An Exact Energy for TRM Theory

Tivon Jacobson Department of Earth and Planetary Science, University of Tokyo, Tokyo, Japan

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Hidenori Aiki Frontier Research Center for Global Change, Japan Agency for Marine–Earth Science and Technology, Yokohama, Japan

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Abstract

A time residual mean (TRM) energy is obtained by averaging a transformation of the energy of the Boussinesq hydrostatic incompressible equations of motion. The transformation is the fundamental TRM transformation between level Cartesian coordinates and coordinates that are the mean positions of density surfaces. The TRM energy consists of a sum of mean kinetic, mean potential, wave kinetic, and wave potential energies. It is shown that the interaction between the mean kinetic energy and mean potential energy can be expressed entirely in terms of mean fields. The wave forcing of the mean TRM momentum equations is expressed as a divergence. An explicit and exact form of the TRM equations, with the transformed pressure term expressed in terms of the mean and wave fields, is also noted. It is suggested that the mean domain for the TRM equations and the Cartesian domain may not be the same, which would have consequences for the TRM boundary conditions.

Corresponding author address: Tivon Jacobson, Department of Earth and Planetary Science, University of Tokyo, Tokyo-to, Bunkyo-ku, Hongo 7-3-1, Tokyo 113-0033, Japan. Email: tivon@eps.s.u-tokyo.ac.jp

Abstract

A time residual mean (TRM) energy is obtained by averaging a transformation of the energy of the Boussinesq hydrostatic incompressible equations of motion. The transformation is the fundamental TRM transformation between level Cartesian coordinates and coordinates that are the mean positions of density surfaces. The TRM energy consists of a sum of mean kinetic, mean potential, wave kinetic, and wave potential energies. It is shown that the interaction between the mean kinetic energy and mean potential energy can be expressed entirely in terms of mean fields. The wave forcing of the mean TRM momentum equations is expressed as a divergence. An explicit and exact form of the TRM equations, with the transformed pressure term expressed in terms of the mean and wave fields, is also noted. It is suggested that the mean domain for the TRM equations and the Cartesian domain may not be the same, which would have consequences for the TRM boundary conditions.

Corresponding author address: Tivon Jacobson, Department of Earth and Planetary Science, University of Tokyo, Tokyo-to, Bunkyo-ku, Hongo 7-3-1, Tokyo 113-0033, Japan. Email: tivon@eps.s.u-tokyo.ac.jp

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