals, Inc.

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“…The temperature equation in pressure coordinates reads 30) which can again be reformulated in terms of the potential temperature as follows 31) where the diffusion in vertical direction however deviates due to the additional pressure function arising when replacing T in terms of θ according to (1.4)…”

Section: 2mentioning

confidence: 99%

“…Setting u 1 = u − Ψ 0 , one can easily check that u 1 is a weak solution to the following elliptic problem 30) where f 1 and ϕ 1 are given by…”

Section: Linear Elliptic Problemmentioning

confidence: 99%

Global well-posedness for passively transported nonlinear moisture dynamics with phase changes

Hittmeir

Klein

et al. 2017

Nonlinearity

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Abstract. We study a moisture model for warm clouds that has been used by Klein and Majda in [21] as a basis for multiscale asymptotic expansions for deep convective phenomena. These moisture balance equations correspond to a bulk microphysics closure in the spirit of Kessler [20] and Grabowski and Smolarkiewicz [15], in which water is present in the gaseous state as water vapor and in the liquid phase as cloud water and rain water. It thereby contains closures for the phase changes condensation and evaporation, as well as the processes of autoconversion of cloud water into rainwater and the collection of cloud water by the falling rain droplets. Phase changes are associated with enormous amounts of latent heat and therefore provide a strong coupling to the thermodynamic equation.In this work we assume the velocity field to be given and prove rigorously the global existence and uniqueness of uniformly bounded solutions of the moisture model with viscosity, diffusion and heat conduction. To guarantee local well-posedness we first need to establish local existence results for linear parabolic equations, subject to the Robin boundary conditions on the cylindric type of domains under consideration. We then derive a priori estimates, for proving the maximum principle, using the Stampacchia method, as well as the iterative method by Alikakos [1] to obtain uniform boundedness. The evaporation term is of power law type, with an exponent in general less or equal to one and therefore making the proof of uniqueness more challenging. However, these difficulties can be circumvented by introducing new unknowns, which satisfy the required cancellation and monotonicity properties in the source terms.

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Some Mathematical Problems in Geophysical Fluid Dynamics

Cited by 98 publications

References 39 publications

Scientific Challenges of Convective-Scale Numerical Weather Prediction

Scientific Challenges of Convective-Scale Numerical Weather Prediction

Global Well‐Posedness of the Three‐Dimensional Primitive Equations with Only Horizontal Viscosity and Diffusion

Global well-posedness for passively transported nonlinear moisture dynamics with phase changes

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