Chapter 7  Some Mathematics: The Equations of Motion
7.2 Coordinate System
Coordinate systems allow us to find locations in theory and practice. Various systems are used depending on the size of the features to be described or
mapped. I will refer to the simplest systems; descriptions of other systems can be found in geography and geodesy books.
 Cartesian Coordinate System is the one I
will use most commonly in the following chapters to keep the discussion as
simple as possible. We can describe most processes in Cartesian coordinates
without the mathematical complexity of spherical coordinates. The standard
convention in geophysical fluid mechanics is x
is to the east, y is to the north, and z
is up.
fPlane is a Cartesian coordinate system
in which the Coriolis force is assumed constant. It is useful for describing
flow in regions small compared with the radius of the Earth and larger than
a few tens of kilometers.
βplane is a
Cartesian coordinate system in which the Coriolis force is assumed to vary
linearly with latitude. It is useful for describing flow over areas as large
as ocean basins.
 Spherical coordinates are used to describe
flows that extend over large distances and in numerical calculations of basin
and global scale flows.
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