Cht. IV The Equation of Motion

Prof. Alison Bridger

10/2011

Overview In this chapter we will:

Develop - using Newton’s 2nd Law of Motion - an equation that in principal - will enable us to predict flows.

Identify the forces which cause air motions.

Adapt the resulting equation to Earth. See how the various forces work. See what the resulting equation(s) look

like in component form.

Introduction In the previous chapter, a flow was

assumed.

In this chapter we will develop an equation to predict the flow of air

This is called the Momentum Equation, or the Equation of Motion.

Introduction In the next chapter, we’ll begin the

process of solving this equation.

NOTE: flow evolution also depends on thermodynamic variables.

For example, we know (MET 61) that flows are forced by pressure gradients which follow from heating variations.

Introduction In other chapter of CAR we will develop

additional equations that forecast the entire evolution of the flow - dynamic and thermodynamic (via the variables V, p, T, etc.).

The entire set of equations are called the Equations of Motion.

Newton’s 2nd Law of Motion Read the full definition on p. 142. We have:

Newton’s 2nd Law of Motion

i

i

d

dt m m

������������������������������������������ V F Fa

Newton’s 2nd Law of Motion We sum over all possible forces (“Fi”) per

unit mass.

If we can solve – integrate over time – we gain knowledge of the future velocity of the air parcel.

Speed & direction.

Newton’s 2nd Law of Motion Here are some steps we need to follow:

identify the forces that affect air parcel motions

formulate how these work (mathematically)

substitute these forms back into the Eqn. above (“F=ma”), and then...

Newton’s 2nd Law of Motion There is a problem (of course!)

Earth is rotating - this makes it a non-inertial frame of reference.

However, Newton’s 2nd Law is formulated for an inertial frame of reference.

Newton’s 2nd Law of Motion Thus we must “adapt” the equation that

we have developed to a non-inertial frame of reference.

This introduces the Coriolis force!

Forces... We can start by identifying

2 basic types of forces:

body forces… affect the entire body of the fluid

(not just the surface) act at a distance

Forces... examples are

gravitational forces (not the same as gravity!)

magnetic forces electrical forces

we ignore the latter two (lower atmosphere only!)

Forces... surface forces…

affect the surface of a fluid parcel caused by contact between fluid

parcel

• examples are

• pressure forces• viscous forces (friction)

Gravitation Newton’s Law of Gravitation…p.137

the magnitude of the force is given by

Ga = GmM / r2

G is the Universal Gravitation Constant m and M are the two masses r is the distance between the two

centers of masses

Gravitation We assume M = Earth’s mass (so MMe)

and m = mass of air parcel (we will soon set m=1).

For the force direction, we assume: Earth is a sphere Earth is not rotating Earth is homogeneous (so the center

of mass is at the Earth’s geometric center)

As a result, we can write:

Gravitation

2e

a

mMG rr

G

Gravitation For the case m=1 (unit mass), Ga ga

and we have:

Gravitation

2e

a

Gr

r g

Gravitation Here, Ge is Earth’s Gravitation Constant

(=GMe).

And note that the lower case (ga) means “per unit mass”.

So - the expression above for ga goes into the RHS of our expression of Newton’s 2nd Law (above).

Friction This topic will be treated in detail in MET

130 etc. In dynamics we typically take one of two

approaches:

ignore friction - it’s usually a “second order” correction to the “important” “first-order” dynamics

assume something very simple for friction

Friction example…we may set friction to

depend linearly on the strength of the existing wind, as in

Friction

-d

dt

VV

with constant. This has the solution:

te oV =V

Friction Here, is a constant that determines the

rate of decay of V with time due to friction.

When we wish to allow for friction in our work, we will often simply add “F” to our Eqn. Of Motion - you should remember that “F” stands for friction.

Pressure Gradient Force The last force important in driving

motions is the pressure gradient force.

In many respects, it is the most important force since it initiates motions!

It is important to remember that it is pressure gradients that matter - not actual pressures themselves.

Pressure Gradient Force Fluid parcels experience pressure forces

due to contact with surrounding parcels.

When these forces are spatially variable, the parcel will experience a net motion.

We need to quantify this...