Vector notations suitable for geophysical fluid dynamics with examples and applications

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  • Masuda Akira
    Research Institute for Applied Mechanics, Kyushu University

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Abstract

In geophysical fluids like the ocean and atmosphere, the horizontal plane and vertical direction have much different properties from each other, partly because of a small aspect ratio and partly because of stable density stratification and rotation of the earth. A useful system of vector notations is proposed for describing geophysical fluid dynamics in a concise manner. We first introduce "¬", a vector operator to be called strophe or turn. It rotates a horizontal vector clockwise at right angle. Likewise "◁ ≡ ¬∇" (blana or alongent) is defined. When operated on a scalar field ψ, blana yields a vector that is as large as ||∇ψ||, but parallel to the isoline of ψ with higher ψ on the left-hand side. A three-dimensional vector is written as u__- = u + w, where u in boldface indicates the horizontal components and w is the vertical component of three-dimensional vector u__-. Examples and applications are presented to show the utility of the present system of notations called GFDVN (Geophysical Fluid Dynamics Vector Notation), which not only simplifies the description, but also gives a clear geometrical image of vectors in oceanography or meteorology. We observe quite symmterical relations between the inner and outer product of two vectors, divergence and rotation of a vector field, through ¬ or ◁. As a focused application, an investigatios is made of the geometrical implication of Lagrange's formula, or the formula of triple vector product. By using GFDVN we find that the formula is none other than a representation of skew coordinates on a plane. Also that formula in two-dimension turned out to have a simple relation with the mixing ratio of three water types on a T-S diagram used in oceanography.

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