METEO 300
Fundamentals of Atmospheric Science

8.3 Describing weather requires coordinate systems.

In meteorology and other atmospheric sciences, we mostly use the standard x, y, and z coordinate system, called the Cartesian coordinate system, and the spherical coordinate system. Let’s review some of the main points of these two systems.

Cartesian Coordinate System

The Cartesian coordinate system applies to three dimensions (as seen in the figure below). The convention is simple:

  • The zero point (x=y=z=0, or (0,0,0)) is arbitrary.
  • x increases to the east; x decreases to the west.
  • y increases to the north; y decreases to the south.
  • z increases going up; z decreases going down.
  • A distance vector extending from the origin to (x,y,z) as L = i x + j y + k z.

Unit vectors (length 1 along standard coordinates) are i (east); j (north); k (up).

Often we will consider motion in two dimensions as being separate from motions in the vertical. We usually denote the horizontal with a subscript H: LH = i x + j y, where L is a distance vector.

We like this coordinate system because it works well over relatively small scales on Earth, perhaps the size of an individual state, where Earth's curvature is not important. However, it does not work so well for large-scale motion on Earth, which is spherical.

Cartesian coordinate system.
Cartesian coordinate system.
Credit: W. Brune
Cartesian coordinate system when placed on the globe
The Cartesian coordinate system when placed on the globe.
Credit: W. Brune

When we place the Cartesian coordinate system on a sphere, note that x always points toward the east, y always points toward the North Pole, and z always points up along the direction of Earth’s radius (as seen in the figure above).

Spherical Coordinate System

Life would be much easier if the Earth were flat. We could then use the Cartesian coordinate system with no worries. But the Earth is a sphere, which implies that to accurately describe motion, we must take the Earth’s spherical shape into account.

globe showing spherical coordinate system
Spherical coordinate system.

We use the following terms:

  • a (sometimes r) = Earth’s radius
  • ΦThis equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. = latitude (-90o to +90o, or - πThis equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. /2 to + πThis equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. /2)
  • λThis equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. = longitude (-180o to 180o, or - πThis equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. to + πThis equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. )
  • State College PA is at Φ =  40.8 o  and λ= 77.9 o This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. .

Note that 1o of latitude is always 111 km or 60 nautical miles, but 1o of longitude is 111 km only at the equator. It is smaller in general and equal to 111 km x cos( ΦThis equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. ). Note that 1 nm = 1.15 miles.

To find the horizontal distance between any two points on Earth's surface, we first need to find the angle of the arc between them and then we can multiply this angle by Earth's radius to get the distance. To find the angle of the arc, ΔσThis equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. , we can use the Spherical Law of Cosines:

Δσ=arccos( sin ϕ 1 sin ϕ 2 +cos ϕ 1 cos ϕ 2 cosΔλ )This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.



where the latitude and longitude of the two points are Φ 1 ,  λ 1  and  Φ 2 ,  λ 2 This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. respectively and Δλ=| λ 1 λ 2 |This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. is the absolute difference between the longitudes of the two points. Note that the angle of the arc must be in radians, where 2π radians =  360 o This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. . To find the distance, simply multiply this arc angle by REarth = 6371 km.

Check Your Understanding

Show that 1o of latitude = 111 km distance.

Click for answer.

Distance = 6371 km * (1/360)*2π = 111.2 km

In summary, we will use a Cartesian coordinate system when our scales of interest are not too large (synoptic scale or smaller), but will need to use spherical coordinates when the scale of interest is larger than synoptic scale.

For another explanation of these two systems, visit this Coordinate Systems website.

Vertical Coordinates

Three different vertical coordinates are used in meteorology and atmospheric science.

We have already introduced the vertical coordinate z, which is a height, usually in m or km. This vertical coordinate is the most commonly used in meteorology and in any process that involves getting off the ground, such as flight. Often pilots talk about flight levels, which are measured in hundreds of feet. So, flight level 330 is about 10 km altitude.

Another useful vertical coordinate is pressure, which decreases with height. Pressure is often a useful vertical coordinate for calculating dynamics. To a good approximation, pressure falls off exponentially with height (p = poexp(-z/H)), so that ln(p) is fairly linear with height. We’ll get into this in greater detail later. For now, consider the following table of typically used pressure heights:

Typically Used Pressure Heights
altitude (km) altitude  (kft) pressure level (hPa or mb)
0 0 1000
1.5 4.4 850
3.0 9.9 700
5.5 18.3 500

A third important vertical coordinate is potential temperature, θThis equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. (Equation 2.58). This quantity is the temperature that an air parcel would have if it were brought to a pressure of 1000 hPa without losing or gaining any energy from its surroundings. This vertical coordinate has a nice property: air parcels tend to move on surfaces of constant potential temperature because moving on such a surface requires no energy. This coordinate is particularly useful in the stratosphere, where the rapid increase with altitude tends to keep air motion stratified.