Chapter 8
Coordinate Systems
(1) Conventional Coordinate Systems
Cartesian: Three axes perpendicular to
each other.
Plane polar coordinates are a natural way to
describe circular motion.
Instead of describing the position of an
object using x and y coordinates, this
system uses two coordinates that
locate the object's position using its
distance from the origin (called "r" for
radius) and its angle (theta) from the
positive x-axis .
Angles measured counterclockwise
from the x-axis are defined to be
positive angles, and angles measured
clockwise from the x-axis are negative
angles. The distance from the origin is
ALWAYS positive.
x  r cos 
y  r sin 
Spherical Coordinate System.
The angle (ϕ) defines the
elevation above (or below)
the plane of (Θ).
Latitude and longitude are
examples of a spherical
coordinate system.
r is the distance from the
center of the Earth.
• Θ is the longitude with the Greenwich Meridian as the
starting point.
• ϕ is the latitude with the equator as the starting point.
• Up, where the elevation angle/latitude = 90o is the north
pole.
• Looking down from the up position, longitude increases in a
counterclockwise direction.
Use of spherical coordinates with radar
The elevation angle, , is
zero along a line parallel to
the horizontal.
The azimuth angle, , begins
with zero at north, and moves
clockwise just as the azimuth
angle for wind.
r is the radial distance from
the radar location.
• Up refers to moving outward directly from the center of the
Earth.
• Looking down from the up direction,  moves in a clockwise
direction just as wind.
Because the Earth rotates, an
object moving in a rotating
coordinate system with the Earth
experiences an apparent force
outward from the axis of rotation
of the Earth, the centrifugal
force, FCN = Ω2R.
That object also experiences
gravity trying to move it toward the
center of the Earth.
The resultant of these two forces can be expressed as a
single force, FEG, effective gravity.
Since the centrifugal force is small compared to gravity, the
direction of the effective gravity force is close to, but not
exactly in the same direction as gravity.
We, air molecules, etc. experience this and call it the force of
gravity, except it is really the result of true gravity and the
Earth’s motion.
An object moving around and
experiencing the same “effective”
gravitational force on it is said to be
moving on a surface of constant
geopotential. A surface of constant
geopotential is a surface along which a
parcel of air could move without
undergoing any changes in its potential
energy. Also known as
equigeopotential surface or a level
surface.
Because the Earth rotates, it bulges at the
equator.
For large scale motions, the spherical
coordinate system works well for identifying
the location of air parcels.
For small scale motions, the Cartesian
coordinate system works well, even though
the axes of the coordinate system must really
change direction as parcels move about in
the atmosphere.
Pressure coordinate. In the atmosphere in the
vertical, pressure is easier to measure than height.
Often, the location of a parcel of air, or other
characteristics of the air, is expressed in terms of the
pressure where it is located rather than the height
above some surface (as sea level).
Isentropic coordinate. An isentrope is a surface
along which the potential temperature is constant.
Parcels remain on isentropic surfaces as long as
conserved quantities (like mixing ratio) don’t change;
i.e., the parcel doesn’t become saturated and start
converting water vapor to liquid or ice.
(3) Hydrostatic equation
The hydrostatic equation represents a
balance of the vertical pressure gradient
force (acting upward) and the gravitational
force (effective gravitational force) acting
down.
When these two forces are not in balance,
there is an acceleration of a parcel of air
either upward or downward.
This is expressed by the equation below from
chapter 6.
Dw
1 P
 g 
Dt
 t
When they are in balance, the right side of
the equation is zero and we have the
P
hydrostatic equation.
  g
z
Using the Ideal Gas Law Equation to get an
P
expression for density
 
Rd T

(as done
in chapter 5) and substitute that in for density
gives.
P
z



P
Rd T
g
P
z
P

g
Rd T
Pg
Divide both sides by
Rd T 1 P
Rd T
 1
g P z

This can be written as:



R d T  ln  P 
g
z
 1
gives.
(4) Hypsometric equation
Integrate from some low level to some high
ln  P 
level.
RdT

z2
2
ln  P1 
g
 ln  P   
 1 z
z1
Multiply both sides by -1, reverse the limits
on the left (gets rid of the - sign on the left)
and perform
the integration on the right.

