# 1 Coordinates

To mathematically describe variables in the ocean, we need three dimensions of space and one dimension in time. With more than one spatial dimension, position becomes a vector of distances along each dimension from some origin (0,0,0).

Often (and almost always in this class) it is convenient to use a local Cartesian coordinate system for a plane tangent to the Earth’s surface. We write:

The Standard Open-Ocean Coordinate System

*x* increases to the East along the local horizontal plane, in the direction of increasing longitude

*y* increases to the North along the local horizontal plane, in the direction of increasing latitude

*z* increases upward in the local vertical direction

*z* = 0 is at the surface of the ocean (to be precise: where the ocean surface would be if there were no motion, i.e., the surface of a resting ocean in equilibrium with Earth’s gravity aka geopotential field).

**Unit vectors** have length = 1. You can create a unit vector in any direction by dividing the vector by its magnitude. In physics, this means that you have divided out the units and so the vector is *dimensionless*.

Special symbols are often given to unit vectors in the directions of the coordinate system axes. In our (*x*,*y*,*z*) = (Eastward, Northward, Upward) coordinate system:

A few details are important about this coordinate system:

*z*will be generally be**negative within the ocean**and positive within the atmosphere.*Depth*is a positive number that represents the distance below the ocean surface. Larger negative numbers are found deeper in the ocean. We can write the*z*value for a water parcel found at a depth of*h*meters below the (resting) ocean surface as- The absolute orientation of this standard local Cartesian coordinate system changes depending on where you are on the Earth (i.e., the direction of “Up” differs if seen from outer space).
- The system isn’t universal in oceanography, but is widely used in studying large-scale systems. Coastal or estuarine problems often use a different local Cartesian coordinate system that matches the important geography of the system. For example,
*x*might point along the coastline, with*y*pointing offshore. - The rate of moving through space in each direction, the velocity, will also be a three-dimensional vector,

** FYI:** Latitude and longitude come from a spherical coordinate system. Longitude is the angle counterclockwise looking down on the north pole relative to 0° passing through Greenwich, England. Latitude is the angle counterclockwise looking down at the Earth from above the equator (0° corresponds to the eastward direction along the equator). Elevation (formally: distance from the center of the Earth) is the third coordinate in this system.

Converting Latitude and Longitude (in Degrees) to Distance Units

- On the surface of the Earth, there are 111 km distance in every degree of latitude.
- The distance between longitude lines decreases towards the poles. You can calculate the distance between 1 degree of longitude as , where is the latitude.

### Media Attributions

- Cartesian Coordinates adapted by Susan Hautala is licensed under a CC BY-SA (Attribution ShareAlike) license