Chapter VII.

Rotating Coordinate Systems

7.1. Frames of References

In order to really look at particle dynamics in the context of the atmosphere, we must
now deal with the fact that we live and observe the weather in a non-inertial reference
frame. Specially, we will look at a rotating coordinate system and introduce the Coriolis
and centrifugal force.

Rotating and Non-rotating Frames of Reference

First, start by recognizing that presence of 2 coordinate systems when dealing with
problems related to the earth:

a) one fixed to the earth that rotates and is thus accelerating (non-inertial), our real
life frame of reference

b) one fixed with respect to the remote "star", i.e., an inertial frame where the
Newton's laws are valid.

Apparent Force

In order to apply Newton's laws in our earth reference frame, we must take into account
the acceleration of our coordinate system (earth). This leads to the apparent forces that
get added to Newton's 2nd law of motion:
r r
dV Freal
= (inertial) (7.1)
dt m
r r r
dV Freal Fapparent
= + (non-inertial) (7.2)
dt m m
r
Fapparent
where are due solely to the fact that we operate / observe in a non-inertial
m
reference frame.
r r
dA dA
We need to relate to .
dt inertial dt non −inertial

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7.2. Time Derivatives in Fixed and Rotating Coordinates

Reading Section 7.1 of Symon.

Using concepts developed in the last chapter, let's introduce some notion that will prove
to be useful.
r
Ω z

y
r
VT
r
R
r
r
Latitude circle α =colatitude

φ = latitude

Equatorial Plane

r
Let r = position vector from the origin (center of the earth). Then, we can find out that
r r r
the instantaneous tangential velocity of a particle at any distance r ( a + z where a =
mean radius of earth and z = altitude above ground) is given by

r r r
VT = Ω × r . (7.1)

r
Here Ω is the vector angular velocity of the earth rotation.
r
We can see this by first looking at the magnitude of VT . According to the above figure,

r r r r r r
| VT |=| Ω | R =| Ω || r | sin α =| Ω × r | (7.2)

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 r
dA
Now, let's try to relate in 2 reference frames:
dt

a) Fixed (absolute) system (iˆ, ˆj , kˆ )

r
b) Rotating ( Ω ) system (iˆ ', ˆj ', kˆ ')

z
z'

y'

r
Ω
y

x x'
r
Now, a vector A is the same vector no matter what coordinate system it is viewed from.
This
r
A = Axiˆ + Ay ˆj + Az kˆ = A ' x iˆ '+ A ' y ˆj '+ A 'z kˆ ' .
r
Given that the primed system is rotating, however, the time derivative of A will be
different if viewed from the 2 systems. Mathematically, we have

'a' for absolute

r
d a A dAx ˆ dAy ˆ dAz ˆ
= i+ j+ k
dt dt dt dt
dA 'x ˆ dA ' y ˆ dA 'z ˆ
= i '+ j '+ k' (7.3)
dt dt dt
diˆ ' djˆ ' dkˆ '
+ A 'x + A 'y + A 'z
dt dt dt

Note that the primed unit vectors vary with time! (Remember our earlier derivation of the
velocity in plane polar coordinates?)

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 iˆ '(t + ∆t )

∆iˆ ' due to rotation of unit vector

iˆ '(t )

r r r r
Recall earlier that VT = Ω × r for a particle rotating at angular velocity Ω . Since
r drr
VT = therefore
dt
r
dr r r
=Ω×r (7.4)
dt

The above is correct only for a vector whose length does not change. The same is true for
the unit vectors, therefore

diˆ ' r ˆ
=Ω×i '
dt
djˆ ' r ˆ
=Ω× j' (7.5)
dt
dkˆ ' r ˆ
=Ω×k '
dt
r
dA
If we now loot at in the reference frame of the rotating (primed) system, then
dt
iˆ ', ˆj ' and kˆ ' appear to be fixed!
r
d r A dA 'x ˆ dA ' y ˆ dA 'z ˆ
= i '+ j '+ k' (7.6)
dt dt dt dt
r
which is a time rate
r of change in A seen from the rotating reference frame and is
d A
analogous to a earlier.
dt