Gives:
Rd
g

ln  P1 
 T  ln  P   z
ln  P 2 
2
 z1
Graphically, integration can be expressed as
“computing the area under a curve.”
For the integration of the term on the right, the
graph would look like the following.
The “integrand” is constant and equal to 1 along
the distance (z2-z1).
Rd
g
ln  P1 
 T  ln  P   z
2
 z1
ln  P 2 
For the left side of the equation, there is the
temperature term, and temperature changes

as we
move vertically in the atmosphere.
However, we can simplify the equation by
using the average value of temperature
between the levels represented by ln(P1) and
ln(P2).
Then, we can write the equation as:
Rd T
g
ln  P1 
  ln P   z
2
 z1
ln  P 2 
RdT
The quantity
g is just a constant by
whichwe are multiplying the integrand
between the limits ln(P1) and ln(P2).

Integrating
the left side between those limits
gives:
R T
d
g

ln  P1   ln  P2  
z 2  z1
Graphically, it would look like:
Geopotential height.
Remember, a surface of constant geopotential
is a surface along which a parcel of air could move
without undergoing any changes in its potential
energy
The geopotential height is the height at which an
object (such as a volume of air) would have the
same geopotential (assuming constant value for
gravity) as it would in the real atmosphere (where
gravity changes).

It is expressed as: Z  z 
g


 g 0 
Virtual Temperature
The real atmosphere has water vapor in it, which
varies. Therefore, Rd is not a good value to use
for the gas constant to express how the volume is
going to behave. An R value should be used that
expresses how each particular volume would
behave based on the particular molecules
(including water vapor of varying amounts) it has
in it.
However, that means calculating a new R value
for every different volume of air, or coming up with
an equation that expresses how R changes with
different types of air.
We can get around that problem with water
vapor and still use Rd by using virtual
temperature.
Virtual temperature is defined as the
temperature a dry parcel of air would have if it
had the same density and pressure as a moist
parcel of air.
Virtual temperature is always greater than
actual temperature of the parcel because
having moisture in the parcel makes it less
dense.
Thus, virtual temperature accounts for the
effect of having water vapor in a volume of air.
Virtual temperature is given by:
T v  T 1  0.611  
Where “ω” is the mixing ratio of the air.
 
mass of vapor kg
mass of dry air kg

Sometimes
it is expresses as g/kg.
Thus,
 by using virtual temperature, we can use
Rd and still account for the effects of water
vapor being in the volume, as long as that
water vapor amount doesn’t change.
So, now we can use gravity as a
constant as long as we realize we are
dealing with geopotential heights (Z),
not true height(z) - as measured with a
ruler.
And, we can use Rd as long as we use
Tv, virtual temperature, to account for
moisture.
Hypsometric equation
This integrated form of the hydrostatic
equation shows that the vertical
distance between two pressure surfaces
in the atmosphere is proportional to the
average temperature between those
two surfaces.
Rd
go

ln  P1   ln  P2 T v
 Z 2  Z1
By knowing pressure and temperature, the
height between pressure surfaces can be
calculated (with only a small error if water
vapor is ignored).
By also knowing the water vapor amount
(which can be obtained from knowing the dew
point) a more accurate determination of the
thickness between pressure surface can be
made.
By knowing the pressure at the ground, and
temperatures up through the atmosphere,
one can work their way up through the
atmosphere determining the thickness of
each layer between pressure surfaces and
thus the height of pressure surfaces.
It’s easy to see that where temperatures are
cold, pressure surfaces are going to be at a
lower height in the atmosphere. And, where
temperatures are warm, pressure surfaces
will be at a higher height in the atmosphere.
(8) Critical thickness
The thickness of a layer is used for
forecasting (e.g., snow) because it gives a
good indicator of the average temperature of
the layer.
The smaller the thickness between the
layers, the colder the air.
Some use 1000-500 mb thickness.
Some use 850-700 mb thickness.
Some use both.
The critical thickness varies with the
elevation of the station and the stability of the
air mass.
Large changes in the thickness across
a short distance (thickness gradient)
can also give an indication of the
strength of a front.
So, on a thickness chart, thickness
contours close together indicate a rapid
change in the average temperature of
the air perpendicular to those thickness
contours.
Homework
Do: 1, 3, 4, 5, 6, 7