1. SO254 – Forces and the equation of motion

2. Why does air move? Imagine a parcel of air that is stationary
Now, imagine that parcel of air starts to move, perhaps toward the northeast at 5 kt
Why did the parcel start to move?
Why is it moving toward the NE?
Why is its velocity 5 kt?
The answer to these questions is found in a fundamental principle that you’ve heard before
The air parcel accelerated (changed its velocity) because one (or more) forces acted on it
An equation relates the acceleration of an air parcel with the sum of the forces acting on it:
This equation is Newton’s second law is the sum of all forces (and is a vector because it has magnitude and direction), m is the mass of the air parcel, and is a vector for the same reason as

3. Forces that act on air There are 5 main forces that act on air:
Pressure gradient
Coriolis
Gravity
Friction
Centrifugal
A sixth, turbulence, is also important, but we will not focus on it in this class
Gravity acts only in the vertical, and we will often combine it with the centrifugal force
We will also consider that the Coriolis force acts only in the horizontal
PGF and friction are often considered in the horizontal and vertical

4. Acceleration due to pressure gradient force
Mathematical formulation
is the density of air (kg m-3)
p is the air pressure (Pa, or kg m-1 s-2)
is the del operator
Physical description:
“The force due to differences of pressure within a fluid mass” –American Meteorological Society Glossary (
When pressure changes over a distance, air at a higher pressure will accelerate toward air at a lower pressure
If PGF is the only force present, then air will move from H  L. But, in the real world, air rarely moves directly from H  L. That is because PGF isn’t the only force present
Components of PGF:

5. Acceleration due to Coriolis force
Mathematical formulation
is the angular velocity of the Earth
There are several ways to write this angular velocity. The simplest, and most common is , where is the latitude of the air parcel. In this simple approach, the angular velocity only has a vertical component (and no horizontal ones)
is the velocity vector of the air parcel,
Coriolis’ mathematical formulation forces several important properties:
Because Coriolis is a cross product between and , it (Coriolis) is perpendicular to both!!
Coriolis acts to the right of parcel motion in the Northern Hemisphere (and to the left in the S. Hem.)
Coriolis only acts when parcels move. When is zero, there is no Coriolis. (Similarly, if the Earth stopped rotating, there would be no Coriolis).
Physical description:
“An acceleration of a parcel moving in a relative coordinate system” –American Meteorological Society Glossary (
An air parcel moving to the east in the N. Hem. will deflect 90° to the right (so it deflects to the south)
An air parcel moving to the southwest in the N. Hem. will deflect to the northwest
Components of Coriolis:

6. Acceleration due to gravity force
Mathematical formulation:
We already saw that gravity can be written as a vector
Physical description:
“The force imparted by the Earth to a mass that is at rest relative to the Earth” –American Meteorological Society Glossary (
As we saw before, gravity varies vertically (less gravity farther from Earth). Because the Earth is a rotating geoid, not a perfect sphere, gravity can be combined with centrifugal force to give effective gravity
Components of gravity:

7. Acceleration due to centrifugal force
Mathematical formulation
is the angular velocity of the Earth, just as before in Coriolis
is the position vector of an air parcel (distance from the axis of Earth’s rotation; units of meters)
Physical description:
“The apparent force in a rotating system, deflecting masses radially outward from the axis of rotation” –American Meteorological Society glossary (
Components of centrifugal force:
Combine centrifugal force with gravity to get effective gravity

8. Acceleration due to friction
Mathematical formulation
, the Greek letter “nu”, is kinematic viscosity (how “sticky” a fluid is) Kinematic viscosity of air at 15°C is 1.48 m2 s-1.
is the Laplacian operator,
is the velocity vector
Physical description
“The mechanical resistive force offered by one medium or body to the relative motion of another medium or body in contact with the first.” – American Meteorological Society glossary (
Components of friction

9. Putting the forces together
Return to Newton’s second law,
Let’s re-write as something called the “material derivative”:
The units of are acceleration (m s-2). The material derivative thus quantifies the acceleration of an air parcel.
Let’s put everything together:
What does each term in this equation represent? Can we translate this equation into a meaningful sentence?

10. Components of the equation of motion
The full vector equation of motion is
Let’s first group centrifugal force with gravity to form effective gravity
Now, each one of the terms (one on the right-hand side, and 4 on the left-hand side) can be broken down into components:
Translate each of these component equations into meaningful sentences, just like you did for the vector form of the equation of motion.