Renormalization group method applied to the primitive equations
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.
scientific article; zbMATH DE number 2134397
scientific article