English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

HOLMBOE and his collaborators (1945) used the natural coordinates extensively in "Dynamical Meteorology ", but their treatise was restricted within the horizontal flow in level surface, casting somewhat important role to geometrical intuition in the process of expressing equations by the natural coordinates. ARAKAWA (1951) tried to derive more consistently the equations of motion in the natuarl coordinates. However, they could not touch on the second derivative because their definition of the natural coordinates is mathematically obscure. In a previous paper, the present author introduced analytically the natural coordinates basing on the extended directional differentiation and referring to the isobaric surfaces, and the geometry and kinematics of this coordinates were treated applying some part of them to the Kuroshio System (KAWAI, 1957). Though the vertical motion was taken into consideration and the second derivative was referred to, in order to simplify the mathematical expression there were adopted three assumptions upon the isobaric surfaces: the steady state of pressure field, the coincidence of isobaric surfaces with the surfaces of equal depth below the sea surface and the neglect of the curvature of isobaric surfaces. In short, this system was a natural coordinates on the isobaric surface, whose curvature is negligibly small. Thus, it is needful to combine more strictly the natural coordinates with isobaric surface or further with any equiscalar surface undulating with time. The quasi-Lagrangian coordinates proposed by STARR (1945) were investigated in detail by ELIASSEN (1949), but the condition of the hydrostatic equilibrium was assumed and the vertical p-velocity divided by gravity force is substituted for the vertical velocity itself, neglecting the time change of the level of isobaric surfaces. The natural coordinates on an equiscalar surface in the present paper seems to be a combination of the natural coordinates with the quasi-Lagrangian coordinates, but it has not the character of the quasi-Lagrangian coordinates because the coordinate replaced for the vertical coordinate in the present system is not a scalar quantity but is a metric coordinate along the normal of the equiscalar surface.

The Natural Coordinates on the Equiscalar Surface Undulating with Time