Renormalization group method applied to the primitive equations
type: Item
instance of: scholarly article
title: Renormalization group method applied to the primitive equations
zbMATH Open document ID: 1080.35090
published in: Journal of Differential Equations
publication date: 2005-02-16T00:00:00Z
review text: The equations describing the dynamics of the ocean and atmosphere, considered by the authors are \[ \begin{cases} {{\partial u}\over{\partial t}}+u\, {{\partial u}\over{\partial x}}+w\, {{\partial u}\over{\partial z}}-{1\over{\varepsilon}}\,v+{1\over{\varepsilon}}\, {{\partial p}\over{\partial x}}=\nu_v\Delta u+S_u;\\{{\partial v}\over{\partial t}}+u\,{ {\partial v}\over{\partial x}}+w\, {{\partial v}\over{\partial z}}+{1\over{\varepsilon}}\,u=\nu_v\Delta v+S_v;\\{{\partial p}\over{\partial z}}=-N\rho;\quad {{\partial u}\over{\partial x}}+ {{\partial w}\over{\partial z}}=0;\\{{\partial \rho}\over{\partial t}}+u\, {{\partial \rho}\over{\partial x}}+w\, {{\partial \rho}\over{\partial z}}- { N\over{\varepsilon}}\,w=\nu_\rho\Delta \rho+S_\rho,\end{cases}\tag{1} \] where \((u,v,w)\) are the coordinates of the velocity, \(p\) is the pressure, \(\rho\) is the density, while \(\varepsilon\) stands for the Rossby number. The quantities \(S_u,S_v\) and \(S_\rho\) stand for source terms, and they usually vanish. Finally, \(N\) is the Burger number, and \(\nu_v,\nu_\rho\) are viscosity coefficients. It is assumed that all the unknown functions are \(\mathcal M\)-periodic, with \({\mathcal M}=(0,L_1)\times(-L_3/2,L_3/2)\), in the space variables \((x,z).\) The system (1) is subject in this paper to various transformations, and a variational formulation of the problem is provided. It is also equivalent to the abstract evolutionary equation \[ dU/dt+AU+B(U,U)+\varepsilon^{-1}LU=S,\tag{2} \] under the initial condition \(U(0)=U_0,\) \(U=(u,v,\rho)\). Existence of global solutions for (2) is discussed in detail. The RG method (renormalization group) is described, in order to obtain approximate (averaged) solutions. Attention is also given to the problem of estimates for the error, and Appendices A, B.1, B.2 are provided for proving some auxiliary results.
reviewed by: Constantin Corduneanu
Mathematics Subject Classification ID: 35Q35
Mathematics Subject Classification ID: 76D03
Mathematics Subject Classification ID: 35C20
Mathematics Subject Classification ID: 76D50
Mathematics Subject Classification ID: 86A05
Mathematics Subject Classification ID: 86A10
zbMATH DE Number: 2134397
zbMATH Keywords: primitive equations of the ocean
zbMATH Keywords: energy estimates
zbMATH Keywords: error estimates
zbMATH Keywords: variational formulation
zbMATH Keywords: existence of global solutions
author: Mădălina Petcu
author: R. Temam
author: D. Wirosoetisno
MaRDI profile type: Publication
full work available at URL: j.jde.2003.10.011
OpenAlex ID: W2053382961
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DOI: 10.1016/J.JDE.2003.10.011
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description: scientific article; zbMATH DE number 2134397