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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Omega equation ： ウィキペディア英語版 Omega equation The omega equation is of great importance in meteorology and atmospheric physics. It is a partial differential equation for the vertical velocity, $\backslash omega$, which is defined as a Lagrangian rate of change of pressure with time, that is, $\backslash omega\; =\; \backslash frac$. The equation reads: : where $f$ is the Coriolis parameter, $\backslash sigma$ is the static stability, $\backslash mathbf\_g$ is the geostrophic velocity vector, $\backslash zeta\_g$ is the geostrophic relative vorticity, $\backslash phi$ is the geopotential, $\backslash nabla^2\_H$ is the horizontal Laplacian operator and $\backslash nabla\_H$ is the horizontal del operator.〔Holton, J.R., 1992, ''An Introduction to Dynamic Meteorology'' Academic Press, 166-175〕 ==Derivation== The derivation of the $\backslash omega$ equation is based on the vorticity equation and the thermodynamic equation. The vorticity equation for a frictionless atmosphere may be written as: : Here $\backslash xi$ is the relative vorticity, $V$ the horizontal wind velocity vector, whose components in the $x$ and $y$ directions are $u$ and $v$ respectively, $\backslash eta$ the absolute vorticity, $f$ the Coriolis parameter, $\backslash omega\; =\; \backslash frac$ the individual rate of change of pressure $p$. $k$ is the unit vertical vector, $\backslash nabla$ is the isobaric Del (grad) operator, $\backslash left(\; \backslash xi\; \backslash frac\; -\; \backslash omega\; \backslash frac\; \backslash right)$ is the vertical advection of vorticity and $k\; \backslash cdot\; \backslash nabla\backslash omega\; \backslash times\; \backslash frac$ represents the transformation of horizontal vorticity into vertical vorticity.〔Singh & Rathor, 1974, Reduction of the Complete Omega Equation to the Simplest Form, Pure and Applied Geophysics, 112, 219-223〕 The thermodynamic equation may be written as: : where $k\; \backslash equiv\; \backslash left(\; \backslash frac\backslash right)\; \backslash frac\; \backslash ln\backslash theta$, in which $q$ is the supply of heat per unit-time and mass, $C\_p$the specific heat of dry air, $R$ the gas constant for dry air, $\backslash theta$ is the potential temperature and $\backslash phi$ is geopotential $(gZ)$. The $\backslash omega$ equation () is then obtained from equation () and () by substituting values: :$\backslash xi\; =\; \backslash frac\backslash nabla^2\; Z$ and :$\backslash hat\; k\; \backslash cdot\; \backslash nabla\backslash omega\; \backslash times\; \backslash frac\; =\; \backslash frac\backslash frac\; -\; \backslash frac\backslash frac$ into (), which gives: : Differentiating () with respect to $p$ gives: : Taking the Laplacian ($\backslash nabla^2$) of () gives: : Adding () and (), simplifying and substituting $gk\; =\; \backslash sigma$, gives: : Equation () is now a linear differential equation in $\backslash omega$, such that it can be split into two part, namely $\backslash omega\_1$ and $\backslash omega\_2$, such that: : and : where $\backslash omega\_1$ is the vertical velocity due to the mean baroclinicity in the atmosphere and $\backslash omega\_2$ is the vertical velocity due to the non-adiabatic heating, which includes the latent heat of condensation, sensible heat radiation, etc. (Singh & Rathor, 1974). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Omega equation」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.