# A Large Spectrum of Free Oscillations of the World Ocean Including the Full Ocean Loading and Self-attraction EffectsTheory and Model

A Large Spectrum of Free Oscillations of the World Ocean Including the Full Ocean Loading and... [The barotropic free oscillations of the global ocean are defined through the linearized homogeneous shallow water equations (e.g. [57]). 1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} \eqalign{ & {{\partial v} \over {\partial t}} + fxv + {{r'} \over D}v + F + g\nabla \zeta + L_{sek} = 0 \cr & {{\partial \zeta } \over {\partial t}}\nabla .(Dv) = 0, \cr} \end{document} where ξ denotes the sea surface elevation with respect to the moving sea bottom, v % (u, v) the horizontal current velocity vector. The undisturbed ocean depth is D, the vector of Coriolis acceleration f % 2ω) sin ýz, the coefficient of linear bottom friction r1 and the gravitational acceleration g. F denotes the vector defining the second-order eddy viscosity term (Fℏ,Fý) % (−AhAΔ −AhΔv) and (h, ý) a set of geographic longitude and latitude values. Lsek is the vector of the secondary force of the loading and self-attraction (LSA), it is derived in section 2.1.1. In spherical coordinates this system of equations is written as: 2.1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $${{\partial u} \over {\partial t}} - 2\omega \,\sin \varphi .v + {{r'} \over D}.U - A_h \Delta Hu + {g \over {R\cos \,\varphi }}{{\partial \zeta } \over {\partial \zeta }} + L_{sek,\lambda } = 0$$ \end{document}2.3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $${{{{\partial v} \over {\partial t}} + 2\omega \sin \,\varphi .u + {{r'} \over D}.D - A_\lambda \Delta _H v + {g \over R}{{\partial \zeta } \over {\partial \varphi }} + L_{sek,\varphi = 0} } \over {}}$$ \end{document}2.4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $${{\partial \zeta } \over {\partial t}} + {1 \over {R\,\cos \varphi }}\left( {{{\partial (Du)} \over {\partial \lambda }} + {{\partial (Dv\cos \varphi )} \over {\partial \varphi }} = 0} \right)$$ \end{document}] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

# A Large Spectrum of Free Oscillations of the World Ocean Including the Full Ocean Loading and Self-attraction EffectsTheory and Model

Part of the Hamburg Studies on Maritime Affairs Book Series (volume 14)
Editors: Müller, Malte
15 pages

/lp/springer-journals/a-large-spectrum-of-free-oscillations-of-the-world-ocean-including-the-FOvmxXZRGG
Publisher
Springer Berlin Heidelberg
ISBN
978-3-540-85575-0
Pages
7 –22
DOI
10.1007/978-3-540-85576-7_2
Publisher site
See Chapter on Publisher Site

### Abstract

[The barotropic free oscillations of the global ocean are defined through the linearized homogeneous shallow water equations (e.g. [57]). 1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} \eqalign{ & {{\partial v} \over {\partial t}} + fxv + {{r'} \over D}v + F + g\nabla \zeta + L_{sek} = 0 \cr & {{\partial \zeta } \over {\partial t}}\nabla .(Dv) = 0, \cr} \end{document} where ξ denotes the sea surface elevation with respect to the moving sea bottom, v % (u, v) the horizontal current velocity vector. The undisturbed ocean depth is D, the vector of Coriolis acceleration f % 2ω) sin ýz, the coefficient of linear bottom friction r1 and the gravitational acceleration g. F denotes the vector defining the second-order eddy viscosity term (Fℏ,Fý) % (−AhAΔ −AhΔv) and (h, ý) a set of geographic longitude and latitude values. Lsek is the vector of the secondary force of the loading and self-attraction (LSA), it is derived in section 2.1.1. In spherical coordinates this system of equations is written as: 2.1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $${{\partial u} \over {\partial t}} - 2\omega \,\sin \varphi .v + {{r'} \over D}.U - A_h \Delta Hu + {g \over {R\cos \,\varphi }}{{\partial \zeta } \over {\partial \zeta }} + L_{sek,\lambda } = 0$$ \end{document}2.3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $${{{{\partial v} \over {\partial t}} + 2\omega \sin \,\varphi .u + {{r'} \over D}.D - A_\lambda \Delta _H v + {g \over R}{{\partial \zeta } \over {\partial \varphi }} + L_{sek,\varphi = 0} } \over {}}$$ \end{document}2.4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $${{\partial \zeta } \over {\partial t}} + {1 \over {R\,\cos \varphi }}\left( {{{\partial (Du)} \over {\partial \lambda }} + {{\partial (Dv\cos \varphi )} \over {\partial \varphi }} = 0} \right)$$ \end{document}]

Published: Jan 1, 2009

Keywords: Spherical Harmonic; Solid Earth; Free Oscillation; Krylov Subspace; Ocean Bottom