Thus, from (7.3), we can write

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 r r
da A dr A r r r
= + A 'x (Ω × iˆ ') + A ' y (Ω× ˆj ') + A ' y (Ω × kˆ ')
dt dtr r
dr A r dr A r r
= +Ω × ( A ' x i '+ A 'y j '+ A 'y k ') =
ˆ ˆ ˆ +Ω× A à
dt dt

r r
da A dr A r r
== +Ω× A (7.7)
dt dt

r r
Let's look at a physical interpretation. If A = r , then we can write
r r r r
dar dr r
= +Ω×r (7.8)
dt dt

Absolute Velocity relative Velocity of the

velocity in to moving coordinate
the inertial (rotating) system itself
(remember
frame Va coordinates (Vr ) r r r
V =Ω×r )

r r r
In another word, Vabs = Vrel + Vcoord .

Think of the earlier example of throwing a baseball towards a person on the ground from
a moving railway car. If you can remember this simple analogy, you will be able to
remember the formula in the box.

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7.3 Equation of Motion in Absolute and Rotating Coordinates

Let's now apply concepts from the previous section to any arbitrary velocity vector of a
particle assuming that the origins of the rotating and fixed (inertial) frames are the same.
In this case, the position vector to any location is the same in both systems.

Our goal is to find the equations of motion in the absolute and rotating coordinates.

Now, we just showed that

r r r r
Va = V + Ω × r (7.9)
r
where V is the relative velocity (we shall drop subscript r from now on).
r
Apply Eq.(7.7) to Va in (7.9) à
r
dVa d r r r r r r r
= V + Ω × r  + Ω × V + Ω × r 
dt dt
r
dV d r r r r r r r
= + Ω × r  + Ω × V + Ω ×  Ω × r 
dt dt
r r r (7.10)
dV d Ω r r dr r r r r r
= + × r + Ω × + Ω × V + Ω × Ω × r 
dt dt dt
r
dV r r r r r
= + 2Ω× V + Ω ×  Ω× r 
dt

Apply the Newton's second law à

r
r r ∑ F
dVa dV r r r r r i
= + 2Ω × V + Ω ×  Ω× r  = i (7.11)
dt dt m

The part in the box is the equation of motion in the rotating coordinate system!
It describes the change of (relative) velocity in time subjecting the net force. The forces
on the right hand side are real forces, and the second and third term on the left arises
because of the coordinate rotation, and there are apparent (not real) forces. We will
discuss them in more details in the following.

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We will look at the last term one the left with respect to an earth oriented coordinate
system:
r
Ω z

y
r
VT
r r
R
r
−Ω2 R
r
Latitude circle α =colatitude

φ = latitude

Equatorial Plane

r r
Note that | R | = | r | cos φ .

Look at the last term in (7.11):

r r r
Ω × Ω × r  .

r r r r
SHOW IT FOR YOUSELF that Ω× Ω × r  = −Ω2 R ! (7.12)

(Hint – use the right hand rule to find out the direction of this vector, then use the
definition of cross product to determine its magnitude)

Using this we can rewrite equation (7.11) as

r
r r ∑ i
F
dVa dV r r r
= + 2Ω × V − Ω2 R = i
(7.13)
dt dt m

Abs. Accel Rel. Accel. Coriolis accel Centripetal accel Net real force

This is a VERY IMPORTANT EQUATION.

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How do we make use of this equation?

Let's look at the forces on the right hand side of (7.13) in the real atmosphere. They are

r
Gravity = g (force per unit mass)

1
PGF (per unit mass) = − ∇p (you have seen this before and you will
ρ
derive this term in Dynamics I)

Friction = …. (neglected for now)

These are real forces (those seen and felt in an inertial / fixed reference frame), so let's
put them into the equation:
r
dV r r r r 1
+ 2Ω × V − Ω 2 R = g − ∇p (7.14)
dt ρ

this is the acceleration that we

see and measure in our earth
coordinate system – not an
inertial frame, but the next 2
terms take that into account.

Rearranging (7.14) à

Coriolis force Centrifugal force

r
dV 1 r r r r
= − ∇p + g − 2Ω × V + Ω 2 R (7.15)
dt ρ

Relative Real forces Apparent forces due

acceleration solely to the rotation
of the coordinate
system

Equation (7.15) is "THE" EQUATION OF MOTION used in meteorology – it is the

backbone of meteorology!

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