Cht. IV The Equation of Motion Prof. Alison Bridger 10/2011

Overview n In this chapter we will: n Develop - using Newton’s 2nd Law of Motion - an equation that in principal - will enable us to predict flows. n Identify the forces which cause air motions. n Adapt the resulting equation to Earth. n See how the various forces work. n See what the resulting equation(s) look like in component form.

Introduction n In the previous chapter, a flow was assumed. n In this chapter we will develop an equation to predict the flow of air n This is called the Momentum Equation, or the Equation of Motion.

Introduction n In the next chapter, we’ll begin the process of solving this equation. n NOTE: flow evolution also depends on thermodynamic variables. n For example, we know (MET 61) that flows are forced by pressure gradients which follow from heating variations.

Introduction n 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.). n The entire set of equations are called the Equations of Motion.

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

Newton’s 2nd Law of Motion

n We sum over all possible forces (“F i ”) per unit mass. n If we can solve – integrate over time – we gain knowledge of the future velocity of the air parcel. n Speed & direction.

Newton’s 2nd Law of Motion n Here are some steps we need to follow: u identify the forces that affect air parcel motions u formulate how these work (mathematically) u substitute these forms back into the Eqn. above (“F=ma”), and then...

Newton’s 2nd Law of Motion n There is a problem (of course!) n Earth is rotating - this makes it a non-inertial frame of reference. n However, Newton’s 2nd Law is formulated for an inertial frame of reference.

Newton’s 2nd Law of Motion n Thus we must “adapt” the equation that we have developed to a non-inertial frame of reference. n This introduces the Coriolis force!

Forces... n We can start by identifying 2 basic types of forces: u body forces… F affect the entire body of the fluid (not just the surface) F act at a distance

Forces... F examples are F gravitational forces (not the same as gravity!) F magnetic forces F electrical forces F we ignore the latter two (lower atmosphere only!)

Forces... u surface forces… F affect the surface of a fluid parcel F caused by contact between fluid parcel examples are pressure forces viscous forces (friction)

Gravitation n Newton’s Law of Gravitation…p.137 n the magnitude of the force is given by u G a = GmM / r 2 u G is the Universal Gravitation Constant u m and M are the two masses u r is the distance between the two centers of masses

Gravitation n We assume M = Earth’s mass (so M  M e ) and m = mass of air parcel (we will soon set m=1). n For the force direction, we assume: u Earth is a sphere u Earth is not rotating u Earth is homogeneous (so the center of mass is at the Earth’s geometric center) n As a result, we can write:

Gravitation

n For the case m=1 (unit mass), G a  g a and we have:

Gravitation

n Here, G e is Earth’s Gravitation Constant (=GM e ). n And note that the lower case (g a ) means “per unit mass”. n So - the expression above for g a goes into the RHS of our expression of Newton’s 2nd Law (above).

Friction n This topic will be treated in detail in MET 130 etc. n In dynamics we typically take one of two approaches: u ignore friction - it’s usually a “second order” correction to the “important” “first-order” dynamics u assume something very simple for friction

Friction u example…we may set friction to depend linearly on the strength of the existing wind, as in

Friction

n Here,  is a constant that determines the rate of decay of V with time due to friction. n 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 n The last force important in driving motions is the pressure gradient force. n In many respects, it is the most important force since it initiates motions! n It is important to remember that it is pressure gradients that matter - not actual pressures themselves.

Pressure Gradient Force n Fluid parcels experience pressure forces due to contact with surrounding parcels. n When these forces are spatially variable, the parcel will experience a net motion. n We need to quantify this...